An Integrated Approach to Aircraft Modelling and Flight ...

An Integrated Approach to Aircraft Modelling

and Flight Control Law Design

Gertjan H.N. Looye

An Integrated Approach to Aircraft Modellingand Flight Control Law Design

An Integrated Approach to Aircraft Modellingand Flight Control Law Design

PROEFSCHRIFT

ter verkrijging van de graad van doctor

aan de T ech nisch e U niversiteit D elft,

op gezag van de R ector M agnifi cu s p rof. dr. ir. J .T . F okkem a,

voorz itter van h et C ollege voor P rom oties,

in h et op enb aar te verdedigen

op w oensdag 1 6 janu ari 2 0 0 8 om 1 0 .0 0 u u r

door

Gertjan Hendrik Nicolaas LOOYE

ingenieu r lu ch tvaart en ru im tevaart

geb oren te M aasland

Dit proefschrift is goedgekeurd door de promotor:P rof.dr.ir. J .A . M ulder

Toegev oegd promotor: Dr. Q .P . C hu

Samenstelling promotiecommissie:

R ector M a gn ifi cus v oorz itterP rof.dr.ir. J .A . M ulder Techn ische U n iv ersiteit Delft, promotorDr. Q .P . C hu Techn ische U n iv ersiteit Delft,

toegev oegd promotorP rof.dr. C .W . S cherer Techn ische U n iv ersiteit DelftP rof. Dr.-In g. P . V orsma n n Techn ische U n iv ersita t C a rolo-W ilhelmin a ,

B ra un schw eig, Duitsla n dP rof. Dr.-In g. R . L uckn er Techn ische U n iv ersita t B erlin , Duitsla n dP rof.dr.ir. M . V erha egen Techn ische U n iv ersiteit DelftDr.-In g. J . B a ls Deutsches Z en trum fur L uft- un d R a umfa hrt,

O b erpfa ff en hofen , Duitsla n dP rof.dr. Z . G urda l Techn ische U n iv ersiteit Delft, reserv elid

C over:

The highly ma n oeuv ra b le a n d post-sta ll ca pa b le X -3 1 A ex perimen ta l a ircra ft:con cepts from this thesis w ere a pplied to this a ircra ft in the V EC TO R progra m.A cry lic on ca n v a s pa in tin g b y R ob ert J a n L ooy e (Private collection Dr. Steinhauser)

C opy right c© 2 0 0 7 b y G .H .N . L ooy e

A ll rights reserv ed. N o pa rt of the ma teria l protected b y this copy right n otice ma yb e reproduced or utilised in a n y form or b y a n y mea n s, electron ic or mecha n ica l,in cludin g photocopy in g, recordin g or b y a n y in forma tion stora ge a n d retriev a lsy stem, w ithout the prior permission of the a uthor.

IS B N / EA N : 9 7 8 -9 0 -5 3 3 5 -1 4 8 -2

Ty peset b y the a uthor usin g the LATEX Documen ta tion S y stem.P rin ted b y R idderprin t O ff setdrukkerij B V , R idderkerk, The N etherla n ds.

To Claudia

Summary

FLIGHT Control Laws (FCLs) make the difference between the dynamic be-haviour of an aircraft of its own, and what it actually fl ies and feels like

to the pilot and passenger. To this end, FCLs consist of dynamic feedback andfeedforward transformations of sensor and command signals into suitable controldefl ections, providing desired manual or automatic fl ying behaviour and passengercomfort, under clear as well as turbulent atmospheric conditions.Flight control laws have to meet a large amount of performance and safety re-q uirements in order to ensure they perform reliably in all fl ight conditions, underadverse operating conditions, in the event of hardware failures, etc. Feedback ofsensor measurements hereby not only infl uences, but also brings about strong in-teraction between aircraft fl ight – , airframe structural – , control system hardware– , sensor – , engine – , and landing gear dynamics, thus involving several engineer-ing disciplines simultaneously. As a conseq uence, the design of fl ight control lawsis a challenging, multi-disciplinary design task.The multi-disciplinary aspect in FCL design is becoming more and more impor-tant. In order to improve over-all effi ciency, aircraft designs are continuouslypushed to physical limits. This results in for example more fl exibility of the air-frame structure and “ just-right” sized actuators and control surfaces. Also, theavailability of control laws is more and more exploited, e.g. to actively reduceloads on the airframe, to provide active stabilisation, and to reduce structuralvibrations. The current design process for fl ight control laws is not well con-figured for an inherently multi-disciplinary approach. The main reason is thatmulti-disciplinary aspects are addressed only after the principal design loop, inanalyses performed by specialist departments. This usually gives rise to req uestsfor fixes, resulting in additional design loops.The objective of this thesis is to develop concepts and methods for a new processthat allows for easy incorporation of multi-disciplinary aspects in fl ight controldesign from the beginning. The prereq uisite for doing this is the availability ofintegrated multi-disciplinary aircraft models that not only include fl ight dynam-ics, but also accurately describe structural dynamics, airframe loading, systemsdynamics, etc. As an example, an integrated model allows control bandwidth tobe adjusted, while keeping a close watch on airframe loading. This will save timeconsuming design iterations with the loads department afterwards.Multi-disciplinary aircraft modelling req uires a generic model structure that al-

viii Summary

lows for straight-forward integration of components from various disciplines, aswell as methods for appropriate incorporation of data sources behind these com-ponents, especially in case overlaps exist. In this thesis, such a model structure isdefined exploiting object-oriented modelling techniques. The difference betweenobject-oriented modelling and contemporary approaches is that implementationis based on “native” physical rather than “simulation-ready” differential equa-tions. This allows model components on the one hand to be implemented usingengineering discipline-specific physical equations, and on the other hand to in-terconnect these components in integrated models based on a common modellinglanguage. The first contribution of this thesis is the development of a new gener-alised structure for multi-disciplinary aircraft models and its implementation inthe object-oriented modelling language M odelica.

Even in case of a multi-disciplinary design, the flight control laws still have togo through discipline-specific analyses, especially flutter, loads, and systems. Itis therefore very important to directly incorporate data sources from these de-partments, so that design iterations due to model incompatibilities are avoided.Particularly in the case of aeroelasticity, this is a challenging problem, since aeroe-lastic and rigid aircraft flight dynamics models have several overlaps. The second

contribution of this thesis is a procedure for merging equations of motion andaerodynamics behind both types of models in an appropriate way.

Simulation requires the model to be available in the form of Ordinary DifferentialEquations (ODEs) or Differential Algebraic Equations (DAEs). A characteristicaspect of object-oriented modelling is that these equations are automaticallygenerated from the implemented physical model equations. To this end, reliablesymbolic algorithms are readily available. Besides suitable models for controldesign analysis, this offers the interesting possibility of automatic model inversion.Various nonlinear synthesis techniques, like Nonlinear Dynamic Inversion, arebased on inversion of nonlinear model equations, resulting in control laws for apart of or the full flight envelope in one shot. The th ird contribution of thisthesis is the idea of combining automatic model inversion and inversion-basedcontroller synthesis for rapid-prototyping. This allows the control designer toexperiment with control command variables, control allocation, requirements forcontrol sizing sizing, etc. in very short design cycles, supporting key decisions thatheavily impact the eventual control structure, or even the over-all control concept.It also allows the control department to release preliminary, but representativecontrol laws to other engineering departments early on in the aircraft designprocess, e.g. allowing for early closed loop assessment of flight loads. In this thesisrapid-prototyping control law design from an object-oriented aircraft model isdemonstrated on manual control laws for an aircraft manoeuvring on the ground.

In case the rapid prototyping design is promising, detailed design follows. In thisthesis this is discussed for nonlinear dynamic inversion-based control laws for in-ner loops of an automatic landing system for a small aircraft. Hereby especiallyfeedback signal synthesis and robustness issues due to differences between the ac-tual aircraft dynamics and the inverted model equations are addressed. In case ofNDI the traditional approach is to achieve robustness in design of the outer-loop

Summary ix

control law. As a fourth contribution, this thesis proposes a different way. Un-certain model parameters, which also appear in the inverse model equations, areused as additional degrees of freedom in parameter synthesis. The combined NDIand outer loop control laws are then tuned simultaneously to meet performanceand robustness specifications using multi-objective optimisation. Robustness ishereby addressed by directly incorporating robustness measures as optimisationcriteria, and by simultaneously addressing nominal and selected worst-case pa-rameter combinations in the optimisation.In the current design process, tuning of control laws is mostly performed man-ually. However, multi-disciplinary design requires a considerably larger set ofcriteria set to be addressed simultaneously. For this reason the use of multi-objective optimisation for tuning of free control law parameters is recommended.The underlying min-max approach allows the large amounts of (usually conflict-ing) multi-disciplinary design criteria and constraints to be addressed efficiently,aiming to achieve best-compromise solutions. Relative importance of criteria isexpressed using scaling functions that have a clear physical interpretation. Theoptimisation problem is not necessarily convex: the main objective however isto automatically search for satisfactory design solutions. Obviously, suitableaircraft dynamics models needed for evaluation of numerical criteria are auto-matically generated from the object-oriented model structure described above.As a fifth contribution this thesis extends the approach of multi-objective optimi-sation to large controller structures that consist of multiple interacting functions.A sequential tuning strategy is proposed in which new controller functions aresequentially added to the synthesis, eventually resulting in optimisation of theover-all control system. This strategy allows inner loop functions to be tunedin combination with various outer loop functions. This may considerably reducecontrol law complexity, since duplication of functionality is avoided.The contributions described above have been combined into a simplified flightcontrol law design process. This process has been applied for the design of anautomatic landing system for a small passenger aircraft. The proposed tuningstrategy allowed the same set of inner loops to be used with all outer loop con-troller functions, like glide slope and localiser tracking, and flare and runwayalignment in case of cross wind. Before flight testing, autoland control laws areextensively tested using Monte Carlo (MC) analysis, often giving rise to new de-sign iterations in a second design loop as described above. In the autoland designMC analysis has been directly incorporated in the tuning process, demonstratingthat this additional loop can be avoided using the proposed design process. Theautoland system was successfully tested in six automatic landings, without theneed for any re-tuning of control law parameters.

x Summary

Contents

Summary vii

1 In tro d uc tio n 1

1.1 Flight control laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 The flight control law design process . . . . . . . . . . . . . . . . . 7

1.3 Future developments in flight control law design . . . . . . . . . . . 10

1.4 Objective of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5 General approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.6 Overview and contributions . . . . . . . . . . . . . . . . . . . . . . 15

2 M ulti-d isc ip lin ary aircraft mo d e l d eve lo p me n t usin g o b je c t-o rie n te d

mo d e llin g te ch n iq ue s 1 9

2.1 Current practice in multi-disciplinary modelling . . . . . . . . . . . 21

2.1.1 Model implementation . . . . . . . . . . . . . . . . . . . . . 23

2.1.2 Model integration . . . . . . . . . . . . . . . . . . . . . . . 25

2.1.3 Aircraft modelling for flight control law design . . . . . . . 27

2.2 The aircraft model structure . . . . . . . . . . . . . . . . . . . . . . 28

2.2.1 The world model . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2.2 The atmosphere model . . . . . . . . . . . . . . . . . . . . . 30

2.2.3 The terrain model . . . . . . . . . . . . . . . . . . . . . . . 31

2.2.4 The airport infrastructure model . . . . . . . . . . . . . . . 31

2.2.5 Rigid and flexible aircraft models . . . . . . . . . . . . . . . 31

2.3 The Modelica Flight Dynamics Library . . . . . . . . . . . . . . . . 37

2.4 Automatic code generation . . . . . . . . . . . . . . . . . . . . . . 39

2.5 Application example: ATTAS . . . . . . . . . . . . . . . . . . . . . 40

2.5.1 Starting the model project . . . . . . . . . . . . . . . . . . 41

2.5.2 Aircraft-specific model components . . . . . . . . . . . . . . 43

2.5.3 Specification of inputs, outputs, and parameters . . . . . . 43

2.5.4 Model translation . . . . . . . . . . . . . . . . . . . . . . . 45

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

xii Contents

3 Integration of rigid and aeroelastic aircraft models using the

residualised model method 5 3

3.1 Review of aircraft flight dynamics models . . . . . . . . . . . . . . 573.1.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . 573.1.2 External forces and moments . . . . . . . . . . . . . . . . . 58

3.2 Review of aeroelastic aircraft models . . . . . . . . . . . . . . . . . 593.2.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . 593.2.2 External forces . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.3 Integration of flight dynamics and aeroelastic models . . . . . . . . 623.3.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . 623.3.2 External forces and moments: the residualised model method 64

3.4 Application example . . . . . . . . . . . . . . . . . . . . . . . . . . 653.5 Coupling using aeroelastic state space models . . . . . . . . . . . . 67

3.5.1 Aeroelastic state space model . . . . . . . . . . . . . . . . . 703.5.2 The RM method in state space form . . . . . . . . . . . . . 713.5.3 Correction of aeroelastic state space models . . . . . . . . . 72

3.6 Application example (Continued) . . . . . . . . . . . . . . . . . . . 733.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4 R apid prototyping using inversion-based control and object-oriented

modelling 8 1

4.1 Object-oriented modelling of aircraft flight dynamics . . . . . . . . 844.1.1 Object-oriented modelling . . . . . . . . . . . . . . . . . . . 844.1.2 Object-oriented modelling of the aircraft-on-ground . . . . 87

4.2 Translation of object-oriented models . . . . . . . . . . . . . . . . . 904.3 Inverse model generation . . . . . . . . . . . . . . . . . . . . . . . . 934.4 A rapid prototyping design process . . . . . . . . . . . . . . . . . . 974.5 Aircraft-on-ground control design application . . . . . . . . . . . . 1014.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5 D esign of robust autopilot control law s w ith N onlinear D ynamic

Inversion 113

5.1 Nonlinear Dynamic Inversion . . . . . . . . . . . . . . . . . . . . . 1165.2 The Aircraft Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.2.1 Nonlinear model equations . . . . . . . . . . . . . . . . . . 1195.2.2 Linearised model equations . . . . . . . . . . . . . . . . . . 120

5.3 Dynamic inversion attitude control laws . . . . . . . . . . . . . . . 1225.3.1 Automatic control law generation . . . . . . . . . . . . . . . 1245.3.2 Implementation aspects . . . . . . . . . . . . . . . . . . . . 125

5.4 Robust parameter synthesis . . . . . . . . . . . . . . . . . . . . . . 1285.4.1 Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.4.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1305.4.3 Design criteria . . . . . . . . . . . . . . . . . . . . . . . . . 1305.4.4 Scaling of criteria . . . . . . . . . . . . . . . . . . . . . . . . 1325.4.5 Parameter synthesis . . . . . . . . . . . . . . . . . . . . . . 133

Contents xiii

5.5 Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345.5.1 Analysis w.r.t. synthesis criteria . . . . . . . . . . . . . . . 1345.5.2 Robust stability analysis with µ . . . . . . . . . . . . . . . 137

5.6 Flight test results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1505.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6 Design of autoland controller functions with multi-objective op-

timisation 155

6.1 The aircraft model . . . . . . . . . . . . . . . . . . . . . . . . . . . 1606.2 The applied design process . . . . . . . . . . . . . . . . . . . . . . 1616.3 The controller architecture . . . . . . . . . . . . . . . . . . . . . . . 1626.4 Optimisation problem set-ups . . . . . . . . . . . . . . . . . . . . . 1686.5 Controller optimisation strategy . . . . . . . . . . . . . . . . . . . . 1746.6 Formulation of the basic optimisation problem . . . . . . . . . . . 1766.7 Controller optimisation results . . . . . . . . . . . . . . . . . . . . 1776.8 Flight test results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1826.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

7 C onclusions 189

7.1 An integrated flight control law design process . . . . . . . . . . . 1917.2 Recent application examples . . . . . . . . . . . . . . . . . . . . . . 1967.3 Lessons learnt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1987.4 Recommendations for future research . . . . . . . . . . . . . . . . . 200

A ppendices 20 3

A Model building for control law design 20 5

A.1 Simulation models for design analysis . . . . . . . . . . . . . . . . . 207A.2 Interactive desktop simulation . . . . . . . . . . . . . . . . . . . . . 210A.3 Computation of initial conditions . . . . . . . . . . . . . . . . . . . 210

A.3.1 Trim computation using the model ODE . . . . . . . . . . . 211A.3.2 Trim computation by model inversion . . . . . . . . . . . . 217

A.4 Linearisation of the aircraft model . . . . . . . . . . . . . . . . . . 218A.5 Parametric models for robustness analysis . . . . . . . . . . . . . . 219

B E quations of motion of a fl ex ible aircraft 221

B.1 Review of structural dynamics . . . . . . . . . . . . . . . . . . . . 223B.1.1 Structural eigenvalue problem . . . . . . . . . . . . . . . . . 223B.1.2 The half-generalised equations of motion . . . . . . . . . . . 224B.1.3 The generalised equations of motion . . . . . . . . . . . . . 225B.1.4 Recovery of physical degrees of freedom . . . . . . . . . . . 226B.1.5 Orthogonality of modes . . . . . . . . . . . . . . . . . . . . 227

B.2 The equations of motion of a flexible aircraft . . . . . . . . . . . . 229B.2.1 Approach for derivation . . . . . . . . . . . . . . . . . . . . 230

xiv Contents

B.2.2 Kinetic energy . . . . . . . . . . . . . . . . . . . . . . . . . 233B.2.3 Floating reference frames . . . . . . . . . . . . . . . . . . . 235B.2.4 Potential energy . . . . . . . . . . . . . . . . . . . . . . . . 238B.2.5 Application of Lagrange’s equations . . . . . . . . . . . . . 238B.2.6 Kinematic equations . . . . . . . . . . . . . . . . . . . . . . 239B.2.7 Application of local forces and moments . . . . . . . . . . . 241B.2.8 Summary of result . . . . . . . . . . . . . . . . . . . . . . . 242

B.3 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 243

Bibliography 245

Nomenclature 259

Index 271

Samenvatting 275

Ack nowledgements 279

Curriculum vitae 281

Chapter 1

Introduction

2 Introduction

Abstract

Control laws as present in the flight control computers of modern aircraftprovide the functionality for safe manual and automatic control in flight.The design of flight control laws (F CL s) consists of the actual designphase, followed by a validation phase that involves extensive simulations,rig tests and, eventually , flight tests. Multi-disciplinary aspects play animportant role, since F CL s not only improve aircraft flight dynamics, butalso aff ect structural dynamics, airframe loading, sy stem dynamics, etc.Currently these aspects are not addressed until the validation phase, fre-quently resulting in late design iterations. In future aircraft design projectsinteractions between flight control and other engineering disciplines willbecome considerably stronger, requiring multi-disciplinary aspects to be ad-dressed from the beginning. This is where this thesis aims to contribute.Three aspects will be addressed in detail: development of integrated multi-disciplinary aircraft models to allow a wide range of design criteria to beevaluated and addressed in synthesis of the control laws, development ofapproaches for handling various types of uncertainty and model variationin tuning of design parameters, and the integration of these methodologiesinto a design process structure that allows complex control law structuresto be handled effi ciently and transparently .

3

IN the early days of aviation, the pilot used to control the aircraft with thehelp of cockpit controls that were directly linked with aerodynamic control

surfaces via mechanical elements such as rods, cables and trolleys. Stabilisation,compensation of coupled responses to steering inputs (e.g. turn co-ordination),rejection of atmospheric disturbances, manoeuvring of the aircraft, and navigationwere performed by the pilot alone, supported by early cockpit instruments only.The current situation in military and civil aircraft is quite different. Althoughmechanical linkage between cockpit control devices and control surfaces and en-gines has remained in designs well into the seventies and eighties of the previouscentury, the tasks of the pilot in military and civil aircraft have been increas-ingly automated. Starting with active dampers – basic autopilot modes, stabilityand command augmentation systems and automatic landing systems have evolvedover decades, leading to full-authority digital control systems as present in today’saircraft. A detailed historical overview and interesting references can be found inRef. [84].Modern civil transport aircraft are flown using the so-called flight guidance andcontrol (FG& C) systems [38, 15], comprising the flight control system (FCS), theflight management system (FMS), the automatic flight control system (AFCS),and mostly also some form of structural control (SC) or active loads control(ALC). In Figure 1.1 the functions of these systems have been ordered accordingto their hierarchical level in the form of a block diagram. Several componentsof the system, to be discussed subsequently, are operated via control devices,displays, and panels in the cockpit (top left). Eventually, actuation systems arecommanded that drive control surfaces (painted black in the aircraft depicted topright) and engines to make sure the aircraft responds in the desired way.

1. flight control (FC). This task is performed by the FCS. This system allowsthe pilot to manually steer the aircraft with the help of the side stick or con-trol column, and pedals. With the help of feedback and feedforward controlalgorithms the FCS allows the (auto-) pilot to directly command aircraftmotion variables, such as body angular rates, accelerations, or combinationsthereof, and makes sure these commands are tracked accurately and inde-pendently (de-coupling), especially under atmospheric disturbances. Ex-tensive protection features prevent the pilot from bringing the aircraft intodangerous flight situations, outside its envelope of safe operation. Velocityand flight path are controlled with the help of throttles and speed brakes. Incase of system failures, the cockpit controls are electronically directly linkedto control surfaces (“direct law”), bypassing all levels of automation. Thepilot uses the FCS to perform manual, short-term steering tasks, mainlyduring take-off, landing, and ground operation.

2. flight guidance (FG ). This task is performed by the AFCS, consisting of theautopilot and autothrottle. The pilot selects modes of operation and enterstarget values for speed, altitude, vertical speed, and heading via the glareshield control unit (see Figure). The autopilot and autothrottle computeappropriate command signals that are executed by the FCS. Stability and

4 Introduction

command augmentation is provided by the FCS, or by the AFCS itself. Thepilot uses the AFCS to perform longer term tasks automatically.

3. flight management (FM). The FMS has extensive capability to predict andoptimise aircraft performance as a function of flight plan parameters and toautomatically execute complete flight plans using autopilot and autothrottlemodes. The pilot may enter and optimise the flight plan or load a storedone via the multi-purpose control and display unit (MCDU, see Figure).The pilot uses the FMS to manage long term tasks that may span nearlythe entire flight.

In modern aircraft, the control surface and engine actuators are commanded withthe help of electronic signals. For this reason, electronic flight control systemsare more commonly known as fly-by-wire (FBW) systems. The FC functionalityis implemented in the flight control computer (FCC). Besides probable weightreduction due to partial replacement of mechanical equipment and considerablereduction in maintenance costs, the greatest virtue of fly-by-wire lies in the FCC.The possibility of tailoring the dynamic aircraft responses to command inputsconsiderably reduces pilot work load and has allowed for the development ofhighly agile combat aircraft with otherwise unflyable aerodynamically unstableand aerodynamically nonlinear airframe configurations. In the civil sector, FBWhas allowed the airliner manufacturer Airbus to develop a complete aircraft familywith a high level of handling commonality, allowing transition of flight crews fromone type to the other with minimum training effort.Flight control, flight guidance, and flight management are so-called primary air-craft control functions, intended for stabilising and manoeuvring of the over-allaircraft. Since the early seventies aircraft are more and more equipped with so-called secondary control functions, addressing the loading and dynamic behaviourof the airframe structure. The airframe is a lightweight construction that has towithstand peak and fatigue loads caused by gusts, manoeuvring, engine and sys-tem failures, etc. Active Loads Control (ALC) functions allow for reduction ofthese loads at critical locations in the airframe, for example by active damping ofstructural modes, or by distributing loads in a more favourable way (e.g. usingailerons to move the lift distribution over the wing in-board, in order to reducebending moments at the root). To this end, ALC functions use strategically dis-tributed accelerometers and available control surfaces. In this way, fatigue life ofthe structure is increased and peak loads are reduced, allowing for reduction ofstructural weight. As an example, the introduction of active loads control avoidedthe need for structural reinforcement of the outer wing of the longer range (500series) derivative of the Lockheed 1011 Tristar [13]. Structural control functionsmay be implemented to dampen resonant airframe structural modes that causepassenger and pilot discomfort. An example is the comfort in turbulence function(CIT) in the Airbus A340 [119].Besides primary and secondary flight control functions, actuators are equippedwith local control systems as well, allowing for accurate positioning of the controlsurface, and with the possibility to detect internal failures. Most engines are

1.1 Flight control laws 5

equipped with a full authority digital engine control (FADEC) system, whichaugments the otherwise highly nonlinear throttle responses and allows for directcommanding of for example the engine pressure ratio (EPR) or fan shaft speed(N1).

A c t u a t i o n* A e r o d y n a - m i c c o n t r o l s u r f a c e s* E n g i n e s

A i r c r a f td y n a m i c sF CF G

F M A P , A T H R

S e n s o r s

A L C , S CS L C

F M F G F C S L CA L CS CA PA T H RS C A

= F l i g h t M a n a g e m e n t = F l i g h t G u i d a n c e= F l i g h t C o n t r o l= S t r u c t u r a l a n d L o a d C o n t r o l = A c t i v e L o a d C o n t r o l= S t r u c t u r a l C o n t r o l= A u t o p i l o t= A u t o t h r o t t l e= S t a b i l i t y a n d C o m m a n d A u g m e n t a t i o n

t r a j e c t o r yf l i g h t p a t h , s p e e d

a t t i t u d e & r a t e s ,a c c e l .

A t t i t u d e r a t e s ,a c c e l e r a t i o n s

F l i g h t g u i d a n c e a n d c o n t r o l s y s t e m s

b a c k - u p" d i r e c t l a w "

S C A

g l a r e s h i e l d c o n t r o l u n i t

c o n t r o l d e v i c e s

d i s p l a y s

M C D U

p r i m a r y c o n t r o l s u r f a c e s

s e n s o r s

123

thro

ttle

s

Figure 1.1: The relation between displays, controls, and control laws in a moderntransport aircraft

1.1 Flight control lawsSubject of this thesis are the algorithms in the various systems that contain theintelligence to perform the various control and guidance tasks discussed in theprevious section: the flight control laws (FCLs). The dynamic behaviour of theflight control system, and therefore the dynamic behaviour of the complete air-craft, is governed by these control laws. Consequently, the FCS and its FCLsare flight-critical and obviously must be available at all times during flight. Theprobability of failure resulting in loss of the aircraft has to be extremely remote,i.e. less than 1 in 1 billion flight hours (< 10−9) (paragraph 25.1309 in [49]).The design of the flight control laws is a very challenging task for a variety ofreasons:

6 Introduction

• Even the most basic control tasks are subject to a large number of designrequirements. For example, handling qualities criteria have to be met, sta-bility must be guaranteed (which may be challenging in case the pilot is inthe loop), and control activity must be limited (especially in turbulent con-ditions). Control law behaviour must further be accepted by pilots, oftengiving rise to additional qualitative and subjective criteria to be met;

• Control laws have to work over the full operating envelope of the aircraft1.Aircraft flight dynamics may be highly nonlinear as a function of the flightcondition and flight attitude, requiring control law scheduling as a functionof these parameters;

• Sensor signals can usually not be used directly, but need processing throughcomplementary filters to reduce the effect of atmospheric disturbances, throughnotch filters to reduce structural dynamics content of signals, through es-timation algorithms to compute signals not directly measurable (e.g. sideslip angle often requires this), etc.;

• Implementation aspects must be kept in mind, like capacity of the FCC andcertification of software;

• Most control law development work is performed well before the first air-craft has flown. This means that the design team has to rely on data fromtheoretical methods, wind tunnel experiments, and extrapolation from pre-vious programs. Consequently, robustness to tolerances in the data is animportant issue in design and clearance of the FCLs;

• Control laws rely on sensors and use actuators to perform their tasks. Thesedevices may fail. The design team has to make sure that the control lawsunder no circumstance make things worse than they are (uncontrollability,instability!). Failures must be handled properly, even if this means thatfunctionality is reduced or completely deactivated (so-called “direct law”,see Figure 1.1);

• As already discussed in the previous section, FCLs consist of many manyfunctions. Each function requires the above issues to be considered individ-ually, but ...;

• ... at the same time, have to be safely integrated in the over-all system;

• Beyond handling of complex aircraft flight dynamics, flight control law de-sign integrates many engineering areas. The reason is illustrated in the formof a block diagram in Figure 1.2. The collection of interconnected blocksconstitutes the over-all dynamics of the aircraft, involving:

– flight dynamics (aerodynamics, propulsion, environment, loading, etc.);

1Actually, even beyond, since the aircraft must be recoverable from the most awkward flight

attitudes

1.2 The flight control law design process 7

– airframe structural dynamics;

– sensor system dynamics;

– actuation system dynamics;

– engine dynamics;

– flight control laws.

By closing the feedback interconnection, FCLs bring about strong inter-action between dynamics of all sub-systems. As already noted, in case ofmanual control the integrated dynamics in turn are influenced by the humanpilot closing an additional loop. These interactions need to be addressedcarefully in the design process and involve close co-operation with otherengineering departments.

Aircraft flight& s tru ctu rald y n am ics

Actu atio ns y s te m & e n gin e

d y n am ics

S e n s o rs y s te m

d y n am ics

E F C Sd e lay s

& filte rin g

F lightC o n tro lL aw s

Au to p ilo t s e ttin gs

co n tro l s u rfaced e fle ctio n s

thru s t

e le ctro n icactu ato r / e n gin e

s ign als

aircraft m o tio n /d e fo rm atio n ,airflo w , e tc.

s e n s o r s ign als ( b u s )

co n tro lco m m an d s

Integrated aircraft dynamics

atmospheric disturbances

D isplay s,motions, ...

P ilot steering

Figure 1.2: Feedback interaction of flight control laws, aircraft dynamics, andsystem dynamics

1.2 The flight control law design processIn order to handle the challenges described in the p rev iou s sections, flight controllaws req u ire a thorou gh and well organised design p rocess. A lthou gh the details ofthis p rocess look diff erently at each aircraft manu factu rer, a general stru ctu re canmostly be recognised. In the frame of the G A R T E U R Flight M echanics A ctionG rou p 0 8 on R obu st Flight C ontrol [7 7 ], Irv ing dev elop ed a model of su ch anindu strial flight control law design p rocess for military aircraft flight control laws[5 3 ]. T his model describes the interactions between the flight control and otherinv olv ed engineering discip lines in detail. Figu re 1 .3 dep icts a more simp lifi edform, based on a scheme p resented by Fielding and L u ckner in R ef. [3 8 ]. S tartingp oint are C u stomer and A irworthiness R eq u irements, the fi nal resu lt is the controllaws integrated within the fu lly q u alifi ed and certifi ed aircraft.

8 Introduction

Ok?

yes

n o

A irc ra ft / S ystem sD a ta & M o d els

D eta iled d esig nreq u irem en ts

1

C u s to m e r & A ir w o r th in e s sR e q u ir e m e n ts

D esig nP h ilo so p h y

B a selin eF C L S tru c tu re

D eta iled F C LD esig n / A n a lysis

F C L In teg ra tio n /C o d in g

A ssessm en t(C lea ra n c e)

R ig T estin g(Iro n B ird )

L o a d s (* * )A n a lysis

A S E (* )A n a lysis

F C S S ystemT est

S p ec ific a tio n

C o m p . / G ra p h .A ssessm en tC riteria , T est

S p ec ific a tio n s

Ok?

2

Ok?

3

H /W & S /W P ro -d u c tio n /In teg ra tio n

F lig h tT estin g

A irc ra ftIn teg ra tio n

F lig h t T estS p ec ific a tio n

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F u lly C e r tifie d a n d Q u a lifie d F lig h t C o n tr o l L a w s

4

C o m p . / G ra p h .D esig nC riteria

yes

n o

yes

n o

yes

n o

Off-line Design

(* ) A S E = A ero serv o ela stic , c o n d u c tedb y a ero ela stic ity d ep a rtm en t

(* * ) C o n d u c ted b y lo a d s d ep a rtm en t

B a s e lin e F C L

D e s ig ns ta n d a r d F C L

F lig h ts ta n d a r d F C L

P r o d u c tio ns ta n d a r d F C L(fin a l u p d a te )

Figure 1.3: Flight control laws design process (based on [38])

1.2 The fl ight control law design process 9

The depicted process has four main iteration loops:

• Loop 1: Off-line design. In this phase actual design of control laws takesplace. B ased on their required functionality, a baseline structure is devel-oped. The choice of control devices, command variables, functional break-down into loops and components, etc. will largely depend on the philosophyof the design team.

The nex t step is detailed design of control law functions, based on the base-line structure and aircraft and system models/ data that are available at thetime. D etailed design requirements are based on customer and airworthinessrequirements, as well as in-house criteria. These in turn may be formulatedin the form of computational and graphical criteria that relate to designmethods that are used within the design group. For ex ample, specific min-imum stability margins may be required, or bounds on time and frequencyresponses may be imposed.

W ithin the controller structure, parameters, complementary filters, estima-tors, nonlinear functions, etc. are defined and tuned to best meet the speci-fications, after which the control laws are ex tensively tested using linear andnonlinear simulations. N owadays, for control law synthesis and analysis awide range of methodologies is available, see for ex ample [77].

• Loop 2 : A ssessm ent a nd c lea ra nce. After integration of designed control lawfunctions, ex tensive assessment is carried out. This assessment may involvereal time simulation in the flight simulator, ex tensive robustness analysisagainst tolerances in model data (especially in the case of military aircraft),performance of analyses as required for certification (e.g. Monte Carlo anal-ysis in the case of automatic landing), etc. In this phase not only control lawperformance and robustness is addressed, but also interaction with airframestructural dynamics and loads. To this end, the control law specification isreleased to the disciplinary departments involved. The aeroelastic depart-ment will perform ex tensive aeroservoelastic analysis in order to make sureflutter margins of the closed-loop system (Figure 1.2 ) are preserved over theaircraft flight envelope, for all weight and balance conditions and configu-rations. Loads analysis is performed to make sure that a.o. manoeuvre andgust loads design envelopes are not ex ceeded due to flight control systemaction.

Any design deficiency that shows up in this phase gives rise to the seconddesign cycle. According to Irving, control laws that passed this phase havereached the maturity of D esign S ta nda rd.

• Loop 3: R ig testing. B efore installation in the first aircraft, all on-boardsystems are integrated in a test rig, the so-called iron bird. As part ofthe over-all FCS, the flight control laws are ex tensively tested in this ironbird in order to validate proper functioning in interaction with the systemshardware. Especially failure cases are investigated that are risky and costly

10 Introduction

to test in flight. The iron bird may be operated in combination with theflight simulator, allowing realistic test scenarios to be performed. Designdeficiencies may arise from over-looked aspects (e.g. unanticipated failurescenarios), or deficiencies in system models. Especially nonlinearities mayadversely affect control law performance. In this case, the model needs tobe updated before adapting the control laws (Figure 1.3, top right).

The flight control laws status after satisfying the hardware test specificationsis designated Flight Standard by Irving.

• Loop 4 : Flight testing. This loop involves extensive validation of the controllaws in the aircraft in flight. O bviously, the number of tests will be limitedbecause of time, cost, safety, and environmental constraints. Therefore,flight test results are partly intended to validate results from the seconddesign phase. In case of design deficiencies, control law updates may berequired. The final version (p rodu ctio n standard) will be certified with theaircraft.

The design cycles may involve various stages in the off-line design process. Designmodifications may be initiated in Detailed FC L Design / Analy sis, in the B aselineFC L Stru ctu re, as well as in the Detailed Design Requ irements. Even a change indesign philosophy may appear to be necessary, but hopefully at an early stage.Especially in the case of military aircraft, the Assessment and C learance phasemay proceed on parts of the flight envelope or specific functions, as FCLs for otherparts, or other FCL functions, enter rig and flight testing.In case of a new aircraft type or derivative, the development of the aircraft models(top right in Figure 1.3) is a process by itself, which runs in parallel with theFCL design. Major updates usually result from new wind tunnel experiments(e.g. after configuration changes) and, in a later stage, from flight tests. Modelimprovements require most analyses to be performed over again and frequentlyforce the control laws to be updated as well. Clearance results as presented to theauthorities for certification must eventually be produced using the most up-to-date model data, since consistency with flight test results has to be demonstrated[37].

1.3 Future developments in fl ight control law de-sign

The design and certification of flight control laws is hardly ever a routine job. Inthe first place, automation in the cockpit progresses continuously from aircraftprogram to aircraft program, and existing functions are expected to deliver moreperformance with each new design (e.g. higher cross-wind limits for automaticlanding). In the near future, new modes such as automatic take-off and drive-by-wire control laws during taxiing will be introduced, further reducing pilot workload and eventually allowing for full automation of the flight from push-back atthe departure airport until parking at the terminal of the destination airport [28].

1.3 Future developments in fl ight control law design 11

Besides the extension of FCS functionality, also aircraft design is progressingrapidly, pushing more and more towards physical limits. Currently, the relativeamount of composites in the structure is about to increase dramatically. Forexample, both the Boeing 787 and the Airbus A350 X WB will feature fuselagesmade out of Carbon Fibre Reinforced P lastic (CFRP ). Composites allow for opti-misation of fibre directions and density, resulting in dramatic weight reduction ofthe over-all airframe, but simultaneously increasing its flexibility (see Figure 1.4).Also system hardware is more and more subject to weight reduction, resulting in

Figure 1.4: H ighly flexible wing design of the Boeing 787 Dreamliner (Imagesource: The Boeing Company, photo nr. K 6 39 6 5-03)

“ just-right” selection of actuators [112].Increased airframe flexibility and “ just-right” siz ing of actuators heavily impactcontrol law performance. The other way around, the flight control system will alsomore and more impact (transport aircraft) airframe design, since it for exampleallows for relaxation of natural stability of the aircraft, resulting in lower fuelconsumption2.From the above it will be clear that flight control laws will become more complexand that their interaction with other engineering areas in aircraft design willbecome more important than ever.

The current industrial design process is not well configured to accommodate this.

The most important reasons are:

1. Even nowadays, too many design deficiencies are sorted out via iterationloops beyond the off-line design phase [38, 76 ]. This causes high costs, since

2Relaxed stability allows the tailplane to be smaller and lighter, and the centre of gravity ofthe aircraft may be moved aft, so that the tailplane will contribu te to over-all lift in eq u ilibriu mcru ise fl ight [1 5 ].

12 Introduction

from loop 2 onwards (Figure 1.3), more and more people, departments, andtest equipment get involved. Depending on the required modification, inter-mediate steps may have to be performed over again, costing progressivelymore engineering time and resources. As can be seen from the figure, loadsand aeroelasticity aspects are not addressed in detail until specialist depart-ments perform flight loads analyses with the FCLs included. The impactof late design iterations along the third loop will increase significantly asactuators are sized more tightly. Finding trade-offs between control lawperformance and, for example, airframe loading via loop 2 will be impracti-cal and expensive;

2. Models used for FCL design allow for evaluation of criteria primarily re-lated to flight dynamics and handling characteristics. Usually, only staticaeroelastic effects (flight shape deformation of the airframe) are taken intoaccount. Analysis of flight loads, detailed analysis of actuation system dy-namics, and assessment of structural dynamics influences require specialmodels, only available within the respective engineering departments;

3. Controller synthesis parameters are mainly tuned by hand. Design require-ments for control laws usually do not translate directly into specificationsfor the applied controller synthesis method. This requires manual iterationbetween controller synthesis and validation against the requirements. Multi-disciplinary design means that additional criteria and constraints must betaken into consideration. Finding trade-offs then quickly becomes a verytedious task;

4. There is a lack of design methodologies that allow for fast adaptation offlight control laws to changes in the aircraft configuration. Such a method-ology allows the flight control design team to contribute and rapidly adaptrepresentative flight control laws to the emerging (preliminary) aircraft con-figuration from the earliest design stages. This will be very valuable as FCLsare to become a more and more integral part of the over-all aircraft designand design optimisation. Furthermore, this allows the FCL design team tomore accurately (less conservatively) formulate specifications for sizing andpositioning of aerodynamic control surfaces and for sizing of control systemhardware.

1.4 Objective of the thesis

The general objective of this thesis is formulated as follows:

Propose a design process (focusing on the off-line phase) and methodologies thatinherently facilitate multi-disciplinary design of fl ight control laws.

In response to the shortcomings listed in the previous section, the following de-tailed objectives are set:

1.5 General approach 13

• Develop a model structure for aircraft flight dynamics that allows for easyand intuitive integration of model components and data from other engi-neering disciplines, allowing for multi-disciplinary design analysis;

• Develop a model integration methodology that specifically allows aeroelasticeffects to be included in flight dynamics models, based on agreed-on modeldata from loads and aeroelasticity disciplines;

• Develop a rapid-prototyping methodology that allows for quick generationof representative control laws for aircraft design analysis in the preliminarydesign phase;

• Extend methodologies that allow for automatic robust tuning of flight con-trol laws, eventually allowing for automatic multi-disciplinary compromisetuning of FCL parameters.

As a principal constraint, the methodologies must be able to fully incorporateexisting know-how, experience and lessons learnt in the flight control department.

1.5 General approachAt the DLR Institute of Robotics and Mechatronics the (off-line) design processfor flight control laws has been a central research area since the mid-nineties. In1997 a process structure as depicted in Figure 1.5 was proposed, developed in theframe of a German national project called First-Shot Approach in flight controllaw design (FSA).The process consists of the following steps:

• Modelling. This involves development of the aircraft model, including allrelevant dynamic effects (flight dynamics, actuator and sensor dynamics,etc.), and modelling of the FCS with the selected controller structure. Themodel depends on varying or uncertain parameters p and tuning parametersT . The formulation of numerical design criteria from the functional designrequirements is another important modelling activity. Modelling work thatwas performed in the frame of the FSA project is described in [91].

• Analysis and Selection. Before tuning of design parameters (T ), open loopdynamics are analysed and a selection of parameter configurations (e.g.flight cases) as well as design criteria is made, based on which the designparameters are to be tuned.

• Tuning and Compromising. This step is mostly performed automaticallywith the help of multi-objective optimisation. To this end, the softwareenvironment MOPS is used [52]. An important means of achieving a ro-bust design is to simultaneously address nominal and worst-case parameterconfigurations (selected in the previous step) in the optimisation.

14 Introduction

Modelling

* c om p onents (p ,T )* a irc ra ft m odel (p ,T )* c ontroller / filters (T )* F C S (p ,T )* c riteria (p ,T )

Ok?

y es

no

Ana ly s is & S elec tion

* E v a lu a tion & s y nth es is m odels* S u b ta s k s & c riteria

A

T u ning & C om p rom is ing

(T -V a ria tion)

As s es s m ent

(p -V a ria tion)

Ok?

y es

no B 1

* F u nc tiona l des ignreq u irem ents

* P h y s ic s* Model da ta

Figure 1.5: FSA Flight control laws off-line design process

• Assessment. Finally, before release for hardware implementation, the con-trol laws are tested extensively as a function of model parameters p in worst-case analyses.

The process consists of two main loops. The first loop (A) is based on opti-misation results where the designer decides upon shifting trade-offs (via criteriascaling), or in some cases, structural changes to the controller. Loop (B) is basedon assessment results and may involve replacement or selection of additional pa-rameter cases to be taken into account in the optimisation in order to improverobustness. The design work based on the process described above is discussed indetail in [50].

The FSA process has two key elements that make it highly suitable for multi-disciplinary flight control law design. Firstly, the adopted object-oriented mod-elling technology is well suited for multi-disciplinary model integration, as will beexplained in the following section. Secondly, multi-objective optimisation is ableto handle large amounts of criteria simultaneously (up to O(102) or more). Thisprovides enough room to address multi-disciplinary aspects.

1.6 Overview and contributions 15

The structure of the FSA process puts less emphasis on control law architecturedesign (in Figure 1.5 referred to as a modelling activity) and on the way howcomplex control laws are handled. These aspects are to be improved upon in theframe of this thesis.

1.6 Overview and contributionsThe structure of this thesis is depicted in Figure 1.6. Each of the five core chaptersaddresses a different aspect of flight control law design and aircraft modelling,and therefore can be read independently. As indicated by the fat flare trajectory,the chapters sequentially add key elements that result in a new design processstructure proposed in Chapter 7.

1.In

troduction

7.C

onclu

sio

ns

2 . M ulti-dis cip lin.A /C m ode lling

3 . F le x . a ircra ftm ode l inte g r.

4 . R a p id F C Lp rototy p ing

5 . N D I-b a s e dF C L de s ig n

6 . A utola nd s y s te mde s ig n

A p p . B

A p p . A

Figure 1.6: Structure of the thesis

Chapter 2: Multi-disciplinary aircraft model developmentusing object-oriented modelling techniquesThis chapter proposes the use of object-oriented modelling techniques to developintegrated aircraft models that allow flight dynamics as well as inter-disciplinaryeffects to be addressed from the beginning in the design process. Object-oriented

16 Introduction

modelling allows model components from various engineering areas to be com-bined in a single model, but still to be represented in their discipline-specific form.The contribution of the chapter is a new physically-oriented generic aircraft modelstructure, suitable for implementation of rigid as well as flexible aircraft, in slowlow up to fast high flight regimes. In addition, the underlying methodology al-lows for easy implementation of model uncertainty and for automatic generationof run-time code for various types of control design analysis.

Chapter 3: Integration of rigid and aeroelastic aircraft modelsusing the residualised model method

One aspect that is absent in contemporary aircraft models for control design isairframe flexibility. The influence of control laws on flutter margins and airframeloading is currently left to the loads and aeroelasticity department, giving riseto design iterations along loop 2 (Figure 1.3) in case problems show up. Takingaeroelastic aspects into account in the flight control design model allows thesedesign iterations to be eliminated. This however requires the integration of rigidand aeroelastic aircraft models, which is not an obvious task due to overlaps be-tween both model types. In Chapter 3 these overlaps are identified and solvedmathematically in the form of the so-called residualised model method. In addi-tion, a procedure is developed that also allows the aeroelasticity department toimprove its models regarding flight mechanical aspects.

Chapter 4: Rapid prototyping using inversion-based controland object-oriented modelling

As depicted in Figure 1.3, two key aspects in the off-line design phase are theadopted control design philosophy, and the baseline controller structure. In caseof standard flight control functions the design choices are strongly influenced oreven imposed by previous aircraft programs. However, in the near future newfunctions will be introduced, like drive-by-wire for taxiing on the ground. Keydecisions like selection of control variables, control allocation and sizing, etc. arestill open. A possibility for rapid prototyping would be an excellent means tostudy the implications of such choices and to support fundamental decisions,both in flight control as well as over-all aircraft design. Even for conventionalcontrol functions, rapid prototyped control laws may be passed on as a preliminarydesign to other departments for aircraft-level design analysis (e.g. control sizing).Such a rapid-prototyping process is enabled by object-oriented modelling. Thismethodology namely allows for automatic generation of nonlinear control lawsbased on inversion of model equations, like Nonlinear Dynamic Inversion (NDI).This will be demonstrated in, and is main contribution of, Chapter 4.

1.6 Overview and contributions 17

Chapter 5: Design of robust autopilot control laws with nonlin-ear dynamic inversionAs has been demonstrated in Chapter 4, Nonlinear Dynamic Inversion (NDI) is anonlinear multi-variable design technique providing decoupled and uniform com-mand responses over the aircraft flight envelope in one shot. In combination withautomatic generation of the control laws from an object-oriented model imple-mentation, NDI is a highly attractive methodology from a design effi ciency pointof view. However, the method may result in design solutions that, if not appropri-ately taken care of, are highly sensitive to modelling errors. Chapter 5 addressesthis issue in two ways. In the first place, it is shown how with the help of multi-objective optimisation a robust design can be achieved by combining robustnessmeasures (like gain and phase margins) as design criteria, and by simultaneoustuning of control law parameters for different uncertain parameter combinations.It is further shown that uncertain model parameters that also show up in theinverse model equations in the controller, may be effectively used as additionaldegrees of freedom to achieve a robust design. The chapter covers a completeFCL design from modelling to flight test, demonstrating the capabilities of NDIand the proposed method for robust control law synthesis.

Chapter 6: Design of autoland controller functions with multi-objective optimisationThis chapter combines and extends Chapters 3 to 5 into an integrated off-line de-sign process, applied to the design of an autoland control system. The proposedprocess allows for structured design of complex control laws that consist of mul-tiple interacting functions. This allows controller complexity to be reduced, sincea single control function can be tuned for use with various outer loop functions.It is shown that multi-objective optimisation allows for direct incorporation of avariety of criteria and assessment methods (including Monte Carlo analysis) intothe tuning process of control law parameters, avoiding design iterations in loop 2(Figure 1.3) afterwards.

Chapter 7: ConclusionsIn this chapter the design process proposed in Chapter 6 is generalised and it isshown how the other contributions of this thesis fit in. Current developments andsome recent applications are discussed. In addition, some lessons learnt will beshared and recommendations for future work are made.

18 Introduction

Chapter 2

Multi-disciplinary aircraftmodel development usingobject-oriented modellingtechniques

20 Multi-disciplinary aircraft model development

Abstract

In this chapter a physically-oriented model structure for complex multi-disciplinary aircraft dynamics models is presented. For implementationthe object-oriented modelling language M odelica is used. Object-orientedmodelling (OOM ) is based on implementation of “ native” physical, ratherthan simulation-ready differential and algebraic equations, allowing vari-ous engineering discipline-specific modelling methods to be combined in asingle model implementation. The proposed aircraft flight dynamics modelstructure comes with the Flight Dynamics Library, a M odelica library con-taining re-usable components, as well es base classes for aircraft-specificcomponents. The model structure allows for implementation of flexible aswell as rigid aircraft, in slow low up to fast-high flight regimes. Anotherimportant advantage of OOM is that various types of runtime models maybe automatically generated from a single implementation. From a controlpoint of v iew, this allows for automatic generation of models for nonlin-ear simulation analysis, inverse models for trimming and inversion-basedsynthesis techniques, linear symbolic models for robustness analysis, etc.In this chapter the main principles of OOM , the new aircraft model struc-ture, and an example aircraft model implementation will be described.

C o n tributio n s

• A new, physically-oriented generic structure for air vehicles andenvironment models, valid for:

– rigid as well as flexible airframe structures ...

– ... of vehicles in slow or fast flight, at low or high altitudes, oranywhere in between.

• The presented modelling methodology is the basis for contributionsin subsequent chapters.

P ublicatio n

G ertjan Looye, Simon H ecker, Thiemo K ier, Christian Reschke, J ohannBals: Multi-disciplinary aircraft model development using object-orientedmodelling techniques, International Forum on Aeroelasticity and Struc-tural Dynamics (IFASD), M unich, G ermany, J une 2 0 0 5 .

2.1 Current practice in multi-disciplinary modelling 21

SIMULATION plays an important role in aircraft design and certification. Wellknown is the role of simulation in the development of flight control laws and

in the analysis of flight loads. Also in specification, testing and integration ofon-board systems, simulation is more and more used to reduce development timeand hardware cost.In the continuous drive to improve efficiency, the aircraft design is more andmore pushed to its physical limits. An obvious result is that interactions be-tween engineering disciplines become stronger and stronger. A classical exampleis increasing airframe flexibility as a result of light weight design, interfering withaircraft flight dynamics, and affecting passenger comfort. For this reason, there isa growing need to address these interactions in simulation (or other model-based)analyses. This in turn requires the involved engineering aspects to be presentin the underlying models, requiring the availability of multi-disciplinary aircraftdynamics models. A major problem hereby is that engineering departments re-sponsible for the various aspects of aircraft design develop models for their specifictypes of analysis, based on modelling methods and tools that are common placein the specific engineering area. Consequently, bringing model implementationsand data together into a multi-disciplinary simulation model is a very challengingtask. In the following section various approaches to achieve this will be reviewedand compared.

2.1 Current practice in multi-disciplinary modellingIn order to put current practice in model integration of physical systems in general,and aircraft dynamics in particular, in perspective, it is helpful to have a look atsteps a model typically goes through from conception to simulation1. These stepsare depicted in Figure 2.1.The first step is to clearly define what is to be modelled. This involves thesystem, its behaviour of interest, and the role of and the bounds with respect toits environment. Of course, this is mostly determined by the intended applicationof the model. For example, for flight loads analysis the dynamic distributionof air loads over the aircraft is of prime interest, whereas the influence of theenvironment is limited to gravity and the (ideal) atmosphere. For piloted flightsimulation, only the total aerodynamic forces and moments acting on the airframeare of interest, whereas the detailed modelling of for example the terrain andEarth’s rotation and curvature are relevant. This first step in the modellingprocess will be referred to as specification level.The second step, which takes place at level referred to a as system level, is thebreak down of the system into components and the specification of their inter-connections. This in most cases is obvious. An important decision to be madeis which data sources will be used. For example, the aerodynamic model maybe obtained from an aerodynamics department, computed in-house using some

1As will be discussed later on, the steps also apply to model applications other than simula-tion.


Physical componentb reak -d ow n

Physical system andmod elling scope

Physical component / connec-tion eq u ations, alg orithms

O rd inary D ifferential /D ifferential A lg eb raic E q n.’s

S imu lation mod el

System level

M a th ema tic a l level

P h ysic a l level

.

1

2

3

4

5R u n time level

.

.

Sp ec ific a tio n level

Figure 2.1: Levels in development of a physical system

CFD method, or obtained from model identification during a planned flight testcampaign.The third step is to actually formulate the equations and algorithms that underliethe model components and to make sure the required data is available. Theequations are based on laws of physics, application rules that come with datasources (e.g. aerodynamics), international model standards (e.g. InternationalStandard Atmosphere), algorithms as implemented in a system’s control unit,etc. Interconnections between model components may for example be constraints,energy flows, or signal flows. The third step in the modelling process will bereferred to as physical level.The fourth step is to collect all equations and to derive mathematical standardforms, like ordinary differential (ODEs) or differential algebraic equations (DAEs),suitable for time simulation of the model. For this reason, the level this step isperformed at will be referred to as mathematical level. For derivation of ODEs orDAEs it is necessary to specify which variables are to be considered as inputs2,as outputs, or as states. This means that from this moment on, the causality ofthe model is fixed.Finally, the model is connected with a simulation algorithm, allowing the actualmodel simulation to be performed. This algorithm strongly depends on the na-ture of the model. For example, whether the model has been brought into theform of ordinary differential or differential algebraic equations (ODEs or DAEs),the extent of stiffness of these equations, whether the simulation model contains

2If any, since systems do not necessarily have inputs. However, physical systems consideredin this thesis (aircraft) in general do.


continuous states, discrete states or both, or whether state events should be han-dled or not. T he im p lem entation m ay involve inline (local) integ ration of m odelstates, a state vector m ay be p assed to an ex ternal integ ration alg orithm , or var-ious com p onents m ay be integ rated seq uentially using diff erent alg orithm s in aseq uence of m ini sim ulation runs. T his fi fth step in the m odelling p rocess is atthe so-called simulation level.

2.1.1 Model implementation

F rom F ig ure 2 .1 an interesting observation can be m ade. A ctual im p lem entationof m odel com p onents, storing into and retrieving from reusable libraries, andinteg ration into a full m odel usually tak es p lace at a m athem atical level (step4 ). A fter sorting and solving the m odel eq uations and alg orithm s based on theunk nown variables, a function or collection of functions is coded. F or a continuoussy stem such a function ty p ically im p lem ents:

x = f(x,u,p)y = h(x,u,p)

(2 .1 )

where x ∈ IR nx is a vector containing m odel states with dim ension nx, the vectoru ∈ IR nu contains m odel inp uts, y ∈ IR ny contains m odel outp uts, and p ∈ IR np

contains constant p aram eters that m ay be set by the user p rior to sim ulation. Inthe p ast, coding of this function was p erform ed using a p rog ram m ing lang uag e lik eC or F O R T R A N . N owaday s, block diag ram s have becom e very p op ular, allowingsubsy stem s in the above form to be g rap hically interconnected via their inp uts andoutp uts. S ince the variables in u and y have been ex p licitly identifi ed as inp utsand outp uts resp ectively , m odel im p lem entation at this level is also referred to as“ causal” m odelling .

In a num ber of eng ineering dom ains im p lem entation at the p hy sical level (step 3 )has becom e com m on p ractice. F or ex am p le, for construction of m ulti-body sy s-tem m odels software is available that off ers com p onent libraries (bodies, joints,hing es, ex ternal forces, etc.) from which a m odel m ay be constructed g rap hi-cally . T he sam e holds for electronic circuits, for which a rang e of p rog ram s andcom p onent libraries is readily available. T he diff erential (alg ebraic) eq uations areg enerated autom atically , ex p loiting the discip line-sp ecifi c m odel structures. Awell k nown m odelling p aradig m that has found ap p lication in various eng ineeringdiscip lines are bond g rap hs, based on energ y and energ y ex chang e between com -p onents [1 6 ]. In 1 9 9 1 W illem s laid the foundation for what has becom e k nown asthe B ehavioural A p p roach to the descrip tion of dy nam ical sy stem s [1 0 2 ], strong lyadvocating m odel descrip tion at the p hy sical level and p roviding a m athem aticalfram ework for describing sy stem behaviour and (control-relevant) sy stem p rop er-ties. In the B ehavioural A p p roach p hy sical com p onent break -down is referred toas “ tearing ” and describing the com p onent behaviour as “ zoom ing ” [1 3 9 ].

Im p lem entation at the p hy sical level has a num ber of advantag es over doing thisat a m athem atical level:


+

G

AC

=220

R1

=1

0

R2

=1

00

C=

0.0

1

L=

0.1

1

S

1

S1/C

1/L

1/R 1

R 2+

-

-

+

+

+

Figure 2.2: Block diagram implementation (right) of an electronic circuit (left).From [8 8 ].

1. In level 3, causality of the model is not fixed yet, whereas at level 4 causalityis inherent to the implemented differential equations. As a result, from animplementation at level three, different runtime models with different setsof (reversed) inputs and outputs may be generated from one and the samemodel.

2. D ifferential equations for simulation are just one form of executable modelcode. Other forms are static equations, symbolic linear state space models,L inear Fractional Transformations (L FTs), etc. From level 3, the form ofthe runtime model is still open.

3. In level 3 model component interconnections may be specified via physicalequations. From step 4 onwards, model components can only be intercon-nected via their inputs u and outputs y. This makes adding new componentsmore complicated and easily results in a non-physical model structure. Thelatter becomes even worse, since within the components all unknowns haveto be brought to the left-hand side, obscuring the original physical equa-tions. This can be illustrated via a (by now) classical example in Figure 2.2.To the left a simple electronic circuit is shown. To the right the same modelis depicted as a block diagram. It will be clear that, when only providedwith the block diagram, quite some reverse engineering effort is requiredto find the underlying physical equations and to picture the original circuitstructure.

In practice, the above restrictions do not pose severe problems, since within mostengineering areas application-specific work-arounds and program libraries havebeen developed that considerably reduce the coding effort. For example, for theblock diagram-based modelling tool Simulink [8 1] libraries for various engineeringdomains are readily available.


2.1.2 Model integration

Multi-disciplinary modelling involves the integration of model components anddata from various engineering disciplines into a single model. In Figure 2.3 itis shown that this may be done at the three candidate levels of implementation.The first problem that nearly always arises is that different engineering disciplinesuse different modelling methods and simulation tools. Looking at Figure 2.3, apragmatic approach would be to couple models at the very lowest level, namelyby means of distributed or co-simulation. This avoids the need of translatingor importing any model from the one simulation tool into the other, and allowsfor geographically distributed simulation. Standards for hierarchical organisationand interfaces for data exchange are available, like for example the H igh-LevelArchitecture (H LA) [26]. The latter is for example used by the U S Departmentof Defence (DoD) to perform battle field simulations with multiple players. Themodelling scope of this chapter however is initially limited to a single aircraft.In this respect, co-simulation (also referred to as “loosely coupled” simulation)is for example used in coupling multi-body packages with engineering-specificenvironments, like FE M software, CFD solvers, etc. [62].

Aircraft models used for fl ight control and loads analysis must be fast in orderto allow ten thousands of simulations to be performed in a reasonable amount oftime (so-called “loop-capability”). An important disadvantage of co-simulationis that it involves multiple simulations-processes only communicating via com-puted trajectories. This does not allow for optimisation of the integrated modeland causes problems as soon as algebraic loops between model components arise.Several control analysis methods require the model to be available in a specificform, like a Linear Fractional Transformation (LFT). Being at the lowest mod-elling level, co-simulation provides the runtime model in one form only. Finally,as soon as more than two or three disciplines are involved, mastering the modeland simulation becomes a complicated task for a single engineer and may becomecostly because of software licenses of involved programs.

Therefore, a more usual approach is to integrate models at level 4. This requires acommon implementation platform, if only at the point of linking compiled modelcomponent software. This approach is greatly facilitated by the fact that manymodelling platforms nowadays allow for model export in the form of runtimesimulation code. A main program may then be written that sequentially calls theinvolved model component codes and provides data exchange in between. Anotherpossibility is to import an external model code into the (designated) main modelby means of a function call. Disadvantages of this approach is that the modelcode from the one engineering department remains a black box to engineers inthe other.

In some cases, model platforms used by different disciplines are identical. Forexample, the block diagram modelling tool Simulink provides dedicated librariesfor various engineering areas and finds ever wider acceptance. For fl ight dynamicsmodelling for example, several libraries are readily available [104, 126, 29]. In-tegration of model components based on a common implementation platform is




S imu lation mod el

System level

M a th ema tic a l level

P h ysic a l level

1

2

3

4

5R u n time level

Sp ec ific a tio n level



Physical component / con-nection eq u ations, alg orithms

S imu lation mod el

1

2

3

4

5

Distributed / Co-simulation

H /W in th e loop simulation

I/O interc onnec tion

P h y sic al interc onnec tion

Disc ip line A Disc ip line B

T rajec tory interc onnec tion



Physical component / con-nection eq u ations, alg orithms

Figure 2.3: Various levels of interconnection between physical system models

relatively easy. Unfortunately, Figure 2.2 clearly shows that this still does notsolve the “black box” problem.Another important problem remains. Model components may only communicatevia data flows and model components must be executed in a given sequence. Thiscauses problems as soon as components algebraically depend on each others data.Such algebraic loops require expensive iterative solving or very short simulationtime steps in case loops are broken artificially.Apparently, level 4 is not necessarily a suitable level to (1) implement models and(2) to integrate models. The problems sketched above disappear when taking themodel implementation one level up, namely to level 3. Since at this level inter-connection equations are still physical, connection of model components can beperformed in the form of physical equations as well. Of course, this implemen-tation approach has an important prerequisite. It must be possible to combinemodels from various disciplines and based on various modelling paradigms into asingle implementation. As it is diffi cult and unnatural to, for example, implementa block diagram in a multi-body package.This problem was recognised by Elmqvist, who in 197 8 proposed a dedicatedmodelling language, called Dymola (Dynamic Modelling Language) [30]. Thebasic philosophy behind this language is:

1. implementation takes place at the level 3: physical objects and phenomenaand their interactions may be implemented as model objects and modelinteractions respectively in a one-to-one fashion;

2. the language serves as a common base for development of discipline-specificcomponent libraries.

The result of the second point is that model components may be composed fromdiscipline-specific libraries. The common language base allows these components


to be combined in a single model. At the time, issues like algebraic loops betweenmodel components and structural singularities in models prevented practical useof the language. By 1991 most of these issues had been resolved [32], resultingin a first release of a modelling and simulation environment based on the Dy-mola language, also called Dymola (Dynamic Modelling Laboratory) [4]. In themean time, several other research groups started developing dedicated physicalmodelling languages and tools. All languages are strongly supported by graph-ical tools, allowing model components to be constructed from discipline-specificlibraries using “drag-and-drop”. Since libraries may in turn be based on discipline-specific modelling paradigms, an electronic circuit (for example) also graphicallylooks as such, and a control law may still be implemented in the form of a block-diagram.In 1996 several developers in the physical modelling field, as well as simulationand modelling experts, started joint work to develop a common, free physicalmodelling language standard, named Modelica. In 2000 the non-profit ModelicaAssociation was founded, with the objective to maintain, develop and promote theModelica language. By now, the Modelica language has achieved a high degree ofmaturity and several modelling tools (“Modelica translators”) have become avail-able, featuring graphical model composition and extensive simulation capabilities.In 2006 Dassault Systemes announced that the systems simulation extension ofits CATIA CAD software will be based on the Modelica language. A wide rangeof libraries has been developed (e.g. multi-body systems, electronics, thermody-namics, block diagrams, power trains, hydraulics, pneumatics, fuel cells, bondgraphs) that on the one hand allow for composition of discipline-specific modelcomponents, while on the other hand these components may be used along side inan integrated multi-disciplinary model. This allows for development of large scale,intuitively structured hierarchical models. P apers on several complex examplesmay be downloaded from the Modelica home page [89].

2.1.3 Aircraft modelling for fl ight control law designFor development of flight control laws, object-oriented model implementation(level 3) is a very attractive approach.

• Flight control law design requires both understanding of control engineering,but more importantly, a thorough understanding of the behaviour of theaircraft. Model visibility is therefore very important. This is best achievedby implementation using physical model equations and based on physicallystructured model components.

• Analysis of flight control laws involves large amounts of simulation runs(∼ 104) to evaluate control law performance at different flight conditions,weight and balance configurations, with different uncertain parameter val-ues, etc. Object-oriented modelling allows for automatic generation ofhighly optimised simulation code from the multi-disciplinary model imple-mentation.


• Various types of control design analysis require the model to be available indifferent forms. For nonlinear simulation, ODEs are required, for robustnessanalysis symbolically linearised parametric state space models are used, foraccurate trim computation static inverse models are very useful. So far, themain focus in this chapter has been on simulation. By taking implementa-tion one level up, object-oriented modelling allows various forms of runtimemodels for various forms of control design analysis to be automatically gen-erated from one and the same model.

In 1995 Moormann developed a first library (at the time, based on the Dymolalanguage) for object-oriented modelling of aircraft flight dynamics [92]. Objectivewas to build a solid basis for constructing integrated dynamic aircraft models,including flight dynamics, detailed on board system dynamics, structural dynam-ics, etc. First applications were a generic transport and a fighter aircraft [77] anda first flexible aircraft shortly thereafter [72]. Since then, the library has beenexpanded and applied to complex aircraft models that include e.g. system hy-draulics and electronics [91, 93]. In the frame of the international projects REAL(Robust and Efficient Autopilot control Laws design, funded by the EU in thefifth frame work programme [110]) and the G ARTEUR action group AG -11 onclearance of flight control laws [39], the automatic generation of inverse models forfast trim computation and nonlinear control laws was applied for the first time.In 2005 , the DLR Flight Dynamics Library was considerably revised. Main rea-son was the development of a new generalised aircraft model structure, which isthe main contribution of this chapter. The versatility of the Modelica languageallowed this structure to be implemented in a one-to-one fashion. In the followingsection the new model structure is discussed (Section 2.2), based on which theFlight Dynamics Library has been organised (Section 2.3). In Section 2.4 the au-tomatic generation of model code for model simulation and analysis is discussed,followed by an example application. Finally, conclusions are drawn in Section 2.6.

2.2 The aircraft model structureThe objective of this section is to introduce and motivate the basic structure ofan aircraft model as it may be composed from the Flight Dynamics Library. Thestructure of this library will be discussed thereafter. The full specification of theunderlying Modelica language is given in [90].In constructing complex models the choice of hierarchy is crucial, since this largelydetermines how model components interact. For the Flight Dynamics Library atop-level model structure as shown in Figure 2.4 has been adopted. It consistof one or more aircraft, and environment objects (The avionics component andinput and triangular-shaped output connectors will be discussed in Section 2.4).The environment objects include a w orld, atmosp h ere, terrain, and airport model.Note that the (in this case, single) aircraft model has no direct link with theenvironment models, which physically makes sense. Using the so-called inner-outer feature of the Modelica language, these models provide field functions. For

2.2 The aircraft model structure 29

Figure 2.4: Top-level of model: aircraft and environment

example, the aircraft may request its surrounding atmospheric conditions fromthe atmosphere model by sending its local inertial position. Any other aircraft(or e.g. sensor) object in the model may do this as well. This is different frommost block-oriented libraries, where an atmospheric model is directly linked to,and thus occupied by, the one aircraft. The ability to easily include multiple airvehicles is useful for applications involving mutual interactions, like towed gliders,wake vortexes, air-to-air refuelling, release of missiles, etc.

2.2.1 The world model

In the following subsections the environment models in Figure 2.4 (world, atmo-

sphere, terrain, airport) will be discussed. These components determine validityof the over-all model to a large extent. Most important is the world model (inthis case, the Earth), since it provides the inertial reference in the form of theso-called Earth-Centred Inertial (ECI) reference frame. Its origin is attached tothe Earth’s centre of mass, its orientation is fixed with respect to reference stars


[55]. In addition, the model component has the following functions:

• Provide the geodetic reference. As indicated in Figure 2.4, to this end theWorld Geodetic System 1984 [7] (WGS84) is used. The object implementsan Earth-Centred Earth-Fixed (ECEF) reference frame, which has the sameorigin as the ECI, but rotates with the Earth. The attitude of the ECEF(w.r.t. ECI) is available in a connector. A set of functions transform ECIand ECEF referenced position vectors into geodetic longitude, latitude, andheight co-ordinates (w.r.t. WGS84 ellipsoid) and vice versa. For a givenlongitude and latitude, another function provides the local undulation ofthe so-called EGM96 (Earth Gravitational Model 1996) geoid with respectto the WGS84 ellipsoid, providing the Mean Sea Level (MSL) reference [68].

• Implement a model of the Earth’s gravitation. The gravitational modelto be used with WGS84 is the Earth Gravitational Model 1996 (EGM96),provided in the form of tables describing equi-potential surfaces a functionof longitude and latitude. Currently, a more simplified height and geocentriclatitude-dependent (Ref. [125] - Eqn.(1.4-16)) and a constant gravity modelare available.

• Implement a model of the Earth’s magnetic field. This field is requiredto compute indications of compass models. The model is based on theUS National Geo-spatial-Intelligence Agency (NGA) World Magnetic Model(WMM), which is published every five years and predicts the time-varyingintensity and direction of the magnetic field as a function of WGS84 longi-tude, latitude and height. The current model covers 2005 till 2010 [83].

Double-clicking on the world object in Figure 2.4 allows a number of parametersto be set, like whether the Earth is rotating or in rest, initial day time, and thetype of gravity model (approximate EGM96, height independent, or constant).The features of the object may be overkill for many applications, but providesufficient generality for use with for example high speed and high altitude flightvehicles. Furthermore, the applied WGS84 ensures compatibility with standardGPS equipment, most flight simulator vision systems, navigation system models,etc. Obviously, any parameter set in the world and other environment modelsapplies to all components in the aircraft model.

2.2.2 The atmosphere model

The second environmental object in Figure 2.4 is the atmosphere. Normally,the International Standard Atmosphere (ISA) as a function of the height aboveMSL is used. Alternatively, parameters for constant atmospheric conditions maybe entered. The air mass is nominally assumed to be in rest with respect tothe ECEF, explaining why a connection with the world ECEF-connector exists.However, the component also foresees implementation of wind fields. Currently,wind components in northern and eastern directions may be entered at a reference


altitude of 100 ft above the Earth surface. A simple Earth boundary layer modellogarithmically reduces the wind velocity to zero on the ground.

2.2.3 The terrain model

To the right in Figure 2.4 a terrain model has been added. A component contain-ing highly detailed, or simple parametrised models of the Earth’s surface may beselected from the library. Depicted in Figure 2.4 is a terrain model as used forautomatic landing control law design and certification, based on EASA CS-AWOspecifications [37]. The location, elevation, direction, and slope of a runway maybe specified, as well as slopes and steps in the terrain below the approach path. Asimple function call from e.g. an aircraft sensor then returns the corresponding lo-cal terrain elevation above MSL or the WGS84 ellipsoid, allowing for computationof for example the radio altitude.

2.2.4 The airport infrastructure model

The airport object implements earth-fixed navigational equipment (e.g. VOR,DME, ILS systems at specified locations). In the figure the ILS equipment ofthe one runway as positioned in the EASA CS-AWO terrain model is included.Specific characteristics like glide slope angle and antenna transmitter positionsmay be specified via parameters. Any other model object may obtain its localglide slope and localiser deviation via a simple function call.

2.2.5 Rigid and fl exible aircraft models

The core of the model structure is of course the component that represents theactual aircraft. The Flight Dynamics Library foresees the implementation of rigidjust as well as flexible ones. A typical model structure for a flexible transportaircraft is shown in Figure 2.5. The components resemble physical parts an aircraftconsists of (airframe, engines, actuators, sensors), and phenomena it is influencedor driven by (kinematics, aerodynamics, wind).

Component interconnections

The one-to-one implementation of the depicted physical objects and phenomenais enabled by the physical nature of the interconnections. Each component hasone or more mechanical connectors. Such a connector defines a physical attach-ment point on the object. For the engines and sensors this is the point at whichit is attached to the airframe. Aerodynamic forces actually act all over the air-frame. However, in flight mechanics usually only the summed effect with respectto some reference point, like the Aerodynamic Centre (AC), is of interest. Also forthe aerodynamic forces in the aeroelastic aerodynamic model in fact only theirsummed effect is considered due to so-called left-generalisation with rigid andflexible eigenmodes, as will be described in Sections 3.1 and B.2.7. Therefore, theaerodynamic models need a single connector only. The variables contained in the


Figure 2.5: Structure of a Flexible Aircraft model in Figure 2.4

connectors are given in Table 2.1. In this table Fl is the local frame of reference

Name Description TypeFl Force vector in Fl flowMl Moment vector in Fl flowR0 Position vector in FE C I acrossTl,0 Direction cosine matrix from FE C I into Fl acrossΩl Rotational speed w.r.t. FE C I , resolved in Fl across

T a b le 2.1 : Variables in the mechanical connector

the connector variables are defined in. The position and orientation of Fl withrespect to FE C I (the Earth-Centred Inertial frame of reference) are defined by R0

and Tl,0 respectively. Remember that by the inclusion of the world model at the


top-level, see Figure 2.4, FECI has been defined as the basic reference frame for allmodel components. Although Ωl in the connector is superfluous (orientation androtational speed are kinematically directly related), the variable is included forefficiency reasons as explained in [98]. The reasons for using Tl,0 to describe theorientation of Fl are that the description does not become singular (as opposed tofor example Euler angles when θ = ±π / 2) and that within the object the matrixis ready for use to transform local variables between the local and ECI referenceframes. Again, a more elaborate explanation is given in [98].The forces and moments in the connector are declared as flow variables, whereaskinematic variables are declared as across variables. The difference between bothtypes is best illustrated via an example, see Figure 2.6. The three numberedobjects are linked at the connector of object 3. The across variables in the threeconnectors are now related as follows:

R01= R02

= R03

Tl,01= Tl,02

= Tl,03

Ωl1 = Ωl2 = Ωl3

This simply implies that the local reference frames in the three objects are merged:they have the same position and orientation. The flow variables are related asfollows:

Fl1 + Fl2 + Fl3 = 0

Ml1 + Ml2 + Ml3 = 0

The basic interpretation is that the sum of inflowing and outflowing quantitiesmust be zero. In this case, the relations basically imply the balance of forces atconnector C3. Note that no direction of variables is assumed: the interconnectionsjust add two types of connector equations.

1 2

3C 1

C 2

C 3

Figure 2.6: Interconnection example of three objects

Besides mechanical interconnections, Figure 2.5 also shows a Databus on top, towhich most components have been connected. This type of interconnection willbe discussed at the end of this section.


Kinematics

The backbone of the model depicted in Figure 2.5 are the k inematics and airframe

components. The first defines a “North-East-Down” (NED) local vertical framewith its origin moving with a fixed position in the aircraft, preferably the centreof gravity. The object also defines a right-handed body-fixed reference frame withits origin at the same location, but with a fixed attitude w.r.t. the airframe (x-axis towards the nose, z-axis down). The attitudes and inertial positions of bothreference systems are available in the two connectors. The one on top representsthe aircraft body reference system (Fl (Table 2.1) is Fb), the one belows representsthe vehicle-carried NED reference frame (Fl is Fv). For the kinematics blockvarious versions are available or in preparation, like 6 Degrees of Freedom (DOF)with Euler angles and WGS84 co-ordinates as states, as well as versions with 3DOF (longitudinal or lateral), Q uaternion states, or Earth-fixed position states.

A irframe

The difference between a rigid and a flexible aircraft is, in fact, only in the airframe

object. In case of a rigid airframe, it contains the standard Newton-Euler forceand moment equations with respect to a body reference system [15] (attitudeand position in lower connector). Although the origin of this reference system ispreferably the centre of gravity (for compatibility with standard flight dynamicsmodels), a fixed point w.r.t. the undeformed airframe shape may be more usefulfor referencing reasons. The local gravity acceleration is obtained by a call to theworld object (Figure 2.4). Note that the computation of gravity depends on themethod that is selected in the world object. In case of a flexible airframe, linearelastic equations of motion in modal form augment the Newton-Euler equations[137, 21, 106]. The body axes system is hereby considered as a so-called meanaxis system. The momentary shape of the airframe is characterised by states inthe form of generalised co-ordinates (also called mode shape multipliers). Theunderlying data (modal mass, damping, stiffness, and mode shape matrices) areautomatically read from a specified file prior to simulation. More details onnonlinear equations of motion of flexible aircraft can be found in Appendix B.

Connection of the airframe object to the k inematics object (see Figure 2.5) makesthat the reference systems in both connectors merge, i.e. from then on the airframeis moving freely with respect to the inertial reference, according to kinematicequations described in the k inematics object.

The airframe object has a second connector on top (see Figure 2.5). This con-nector may contain a different reference frame with a constant offset, or maysimply be identical to the body frame (to be specified via an offset parameter). Itis intended for interconnection of for example external force model components,sensor models, etc. In case of a flexible airframe also generalised co-ordinatesand generalised forces are contained in addition to the variables in Table 2.1.Note that the kinematic equations that describe the position and orientation ofthe local reference frames are contained in the k inematics object. Applicationof Newton’s second law in case of a constant airframe mass relates the external


forces with the acceleration of the airframe by:

∑Fb = Tb,0m

d2R0

dt2(2.2)

where∑

Fb is the sum the external forces w.r.t. body axes acting on the airframevia both connectors. It is sufficient to implement the equation in this form: themodel compiler will symbolically differentiate R0. For more details, see Ref. [98].

External forces and moments

The airframe equations of motion are primarily driven by aerodynamic and propul-sion forces and moments. These are computed in corresponding model compo-nents in Figure 2.5. These components often need to be prepared for each aircrafttype individually, since application rules and data (sources) behind aerodynam-ics and propulsion models may strongly differ. For this reason, a base class isavailable that already defines interfaces and the connector, as well as equationsfor computation of key variables like the angle of attack, side slip, and true air-speed. Local wind velocities are hereby requested from the atmosphere model inFigure 2.4. The user may develop own model components, inheriting this baseclass.Besides the airframe, each component may be developed around its own local ref-erence frame. In case of aerodynamics, these may for example be the stability orwind axes. Interconnection with the airframe follows via a transformation object(e.g. AeroR ef in Figure 2.5). This object has two connectors representing tworeference systems. The offset (position, orientation) in between may be specifiedvia parameters that become visible and can be edited by double-clicking on theobject. The object also relates the forces and moments that act along the con-nector reference systems. When connecting a model with the airframe object, thetransformation object makes sure that the kinematics between the local compo-nent and the airframe reference systems are correctly related, as well as forcesand moments are applied correctly.The aircraft model in Figure 2.5 has two aerodynamics models (right hand side).The upper one (Aero) contains forces and moments as induced by the over-allmotion of the aircraft (“rigid aerodynamics”), usually also corrected for quasi-steady deformation of the airframe. The underlying model may be based oncomplex application rules, table look-ups, etc. In case a data set is not avail-able or incomplete, computational tools as described in [58] are available. Thelower aerodynamics component (aerodynamicsF lex) computes unsteady (gener-alised) forces and moments as induced by flexible deformation of the airframe.For this component extensive pre-processing tools have been developed, involvingapplication of the Doublet-Lattice Method, axis transformations, Rational Func-tion Approximation and removal of quasi-steady effects (already accounted forin the rigid aerodynamics model), see Chapter 3 and [59] for more details. Theunsteady aerodynamic data are read from a user-specified data file at simulationstart.Note that the Aero component is connected to the lower airframe connector via


the AeroRef object, whereby the latter describes the offset between the airframebody axes and the aerodynamic reference system. The upper aerodynamics com-ponent is directly connected with the upper airframe connector, making use ofgeneralised co-ordinates declared therein. In case kinematics and the balancebetween aerodynamic and actuation forces are relevant, a direct interconnectionbetween the actuators and aerodynamics models may be added.

The engine models (top left) are connected to the airframe via a slightly differenttype of transformation. Instead of an offset, the number of a structural gridpoint, where the object is to be attached, may be specified. At simulation start thetransformation object requests the rows of the modal matrix that apply to the gridpoint from the airframe object, allowing it to continuously compute the kinematicrelation and force balance between its connectors as a function of the offset fromthe airframe reference and the local deformation, see also Appendix B. This forexample implies that directional thrust variations due to local deformation at theengine attachment point are automatically taken into account.

S ensor models

The very same principle as used for interconnecting engine models with the air-frame structure also applies to the sensor models, located in the top-right cornerof Figure 2.5. A set of sensor types is available in the library. For example,accelerometers compute local accelerations at their point of attachment (speci-fied via grid point number, or offset) as a function of the inertial motion of theairframe, its position in the airframe reference, as well as the local airframe de-formation.

The ILS, GPS, and radio altimeter sensors obtain their values by making a func-tion call to the airport, world, and terrain environment models respectively (Fig-ure 2.4), passing on their momentary inertial position as an argument. In thisway, for example multiple GPS sensor objects may be included at various loca-tions on the airframe. Each object can request its very local co-ordinates fromthe world object.

L ocal wind eff ects

As already discussed in Section 2.2.2, mean winds are computed in the atmo-sphere block at the top level of the model in Figure 2.4. However, turbulencemodels are usually described in aircraft body axes, whereby delays as gusts travelalong the airframe, are taken into account. This is described in the localW ind ob-ject (lower left, in this case based on EASA CS-AWO specifications for autolandassessment). Random turbulence velocities are obtained from dedicated filters(Dryden, K arman) that use white noise signals as inputs. This noise is providedvia an external connector.

S ystems

On-board systems are included in the actuators component. This componentmay describe actuators and hydraulic / electric systems using simple transfer

2.3 The Modelica Flight Dynamics Library 37

functions, as well as highly detailed physical models, constructed from hydraulicsand electronics libraries. The library currently only provides the first variant, sincedetailed on-board system models are unique for each type or family of aircraftand are usually provided by systems specialists. A recent example of on-boardsystem model implementation using Modelica can be found in [11].

Avionics bus

Finally, the thin bar at the top of Figure 2.5 represents a so-called data bus. Thisbus includes signals that one would typically find on avionics buses in the aircraft,like the readings of all sensors, command signals to engine and control surfaceactuators, gear status, etc. For this reason, the sensor, actuator, and enginemodels have been attached to the bus object. The bus is also accessible fromoutside and allows direct connection to elements from the Modelica block diagramlibrary. This enables a control system composed using this library to directlycommunicate with the aircraft data bus. When connecting a model componentthe user is asked which variable is to be inserted into, or retrieved from the databus. A large amount of commonly used variables has been pre-defined, but theuser is of course free to add more. It is important to point out that in case ofa bus, any signal connector may be connected. For example, the G P S and ILS

objects via their connections contribute their respective outputs to the D atabus.In case of standard connectors however, both connectors have to be of the sametype (e.g. mechanical as defined above).

2.3 The Modelica Flight Dynamics Library

The top-level structure of the Flight Dynamics Library is depicted in Figure 2.7.The Modelica branch in the depicted tree (top) contains the principal standardlibraries delivered with Modelica (as listed in Section 2.1). The branches of theFlight Dynamics Library will be briefly described below:

• ProjectTemplate contains a basic library structure for an aircraft. Eachaircraft type has its own models for aerodynamics, propulsion, systems,landing gears, etc. These models are built on base classes that already com-pute all basic variables (e.g. for aerodynamics, angle of attack, calibratedairspeed, etc.) and are stored in this structure. The project template con-tains a very simple, but readily working aircraft model. The user may copythis template into an own project and start implementing aircraft-specificcomponents, or add components from the Flight Dynamics or any otherModelica library.

• A erod y n amics contains example aerodynamic models for use in rigid andflexible aircraft models, as well as base classes that the user may extend(inherit) to develop his own aircraft-specific model components. Each air-craft type or family namely tends to use unique application rules. For this


reason, newly implemented aerodynamics components are stored within theaircraft project (see ProjectTemplate).

• Airframes contains rigid and flexible airframe model objects. The rigidones may have constant mass and inertia tensor (entered via parameters),or these may change at a given rate (e.g. as a function of fuel consumption).The flexible airframe component loads its mass, and modal data from anexternal file (usually a Matlab mat-file [80]).

• E nv ironment contains all environment-related models as described at thebeginning of Section 2.2.

• E x amples contains actual implementations of aircraft models, as for ex-ample depicted in Figures 2.4 and 2.5.

• G ear currently contains a simplified landing gear model for which basicproperties may be set and which may be attached to the airframe object inFigure 2.5. A base class containing a standard interface for interconnectionwith the airframe is provided for implementation of detailed landing gearmodels, e.g. composed with help of the multi-body library by specialists inthe field.

• Interfaces contains all library-specific connector types, as well as the databus that was discussed Section 2.2.5.

• K inematics contains the Kinematics object as described in the previoussection.

• Propu lsion contains, as for the aerodynamics, example engine model im-plementations, as well as base classes that allow the user to implement hisown propulsion models.

• S ystems mainly contains sensor models (accelerometers, ILS, GPS, etc.)with time constants and noise if desired, and simple transfer function-basedactuator models.

• Transformations contains standard transformations between reference sys-tems.

• Types contains type definitions for internal variables, to which the usermay add his own.

• U tilities contains miscellaneous functions e.g. for reading external datafiles.

Within a dedicated Modelica modelling and simulation environment, like Dymola(Dynamic Modelling Library [4]) aircraft models may be composed from the li-brary using drag and drop, see Figure 2.8. After copying a component into the

2.4 Automatic code generation 39

Aircraft project library

M od elicaF lig h t D yn am ics L ibrary

M od elica s tan d ardm u lti-en g in eerin glibrary

Figure 2.7: Top-level structure of the Flight Dynamics Library

mod el, its parameters may be ed ited by d ouble-click ing on the object. For ex am-ple, in case of the fl ex ible airframe the number of mod es to be consid ered maybe altered , as w ell as the w ay of hand ling remaining mod es (e.g. truncation orresid ualisation).

2.4 Automatic code generationA fter mod el composition has been fi nished , a mod el translator sorts and solves allmod el eq uations accord ing to specifi ed inputs and outputs into O rd inary Diff er-ential E q uations (O DE ’s) or Diff erential A lgebraic E q uations (DA E ’s), suitablefor use in simulation. This is thus the automated transition from level 3 to level4 in Figure 2 .1 . A mod elling tool that is w ell capable of d oing this is Dymola [4 ].B esid es a graphical mod elling environment and ad vanced symbolic algorithms,the tool off ers ex tensive simulation and d ata analysis capabilities. H ow ever, themod el cod e may be used in other engineering environments and simulation toolsas w ell, lik e for ex ample M atlab/ S imulink [8 1 ]. For this environment an ad d i-tional tool set has been d eveloped that automatically generates trimming andlinearisation scripts, allow ing the user to easily specify and accurately computeinitial cond itions prior to simulation, see A ppend ix A .A simple w ay of specifying mod el inputs and outputs is show n at the top of Fig-ure 2 .4 . H ere a so-called avionics block has been connected to the bus connectorof the aircraft. A t this main mod el level, also input and output connectors havebeen d efi ned . The avionics block injects pilot throttle and control surface inputcommand s (from Throttle, C ontrols connectors) into the d ata bus. O utput vari-ables of interest, in this ex ample case navigation and aird ata signals are read from


Figure 2.8: Graphical aircraft model composition using drag and drop from thelibrary

the bus and passed on to output connectors NavOut and A ird ata respectively.

More details on model translation and automatic generation of inverse modelswill be discussed in Chapter 4.

2.5 Application example: ATTAS

This section discusses the implementation of an example model using the FlightDynamics Library. The aircraft is DLR ’s fly-by-wire test bed ATTAS (AdvancedTechnologies Testing Aircraft System [17 ]), a V FW -6 14 small passenger aircraft(30 passengers) with two turbofan engines mounted on top of the wings, seeFigure 2.9 . This model was originally developed in the frame of the EU projectR EAL [110 ] for automatic landing control laws design and is the bases for forflight control law designs described in Chapters 5 and 6 of this thesis.

2.5 Application example: ATTAS 41

Figure 2.9: DLR’s VFW-614 Advanced Technologies Testing Aircraft System(ATTAS)

2.5.1 Starting the model projectFrom the Flight Dynamics Library a new aircraft model project is started bycopying the aircraft model template (Figure 2.7) and saving it under a new name,in this case ATTAS. This new project is a small library by itself, containing allaircraft specific model components as well as a first integrated model, ready fortranslation. For ATTAS, the following model components need to be developedor adapted from default implementations:

• Aerodynamics: based on a base class that defines equations for typical aero-dynamic variables like angle of attack, angle of side slip, true airspeed, Machnumber, etc. ATTAS-specific application rules and aerodynamic coeffi cientdata are implemented [105]. In addition, for the aerodynamic model multi-plicative uncertainty is added to the coeffi cients in order to cover potentialdifferences with the actual aerodynamics;

• P ropulsion: the ATTAS is equipped with Rolls-Royce turbofan engines.Starting from an engine base-class, application rules for thrust computa-tion and equations of motion for engine shaft dynamics are implemented,including effects such as hysteresis in the fuel control unit;

• Sensors: the ATTAS is equipped with a range of sensors for air data, in-ertial measurements (accelerometers, etc.), guidance (ILS, Radio altitude),etc. These sensors are combined into a sensor system model, composed ofcomponents readily available in the Flight Dynamics Library.

• Actuators: the ATTAS is amongst others equipped with ailerons, rudder,elevators, and trimable horizontal stabiliser. These surfaces are driven by


actuators, which have been modelled as first order transfer functions withrate and position limits.

N o landing gear model is included, since within the scope of the REAL projectsimulations are stopped as soon as the first main gear touches the ground. Thecomposed model is shown in Figure 2.10. N ote that the structure is very similarto Figure 2.5. ATTAS only has two engines and is modelled as a rigid airframe,so that a single rigid body suffices to represent its inertial properties. The enginesof the VFW-614 are mounted on top of the wings, resulting in strong influence onthe air flow over the horizontal tailplane. This explains the direct links betweenthe engine and aerodynamics model components in Figure 2.10.

Figure 2.10: Implementation of the ATTAS model


2.5.2 Aircraft-specific model components

Sensors available for use by the control laws are collected in the object in the top-right corner. The contents of this component are shown in Figure 2.11. To theright, two signal vectors are built: yMeas, containing signals available for feedback,and ySim containing signals for analysis only. All signals have been specified andgrouped accordingly in Table 2.2. Both signal vectors are put on the bus (top andto the right). Most of the individual signals are readily available on the bus (e.g.,true and calibrated airspeed are released to the bus by the Aerodynam ics object)and are sent through first-order filters with time constants between 0.1 and 0.3seconds, representing simplified sensor dynamics. A more physical approach isused for the ILS and altitude sensors. For example, the glide slope sensor (topIL S object) is attached near the nose of the aircraft. The grey connector F ref

to the left is attached to an airframe reference point (Figure 2.10). The objectG sR ef describes kinematic relations and force balance between the reference pointand the sensor attachment point (this component is from the Modelica MultibodyDynamics Library [98]). The glide slope sensor in this way knows its local inertialposition and attitude and submits these to an airport environment model, whichwill be discussed shortly. The latter returns the local glide slope signal, which ispassed on by the sensor model to the yMeas vector. In order to represent sensorlag and disturbances, the signal is filtered and corrupted with noise as specified inEASA Certification Specifications – All Weather Operations [37]. Similarly, thelocaliser sensor obtains the signal in the rear of the aircraft. The guidance sensormodels are aircraft independent and readily available from the Flight DynamicsLibrary.

The top-level of the model is depicted in Figure 2.12. The ATTAS componentrepresents Figure 2.10. Below are the environment models. As discussed before,the w orld model provides the inertial reference, gravity models, etc. For this modelthe Earth is assumed to be flat and non-rotating, which is specified via a simplemenu, see Figure 2.13. The terrain model is a simple parametrised one, whichagain is specified for use in automatic landing certification [37]. Any componentof the aircraft model may interrogate the terrain model to obtain for example itslocal height above the Earth surface (e.g. radio altimeter). The airport modelcontains runway properties, like direction, slope, length, etc., as well as navigationequipment, like ILS transmitters. As mentioned before, ILS sensor models submittheir local inertial position and become their very own local ILS indications inreturn.

2.5.3 Specification of inputs, outputs, and parameters

The model depicted in Figure 2.12 can, after translation, be used for nonlinearsimulation. The inputs are shown to the left. These include noises (n), com-manded control surface and throttle deflections (uc), and other inputs (uo, e.g.stabiliser setting). The C ontrols component collects all inputs and puts themon the data bus. As can be seen from Figure 2.10, the engines and systems are


Figure 2.11: Implementation of the ATTAS sensor suite

connected to the data bus as well, allowing them to grab the appropriate inputs.The avionics component in Figure 2.12 collects sensor and other signals of interestfrom the data bus and makes them available as model outputs (right). Completelists of inputs and outputs of the ATTAS model are provided in Tables 2.2 and 2.3for future reference.

The data behind the various model components has been obtained from compu-tational methods and (flight test) experiments. Many of the parameter valueshave tolerances, even up to 30% . Also, the runway and terrain models are ofparametric nature to allow for Monte Carlo analysis-based control law certifica-tion. For future reference, all parameters are listed in Tables 2.4 and 2.5. Bydeclaring these parameters at the top level of the model, they become accessiblefrom outside and may be modified prior to simulation.


Figure 2.12: ATTAS model for nonlinear simulation

2.5.4 Model translation

Once inputs, outputs and parameters have been specified, the model can be trans-lated into differential equations for simulation (Figure 2.1). The inputs and out-puts as selected above will result in the following set of ordinary differential andalgebraic equations:

x = f(x,u,p ) (2.3)

yMeas = hMeas(x,u,p )

yS im = hS im (x,u,p )

(2.4)

where x is the state vector (states are automatically selected, but the user mayenforce specific variables to be assigned as such), u is the input vector, contain-ing the entries of Table 2.3, yMeas and yS im are the aforementioned vectors of


Figure 2.13: Selection of parameters for the world model

measurement and analysis signals. The model compiler may provide the aboveequations in symbolic as well as algorithmic form (e.g. c-code).

From a control design point of view, numerical integration of the equations 2.3allows for nonlinear simulation analysis of the aircraft with control laws. However,this is not sufficient. For example:

• Accurate computation of initial conditions is extremely important for timedomain evaluation of the closed-loop system (e.g. step responses). There-fore, highly accurate methods and tools are required for trimming the modelin a given flight condition and for a given weight and balance configuration;

• Most control analysis methods are linear, requiring the model to be availableaccordingly. The nonlinear model equations may be linearised numericallyor symbolically.

Some control design-relevant aspects of aircraft modelling are discussed in Ap-pendix A, with examples based on the ATTAS model.

Besides regular simulation models, inverse models of ATTAS have been generatedas well. In Chapter 5 an application to generation of inversion-based control lawsis described.

2.6 Conclusions 47

2.6 ConclusionsIn this chapter a new physically-oriented structure for aircraft flight dynamicsmodels has been proposed. This structure applies to flexible just as well as rigidaircraft. Furthermore, the environment models allow for implementation of flightvehicles flying at a wide range of flight conditions from low to high speed andfrom ground level to large altitudes. For example, Earth curvature and rotation,may be fully taken into account. Based on the object-oriented modelling lan-guage Modelica, this structure could be implemented one-to-one in a graphicalmodelling environment. Re-usable components and base components have beenorganised in a library structure, allowing for drag-and-drop model construction.Modelica comes with a large array of discipline-specific libraries for electronics,hydraulics, multi-body systems, etc. Due to the common language base, modelcomponents can be constructed from these libraries and integrated in a singlemulti-disciplinary aircraft model.From a flight control point of view, various kinds of run time models for designanalysis can be generated from one and the same aircraft model implementa-tion. Examples, like nonlinear simulation models, (symbolic) linearised models,dynamic and static inverse models, etc. are given in Appendix A. As will be seenin the following chapters, this automation aspect saves considerable engineeringtime and guarantees consistency between various model forms.


Nr. Symbol Name Unit

M easu red (yMeas)

1 Vc as calibrated airsp eed m/s2 Vtas tru e airsp eed m/s3 Vg g rou nd sp eed m/s4 pb roll rate ra d /s5 qb p itch rate ra d /s6 rb yaw rate ra d /s7 nx long itu d inal total load factor∗ g8 ny lateral total load factor∗ g9 nz v ertical total load factor∗ g

10 φ roll ang le ra d11 θ p itch ang le ra d12 ψ h ead ing ang le (0 ≤ ψ < 2π) ra d13 χ track ang le (0 ≤ χ < 2π) ra d14 α ang le of attack ra d15 γ fl ig h t p ath ang le ra d16 VZ v ertical sp eed (p ositiv e d ow n) m/s17 εgld g lid e slop e d isp lacement sig nal A18 εlo c localiser d isp lacement sig nal A19 Hr a rad io altitu d e sig nal m20 Hbar o baro altitu d e sig nal m21 m total airc raft mass kg22 xC G x -p osition of C oG in Fac m

23 N1 mean of fan sh aft sp eed s %

Simu lation analysis (yS i m )

1 β sid eslip ang le ra d2 M M ach nu mber –3 F total th ru st N4 d F th ru st d iff erence betw een both eng ines N5 Hlg h eig h t of land ing g ears abov e g rou nd m6 D x -D istance from A / C R E F to ru nw ay th resh old m7 y y-off set of airc raft c g from centre line m

∗ L oad factors are p ositiv e in n ega tive d irections of Fb

Table 2.2: Output definitions

2.6 Conclusions 49

Nr. Symbol Name Unit in

1 δAcCommanded aileron deflection rad uc

2 δEcCommanded elevator deflection rad uc

3 δRcCommanded rudder deflection rad uc

4 δT T ailplane deflection rad uo

5 δT h r1cT hrottle command for engine 1 – uc

6 δT h r2cT hrottle command for engine 2 – uc

7 w nT u rbxW hite noise for x-turbulence in Fe – n

8 w nT u rbyW hite noise for y-turbulence in Fe – n

9 w nT u rbzW hite noise for z -turbulence in Fe – n

10 w ngld W hite noise for glide slope disturbance – n

11 w nloc W hite noise for localiser disturbance – n

12 αdist (Sinusoidal) alpha sensor disturbance rad n

13 N1dist W hite noise for engine fan shaft speed % n

Table 2.3: Input definitions


Nr. Symbol Name Unit in1 m total aircraft mass

(k now n p arameter: ou tp u t 21 in T able 2.2) kg pa

2 xC G C entre of g rav ity x -p osition in Fac

(k now n p arameter: ou tp u t 22 in T able 2.2) m pa

3 T0 T emp eratu re at mean sea lev el (M SL ) K pe

4 WX3 3 W ind (33ft abov e g rou nd ) in Fe x -d irection m/ s pe

5 WY 3 3 W ind (33ft abov e g rou nd ) in Fe y-d irection m/ s pe

6 HW S Starting h eig h t of w ind sh ear abov e ru nw ay m pe

7 WSx Sp ec ifi c w ind sh ear in Fe x -d irection m/ s pe

8 TWSxT ime constant for sp ec ifi c w ind sh ear s pe

9 xP O Srw y Inertial x -p osition of ru nw ay th resh old m pe

10 Hrw y H eig h t of ru nw ay th resh old abov e sea lev el m pe

11 ψrw y H ead ing of ru nw ay ra d pe

12 γrw y Slop e of ru nw ay ra d pe

13 Lrw y L eng th of ru nw ay m pe

14 γg ld IL S g lid e slop e ang le ra d pe

15 hg g ld G lid e slop e reference h eig h t m pe

16 ∆ εlo c D isp lacement of localiser sig nal A pe

17 γt r Slop e of terrain before ru nw ay – pe

18 Ht r T errain h eig h t step before th resh old m pe

19 Xt r x -p osition of h eig h t step w .r.t. th resh old m pe

20 ∆ CD u ncertainty lev el of CD – pu

21 ∆ CLhu ncertainty lev el of CLh

– pu

22 ∆ CY u ncertainty lev el of CY – pu

23 ∂ ε∂ α unc

u ncertainty lev el of ∂ ε∂ α

– pu

24 ∂ ε∂ α g runc

u ncertainty lev el of ∂ ε∂ α g r

– pu

25 ∆ CLg r u ncertainty lev el of CLαg rand CLF , g rh

– pu

26 ∆ Cl0u ncertainty lev el of Clβ

, Clβ,lg, and Clβ, g r

– pu

27 ∆ Cm0u ncertainty lev el of Cm0

, Cmlg, CmM a

, Cm0, g r, and Cmβ

– pu

28 ∆ Cn0u ncertainty lev el of Cnβ

, Cnβ,lg, and Cnβ,g r

– pu

29 ∆ Clp u ncertainty lev el of Clp and Clp, M a– pu

30 ∆ Clr u ncertainty lev el of Clr and Clr,α – pu

31 ∆ Cmq u ncertainty lev el of Cmqw band Cmq,lg

– pu

32 ∆ Cnp u ncertainty lev el of Cnp and Cnp,α – pu

33 ∆ Cnr u ncertainty lev el of Cnr , Cnr,lg, and Cn

β– pu

34 ∆ Clδau ncertainty lev el of Clδa

and Clδa,α– pu

35 ∆ Clδru ncertainty lev el of Clδr

– pu

36 ∆ CLδehu ncertainty lev el of CLδeh

– pu

37 ∆ Cnδau ncertainty lev el of Cnδa

– pu

38 ∆ Cnδru ncertainty lev el of Cnδr

– pu

39 ∆ Ix u ncertainty lev el of Ix – pu

40 ∆ Iy u ncertainty lev el of Iy – pu

41 ∆ Iz u ncertainty lev el of Iz – pu

42 ∆ Ixz u ncertainty lev el of Ixz – pu

43 b lpl back lash of p ow er lev er ra d pu

44 δAmax max imu m aileron rate ra d / s pa

45 δAmax max imu m aileron d efl ection ra d pa

46 δEmax max imu m elev ator rate ra d / s pa

47 δEmi nminimu m elev ator d efl ection ra d pa

48 δEmax max imu m elev ator d efl ection ra d pa

49 δRmax max imu m ru d d er rate ra d / s pa

50 δRmax max imu m ru d d er d efl ection ra d pa

Table 2.4: ATTAS parameter definitions

2.6 Conclusions 51

Parameter Minimum Default Maximum Unit1 m 16500 18500 20500 kg2 xCG -0.6950 (0.22 c) -0.7898 (0.25 c) -0.8845 (0.28 c) m3 T0 219.15 (-54C) 288.15 (15C) 328.15 (55C) K4 WX33 -15.432 (30kts Head) 0 5.144 (10kts Tail) m/s5 WY 33 -10.288 (20kts R ight) 0 10.288 (20kts Left) m/s6 HW S 3.048 (10ft) 9.144 (30ft) 30.34 (= 100ft) m7 WSx ± 2.572 0 ± 5.144 m/s8 TWSx

2 6 10 s

9 xPOSrwy 0 0 0 m10 Hrwy 0 0 2800 m11 ψrwy 0 0 2π rad12 γrwy -0.01 0 0.01 rad13 Lrwy 1700 4000 4500 m14 γgld 2.85 π

18 03 π

18 03.15 π

18 0rad

15 hggld 15.24 (= 50ft) 16.5 (= 54.1ft) 18.288 (= 60ft) m

16 ∆εloc -5·10−6 0 5·10−6 A17 γtr -0.02 0 0.15 –18 Htr 0 0 12.192 m19 Xtr 0 0 200 m20 ∆CD -0.1 0 0.1 –21 ∆CLh

-0.1 0 0.1 –22 ∆CY -0.3 0 0.3 –23 ∂ε

∂α unc-0.1 0 0.1 –

24 ∂ε∂α grunc

-0.1 0 0.1 –

25 ∆CLgr -0.3 0 0.3 –26 ∆Cl0

-0.3 0 0.3 –27 ∆Cm0

-0.1 0 0.1 –28 ∆Cn0

-0.3 0 0.3 –29 ∆Clp -0.3 0 0.3 –30 ∆Clr -0.3 0 0.3 –31 ∆Cmq -0.3 0 0.3 –32 ∆Cnp -0.3 0 0.3 –33 ∆Cnr -0.3 0 0.3 –34 ∆Clδa

-0.3 0 0.3 –

35 ∆Clδr-0.3 0 0.3 –

36 ∆CLδeh-0.1 0 0.1 –

37 ∆Cnδa-0.3 0 0.3 –

38 ∆Cnδr-0.3 0 0.3 –

39 ∆Ix -0.1 0 0.1 –40 ∆Iy -0.1 0 0.1 –41 ∆Iz -0.1 0 0.1 –42 ∆Ixz -0.3 0 0.3 –43 blpl 3.5 π

18 03.5 π

18 03.5 π

18 0rad

44 δAmax 75 π18 0

75 π18 0

75 π18 0

rad/s45 δAmax 25 π

18 025 π

18 025 π

18 0rad

46 δEmax 30 π18 0

30 π18 0

30 π18 0

rad/s47 δEmin

−20 π18 0

−20 π18 0

−20 π18 0

rad48 δEmax 15 π

18 015 π

18 015 π

18 0rad

49 δRmax 25 π18 0

25 π18 0

25 π18 0

rad/s50 δRmax 25 π

18 025 π

18 025 π

18 0rad

Table 2.5: ATTAS parameter ranges


Chapter 3

Integration of rigid andaeroelastic aircraft modelsusing the residualised modelmethod

54 Integration of rigid and aeroelastic aircraft models

Abstract

This chapter describes a procedure for development of an integrated model

of a fl ex ible aircraft by combining available fl ight dynamics and aeroelastic

models. B oth types of models largely complement each other. H owever,

two overlaps are typically present that should be properly taken care of.

In the fi rst place, rigid aircraft fl ight dynamics models usually already

take the deformation of the airframe quasi-statically into account in the

aerodynamics computation. Second, aeroelastic models also contain so-

called rigid-body modes that correspond with the fl ight dynamics degrees

of freedom. The adopted solution is to leave the fl ight dynamics model un-

changed, and to remove rigid modes and the quasi-static infl uence on the

fl ight dynamics from the aeroelastic model. This so so-called “ residualised

model method” is extended to properly handle so-called unsteady aerody-

namic lag states. As a spin-off result, it will be shown how the proposed

procedure can be used to correct rigid body dynamics in an aeroelastic state

space model.

C o n tributio n s

• A general procedure for integrating fl ight mechanics and aeroelastic

models of aircraft, based on available industry-standard data sources;

• P roper handling of unsteady aerodynamic eff ects in aeroelastic mod-

els for integration with flight mechanics models;• A s a sp in-off resu lt: a correction method to imp rove the flight me-

chanics degrees of freedom within u nsteady aeroelastic state spacemodels u sing a linearised flight mechanics model.

Publication

G ertjan L ooye: Integration of rigid and aeroelastic aircraft models using

th e residualised model meth od, International F oru m on A eroelasticityand S tru ctu ral D ynamics (IF A S D ), M u nich, G ermany , J u ne 2 0 0 5 .

Integration of rigid and aeroelastic aircraft models 55

FLIGHT dynamics and aeroelastic models play an important role in the air-craft design process. The fi rst are among others used for dev elopment of

the fl ight control law s, handling q uality assessment and, in comb ination w ith dis-trib uted aerodynamic force and mass models, for manoeuv re loads analysis. Infl ight dynamics (F D ) models the airframe is mostly assumed to b e rigid, or todeform q uasi-statically as a function of the fl ight state and control surface defl ec-tions. A eroelastic (A E ) models describ e complex unsteady interactions b etw eenairframe structural dynamics and the airfl ow . These models are among othersused for computation of fl utter margins and gust loads analysis, as w ell as designof structural control law s. A lthough fl ight dynamics and aeroelastic models aredev eloped b y separate engineering disciplines, there has long since b een a grow inginterest in dev elopment of integrated models. The references [1 3 7 , 4 1 , 1 1 3 , 6 7 , 4 5 ]as w ell as the impressiv e w ork b y N A S A -Langley in this fi eld [2 1 , 1 8 , 2 0 , 8 ] areonly a few ex amples. O ne reason is that aircraft tend to get more and more fl ex -ib le due to lighter and increasingly slender airframe designs. A s a conseq uence,neglecting of structural dynamics on fl ight dynamics and loads, or measures lik eapplication of low -pass fi lters on sensor signals used b y the fl ight control system(F C S ), may no longer do. A nother reason is that tak ing structural dynamics intoaccount early on in the design of the fl ight control law s allow s direct trade-off s tob e made b etw een e.g. control b andw idth and fl ight loads or fl utter margins. Thisw ill av oid the possib le need for re-tuning of control law s on req uest of the loadsand fl utter departments afterw ards. Integrated models further allow for design ofintegrated aeroelastic and fl ight control law functions.

A s a b asic principle, dev elopment of integrated aircraft models should use agreed-on components from the inv olv ed engineering disciplines, rather than each disci-pline to dev elop its integrated models from scratch. This not only sav es eff ort,b ut more importantly, it prev ents inconsistencies b etw een v arious models, w hichmay hamper rather than improv e the ov er-all design process. This b asic principleis the motiv ation for the w ork as describ ed in this chapter.

The prob lem is, giv en a fl ight dynamics and an aeroelastic model, how to properlycomb ine these into an integrated aircraft model. C urrent practice is that fl ightdynamics models are b ased on nonlinear N ew ton-E uler eq uations of motion for arigid b ody and aerodynamic forces and moments are computed using coeffi cienttab les and application rules as a function of the aerodynamic fl ight state. The co-effi cient tab les are ob tained from handb ook methods, C F D analyses, w ind tunnelex periments, and ev entually, from fl ight test. A eroelastic models are b ased on lin-ear structural eq uations as resulting from modal analysis of a fi nite element model.C omputation of unsteady aerodynamic loads is mostly done in the freq uency do-main for the harmonically deforming airframe using the D oub let-Lattice method(D LM ) [6 ]. S ince results are ob tained in the freq uency domain, sev eral methodsare av ailab le for approx imation in the Laplace domain, allow ing for use in timesimulation [5 7 , 1 0 9 , 1 3 3 ]. A lthough aeroelastic and fl ight dynamics models arecomplementary, there are tw o major ov erlaps. In the fi rst place, the fl ight dynam-ics aerodynamic model nearly alw ays already tak es the q uasi-steady deformationof the airframe as a function of the fl ight state into account. To this end aero-


dynamic coefficients are corrected using so-called flex factors. Aeroelastic modelson the other hand, usually contain so-called rigid body modes as resulting frommodal analysis of the unrestrained airframe structure. Including these modes inthe Doublet-Lattice computation results in an aerodynamic model for rigid andflexible airframe motion, as well as interactions in between. Consequently, theAE model includes flight dynamics-related states and rigid aircraft (unsteady)aerodynamics.

Simply leaving out the flex factors and rigid modes from the flight dynamics andaeroelastic models respectively is generally not the best option. The flex factorsmostly result from trim computations using CFD methods that give more accu-rate results in steady flow conditions as compared with DLM. Furthermore, flightdynamics models, including the flex factors, are eventually matched with flighttest data as soon as these become available. Changing these models thereaftershould be avoided. Finally, leaving the quasi-steady effects to the aeroelasticmodel builds in a dependency on the number of modes that is included in thestructural model. On the other hand, rigid aerodynamics in AE models as re-sulting from DLM are not as accurate [113], since the method is less suited forpredicting drag and steady forces.

Alternative approaches have therefore been proposed by W inther et al [140] andby K onig and Schuler [114]. B oth methods have in common that, besides an add-on term in case of the first approach, the flight dynamics model is left untouched.The difference is in the underlying assumptions and in the implementation. An in-depth analysis and comparison is given in R ef. [108]. The K onig-Schuler methodapplies a modal transformation that replaces rigid modes in the AE model withthe quasi-flexible flight dynamics model states. The integration method as pro-posed in this chapter originates from the work by W inther. The methodologybasically states that the flight dynamics model is a fully flexible one with allaeroelastic dynamics quasi-statically residualised as a function of flight dynamicstates and control inputs. This implies that when “ re-implementing” aeroelasticdynamics, only dynamic increments – without the quasi-steady contribution– needto be added. Therefore, as a proper name for this approach Residualised Model(RM) method is proposed.

As presented in R ef. [140] the method has two limitations: (1) the presentedaeroelastic model structure does not include aerodynamic lag states resultingfrom the aforementioned R ational Function Approximation (R FA) of the unsteadyaerodynamics, and (2) the aeroelastic model is treated in its state space form. Thelinear nature of the latter limits validity of the integrated model. For example,the dynamic pressure in the AE model remains fixed and inertial coupling ef-fects are not easily included. Therefore, in this chapter the model integrationwill first be discussed at the level of aerodynamic models and the equations ofmotion, facilitating future incorporation of detailed interaction effects. At theend the application to aeroelastic state space models in line with R ef. [140] willbe discussed. Consequently, the chapter is structured as follows. In Sections 3.1and 3.2 some relevant basics of flight dynamics and aeroelastic models will bereviewed. In Section 3.3 the integration of equations of motion and aerodynamic

3.1 Review of aircraft fl ight dynamics models 57

model components will be discussed. In Section 3.5 the application to state spacemodels is described, including a spin-off result for correcting rigid body modesin aeroelastic state space models. Application examples are given in Sections 3.4and 3.6. Finally, conclusions are drawn in Section 3.7.

3.1 Review of aircraft fl ight dynamics models

This section is intended to give a brief overview of the structure of aircraft flightdynamics models, and to introduce relevant symbols and notations simultane-ously.

3.1.1 Equations of motion

Flight dynamics models are mostly based on Newton-Euler six degrees-of-freedomequations of motion for a rigid body. The equations consist of force and momentequations with respect to a body-fixed reference system Fb, preferably with itsorigin in the centre of gravity (see for example Ref. [124]):

m

Vb + Ωb × Vb − Tbv(Θ)Gv

= FAb+ Fother (3.1)

IbΩb + Ωb × IbΩb = MAb+Mother (3.2)

where m and Ib are the aircraft mass and inertia tensor respectively, Vb and Ωb aretranslational and rotational velocities along the body axes. The components ofthe gravity acceleration vector Gv are transformed from a vehicle-carried vertical(Fv) into body axes (Fb) using the cosine matrix Tbv. This matrix is a function of

the Euler attitude angles Θ = [φ , θ , ψ]T. The equations are driven by external

aerodynamic (FAb, MAb

) and other (Fother, Mother, like thrust, gear) forces andmoments respectively. For notational simplicity it is here assumed that an earth-fixed axis system Fe may be considered as an inertial reference. The vehiclecarried frame Fv is parallel to Fe, but its origin moves with the aircraft centre ofgravity (CoG).

Kinematic relations between Euler angular rates and body rates are given by:

ΘT =[

φ , θ , ψ]T

= Tφ b(φ , θ ) Ωb (3.3)

where the matrix Tφ b depends on the momentary attitude. Alternatively, quater-nion equations may be used [124]. The velocity vector in Fb and Fv are relatedby:

Re = TTbv(φ , θ ,ψ)Vb (3.4)

where Tbv is the rotation matrix from Fv into Fb and Re is the position vector ofthe aircraft CoG in Fe.


3.1.2 External forces and momentsIn (3.1) the external forces and moments have been grouped into aerodynamicand other ones. P art of the latter are thrust forces. These are computed fromengine models, which complexity may vary from a constant scaling on throttlecommands to accurate thermodynamic and mechanics equations. The momentcontribution of thrust mainly arises due to offsets of the engines from the CoG.The aerodynamic forces and moments are of most interest here. Although eachflight dynamics group tends to have its own standard for implementation, mostaerodynamic models have a form like:

FAb= qS Tba(α, β )

CX

CY

CZ

a

MAb= qSc Tba(α, β )

Cl

Cm

Cn

a

+rar p ×FAb(3.5)

where q is the dynamic pressure, S is the wing reference area, c is a reference length(e.g. mean aerodynamic chord, or half wing span), CX , CY , CZ are dimensionlesstotal force coefficients in the direction of the aerodynamic reference system Fa

(e.g. wind or stability axes), and likewise, Cl, Cm, Cn are the moment coefficients.The origin of the aerodynamic reference system Fa is the Aerodynamic ReferenceP oint (ARP ). The vector rar p is the position vector of the ARP with respect tothe CoG and Tba transforms the forces and moments from Fa into Fb.The aerodynamic coefficients are usually computed from polynomials with theaerodynamic flight state (Vtas , α, β , ...), Mach number M , body angular rates,control surface deflection, etc. as variable parameters. The polynomial coefficientsare obtained from look-up tables, as a function of e.g. the angle of attack andMach number. The application rule for CZ may look something like:

CZ = CZ0(α,M ) + CZq (α,M )

qc

Vt a s

+ CZUU + CZ

UU + CZα

(α,M )αc

Vt a s

+ ... (3 .6 )

where q and U are pitch rate and control deflections respectively. Wind andturbulence influences (VW ) are taken into account via the aerodynamic flight statevariables, which are airflow referenced. The coefficients either implicitly includethe influence of quasi-steady deformation of the airframe (e.g. when tuned tomatch flight test data), or coefficients like CZ0

and CZUare explicitly corrected

using so-called flex factors. These factors in turn may have been formulated as afunction of vertical load factor (nz), dynamic pressure, etc [36].The dependency of (3.6) on rates of angle of attack (α) and control deflections(U) represents unsteady flow effects. The most important is that changes in thedown wash angle from the wing and wind disturbances are felt with a delay atthe horizontal tail plane, see for example Ref. [36].Finally, in this chapter the state vector of the flight dynamics model will be calledxR:

xTR =

[

xTRb, xT

Re

]

with: xTRb

=[

V Tb , ΩT

b

]

and xTRe

=[

RTe , ΘT

]

(3.7)

3.2 Review of aeroelastic aircraft models 59

3.2 Review of aeroelastic aircraft modelsThis section is intended to give a brief overview of aeroelastic aircraft models.Aspects that are relevant for model integration in the following section will behighlighted. A more elaborate version can be found in Appendix B.

3.2.1 Equations of motion

xs

zs

xb

zb

C o G

Vta s

z

A R P

ra rp

Gv

Fs

Fb

w

Figure 3.1: Orientation of body and structural x- and z-axes in AE model

A structural airframe model is assumed to be available, consisting of elasticallyinterconnected grid points with condensed lumped masses / inertias attached tothem. The degrees of freedom of all grid points (g-set) are assumed to be w.r.t.the structural reference system Fs, see Figure 3.1, and are collected in the vectorxg (dimension ng). The linear equations of motion are initially written in thefrequency domain as follows:

−ω2Mgg +Kgg

xg = FgA+ Fgother

(3.8)

where Mgg and Kgg are mass and stiffness matrices for the system of intercon-nected grid points, and FgA

and Fgotherare aerodynamic and other external forces

applied to these points. Based on this equation modal analysis is performed invacuo (FgA

= Fgother= 0) and without constraints (i.e. “free-free”), resulting

in nh eigenfrequencies and associated mode shape eigenvectors. Nearly alwaysnh < ng, since the computation is halted as soon as eigenfrequencies reach valueswell beyond the frequency range of interest. The eigenvectors are collected in thecolumns of the modal matrix Φgh. Six of the eigenfrequencies are zero, corre-sponding with the so-called “rigid-body” modes. The vector xg may be writtenin modal co-ordinates ηh (h-set) as follows:

xg ≈ Φghηh =[

ΦgR, ΦgE

]

[

ηR

ηE

]

(3.9)

where the subscripts R and E refer to rigid and elastic modes respectively. The≈-sign is used while mostly nh < ng, but will be considered as an equal sign in


the remainder of this chapter. If necessary, the rigid modes are modified as follows:

(1) Translational modes are corrected to involve unit translations (1 m), and,in case the corresponding generalised co-ordinates ηx, ηy, ηz are one, the air-frame moves forward, downward, and to the right respectively;

(2) Rotational modes are corrected to involve unit rotations (1 rad) around theCoG, and, in case the corresponding generalised co-ordinates ηφ, ηθ, ηψ are one,the airframe rolls right-wing down, pitches nose-up, and yaws nose to the rightrespectively.

In this way, positive entries of ηR make the aircraft move in the directions ofcorresponding axes of Fb (Figure 3.1).

Substitution of (3.9) into the linear structural equations of motion (3.8) resultsin the following right-generalised form:

−ω2 [MgR, MgE ]︸︷︷︸

Mgh

+jω [BgR, BgE ]︸︷︷︸

Bgh

+ [KgR, KgE ]︸︷︷︸

Kgh

[ηRηE

]

= FgA+ Fgother

(3.10)

where Mgh, Bgh, Kgh are the half-generalised mass, damping and stiffness ma-trix respectively, and have been partitioned into columns that apply to rigid andflexible mode shape co-ordinates. Damping is usually assigned per mode, basedon estimated or experimental values. It is common practice to transform theequations from structural into body axes, giving rise to linearised Euler and grav-ity terms in the otherwise zero-valued partitions BgR and KgR respectively. Theright-generalised form is important for loads analysis, since it describes the dis-tribution of the various forces over the airframe. The generalised co-ordinatesare solved from the modal differential equation as obtained by left-multiplicationwith ΦTgh (partitioned as in (3.9)):

−ω2

[MRR 0

0 MEE

]

︸︷︷︸

Mhh

+j ω

[BRR 0

0 BEE

]

︸︷︷︸

Bhh

+

[KRR 0

0 KEE

]

︸︷︷︸

Khh

[ηR

ηE

]

=

[

ΦTgR

ΦTgE

]

(FgA+ Fgother

)

(3 .1 1 )

The matrices MEE , BEE an d KEE are d iag o n al d u e to o rtho g o n ality o f the mo d esw ith resp ect to the mass an d stiff n ess matrices Mg g , Kg g . F o r the same reaso n ,the o ff -d iag o n al b lo ck s in Mh h an d Kh h are zero . D u e to the ad ap ted rig id mo d eshap es the p artitio n MR R has the to tal aircraft mass an d in ertia ten so r o n itsd iag o n al: MR R = d iag (m I3, Ib).

3.2 Review of aeroelastic aircraft models 61

3.2.2 External forcesAs for the FD model, the external forces in (3.10) have been divided into aerody-namic and other ones, like local gear and thrust loads. The aerodynamic forcesare assumed to be in the following form:

FgA= q [QgR(k, M), QgE(k, M)]

︸︷︷︸

Qg h(k,M )

[ηR

ηE

]

+ qQgX(k, M)ηX (3.12)

The matrices Qgh(k, M) and QgX(k, M) are unsteady aerodynamic loads on thegrid points, induced by airframe motion and deformation, and control defl ectionsηX

1 respectively. For notational brevity, gust inputs are considered to be in-cluded in ηX . These loads have been computed for a limited number of reducedfreq uencies k = ωc/ V ta s , and at a given M ach number M using the Doublet L at-tice method. For later use in the model integration process it is important thatηR is body referenced (see remark below (3.10)). This has some implications onQgR(k, M), as explained in S ection 3.A.In order to allow for time simulations, the aerodynamic loads are fi rst approxi-mated in the L aplace domain using a rational function approximation (R FA). Thisinvolves least-sq uares fi tting of a transfer function form as proposed in e.g. R efs.[5 7 , 109 , 133] to available matrices Qgh(ki) and QgX(ki) (with i = 1, 2, · · · , n k)2.For later use it will be most helpful to perform the approximation for QgR

andQgE

either separately, or to partition the R FA result afterwards. This is explainedin S ection 3.B . The aerodynamic loads eventually end up in the form3:

QgR(s) ≈ A1g R

cVt a s

s + A2g R

c2

V 2

t a s

s2

+ DgR(sI − RRVt a s

c)−1ERs

QgE(s) ≈ A0g E+ A1g E

cVt a s

s + A2g E

c2

V 2

t a s

s2

+ DgE(sI − REVt a s

c)−1EEs

QgX(s) ≈ A0g X+ A1g X

cVt a s

s + DgX(sI − RXVt a s

c)−1EXs

(3.13)

N ote that A0g Rshould be zero, since attitude angles and position physically do

not induce aerodynamic forces (see S ection 3.A). Also, A2g Xhas been left out from

the approximation here, since second derivatives of control and wind inputs arenot always available from the actuator or wind models respectively. Dependingon the applied method, the diagonal of R.. may contain a single set of so-calledlag-poles and a full E -matrix [5 7 ], or the same set of poles repeated as many timesas the column dimension of Qg... [109 , 133].

1The apparently strange notation ηX originates from the aeroelastic ity d om ain: here itis com m on practice to consid er control su rface d efl ections ju st as a spec ial m od e of airfram ed eform ation.

2R ecently, a far m ore effi c ient approach has b een d ev eloped , inv olv ing a single R F A only, seeR ef. [5 9 ].

3The approx im ation is v alid for the M ach nu m b er M at w hich Qgh / gX(k ,M) w as com pu ted .This d epend ency is left ou t for notational sim plic ity.


The expressions in (3.13) may be generalised by left-multiplication with ΦgR andΦgE , resulting in RR, RE, ER, RX and EX partitions of generalised aerody-namic forces Qhh(s) and QhX(s) respectively. For example:

QEE(s) =

A0EE+ A1EE

c

Vtas

s + A2EE

c2

V 2tas

s2 + DEE(sI −

Vtas

cRE)−1EEs

(3.14 )

Note that since A0gR= 0 (eq. (3.13)) A0RR

and A0ERwill be zero as well.

Finally, for use in the following section the right most terms in (3.13) are realisedin state space form as follows:

xLE= RE

Vtas

cxLE

+ EE ηE so that: QgElag= DgExLE

(3.15)

where xLEare so-called lag states. The realisations of QgR and QgX are similar.

3.3 Integration of fl ight dynamics and aeroelasticmodels

In the previous sections rigid aircraft flight dynamics (FD) and aeroelastic (AE)airframe models, as developed and applied in the respective engineering disci-plines, have been reviewed. The objective of this chapter is to develop an in-tegrated model of the flexible aircraft, by combining data and components fromthese models4. The advantage of this integration approach is that agreed-on com-ponents from both disciplines are used. A problem is that, as discussed in theintroduction, a number of overlaps has to be taken care of. In the previous sectionsequations of motion, and external forces and moments have been discussed indi-vidually. Accordingly, in this section the integrated equations of motion will bediscussed first. Then the integration of external (especially aerodynamic) forcesis described, using the so-called residualised model (RM) method.

3.3.1 Equations of motionIn Appendix B and Refs. [137, 21] it is shown that the nonlinear equations ofmotion of a flexible aircraft may be obtained by simply combining (3.1) and thelower partition of (3.11):

m

Vb + Ωb × Vb − Tbv(φ , θ , ψ )Gv

= FAb+ QREt

+ Fo th e r

IbΩb + Ωb × IbΩb = MAb+ QREr

+ Mo th e r

MEE ηE + BEE ηE + KEEηE = QER + QEX + QEE + QEother

4It is assumed that, at least at initial conditions, the AE and FD models are valid for thesame flight condition, aircraft confi guration, weight and balance, etc.

3.3 Integration of fl ight dynamics and aeroelastic models 63

(3.16)

where QER, QEX and QEE are time-domain versions of generalised aerodynamicforces as in (3.14). The terms QREt

and QRErare deformation-induced forces

and moments affecting translational (subscript t) and rotational (subscript r)dynamics of the aircraft. These will be discussed in the following sub-section.Finally, QEother

= ΦTgEFgother

.Note that the equations only couple via external forces and moments. This isachieved by the choice of mean body axes, which attitude and translation describethe mean motion of the airframe, and by making the additional assumptions thatthe inertia tensor Ib is constant and that deformation and deformation rates arecollinear [137]. In case these assumptions are not valid, inertial coupling termsmust be included as a.o. derived by Buttrill et al. [21] and Reschke [106].Mean body axes are located such that relative momentum due to deformation ofthe airframe is zero at all times. This nonlinear constraint is diffi cult to satisfy.H owever, in case free-free modal analysis has been performed, a body referencesystem with its origin attached to the momentary C oG and with its axes in thedirections of the translational rigid mode shapes, automatically fulfils so-called“ practical” or linearised mean axis constraints [22]. The reason is that thesepractical constraints are basically equivalent to orthogonality of flexible w.r.t.rigid modes. This is the case in the models as described in Section 3.2, causingthe off-diagonal partitions of Mhh and Khh in Eq. (3.11) to be zero. The rigidmodes were re-oriented in the directions of the body axes Fb, without affectingorthogonality with respect to flexible modes. As a result, the body axes Fb asused in the FD model may simply be regarded as mean axes, and the state vectorxR of (3.1) describes the mean over-all motion of the aircraft. Furthermore, xRb

and ηR may be related as follows:

ηR = δxRb=

[δVb

δΩb

]

ηR = δxRb=

[δVb

δΩb

]

(3.17)

where δ means a deviation from the initial condition. Similarly, control and windinputs and external forces in both models are related as:

ηX = δU ηX = δ ˙U QRother= ΦT

gRFgother=

[δFother

δMother

]

(3.18 )

W ind inputs are not included explicitly here, but these are handled exactly thesame way as ηX . AE models may have many more wind inputs than FD models,allowing wind disturbances to hit (groups of) DLM panels at different delayedtime instances [56].


3.3.2 External forces and moments: the residualised modelmethod

In the previous sub-section the overlap of rigid degrees of freedom between FDand AE models has been removed by combining the FD equations of motionand the elastic degrees of freedom of the AE generalised equations of motion.Compared with (3.1) in (3.16) the elastic-rigid coupling terms QREt

and QREr

have been added. At this point a more subtle overlap between the AE and FDmodels arises, since FAb

and MAbalready take quasi-steady deformation of the

airframe into account. For reasons given in the introduction, the intention is toleave the FD model unchanged as far as possible. The key in thus removing theoverlap from the AE model is in the following assumption:

The quasi-steady effects in FAband MAb

are equivalent to

statically residualised aeroelastic dynamics.

This implies that the quasi-steady contributions of QREt,rshould be computed

and subtracted. This idea was first published by Winter et al. in Ref [140]. Inthis chapter an extension is made to AE models that contain aerodynamic lagstates resulting from an RFA.

As a first step, the quasi-static deformation of the airframe as a function of rigidmotion, control deflections, and non-aerodynamic external generalised forces iscomputed from (3.11) and (3.14 – including remarks), with sηE0

= s2ηE0= 0:

KEEηE0= QEother

+ qA0EEηE0

+ [ A1ER

cVtas

s + A2ER

c2

V 2tas

s2

+ DER(sI − Vtas

cRR)−1ERs ]ηR

+ [A0EX+ A1EX

cVtas

s + DEX(sI − Vtas

cRX)−1EXs ]ηX

(3.19)

Note that O NLY the quasi-steady value of ηE is sought. Consequently, theunsteady term driven by elastic deformation disappears (sηE0

= 0), but thosedriven by rigid motion and control deflections (including wind disturbances) have

to be retained. Solving for η0, substituting (3.17) and (3.18), and rewriting in thetime domain with the help of (3.15) then gives:

ηE0= (KEE − qA0EE

)−1 · (QEother(3.20)

+ qA1ER

c

Vtas

δxRb+A2ER

c2

V 2tas

δxRb+DERxLR

(3.21)

+ qA0EXδU +A1EX

c

Vtas

δU +DEXxLX)

where xLRand xLX

are lag states driven by rigid dynamics, and control surfaceand wind inputs respectively. In (3.11) the flexible to rigid coupling is then given

3.4 Application example 65

by:

QRE(s)ηE = A0REηE +A1RE

c

Vtas

sηE +A2RE

c2

V 2tas

s2ηE (3.22)

+ DRE(sI −Vtas

cRE)−1EEsηE

Subtracting the effect of ηE0the coupling terms in (3.16) are now formulated as

follows:

QRE =

[QREt

QREr

]

= q

A0RE(ηE − ηE0

) +A1RE

c

Vtas

ηE +A2RE

c2

V 2tas

ηE +DRExLE

Here xLEare lag states resulting from state-space realisation of the unsteady term

in (3.22) using (3.15).At this point it may have become clear why it is helpful to have elastic dynamics todrive a separate set of lag states xLE

. The coupling term DRExLEnow involves

unsteady flow effects as caused by structural dynamics only. U nsteady effectsinduced by rigid motion and control deflections are namely already accounted forin MAb

and FAbin the FD model, for example using the α and β derivatives in

(3.6). Furthermore, the computation of ηE0in (3.20) may use the very same lag

states as applied for QER and QEX respectively.The dependency of the unsteady aerodynamics forces on the Mach number hasbeen ignored in this section. In nonlinear simulation however, this effect shouldbe accounted for. As already indicated in Equation (3.12), the aerodynamicforces are computed over a grid of Mach numbers and reduced frequencies. Thedependence on the latter is accounted for by the Rational Function Approximationat a given Mach number (Eqn. (3.13)). The recommended approach is to developinterpolation schemes between Rational Function Approximations of Qgh(k,M)and QgX(k,M) at various values of M .A summary of the integrated model equations is presented in Table 3.1. Thevery same principle as sketched above may be applied to augment flight loadsmodels with aeroelastic dynamics. To this end the (elastic part of) the AE modelin right-generalised form as in (3.10) and (3.13) is required. The aerodynamiccontribution QgE is similar to (3.23).

3.4 Application exampleThe Residualised Model method has been applied to various large transport air-craft to develop fully flexible simulation models for loads analysis and real-timeflight simulation. The results however are confidential. For this reason, in thissection the method will be applied to an example aircraft for which model datais publicly available, see Ref. [137]. The model has been implemented in theobject-oriented modelling language Modelica, using the Flight Dynamics Library


Equations of motion:

m

Vb + Ωb × Vb − Tbv(φ , θ , ψ)Gv

= FAb+QREt

+Fother

IbΩb + Ωb × IbΩb = MAb+QREr

+Mother

MEE ηE +BEE ηE +KEEηE = QER + QEX +QEE +QEother

G ene ralise d ae rod y namic forc e s:

QRE = q

A0RE(ηE − ηE0

) +A1RE

c

Vt a sηE +A2RE

c2

V 2t a s

ηE +DRExLE

QEE = q

A0EEηE +A1EE

c

Vt a sηE +A2EE

c2

V 2t a s

ηE +DEExLE

QER = q

A1ER

c

Vt a sηR +A1ER

c2

V 2t a s

ηR +DERxLR

QEX = q

A0EXηX +A1EX

c

Vt a sηX +DEXxLX

ηE0= (KEE − qA0EE

)−1

QER + QEX +QEother

FAb, MAb

computed from (3.5)Fother, Mother and QEother

computed from engine, gear models, etc.

L ag state s:

xLR= RR

Vt a s

cxLR

+ ERηR

xLE= RE

Vt a s

cxLE

+ EE ηE

xLX= RX

Vt a s

cxLX

+ EX ηX

N ote: R.. and E.. are usually sparsely occupied, which should be exploited in the imple-mentation.

K inematic re lations:

[

φ , θ , ψ]T

= Tφ b(φ , θ )Ωb Re = TTbv(φ , θ , ψ)Vb ηR =

[δVb

δΩb

]

ηR =

[δVb

δΩb

]

O th e r re lations:

ηX = δU ηX = δ ˙U QRother= ΦT

gRFgother=

[δFother

δMother

]

Table 3.1: Combined equations of the integrated flexible aircraft model

3.5 Coupling using aeroelastic state space models 67

described in Chapter 2. For the flight dynamics model an aerodynamic modelincluding static aeroelastic eff ects (i.e. flight shape) is av ailable, based on look -uptables for coeffi cients and according application rules5. A lso for the engines andactuation systems detailed models hav e been implemented, see Figure 2.5 . H ow -ev er, in order to see the eff ect of step control inputs, the actuators are by-passedin the follow ing analyses. For the airframe aeroelastic data are av ailable in theform of modal data (4 symmetrical modes), as w ell as an unsteady aerodynamicmodel (w ithout lag states, though)6. T he integrated model has been implementedaccording to T able 3 .1 .For v alidation of the R M method, the follow ing trim condition is chosen:

• T rim: straight and lev el flight;

• A ltitude: 1 5 0 0 m abov e M S L ;

• M ach number: 0 .5 0 .

In the follow ing analysis, only longitudinal dynamics w ill be considered. T heaircraft has elev ators for pitch control, as w ell as v anes near the cock pit, w hichare used for structural control (especially to reduce fuselage bending), see forexample [1 4 1 ].T he aircraft responses due to elev ator excitation (step input) are depicted inFigure 3 .2. Clearly, the flight dynamical behav iour of the aircraft w ith respectto its centre of grav ity is hardly changed by addition of the aeroelastic dynamics.T he short period mode is relativ ely w ell damped at this flight condition. O fcourse, it is more interesting to look at airframe-referenced v ariables. Figure 3 .3depicts the v ertical load factor and pitch rate responses of an airframe structuralpoint near the cock pit. Fuselage bending is clearly excited. T he low er plot inthis fi gure show s one of the excited modes w ith its mean value, w hich eff ect isremov ed from the flex-to-rigid coupling, see E qn. (3 .23 ).T he same responses for control v ane inputs are show n in Figures 3 .4 and 3 .5 . T hev ertical load factor response in the cock pit prov ides a qualitativ e check of themodel. S ince the control v ane produces a small upw ard force, the front fuselageshould accelerate upw ard as w ell, after w hich the response due to pitch accelera-tion and ov er-all v ertical acceleration follow . T his is clearly the case.

3.5 Coupling using aeroelastic state space models

In this section the R M method w ill be applied to state space A E models, in linew ith R ef. [1 4 0 ]. T his is useful in case the A E model is at hand in this form only.A lso a simple procedure can be deriv ed to match the rigid body dynamics w ith alinearised FD model.

5These tables can be found in [138]. Static aeroelastic effects are incorporated by enforcingη = η = 0 in the aeroelastic m odel eq uations

6This aeroelastic com ponent is based on slightly m odifi ed data from [137 ].


0 2 4 6 8 10 12 14 16 18 20−15

−10

−5Elevator input

δ E (d

eg)

0 2 4 6 8 10 12 14 16 18 20−10

0

10Pitch rate

q b (deg

/s)

Flexible ACRigid AC

0 2 4 6 8 10 12 14 16 18 200

10

20Angle of attack

α (d

eg)

0 2 4 6 8 10 12 14 16 18 200

1

2Load factor at CoG

time (s)

n z (−)

Figure 3.2: Aircraft responses due to elevator input: flight mechanical parameters

0 2 4 6 8 10 12 14 16 18 200

1

2

3Vertical acceleration near cockpit

n z (−)

Flexible ACRigid AC

0 2 4 6 8 10 12 14 16 18 20−0.1

0

0.1

0.2

0.3Pitch rate near cockpit

q (−

)

0 2 4 6 8 10 12 14 16 18 20−0.1

0

0.1

0.2

0.3Generalised co−ordinate 1

time (s)

η 1 (−)

η1

η10

Figure 3.3: Aircraft responses due to elevator input: structural dynamics


0 2 4 6 8 10 12 14 16 18 200

5

Control vane input

δ CV (d

eg)

0 2 4 6 8 10 12 14 16 18 20−5

0

5Pitch rate

q b (deg

/s)

Flexible ACRigid AC

0 2 4 6 8 10 12 14 16 18 200

5

10Angle of attack

α (d

eg)

0 2 4 6 8 10 12 14 16 18 200.5

1

1.5Load factor at CoG

time (s)

n z (−)

Figure 3.4: Aircraft responses due to control vane input: flight mechanical pa-rameters

0 2 4 6 8 10 12 14 16 18 200

1

2

3Vertical acceleration near cockpit

n z (−)

Flexible ACRigid AC

0 2 4 6 8 10 12 14 16 18 20−0.05

0

0.05

0.1

0.15Pitch rate near cockpit

q (−

)

0 2 4 6 8 10 12 14 16 18 200

0.05

0.1

0.15

0.2Generalised co−ordinate 1

time (s)

η 1 (−)

η1

η10

Figure 3.5 : Aircraft responses due to control vane input: structural dynamics


3.5.1 Aeroelastic state space modelStarting from (3.11) the AE model is first written in descriptor state space formas follows.

MRR 0 −A2RE0 0 0 0

0 I 0 0 0 0 0

−A2ER0 MEE 0 0 0 0

0 0 0 I 0 0 00 0 0 0 I 0 00 0 0 0 0 I 00 0 0 0 0 0 I

ηR

ηR

ηE

ηE

xLR

xLE

xLX

=

QRother

0QEother

0000

+ (3.2 3)

−BRR −KRR A1REA0RE

DRR DRE DRX

I 0 0 0 0 0 0

AER10 −BEE −KEE DER DEE DEX

0 0 I 0 0 0 0

ER 0 0 0 RR 0 0

0 0 EE 0 0 RE 0

0 0 0 0 0 0 RX

ηR

ηR

ηE

ηE

xLR

xLE

xLx

+

A1RXA0RX

0 0

A1EXA0EX

0 00 00 00 EX

[

ηX

ηX

]

with augmented mass, damping and stiffness matrices:

MEE = MEE − A2EEMRR = MRR − A2RR

BEE = BEE − A1EEBRR = BRR − A1RR

KEE = KEE − A0EEKRR = KRR − A0RR

R.. = R..Vt a s

cD.. = q D..

A0. .= q A0. .

A1. .= q A1. .

cVt a s

A2. .= q A2. .

c2

V 2t a s

(3.2 4 )

Again, it has to be made sure that elastic modes use a separate set of lag states. Ifnecessary the matrices are easily adapted to achieve this. The state space systemis completed by bringing all matrices to the right hand side. Having explainedthe structure of the system, in the following a simplified notation will be used:

ηR

ηR

ηE

ηE

xLR

xLE

xLX

=

ARR1 ARR0 ARE1 ARE0 ARLR ARLE ARLX

I 0 0 0 0 0 0AER1 0 AEE1 AEE0 AELR AELE AELX

0 0 I 0 0 0 0

ER 0 0 0 RR 0 0

0 0 EE 0 0 RE 0

0 0 0 0 0 0 RX

ηR

ηR

ηE

ηE

xLR

xLE

xLX

+

BR1BR0

0 0BE1

BE0

0 00 00 0

EX 0

[

ηX

ηX

]

+

BRQ 00 00 BEQ

0 00 00 00 0

[

QRother

QEother

]

(3.2 5 )


The RM method can now be applied the same way as in Section 3.3. First ηE0is

computed from the third row by setting ηE = ηE = 0:

ηE0= −A−1

EE0AER1ηR + AELRxLR

+ AELXxLX+ (3.26 )

BEX0ηX + BEX1ηX + BEQ QEother

The aeroelastic dynamics are described by the lower part of (3.25):

ηE

ηE

xLR

xLE

xLX

=

AEE1 AEE0 AELR AELE AELX

I 0 0 0 0

0 0 RR 0 0

EE 0 0 RE 0

0 0 0 0 RX

ηE

ηE

xLR

xLE

xLX

(3.27)

+

BE1BE0

0 00 00 0

EX 0

[

ηX

ηX

]

+

BEQ

0000

QEother+

AER1

0ER

00

δ xRb

where δ xRbfrom the FD model, instead of ηR is now used. The flexible to rigid

coupling then becomes a simple output equation of the state space system (3.27 ):

yRE =[

ARE1 ARE0 ARE0A−1

EE0AELR AELE ARE0A−1

EE0AELX

]

ηE

ηE

xLR

xLE

xLX

+ ARE0A−1

EE0

[

BEX1 BEX0

]

[

ηX

ηX

]

+ ARE0A−1

EE0BEQ QEother

+ ARE0A−1

EE0AER1δ xRb

(3.28)

where yRE is related to QRE in Eqn. (3.23) as follows:

yRE = M−1

RRQRE

N ote that there is no need to compute ηE0explicitly. In fact, provided that the

rigid degrees of freedom, as determined by ηR match the flight mechanical ones,as described by δ xRb

(see Section 3.2), the AE model only needs to be partitionedas in (3.25). Obtaining the equations (3.27 ) and (3.28 ) is then straight forward.

3.5.2 The RM method in state space formW ith the state space model (3.25) at hand, it is fairly easy to bring about acoupling with an existing FD model, requiring only minor modifications to thelatter. The interaction thus takes place at a higher level, between available FD andAE models, as sketched in Figure 3.6 . In Ref. [108 ] this coupling is compared withan alternative method as described in [114]. In case the addition of an aeroelastic


increment to the FD model is not desirable or possible, one approach is to includea linearised FD model in AE state space model (the procedure is described in thefollowing), and to use this to approximately compute the increments to the FDstate vector. This will not be further discussed in detail.

eq. (3.27,3.28)yRE

Nonlinear FlightM ec hanic s M od el(inc l. s tatic -flex ib ility )

L inearA eroelas tic

M od el

xR

.

C om p u tation of:- L oad s- A c c elerations- S ens or s ignals

U.

U

U0

.

U0

-

U.

U

-

, xR

xRb

xRb

xRb 0

E la s tic a n dla g s ta tes

Figure 3.6: High-level interconnection of AE and FD models

3.5.3 Correction of aeroelastic state space modelsIt is interesting to residualise (3.25) using (3.26), only leaving the rigid flightdynamics (including involved lag states) in the AE model:

ηR

ηR

xLR

xLX

=

BR1− ARE 0A

−1

E E 0BE X0 BR0

− ARE 0A−1

E E 0BE X0

0 00 0

EX 0

[

ηX

ηX

]

+

BRg

000

Fgo th er+

ARR1 − ARE 0A−1

E E 0AE R1 ARR0

I 0ER 00 0

(3 .2 9 )

ARLR − ARE 0A−1

E E 0AE LR ARLX − ARE 0A

−1

E E 0AE LX

0 0

RR 0

0 RX

ηR

ηR

xLR

xLX

In c a se a lin e a rise d fl ig h t d y n a m ic s m o d e l is a t h a n d , (3 .2 9 ) m a y b e u se d to c o rre c tth e A E m o d e l (3 .2 5 ) to m a tch th e rig id d y n a m ic s to th e F D m o d e l. S u p p o se th elin e a rise d F D m o d e l is g iv e n b y :

[

δxRb

δxRe

]

=

[

AF D 1 AF D 0

Ak in 1 Ak in 0

] [

δxRb

δxRe

]

+

[

BF D 1 BF D 0

0 0

] [

δU

δU

]

+

[

BF D o th er

0

] [

δFo th e r

δM o th e r

]

3.6 Application example (Continued) 73

(3.30)

w here the sparsely populated matrices Akin1 and Akin0 describe linearised k ine-matic relations. P rovided that (3.1 8 ) holds, comparison betw een (3.29) and (3.30)then leads to the follow ing corrections in (3.25):

A∗

RR1= AFD1 + ARE0A

−1

EE0AER1

A∗

RR0= AFD0

A∗

RLR = ARE0A−1

EE0AELR

A∗

RLX = ARE0A−1

EE0AELX

B∗

R1= BFM 1 + ARE0A

−1

EE0BE1

B∗

R0= BFM 0 + ARE0A

−1

EE0BE0

B∗

R0= BFM 0 + ARE0A

−1

EE0BE0

B∗

RQ = BFDother

B∗

REQ = ARE0A−1

EE0BEQ

So that eventually, the corrected aeroelastic state space system look s lik e:

δxRg

δxRe

ηE

ηE

xLR

xLE

xLX

=

A∗

RR1AF D 0 ARE1 ARE0 A∗

RLRARLE A∗

RLX

Ak i n 1 Ak i n 0 0 0 0 0 0AER1 0 AEE1 AEE0 AELR AELE AELX

0 0 I 0 0 0 0

ER 0 0 0 RR 0 0

0 0 EE 0 0 RE 0

0 0 0 0 0 0 RX

δxRg

δxRe

ηE

ηE

xLR

xLE

xLX

+

B∗

R1B∗

R0

0 0BE1

BE0

0 00 00 0

EX 0

[

ηX

ηX

]

+

BF D otherB∗

REQ

0 00 BQ E

0 00 00 00 0

[

QRother

QEother

]

(3 .3 1 )

T his model is w ell suited for aeroservoelastic analysis, since the rigid states di-rectly match the flight dynamics ones.

3.6 Application example (Continued)T he procedure for correcting the flight dynamics modes w ithin an aeroelasticstate space model is illustrated on the same aircraft ex ample as presented inSection 3.4 . Figure 3.7 show s the eigenvalues of three models: (1 ) the originalAE model (mark er ’x ’, Eq n. (3.25)), (2) the corrected AE model (mark er ’o’,Eq n. (3.31 )), and (3) the original FD model that w as used for correction of thefi rst (mark er ’∆ ’, Eq n. (3.30)). It is immediately clear that the correction ofthe AE model has no eff ect on the location of the eigenvalues associated w ithstructural dynamics (’x ’ and ’o’ mark ers match for poles w ith imaginary partslarger than 5). Figure 3.8 show s close ups around the origin. From this plot it is


clear that the eigenvalues of the AE model, associated with the flight dynamicsmodes, have moved to those of the linearised FD model (’o’ and ’∆’ markersmatch), as expected. As a consequence, the phugoid mode (right) is now correctlyrepresented by the integrated model, whereas in the AE model it was only presentin the form of two real poles very close to the origin. The main reason is thatdrag is not accounted for in the AE model.

The correction of the AE model becomes even more apparent from the examplefrequency response from elevator to pitch rate at a structural point near thecockpit (Figure 3.9). The left most peaks of the AE and FD models clearly donot match. From Figure 3.8 it can be seen that the underlying modes are theshort period and phugoid. As noted before, an important reason is the absence ofdrag in the AE model. The integrated model (Eqn. (3.31)) matches both peaksof the FD model (Eqn. (3.30)) and the structural dynamic peaks of the originalAE model (Eqn. (3.25)). It will be clear that the corrected AE model will bemore reliable for use in for example the design of structural or flight control lawfunctions.

−25 −20 −15 −10 −5 0

0

5

10

15

20

25Eigenvalues of original and corrected AE model

Real

Imag

originalcorrectedlin. FM

Figure 3.7: Eigenvalues of the (longitudinal) linear models

3.6 Application example (Continued) 75

−1 −0.5 0 0.5−0.5

0

0.5

1

1.5

2Eigenvalues of original and corrected AE model

Real

Imag


−0.02 −0.01 0 0.01 0.02−0.02

0

0.02

0.04

0.06

0.08

0.1Eigenvalues of original and corrected AE model

Real

Imag


Figure 3.8: Eigenvalues of longitudinal FD modes of the linear models

10−2

10−1

100

101

102

103

10−3

10−2

10−1

100

101

102

Frequency (rad/s)

mag

(q)

Frequency response: elevator to pitch rate in cockpit

Aeroelastic modelFlight mechanics modelIntegrated model

Figure 3.9 : Frequency responses of the linear models: elevator to pitch rate incockpit


3.7 Conclusions

In this chapter a method for integration of available flight dynamics and aeroe-lastic aircraft models has been proposed. The most important contribution hasbeen in the combination of the aerodynamics models, whereby only dynamic incre-ments are added to the otherwise unchanged flight dynamics equations of motion.Since the underlying assumption is that flight dynamics models include staticallyresidualised aeroelastic effects, the approach is called R esidualised M odel method.O riginally proposed in R ef. [140], the main contribution of this chapter is in thehandling of unsteady aerodynamic lag states. It is hereby important to makesure that no rigid dynamics unintentionally couple into the FD model via thesestates. As a secondary contribution, the integration of models has been discussedfor equations of motion and aerodynamics separately, allowing more detailed (e.g.inertial) coupling effects to be included later on. The approach is directly appli-cable to modelling for loads analysis.

A practical application and validation of the described modelling approach ispresented in the companion R eference [108], where also a detailed comparisonwith the so-called K onig-Schuler method [114] is made. The modelling approachhas been applied to several industrial aircraft models for flight loads analysis andforms a basis for model construction in DL R ’s M odelica Flight Dynamics L ibrary(C hapter 2).

It is important to remind that the validity of the integrated model is limited to thescope of its components. For example, the AE aerodynamic model is more or lessonly valid around the (subsonic) M ach number it has been computed at. Dynamicpressure variation is taken into account in the AE model, so that some form ofinterpolation between R FA matrices derived for a grid of M ach numbers will do.O ther limitations are that the DL M assumes that the angle of attack remainssmall. Since modal analysis is performed for a specific aircraft configuration andloading, the aircraft mass and C oG location should stay approximately constantduring simulation.

From the chapter it may have become clear that aeroelastic models may involvetremendous amounts of data. For this reason, a lot of effort has been and will beinvested in more effi ciently and transparently handling of this data, as well as thepre-processing for model use. This has resulted in the Dynamic Aircraft M odelIntegration Process (DAM IP), described in R ef. [59].

For now, the idea behind the model has been to leave the FM model unchanged asfar as possible, assuming that the FD model has been matched with experimentaldata and that the quasi-flexible influences in the FD model are equivalent toresidualised aeroelastic dynamics. A next step is to use aeroelastic methods andmodels to improve FD models in other aspects. This requires model integrationearlier on in the development process, before validation work has started.

3.7 Conclusions 77

Appendix 3.A: Rigid motion-induced unsteady aero-dynamic loadsAerodynamic models in aeroelasticity are often provided in a geodetic referencesystem. For use in integrated flexible aircraft models, a transformation intobody axes is then necessary. This transformation involves some inelegant “ re-engineering” and is derived in this appendix. In case the user has the possibilityto influence the modelling process at an earlier stage, it is strongly recommendedto directly generate body-referenced aerodynamics models from the start [59],making the approach described below obsolete.From Doublet Lattice computation a set of so-called Aerodynamic Influence Co-efficient matrices (AICs) at a number of reduced frequencies k is obtained. V iaso-called spline matrices the degrees of freedom of the grid points (g) are trans-ferred to control points on the panels, and the resulting aerodynamic forces onthese panels are applied to the grid points again. In the following, it is assumedthat these spline matrices have been included in the AIC, therefore writing AIC gg.Right multiplication with the modal matrix results in the half-generalised un-steady aerodynamic force matrix as a function of unit deflections of each mode:

Qgh (k, M) = [QgR(k, M), QgE(k, M)] = AIC gg(k, M) [ΦgR, ΦgE ] (3.32)

H owever, both the pitch (φy) and yaw (φz) mode shapes have to be corrected,such that only pitch and yaw effects, but no angle of attack respectively sideslip arise. These are namely already accounted for by the heave (φz) and lateraldisplacement (φy) mode shapes. In the following it is important to remember theadopted forms of the rigid mode shapes as described in Section 3.2.The pitch mode shape should be combined with the heave one, such that the heavevelocity in the CoG (ARP), and thus the angle of attack, equals zero. In case ofoscillatory pitching with magnification of the mode shape vector with ηθ(ω), thelocal angle at the CoG equals αp itc h (ω) = φy0

ηθ(ω) = ηθ(ω), with φy0= 1 rad

the corresponding angle in the mode shape vector. For notational brevity, it ishere assumed that the static angle of attack equals zero (if this is not the case,the derivation is very similar). The angle of attack due to heaving equals:

αh e a v e (ω) = −w(ω)

Vta s

= −jωz(ω)

Vta s

= −jωz0ηz(ω)

Vta s

=jω

Vta s

ηz(ω) (3.33)

where z0 = −1 is the z -displacement in the heave mode shape vector (positiveupwards, see Section 3.2), ηz is the mode shape multiplier, and w = z (in frequencydomain: w(ω) = jωz(ω)) the vertical velocity of the CoG in the reference frameFs, as depicted in Figure 3.1. The combined angle of attack at the CoG due topitching and heaving then must be zero:

αc o m b ine d (ω) = αh e a v e (ω)+αp itc h (ω) = 0 ⇐ ⇒jω

Vta s

ηz(ω)+ηθ(ω) = 0 (3.34)


so that the required amount of heave compensation as a function of the reducedfrequency equals:

ηz(ω) = −Vtas

jωηθ(ω) = −

c

jkηθ(ω) (3.35)

The same may be derived for the yawing motion, resulting in ηy = cj k

. The rigidmode shape matrix AICgg is to be right-multiplied with, becomes:

ΦR =

[

Φx, Φy, Φz, Φφ, Φθ −c

jkΦz, Φψ +

c

jkΦy

]

= ΦR

1 0 0 0 0 00 1 0 0 0 c

j k

0 0 1 0 −cj k

0

0 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

The matrix to the right may also be applied in case Qgh is already at hand. Inthis chapter with QgR(k, M) always the transformed version is meant. N ote thatthe derivation above may just as well be done with respect to the ARP instead,requiring additional heave and lateral components in the pitch and yaw modevectors to be compensated for, due to the offset rar p from the CoG.

Appendix 3.B: Separation of lag statesFor integration of flight dynamics and aeroelastic models it is necessary to makesure that structural dynamics drive a separate set of lag states xLE

. This canbe ensured by performing a separate RFA for QgE(k), or by partitioning theRFA result of Qgh(k) afterwards. In case of e.g. Roger’s approximation this iseasy, since each mode contributes its own lag states anyway. However, Karpel’sminimum state method uses a single set of lag states xLh

for all modes. In thatcase a separate set of lag states for the elastic modes can be added as follows.In the Laplace domain the unsteady contribution to the aerodynamic loads onthe grid points g is:

Qghlag= Dgh(sI −

Vtas

cRh)

−1 [ER, EE ] s

[

ηRηE

]

This equation is rewritten as follows:

Qghlag=

[

DgR DgE

]

(3.36 )[

(sI −Vt a s

cRR)−1 0

0 (sI −Vt a s

cRE)−1

] [

ER 00 EE

]

s

[

ηRηE

]

with RR = RE = Rh and DgR = DgE = Dgh. N ote that nothing has changed,except that in the state space realisation the elastic and rigid modes now use

3.7 Conclusions 79

independent lag states:[

xLR

xLE

]

=

[

Vtas

cRR 0

0 Vtas

cRE

] [

xLR

xLE

]

+

[

ER 00 EE

] [

ηRηE

]

QgRlag= DgRxLR

QgElag= DgExLE


Chapter 4

Rapid prototyping usinginversion-based control andobject-oriented modelling

82 Rapid prototyping using inversion-based control

Abstract

Object-oriented modelling allows for efficient construction of multi-

discip linary sy stem models in a p h y sically -oriented way . A s a unique

feature, th e modelling ap p roach allows for automatic generation of regu-

lar, static, as well as inverse simulation models. Inverse models form th e

basis of various commonly used nonlinear controller synth esis meth ods,

such as N onlinear D ynamic Inversion. T h e resulting multi-variable con-

trol laws are easily tuned to meet performance specifi cations and avoid th e

need for gain sch eduling. In combination with automatic inversion, th ese

synth esis meth ods are ideally suited for control law rap id-p rototy p ing, al-

lowing experimentation with command variables, control selection and al-

location, etc. in very sh ort automated design cycles. A fter selection of th e

fi nal arch itecture, th e control laws may be developed furth er in a detailed

design stage. T h is is demonstrated on a nonlinear control p roblem for an

aircraft manoeuvring on th e ground.

C o n tributio n s

• A general p rocedure for automatic generation of N onlinear D ynamic

Inversion fl igh t control laws from a symbolic aircraft model;

• A N onlinear D ynamic Inversion-based design p rocedure for rap id

p rototy p ing of fl igh t control laws in early design stages of a new

aircraft, or for rap id-p rototy p ing of new ty pes of fl igh t control law

functions.

P ublicatio n

G ertjan L ooye: Rapid prototyping using inversion-based control andobject-oriented modelling. In: N onlinear A naly sis and S ynth esis T ech -

niques for A ircraft C ontrol, series L ecture N otes in C ontrol and Informa-

tion S ciences, V ol. 3 6 5 , S p ringer V erlag, B erlin, A ugust 2 0 0 7 .

Rapid prototyping using inversion-based control 83

MANY nonlinear control law synthesis methods, like Nonlinear DynamicInversion (NDI), Feedback Linearisation (FL), Model Following Control

(MFC) and Inverse Feedforward Compensation (IFFC) are based on inverse modelequations. These equations directly compensate for nonlinear dynamic behaviourof the system, whereas desired command response behaviour is imposed using alinear outer loop feedback controller and command filters. The main advantage ofthese methods, disregarding any modelling errors, is that the control laws auto-matically achieve full decoupling of command variables (CVs) and automaticallyadapt as a function of all known parameters (e.g. flight condition), avoiding theneed for gain scheduling. This implies that, besides the effort of model inver-sion and coding, functional control laws are obtained for the complete operatingenvelope in one shot.

During the last decade, the methodology of object-oriented modelling for im-plementation of multi-disciplinary models has matured and found application inmany fields of engineering, such as mechatronics, electronics, automotive andaerospace engineering. The main point in object-oriented modelling is that phys-ical objects and phenomena, and their interactions, may be implemented one-to-one into model components, and components interconnections respectively. Modelimplementation takes place at a physical level, rather than in the form of sortedand solved differential (algebraic) equations, see Figure 2.1. The step from user-implemented model equations to differential equations for simulation is performedautomatically by a model compiler. To this end, reliable algorithms and tools arereadily available. The way differential equations are generated depends on theinputs and outputs selected by the user. This implies that the model implemen-tation is independent of its causality in simulation. In the frame of this work,the most interesting aspect is that this automation step allows for generation ofregular just as well as inverse simulation models.

The straight-forward approach of inversion-based control design and the capabilityof automatic model inversion allow for a highly interesting approach in early stagesof flight control law design: rapid prototyping. This is of great interest in variousaspects:

• early availability of an initial set of control laws, as soon as a first model isavailable;

• quick comparison of different command variables, up to the stage of pilot-in-the-loop flight simulations;

• quick re-design in case of model data updates (as is frequently the case inthe aircraft pre-design phase);

• early release of initial control laws to other engineering departments (e.g.loads);

• possibility of simulation-based formulation of requirements for control sizingin preliminary aircraft design;


• testing of individual control law functions in the detailed design phase withinor around a preliminary, but fully functional control system.

Since synthesis techniques like Feedback Linearisation already have proved theirvalue in many applications, these control laws may be naturally developed furtherin the detailed design phase, once key decisions like control variable selectionand control allocation have been made, and the aircraft configuration has beenfinalised.In this chapter the above approach is described and demonstrated on an aircraft-on-ground (i.e., taxiing, roll-out, and take-off run) nonlinear control problem. InSection 4.1 object-oriented modelling of flight dynamics is discussed. Sections 4.2and 4.3 describe the generation of simulation and inverse models, as well as howthe latter may be applied within various control law structures. Section 4.4 de-scribes the proposed rapid-prototyping design steps. Section 4.5 describes theirapplication to lateral-directional ground control laws. Conclusions are drawn inSection 4.6.

4.1 Object-oriented modelling of aircraft fl ight dy-namics

In this section a brief overview of object-oriented modelling and its application tomodel development for the aircraft-on-ground is given.

4.1.1 Object-oriented modellingImplementation of models and model components is usually performed in the formof ordinary differential equations (ODEs). This is done in software code (e.g. C,FORTRAN), but also more and more in the form of block-diagrams. The latterallows for easy organisation of generic components in libraries from which modelsmay be constructed graphically. The ODEs originate from physical equationsfrom which derivatives of the selected state variables and output variables havebeen solved. System components are interconnected using input-output-basedsignal flows.The basic principle of object-oriented modelling is that model implementationalready takes place at the level of the original physical equations behind systemcomponents, rather than at the level of sorted and solved differential equations,see Figure 2.1. This approach has the following advantages:

• system components may be interconnected according to physical interac-tions (energy flows, constraints), and are not limited to input-output rela-tions. This allows for true one-to-one implementation of physical objectsand phenomena into model software objects;

• the above advantage allows for development of engineering discipline-specificcomponent libraries, but still based on a common language base;

4.1 Object-oriented modelling of aircraft fl ight dynamics 85

• model code in executable form for simulation (i.e. differential equations) isgenerated automatically, based on user-specified inputs and outputs at thehighest hierarchical level of the model.

The second advantage provides an ideal basis for multi-disciplinary modelling,since each component of a model may be constructed using methods that arecommon place within the involved engineering area (i.e. an electronic circuit canbe implemented as a circuit diagram, a mechanical systems can be constructedusing multi-body formalisms, a control law may be implemented as a block dia-gram, etc.). This greatly improves model visibility, especially to engineers fromother disciplines. The third advantage implies that model implementation doesnot yet freeze model causality. This feature will be exploited in this chapter.

Example 4.1: The bicycle model

Throughout this chapter the concepts behind object-oriented modelling and automaticeq uation solv ing w ill be illustrated on a highly simplifi ed bicy cle approx imation of anaircraft tax iing on the ground [4 7 ]. This approx imation neglects the roll and pitchdegrees-of freedom and aerodynamic loads. F urthermore, the main gear of the aircraftis concentrated below the centre line of the fuselage and ty re forces are linear functionsof the ty re slip angles.The bicy cle model may be div ided into three components: the frame, the rear w heel,and the steerable front w heel. In order to k eep the example simple, also the acceler-ation in body x -direction is neglected (i.e. ub is constant) and mass is assumed to beconcentrated in the frame, resulting in the follow ing set of eq uations:

m (vb + ubrb) − Fyext= 0 (4.1)

Izz rb − Mzext= 0 (4.2)

w here Fyextand Mzext

are the ex ternal force and moment along the body yb andzb ax es respectiv ely. F or both w heels the eq uation for the lateral ty re force has thefollow ing structure:

Mzty r e= Fyty r e

xw heel (4.3)

Fyty r e= Gtyreβtyre, w ith: (4.4)

βtyre =vb + xw heelrb

ub− θsteer (4.5)

w here Gtyre is the (linearised) cornering gain for a giv en ty re type, v ertical loadand tax i w ay surface condition, βtyre is the ty re slip angle and xw heel is the bodyx -coordinate of the w heel ax le. In case of object-oriented modelling, theseeq uations are suffi cient to start implementing the bicy cle model. F igure 4 .1 show sthe basic model structure. The frame eq uations of motion (4 .1 ), (4 .2 ) are codedliterally into the object Frame. The equations (4.4) and (4.5) can first be coded in a


class Wheel. This class is then instantiated as a rear and front w heel in the m odel bysetting ap p rop riate v alues for Gtyre, xw h eel, and θs teer (for the rear w heel θs teer = 0,for the front w heel, θs teer = θN W ).The connectors contain tw o so-called fl ow v ariables (Mz and Fy) and tw o acrossv ariables: rb and vb (in p ractice, inertial p osition and the direction cosine m atrix ofthe local body ax es are used instead, S ection 2 .2 ). W hen connecting com p onents, fl owv ariables are sum m ed to zero and across v ariables are set equal betw een connectorsinvolv ed. This is autom atically done by the m odel com p iler, as w ill be discussedfurther on. A s show n in F ig ure 4.1 , this ap p roach allow s for v ery easy addition ofother com p onents. E v en if the m ass is no long er concentrated in the fram e (i.e.distributed over the w heels as w ell), the m odel structure rem ains the sam e and onlyap p rop riate equations of m otion hav e to be added to the Wheel m odel class.

Figure 4.1: B ic y c le m o d e l in o b je c t-o rie n te d fo rm .

A s c a n b e se e n fro m th e e x a m p le , o b je c t-o rie n te d a llo w s fo r m o d e l im p le m e n ta -tio n a t a n e a rlie r sta g e in th e m o d e llin g p ro c e ss in th e fo rm o f p h y sic a l e q u a tio n s,b e fo re th e se a re tra n sfo rm e d in to d iff e re n tia l e q u a tio n s. In c a se o f im p le m e n ta -tio n in O D E o r b lo ck d ia g ra m fo rm , th e u se r h a s to d o c o n sid e ra b ly m o re w o rkb e fo re im p le m e n ta tio n c a n sta rt, lik e d e riv in g d iff e re n tia l e q u a tio n s, tra n sla tin ge q u a tio n s in to d e c la ra tiv e fo rm to m a k e su re a ll u n k n o w n s a p p e a r o n th e le ft h a n dsid e , a n d so rtin g th e m in th e rig h t o rd e r.O b je c t-o rie n te d m o d e llin g is fre e ly a v a ila b le in th e fo rm o f th e m o d e llin g la n g u a g eM o d e lic a . In itia te d in 1 9 9 6 , a n d sp e c ifi e d a n d m a in ta in e d b y th e in te rn a tio n a lM o d e lic a a sso c ia tio n [8 9 ], M o d e lic a is e sp e c ia lly su ita b le fo r im p le m e n ta tio n o fla rg e -sc a le m u lti-e n g in e e rin g m o d e ls. C o m p o n e n t lib ra rie s fo r m u lti-b o d y sy s-te m s, b lo ck d ia g ra m s, e le c tro n ic s, h y d ra u lic s, p o w e r-tra in s, v e h ic le d y n a m ic s, e tc .a re a v a ila b le1, a s w e ll a s c o m m e rc ia l e n v iro n m e n ts fo r g ra p h ic a l c o n stru c tio n a n d

1For a more complete list, see [89]


compilation of models. In the frame of this work, Dymola (Dynamic modellinglaboratory [4 ]) has been used.In aircraft dynamics modelling there is a strongly growing need for multi-disciplinarymodel integration, since interactions between various disciplines, like fl ight me-chanics, structural dynamics and systems, tend to grow with each new aircraftdesign. F or this reason, DL R has developed the F light Dynamics L ibrary (F light-DynL ib). T his library, described in Chapter 2 , is fully compatible with Model-ica libraries for other engineering disciplines, thus allowing the multi-disciplinaryideas behind object-oriented modelling to be fully exploited. In constructingthis library, a new physically-oriented generic aircraft model structure has beenadopted.

4.1.2 Object-oriented modelling of the aircraft-on-ground

As already indicated in the introduction, the main example application of themethodology developed in this chapter will the development of preliminary controllaws for a transport aircraft manoeuvring on the ground (taxiing, landing roll-out, take-off run). Model details and control law specifications can be found inR efs. [4 8, 2 8]. T he model for the specific aircraft has been implemented usingcomponents from the F light Dynamics L ibrary. T he main tree of the libraryis depicted to the left in F igure 4 .2 . G eneric components, like environment orsensor models, are readily at hand. F or aircraft-specific components base classesare available that provide standard interfaces and variables. T hese base classesmay be inherited by components describing aircraft-specific model equations, likeapplication rules in the aerodynamic model.T he basic model structure is depicted to the right in F igure 4 .2 . As discussed inS ection 2 .2 , the environment objects (lower right) include a world (in this case,Earth), a tm osp h ere , and terra in model. T he world model provides the Earth-Centred Inertial (ECI) [12 5 ] as an inertial reference frame for all model compo-nents, as well as a geodetic reference in the form of the W orld G eodetic S ystem1984 (W G S 84 ), based on the Earth-Centred Earth-F ixed (ECEF ) frame. As de-scribed in S ection 2 .2 , the world model contains functions to compute the gravityacceleration, the EG M-96 Mean S ea L evel (MS L ) surface, and W MM-2 005 mag-netic field as a function of W G S 84 co-ordinates. Double-clicking on the objectallows a number of parameters to be set, like whether the Earth is rotating orassumed to be at rest, the type of gravity model to be used, etc.T he a tm osp h ere model in F igure 4 .2 in this case represents the MS L -referencedInternational S tandard Atmosphere (IS A). Again, functions are provided thatcompute atmospheric conditions and geodetic wind fields with respect to theEarth surface.T he depicted terra in model contains a parametrised model of the Earth’s surface,based on EAS A CS -AW O specifications [3 7 ]. T his model is normally used for au-tomatic landing certification. More detailed models may be linked from simulatorterrain databases in the future. Earth-fixed navigational equipment (e.g. V OR ,DME, IL S systems at specified locations) are incorporated in separate objects


Figure 4.2: Structure of the aircraft-on-ground model, along side the Flight Dy-namics Library top level structure (left).

(not shown), which are naturally attached to the terrain model.

N ote that the environment models have no connection with the aircraft model,which physically makes sense. This allows multiple aircraft to be implemented,all using the same environment objects. Components of the aircraft model mayindividually call to the environment models. For example, each landing gear mayobtain its own height above terrain, or its compression, by a simple function callto the terrain model. The environment models also make sure that all compo-nents use the same field functions and parameters. For example, clicking on theatmosphere model, the ISA standard atmosphere may be selected with a higherthan nominal sea level temperature T0. All model components that request theirlocal atmospheric conditions are then provided with ISA conditions based on thisT0.

The core of the model structure is of course the component that represents theactual aircraft, see Figure 4.2. The backbone of the aircraft model are the k ine-

matics and airframe blocks. The first defines a “ N orth-East-Down” (N ED) localvertical frame with its origin moving with a fixed position in the aircraft, prefer-ably the centre of gravity (CoG). The object further defines a body-fixed referenceframe (x-axis towards the nose, z -axis down, origin in CoG) to which the airframeis attached. The attitudes and inertial positions of both reference systems are


available in the two connectors (top: body reference frame, below: NED refer-ence frame). Attitude descriptions may be based on quaternions or Euler angles,and position integration may be based on a flat, round, or elliptic Earth.

The airframe object basically describes the mechanical equations of motion. Incase of the on-ground model, the airframe is assumed to be a rigid-body. H owever,the airframe may be flexible as well (Chapter 2), or may be constructed from theModelica multi-body systems (MBS) library. Connection between the airframe

and kinematics objects (see Figure 4.2) cause the reference systems in both con-nectors merge, i.e. from then on the airframe is moving freely with respect to theinertial reference, with its kinematics described in the corresponding object. Theairframe object has a second connector on top, intended for interconnection of forexample external force model components, sensor models, etc. Besides kinematicvariables, each connector also describes (generalised) forces and moments alongthe local reference systems axes, declared as flow variables, see Section 2.2.

The airframe equations of motion are primarily driven by aerodynamic, propul-sion, and gear forces and moments. These are computed in corresponding modelcomponents in Figure 4.2. These components are aircraft-specific, in this casecontaining the neural-network based model described in Ref. [48]. Computationof key variables like the angle of attack, side slip, true airspeed, etc. is inheritedfrom an aerodynamics base class. Local mean wind and turbulence are obtainedfrom the localW ind object.

Besides the airframe, each component is described in its own local reference frame.An attachment object, as for example between an engine and the airframe models,transforms kinematics and forces and moments between its connectors. In thiscase of a rigid attachment, relative positions and attitudes may be entered asparameters. In case of a flexible airframe it is suffi cient to enter the grid pointnumber on the structural model. The attached components then automaticallymove and rotate with the local structure point.

The actu ation sy stem component in Figure 4.2 describes actuators and hydraulic/ electric systems. These may be constructed from hydraulics and electronicslibraries (see for example [11]), but for the aircraft-on-ground model, only theco-ordination of control surface movements is described.

Finally, the thin bar at the top of Figure 4.2 represents a so-called data bus.This bus includes signals that one would typically find on avionics buses in theaircraft, like the readings of all sensors, command signals to engine and controlsurface actuators, gear status, etc. For this reason, the sensor, actuator, andengine models have been attached to the bus object. The bus is also accessiblefrom outside and allows direct connection to elements from the Modelica blockdiagram library. This enables a control system composed using this library toeasily communicate with the aircraft components.

L anding gears

Of course, the most interesting aspect of the aircraft-on-ground model is the land-ing gear. The physical equations are provided in Refs. [48, 28]. Figure 4.3 showsthe gear constructed as a simple multi-body system. Most important is the gear


compression, which determines internal gear load, which in turn heavily influencestyre forces. The gear model uses the terrain model to the right in Figure 4.2, sothat the compression of each gear results from location and orientation of theairframe, and the local terrain elevation above the Earth ellipsoid as defined inthe World Geodetic System 1984 (WGS84 [7]). The principle of computation isillustrated in Figure 4.4. The centre line of the gear in its uncompressed positionis extended to the ground:

Rterrain = Rg ear + T0,g ear [0, 0, lg ear]T

(4.6)

h = fWG S 8 4 (Rterrain) (4.7)

hterrain = fterrain(Rterrain) (4.8)

0 = h − hterrain (constraint) (4.9)

whereby Rg ear is the inertial position of a fixed reference point on the gear (inthis case, the bottom of the unloaded gear), Rterrain is the inertial position of thecrossing between the gear extension and the ground, lg ear is the length from thegear reference point to the ground. In case lg ear < 0 this variable is the amountof compression (tyre compression is neglected, see Ref. [48]). The transformationmatrix from gear body into inertial axes is T0,g ear. The geodetic height h (inWGS84 co-ordinates) at Rterrain is computed from the function fWG S 8 4 , whichis available in the world model. The terrain elevation at Rterrain is hterrain,computed from the terrain model function fterrain (in the terrain model). Byapplying the constraint (4.9), lg ear can be computed. The equations have beenimplemented as such in the G round R eference object in Figure 4.3. The modelcompiler will automatically build in a numerical nonlinear equation solver to solvefor lg ear.

4.2 Translation of object-oriented modelsIn object-oriented modelling, model objects are defined with the help of physicalequations. Since the equations are not declarative, as in programming languages,the final model comprises one large system of (nonlinear) equations that may becollected from all components and interconnections into the following form:

0 = F (X(t), X(t), w(t), p, t) (4.10)

Here X(t) ∈ IR nX are variables of which time derivatives X(t) occur in themodel, and w(t) ∈ IR nw contains any other variables, like local ones, inputs, andoutputs. The vector p ∈ IR np contains any parameters that may be set prior to,but remain constant during, simulation. In the sequel of this chapter, the time(t) argument will be omitted if time dependency is obvious.Numerical integration algorithms as used in simulation tools require the modelequations to be translated into an algorithm that computes state derivatives x(t)and the unknown part of w(t) (these are outputs and intermediate variables)from the state vector x(t) and the known part of w(t) (these are inputs). The

4.2 Translation of object-oriented models 91

Figure 4.3: Landing gear model, constructed from multi-body system components

hte r r a in

W G S - 8 4 e llip s o id

hte r r a in

a ir f r a m e ( o n g r o u n d )

a ir f r a m e ( a ir b o r n e )

lg e a r

lg e a rRg e a r

Rte r r a in

Figure 4.4: Computation of landing gear compression


state variables are automatically selected or may be imposed by the user (usuallyfrom X(t)). The algorithm must be provided in the form of computer code, likeC or FORTRAN. After compilation this code may be called by the numericalintegration algorithm.

The way how the equations are solved depends on which variables in w(t) areknown or unknown. This is fixed at the moment the user has specified the modelinputs (u ∈ IR nu) and outputs (y ∈ IR ny ). Mathematically, the system of equa-tions (4.10) is then to be translated into the form of an ordinary differentialequation (ODE) in state space form:

x(t) = f(x(t), u(t), p, t)y(t) = h(x(t), u(t), p, t)

(4.11)

For simplicity of notation it is here assumed that intermediate variables in w(t)have been eliminated. Although the second equation is an algebraic one, it onlycomputes output variables y(t), on which the first equation does not depend.

Sometimes it is not possible to explicitly solve for outputs and state derivatives.The equations are then translated into the form:

0 = F1(x(t), x(t), y(t), u(t), p, t) (4.12)

If x(t) and y(t) can be solved for given u(t) and p, this equation is a so-calleddifferential algebraic equation (DAE) of differential algebraic index 1 [97], whichmay be time integrated using a dedicated DAE solver. Nowadays powerful sym-bolic algorithms are available that allow the translation of a physical model intothe form of (4.11) or (4.12) to be performed automatically [31, 32, 99].

Example 4.2: The bicycle model (cntd.)

In order to simulate the bicycle model, the inputs and outputs need to be selectedfirst. The input obviously is the nose wheel steering angle, as output the yaw rate ischosen:

θNW = u (4.13)

y = rb (4.14)

The model compiler will first collect all equations and identify unk nown variables.

4.3 Inverse model generation 93

The collected equations are as follows:

0 = m (vb + ubrb) − Fyext(from frame)

0 = Izz rb − Mzext(from frame)

0 = FyN W− GNW βNW (from nose gear)

0 = FyM L G− GM L GβM L G (from main gear)

0 = MzN W− FyN W

xNW (from nose gear)0 = MzM L G

− FyM L GxM L G (from main gear)

0 = βNW − (vb + xNW rb) / ub − θNW (from nose gear)0 = βM L G − (vb + xM L Grb) / ub (from main gear)0 = (−Fyext

) + FyN W+ FyM L G

(connector equation)0 = (−Mzext

) + MzN W+ MzM L G

(connector equation)0 = y − rb (output equation)0 = θNW − u (output equation)

(4.15)

C omparing this system of equations with (4.10 ), the vector arguments are:

X = x = [vb, rb]T

(4.16)

x = [vb, rb]T

(4.17)

w = [y, u, θNW , Fyext, FyN W

, FyM L G, Mzext

, MzN W, MzM L G

,

βNW , βM L G]T

(4.18)

p = [ub, m, Izz, GNW , GM L G, xNW , xM L G]T

(4.19)

Since u and x are known from input and integration of x respectively, 12 variables(2 in x, the remaining 10 in w) need to be solved from the 12 equations above. Inthis case, this is very easily done by the model compiler using graph-based methods,resulting in equations of the form (4.11).

4.3 Inverse model generationAs already remarked in Section 4.1, in object-oriented models causality is not yetfixed in the implementation. This allows for automatic generation of inverse sim-ulation models, where physical inputs and outputs have been reversed. Startingwith the bicycle model, this section sketches the underlying procedures for modelinversion. More details and examples can be found in [75, 127].


For inversion of the bicycle model, the input and outputs are now selected as follows:

rb = u (4.20)

y = θNW (4.21)


The collected model equations (4.15) remain the same, only the last two are replacedwith (4.20) and (4.21). The number of unknowns and equations does not change.H owever, equation (4.20) no longer contains unknowns, since rb is a state variable.The model compiler will therefore diff erentiate the equation:

rb = u (4.22)

Since this requires availability of the derivative of the new input u, the user is requestedto facilitate this. O ne option is to declare u as input, rather than u, or to generate adiff erentiable u from e.g. a filter with relative degree 1. In this example, the followingfilter is added:

rc = −1/τurc + 1/τuucm d (4.23)

u = rc so that: (4.24)

u = rc (4.25)

where τu is the filter time constant, rc is the commanded yaw rate, and ucm d isthe commanded input. The diff erentiated equation (4.22) and the filter add threeequations, but just two unknowns: u and rc. O ne solution now is to introduce aso-called dummy-derivative for u [8 2]: instead of u, u′ is used, making u and u′independent variables:

rb = u′ (4.26)

u′ = rc (4.27)

The total number of equations now amounts 15, from which 15 unknown variablescan be solved, resulting in the following set of diff erential equations:

x = f(x, ucm d , p) (4.28)

θNW = h(x, ucm d , p) (4.29)

where x = [vb, rb, rc]T. The obtained inverse model may be connected with the

original model, as depicted in Figure 4.5. B ecause of (4.26 ), the depicted relation be-

bicyclem o d el

in v ers ebicyclem o d el (d yn .)

co m m a n dfilter

ucmd u'

N Wr

b

1

s

1

s+ 1u

rc

.

rc

Figure 4.5: Controlled bicycle model

tween u′ a nd th e o u tp u t rb is a n integ ra tio n. In c o m b ina tio n with th e c o m m a nd fi lter(with d iff erentia b le o u tp u t rc), th e tra nsfer fu nctio n b etween ucm d a nd rb b ec o m es:

rb =1

τus + 1ucm d (4 .3 0 )

4.3 Inverse model generation 95

The selection of τu thus determines the command response behav iour, which is inde-pendent of the parameters p. In F igure 4 .6 an example response of the configurationin F igure 4 .5 is shown. To the left the command response of rb perfectly matchesthe response of the command filter rc to the step input on ucmd. To the right thelateral acceleration (ny = − (vb + ubrb) / g, g = 9 .8 1 m/ s2) response in the C oG andthe defl ections of θN W are shown for future reference.

0 2 4 6 8 100

0.005

0.01

0.015

0.02

0.025

time (s)

−ny

(−),

θN

W (r

ad)

Lateral acceleration, NWS input

−ny

θNW

0 2 4 6 8 100

0.002

0.004

0.006

0.008

0.01

time (s)

r c, rb (r

ad/s

)

Yaw rate

uc

rb

rc

Figure 4.6: E x a mp le time resp onse of th e stru ctu re in F ig u re 4.5

In order to ma k e a comp a rison w ith inv ersion-ba sed controller synth esis meth ods,th e tra nsla tion for inv erse models is sk etch ed sta rting from (4.11), ra th er th a n(4.10):

x = f(x,uc,uo,p) (4.31)

ycmd = hcmd(x,p) (4.32 )

T h e inp u ts u h a v e been div ided into uc (a v a ila ble to th e control la w s) a nd uo

(a ny oth er inp u ts, lik e distu rba nces, fl a p setting s, etc.). T h e ou tp u ts y h a v e beeng rou p ed into ycmd (comma nd v a ria bles, to be tra ck ed by th e controller), a nd oth erou tp u ts yo (sensor ou tp u ts, oth er v a ria bles for a ssessment, etc.). T h e ou tp u ts yo

a re comp u ted from:

yo = ho(x,u,p) (4.33)

M odel inv ersion inv olv es a ssig ning ycmd a s inp u ts a nd uc a s ou tp u ts. F or now itis a ssu med th a t th e dimensions of both v a ria bles a re th e sa me a nd, a s a lrea dya p p a rent from (4.32 ), do not a lg ebra ica lly dep end on ea ch oth er. A s in th e bicy-cle ex a mp le a bov e, immedia tely th e p roblem a rises th a t (4.32 ) no long er conta insu nk now ns. M a nu a l nonlinea r controller synth esis meth ods lik e F B L , a nd a modelcomp iler both a ddress th is p roblem in th e sa me w a y a s in ex a mp le 4.3: by dif-ferentia ting ea ch of th e indiv idu a l ou tp u t eq u a tions. T h is ca n be described u sing

L ie deriv a tiv es [12 1, 46 ] for th e ith entry a s follow s:

ycmdi=

∂ hcmdi(x,p)

∂ xT· f(x,uc,uo,p) = Lfhcmdi

(x,p) (4.34)


In case Lfhcmdi(x, p) does not explicitly depend on one or more control inputs,

diff erentiation proceeds:

y(ri)cmdi

= Lri

f hcmdi(x, p) (4.35)

where (ri) is the rthi derivative. This procedure is repeated for each of the entriesin ycmd until each of the entries of uc can be solved. If done manually, thisprocedure may be quite tedious, especially if relative orders of ycmd are high. Incase of object-oriented modelling, the procedure is performed automatically. Thealgorithms by P antelides [99] are used to determine the minimum number of timeseach of the equations (4.32) has to be diff erentiated until uc can be solved for.Eventually, the model diff erential equations can be written in the following form:

x = finv (x, ν, uo, p) (4.36)

uc = hinv (x, ν, uo, p) (4.37 )

where finv arises from f by substitution of uc (in the case that uc cannot be solvedexplicitly, a solution is found numerically). The vector ν contains diff erentiatedentries of ycmd:

ν =[

y(r1)cmd1

, y(r2)cmd2

, · · · , y(rny c )

cmdny c

]T

(4.38)

The model translator will make the user aware that derivatives of ycmd are ex-pected as inputs, rather than ycmd directly. These derivatives may then for ex-ample be generated using command fi lters with relative degrees of r1, ..., rny c

respectively.The inversion procedure described above is very similar to the one for synthesis ofFeedback Linearisation control laws [121, 46]. The principal diff erence is that inthese references the diff erential equations depend on the controls uc in an affi neway.In equations (4.36) and (4.37 ) the diff erential equations for x(t) have been re-tained. In order to make sure the inverse model is stable, finv must be investi-gated. For this, methods as described in [121, 46] may be used.S o far, the inverse and forward systems connected in series are still two indepen-dent entities, each including diff erential equations for nearly the same selectionof state variables. Modelling errors, disturbances, integration errors, etc. willeventually cause the internal states x in (4.37 ) to deviate from the actual state

variables of the system and worse, the rthi derivative of the system output ycmdi

to deviate from the commanded νi. For this reason, the inverse control lawsare usually combined with a feedback controller and inverse model states are ob-tained from measurement or a reference model. The diff erences between variousinversion-based synthesis methods, like FBL, MFC and IFFC, is mainly in theseaspects. This will be shown for the bicycle model.

4.4 A rapid prototyping design process 97


Figure 4.7 shows various control structures around the core of reference filter/model,inverse model, and system (Figure 4.5). The first variant adds a feedback controllaw, regulating the error between the actual yaw rate rb and rc, commanded by thereference filter. The inverse model includes its own state eq uations. In fact, thisstructure is comparable with inverting a (proper) transfer function of the linearisedsystem and to add this as a dynamic compensator. The integrated states in theinverse model eq uations will easily drift away from the actual values in the controlledsystem. For this reason, the next variant generates reference values for states in thecommand model, extending the command filter with desired dynamics for the lateralvelocity vb, using a state eq uation for vbc

. In order to make sure vbc≈ vb, the error

in between is also regulated by the feedback controller. The advantage is that theinternal or zero dynamics of the system (in this case, vb) are not left on their own.B oth controller structures have the disadvantage that the feedback controller still mayreq uire scheduling as a function of parameters p (4.1 9 ) or, in case of an aircraft, theflight condition. The third variant therefore uses feedback around both the systemand the inverse model. Furthermore, the internal states are directly obtained frommeasurement. The latter provides the most accurate inversion, but req uires the fullstate vector to be measured, or estimated. The latter structure is the one behindFeedback L inearisation (FB L ) or N onlinear D ynamic Inversion (N D I).

As can be seen from the example, states are preferably not integrated within theinverse model itself, but rather obtained from the reference model (MFC) [42], orfrom measurement (FBL, N D I). In order to facilitate this, a model compiler caneasily declare states as inputs instead. The model differential equations are thenautomatically removed.

4.4 A rapid prototyping design processAs discussed in the previous section, once the system model is available in anobject-oriented form the generation of an inversion-based controller requires littleeffort, since algebraic and coding work is automated. O f course, the generatedcontrol laws are not ready for hardware implementation, but they are fully func-tional over the operating envelope and can (apart from the selection of a fewfeedback gains) readily be used as a prototype design in a manned or off-linesimulation environment. In the introductory section several examples of practicaluse in flight control law design have been discussed.Figure 4.8 shows the basic steps for inversion-based design up to detailed design.Although the model is assumed to be available, model construction has beenadded as a first step. The reason is that not necessarily all dynamics in themodel are to be inverted. For example, actuator models are often representedby low order transfer functions. The internal states are usually not available as


bicyclem o d el

in v ers ebicyclem o d el (s ta t.)co m m a n d

filter

ucm d

u' N Wrb

x

N D I , F B L

rc

.

rc

K (s )

bicyclem o d el

in v ers ebicyclem o d el (s ta t.)referen ce

m o d el

ucm d

u' N Wrbrc

.

K (s )

x

xc

+

+

xc

+ -M o d e l f o llo w in g

bicyclem o d el

in v ers ebicyclem o d el (d yn .)co m m a n d

filter

ucm d

u' rbrc

.

K (s )

+

+ -

rc

N W

I n v e rs e f e e d -f o rw a rd

+ -

Figure 4.7: Controlled bicycle model: various inverse controller structures

measurements from the real actuator, or sometimes not even of physical origin.It is common practice to residualise the transfer function to a static gain, sincethe bandwidth is usually considerably higher than that of the aircraft dynamics.Also, it should be kept in mind that model complexity directly influences inversecontrol law complexity. For this reason, if complex features can be reasonablyreplaced with simplified versions, this should be taken into consideration.The next step in Figure 4.8 is command variable selection. In most cases thisselection is not based on free choice, since commonality with other aircraft seriesis required. In new applications, like manual on-ground control, this section maynot be obvious from the start. The same holds for selection of control effectorsto be used. In case of redundancy, equations or algorithms have to be addedin order to allow the model compiler to appropriately allocate and co-ordinatecontrol deflections.The type of nonlinear control law that is generated depends on the way the linearfeedback control law and command filters are implemented (see for example Fig-ure 4.7). In this chapter, always NDI is used, avoiding the need for gain schedulingof the linear control law as a function of flight condition and other known param-

4.4 A rapid prototyping design process 99

eters (e.g. landing gear position). In the Flight Dynamics Library some controlstructures are available that may be copied to the model and interconnected priorto inverse model generation. After the inverse model code has been generated,it may be implemented in the preferred simulation environment with the aircraftsimulation model. The preliminary design may be evaluated in batch or pilot-in-the-loop simulations. Once the basic configuration has been frozen, detaileddesign may start, involving feedback signal synthesis (estimation of all states usedin the inverse model equations), tuning of free controller parameters, robustnessassessment, etc. In this phase also the coding must be reviewed. For example,variables computed from environment models are to be replaced with measuredones, iterative equation solving should be prevented, etc. An application exampleof the rapid-prototyping process up to manned simulations is described in [123].Chapter 5 of this thesis describes a detailed design for a (flight tested) automaticlanding system, with emphasis on robustness aspects.

Selection of command variables ( )yc m d

Allocation of controls ( )uc

Ad d line ar controlle r /com m and filte rs (from lib rary )

Inv e rsion of A/C m od e l

B asic d e cisions

Au tom atic

Working design

P re lim inary analy sis(v alid ate /com p are b asic d e cisions)

okno

y e s

D e taile d d e sig n

W ork ing p rototy p e foroth e r d e p artm e nts

B atch / re al-tim esim u lation

C onstru ct aircraft m od e l

.

S e le ction of com m and v ariab le s ( )ycm d

Allocation of controls ( )uc

Ad d line ar controlle r /com m and filte rs (from lib rary )

Inv e rsion of A/C m od e l

B asic d e cisions

Au tom atic

Working design

P re lim inary analy sis(v alid ate /com p are b asic d e cisions)

okno

y e s

D e taile d d e sig n

W ork ing p rototy p e foroth e r d e p artm e nts

B atch / re al-tim esim u lation

C onstru ct aircraft m od e l

.

.

Figure 4.8: Flight control law design process from rapid-prototyping to detailed

design

T he b asic decisions in Figu re 4 .8 req u ire physical insight b y the designer. In the

fi rst place, selected control eff ectors shou ld actu ally b e su itab le to perform the

track ing of command variab les. For ex ample, for roll rate track ing control eff ectors

shou ld b e u sed that primarily generate moments arou nd the aircraft longitu dinal

ax is. A lso, when selecting the lateral load factor as a command variab le, the


designer should be aware that the rudder generates a yawing moment, as well asa small side force, reducing the relative degree (to be introduced in C hapter 5 ) ofthe input-output relation to zero. The model compiler will exploit the small sideforce when generating inverse model equations. Although tracking performancemay look good at first sight, internal dynamics will be unstable, since (at leastin the C oG ) the lateral acceleration response to rudder input is non-minimumphase2. In that case, the designer has to adapt the aerodynamic model to makesure the rudder no longer infl uences side force, or the command variable selectionmust be reviewed. S imilar practical issues are discussed in R ef. [1 2 3 ].


Making an improper selection of command variable can once more be nicely illustratedon th e bicycle model. S uppose th e lateral load factor in th e C oG is ch osen:

θNW = u

y = −ny = (vb + ubrb) / g (4.3 9 )

w h ere g is th e gravity acceleration. T h e – sign h as been added in order to obtaincommand responses in th e same direction as for yaw rate control. T h e model compilerw ill q uickly sh ow th at from th e connector eq uation for lateral force it follow s th at:

(vb + ubrb) / g = (GNW βNW + GM L G βM L G ) / m g (4.40 )

S ince βNW depends on th e control eff ector θNW , inversion may be done w ith out dif-ferentiation of th e eq uation. F igure 4 .9 sh ow s th e time responses to a step commandon nyc

. T o th e left it appears th at ny is perfectly follow ing its command. T o th erigh t, th e response of rb and θNW are sh ow n. It immediately becomes clear th at th einternal dynamics of th e combined system are badly damped. T h e reason is q uiteobvious and could h ave been th ough t of in advance. B eing positioned at th e front ofth e aircraft, th e nose w h eel steering is primarily intended to generate yaw ing momentsaround th e vertical ax is to initiate and to maintain a turn. T h e lateral force is neces-sary to ach ieve th is, but at th e same time infl uences th e total lateral force balance.T h e inverse controller is just using th is eff ect to ach ieve its tracking task, th erebygenerating large yaw ing moments th at make th e yaw response go w ild. O ne solutionin th is case is to ch ange th e command variable (e.g. ny ≈ −ubrb/ g ), as is depicted inF igure 4 .1 0 . N ote th at a fi rst order command fi lter is req uired and th at th e responseof course closely resembles F igure 4 .6 . T h e direct feed th rough of th e lateral forcegenerated by θNW can be seen in th e ny response to th e left. In th e mean time, th eyaw rate increases, taking over th e main eff ect on ny due to centrifugal acceleration.

2A nice example is given in [85].

4.5 Aircraft-on-ground control design application 101

0 2 4 6 8 100

0.002

0.004

0.006

0.008

0.01

0.012

time (s)

−ny, −

n y c (−)

Lateral acceleration

−ny

c

−ny

0 2 4 6 8 10−0.01

−0.005

0

0.005

0.01

0.015

0.02

time (s)

r b (rad

/s),

θN

W (r

ad)

Yaw rate, NWS input

rb

θNW

Figure 4.9: Example time response of the IFFC bicycle model with ny as com-mand variable

0 2 4 6 8 100

0.002

0.004

0.006

0.008

0.01

time (s)

−ny

(−)

Lateral acceleration

−ny

c

−ny (appr.)

−ny

0 2 4 6 8 100

1

2

3

4

5x 10−3

time (s)

r b (rad

/s),

θN

W (r

ad)

Yaw rate, NWS input

rb

θNW

Figure 4.1 0 : Example time response of the IFFC bicycle model with approximateny as command variable

4.5 Aircraft-on-ground control design applicationIn this section the rapid-prototyping design process is applied to develop basiccontrol laws for an aircraft manoeuvring on the ground. D esign specificationsand model data have been provided in the frame of GARTEU R Flight M echanicsAction Group 17 on nonlinear flight control and are provided in Ref.[48].First step in development of on-ground control laws of course is to prepare theaircraft simulation model. To this end, inputs and outputs identical to those ofthe Aircraft block on the right in Figure 4.11 are added to the object-orientedimplementation in Figure 4.2. This is done by extracting and inserting signalsfrom and into the data bus. The model compiler then produces simulation code,which replaces the aforementioned Aircraft block in the original implementation.In order to properly initialise the simulation model, trim computation is needed.As described in Appendix A, this can be done in two ways: by generating staticsimulation code from the object model after specifying trimming conditions asinputs and unknown control deflections as outputs, or by automatically generatinga script based on the differential equations generated for simulation. This script


Figure 4.11: Generation of simulation code for the aircraft on ground model.N ote that inputs and outputs match those in the Simulink environment and areinserted into and taken from the bus via the avionics object.

specifies which entries of x, x, y and u are known or unknown (making sure thenumbers of equations and unknowns fit) and calls a nonlinear equation solver tofind the trim settings [72]. For the aircraft-on-ground model the latter version wasused, allowing for accurate computation of equilibrium conditions on the groundas well as in the air.The preliminary design of on-ground control laws will be discussed along the linesof Figure 4.8. The first step involves composition of an appropriate model forinversion. The following changes as compared with the original model are made:

1. Removal of actuator dynamics on the nose wheel steering system. It isassumed that

θNWc= θNW

i.e. the nose wheel steering angle θNW directly follows its commanded valueθNWc

. In the simulation model, from which the actuator dynamics of courseare not removed, the command input is the servo valve control current(ISV NW ). For computing the appropriate input from θNWc

the nose wheelsteering control law provided within the B SCU is used (see Ref. [48]);

2. The computation of the lateral tyre force of (only) the nose wheel is simpli-fied as follows:

FyN W≈ GyN W

(t)βNW

where GyN W(t) is the current cornering gain of the combined nose wheels

(depends on momentary vertical loading, see Ref. [48]). The reason is thatthe actual nonlinear function has a maximum at βNWo p t

, causing inversionproblems if the demanded side force FyN W

is larger than FyN W m a x(Ref.

[48]);


3. L imits on control inputs that are to be computed must be removed from themodel to be inverted. O therwise, no solution exists once control deflectionsbeyond these limits are required;

4. W ind and other disturbance models are removed.

The next step in Figure 4.8 is to make a selection of command variables. Basedon the objective to control lateral dynamics only, a number of options may beconsidered: (1) yaw rate, (2) lateral acceleration in the CoG, and (3) lateralacceleration in the cockpit. In this section, the yaw rate design will be elaborated.The other options have been evaluated in Ref. [95].

For turning the aircraft, the two most obvious controls are rudder δR and nosewheel steering θNW . The first is most effective at higher dynamic pressure, thesecond most at lower speeds. O ther means, like differential braking are not con-sidered for the moment. In order to invert the model, some rule must be providedhow to distribute the available control power between both effectors. As a firststep, the distribution is based on calibrated airspeed, as shown in Figure 4.12. IfVca s < Vlo w = 30 m/ s only nose wheel steering is used, if Vca s > Vh ig h = 80 m/ sonly rudder is used for yaw control, in between a linear blend based on generatedyawing moments is applied. O f course, the values of Vlo w and Vh ig h can be op-timised or a more advanced allocation structure may be selected in the detaileddesign phase.

Figure 4.12: Initial allocation of nose wheel steering and rudder as a function ofairspeed

In order to implement the control architecture, a structure as depicted in Fig-ure 4.13 has been constructed. To the left, the control allocation can be recog-nised. The outputs are δR and θNW . The allocation block obtains its requiredvariables from the aircraft data bus. To the right the linear P I control law as wellas the command filter can be recognised. As in the case of the bicycle model ex-ample, the yaw rate has relative order one after removal of actuator and hydraulicsystem dynamics. Initially, the yaw rate rb = rc is set as input and inserted intothe data bus. After connection with the aircraft model, the compiler will notify


the designer that rb needs to be differentiated once in order to solve the inversionp roblem . A s in case of ex am p le 4 .3 , the com m and shap ing fi lter:

rc =1

τrs + 1ucm d (4 .4 1)

is added, w here ucm d is the new com m and inp u t. T he m odel com p iler m ay thendifferentiate rc. T he feedback controller is added as is done in N D I or F eedbackL inearisation, see F ig u re 4 .7 . T he error betw een com m anded and actu al y aw rateis m inim ised u sing a P I controller. S ince this sy stem is to p rodu ce references forthe differentiated com m and variable, a sm all trick is ap p lied in F ig u re 4 .13 : thederivative of the y aw rate com m and is tak en from the com m and fi lter and fed tothe data bu s via an integ rator. T he P I controller ou tp u t m ay then be added to rc

before the integ rator inp u t. D u ring com p ilation, this integ rator is au tom aticallyrem oved w hen differentiating rb, since the latter variable is an enforced state inthe im p lem ented airfram e eq u ations of m otion.

1

S1/

r

+

-

1

S

P I

+

-

+

command filterlinear

controller

ucm d

rs e n s

u= rb

u u'.

rc

rc

.

A ircraft b u s acces s

Vca s

1

N W

R

N W

R

C ontrol allocation: N W S / ru dder

C onnect w ith A /C b u s

Figure 4.13: Im p lem entation of inverse control law in M odelica

T he com p lete block in F ig u re 4 .13 is added to the p rep ared aircraft-on-g rou ndm odel and connected to the aircraft data bu s (F ig u re 4 .2 ). N ote that the stru c-tu re itself is aircraft indep endent. F or this reason it has been added to the F lig htD y nam ics L ibrary , along side a sim ilar block for inversion-based pb, qb and β con-trol in fl ig ht. O f cou rse, p aram eters m u st be set dep ending on the sp ecifi c aircraftty p e and control sp ecifi cations. A s show n in the F B L variant in F ig u re 4 .7 , m odelstates are obtained from m easu rem ent. T o this end, the m odel com p iler sim p lyrep laces aircraft states w ith inp u ts, w hile rem oving u nnecessary state eq u ations.T he integ rator in the P I controller as w ell as in the com m and in F ig u re 4 .13 haveto be retained. S ince the control law is to be em bedded in the discrete overallcontrol sy stem (R ef. [4 8 ]), the integ rator state fu rther needs to be discretised.T his is done by the m odel com p iler by adding an in-line integ ration alg orithm [3 3 ].In this case, a sim p le ex p licit E u ler integ ration m ethod w ith a 4 0 m s sam p lingp eriod has been inclu ded.


After automatic model inversion and coding, first simulation runs can alreadybe performed. To this end, step inputs on the yaw rate command (ucmd) aregiven starting at four different trimmed airspeeds: well below Vlo w (10 m/ s), wellover Vh ig h (85 m/ s), and two in between (40 m/ s and 6 0 m/ s), see Figure 4.12.The yaw rate responses are the most interesting. As depicted in Figure 4.14, theymatch the expected first order behaviour as in Figure 4.6 quite well. At this point,only hand-picked values have been set for the linear control law and commandfilter parameters. In following design steps, these values may be tuned to optimiserobustness and performance, for example using multi objective optimisation, seeC hapter 5 and Ref. [9 5 ].

Next, it is interesting to look at the distributions between aerodynamic and nosewheel steering moments (Figs. 4.15 – 4.18). At 10 m/ s the aerodynamic momentis zero, the turn is initiated and maintained using the nose wheel steering only.At 40 m/ s some moment is generated aerodynamically, at 6 0 m/ s the distributionis relatively balanced, whereas starting from 85 m/ s steering is only performedaerodynamically. Since no caster has been included in the original model (Ref.[48]), the control law is actively turning the nose wheel to follow its local track.Note that at low speeds the rudder slightly deflects, see Figure 4.19 . This iscaused by the fact that the rudder is compensating for a small aerodynamic yawingmoment caused by slipping and yaw rate, so that the total aerodynamic momentis zero. Alternatively, the moment distribution may be applied to the yawingmoment delivered by the rudder only. H owever, in the case of the aircraft-on-ground model the aerodynamics have been provided as a “ black box” from whichthe explicit rudder contribution can only be derived indirectly, which has not beendone for this first design. In Figs. 4.20 – 4.23 the actual and commanded nosewheel steering deflections are shown. Note that the curves match well, justifyingthe removal of NW S actuator dynamics from the inverse model. H owever, in caseof high command inputs measures have to be taken in order to prevent saturationof the nose wheel steering system. In Figs. 4.20 – 4.23 an interesting effect in themodel can be observed. Starting from 10 m/ s, the nose wheel steering angle isinitially positive and remains so in order to maintain the turn. The correspondingslip angle of the nose wheel has opposite sign. At higher speeds the nose wheelsteering angle becomes strongly negative in the longer term. This is explained bythe over all side slip angle of the aircraft, required to generate the centripetal forcethat is necessary to turn. In order to generate the required yawing moment, thisslip angle is partly compensated for, so that the nose wheel is moving outwardwith respect to the turn. Above 80 m/ s the nose wheel should rotate with thelocal velocity vector at the nose of the aircraft. In Figure 4.23 it can be observedthat the NW S angle starts with a small offset. This offset was obtained from trimcomputation and compensates a slight asymmetry in the aerodynamics.

As advertised in the process picture in Figure 4.8, the control law obtained fromautomatic inversion is working properly. This has been proved at a range oftaxi speeds and is further confirmed by interactive real-time simulations using adesktop simulator [70]. This simulator allows the aircraft to be taxied around onan airport. The control laws in their present form for example allow for take-off


0 2 4 6 8 10 12 14 16 18 20−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (s)

yaw

rate

(deg

/s)

Yaw rate step response

10 m/s40 m/s60 m/s85 m/s

Figure 4.14: Yaw rate responses for different velocities

0 2 4 6 8 10 12 14 16 18 20

0

Time (s)

Mome

nt (N

m)

Aero and NWS moments: Vground = 10 m/s

NWSAero

Figure 4.15 : Moment distribution at 10 m/s


0 2 4 6 8 10 12 14 16 18 200

Time (s)

Mome

nt (N

m)


NWSAero

Figure 4.16: Moment distribution at 40 m/s

0 2 4 6 8 10 12 14 16 18 200

Time (s)

Mome

nt (N

m)


NWSAero

Figure 4.17 : Moment distribution at 60 m/s


0 2 4 6 8 10 12 14 16 18 20

0

Time (s)

Mome

nt (N

m)


NWSAero

Figure 4.18: Moment distribution at 85 m/s

0 2 4 6 8 10 12 14 16 18 20

0

Time (s)

rudd

er (d

eg)

Rudder deflection

10 m/s40 m/s60 m/s85 m/s

Figure 4.19 : Rudder deflections at various airspeeds


0 2 4 6 8 10 12 14 16 18 20

0

Time (s)

Angle

(deg

)

Nose wheel steering: Vground = 10 m/s

NWS angleNWS angle commandNW slip angleSideslip angle CoG

Figure 4.20: Nose wheel steering and slip angles at 10 m/s

0 2 4 6 8 10 12 14 16 18 20

0

Time (s)

Angle

(deg

)





0 2 4 6 8 10 12 14 16 18 20

0

Time (s)

Angle

(deg

)




0 2 4 6 8 10 12 14 16 18 20

0

Time (s)

Angle

(deg

)




4.6 Conclusions 111

simulations from brake release until the aircraft is airborne, even under strongand turbulent cross-wind conditions. This provides a good basis to start detaileddesign (Figure 4.8) from.

4.6 ConclusionsIn this chapter it has been shown that inversion-based control methods, in com-bination with an object-oriented model implementation allow for automatic syn-thesis of fully functional preliminary control laws. The resulting fast design cyclecan be exploited to make quick simulation-based comparisons between for examplevarious control configurations. The preliminary control laws are easily adaptedas more model data becomes available and can be used as a platform for detaileddesign of individual functions. The methodology has been demonstrated on anaircraft-on-ground problem, showing functionality of the preliminary control lawsin manoeuvring over the full speed range from brake release at stand-still to take-off.Once key decisions have been made and the aircraft configuration is frozen, de-tailed design may proceed based on the same methodology. In this phase issueslike robustness, feedback signal processing, system failures, coding, etc. are ad-dressed. This is subject of Chapter 5.It is important to note that key decisions, especially in the first steps in Fig-ure 4.8, are made by the designer. As demonstrated on a simple example, not thesuccessful inversion by the model compiler, but the designer’s physical insight isthe key to a successful control law design.


Chapter 5

Design of robust autopilotcontrol laws with NonlinearDynamic Inversion

114 Design of control laws with NDI

Abstract

The application of Nonlinear Dynamic Inversion for the design of a robustattitude controller for a civil aircraft is discussed. The main function ofthe controller is to improve the fl ying qualities, including stability, of theaircraft dynamics. F or parameter synthesis multi-objective optimisationis used. The required robustness is achieved via a multi-model approachand local robustness criteria. In addition to the feedback gains, physi-cal parameters in the inverse model that are considered uncertain in thedesign model, are used as synthesis parameters. The control law s are au-tomatically generated from a symbolic aircraft model in the object-orientedmodelling language M odelica. The design w as used as a stability and com-mand augmentation function in an automatic F ly-by-W ire landing systemand w as successfully fl ight tested.

C o n tributio n s

• A structured approach for incorporating local and global robustnessconcepts in multi-objective control law design;

• A n optimisation procedure for searching a best-compromise locationfor model inversion in the uncertain parameter space.

P ublicatio n

G ertjan L ooye: Design of Robust Autopilot Control Laws with NonlinearDynamic Inversion, at – A utomatisierungstechnik V ol. 4 9 (1 2 ), O lden-bourg V erlag, M unich, 2 0 0 1 , pp. 5 2 3 – 5 3 1 .

Design of control laws with NDI 115

THE design of flight control laws is a demanding task due to the fact thatmany aircraft configurations and a considerable operating envelope have to

be taken into account. In order to achieve satisfactory performance in all flightconditions, gain-scheduling is usually required.

Nonlinear Dynamic Inversion (NDI) is an attractive alternative to the gain-sche-duling approach. In NDI the nonlinear dynamics are compensated via inversemodel equations in the control laws. The methodology has found many applica-tions, especially for the design of flight control laws for post-stall flight of highperformance fighter aircraft [5, 122, 123].

In this chapter the application of NDI to design attitude control laws for a civilaircraft is described. These are used for stability and command augmentation inan automatic landing system.

NDI is a special case of Feedback Linearisation [121], where the model outputequations of the controlled variables need to be differentiated only once to arriveat an analytical relation with the control inputs that can be inverted. The closed-loop system is reduced to a set of integrators and the desired dynamics can beimposed using a linear outer loop controller.

An important drawback of Nonlinear Dynamic Inversion is that the resultingcontrol laws may show poor robustness to uncertainties in the design model [34,10]. Furthermore, the derivation, coding, and verification of the control laws canbe tedious and error-prone, especially when the model is complex.

U sually, robustness is addressed in the design of the linear part of the controller[34, 5, 10]. In this contribution, it is shown that physical parameters in the inversecontrol laws that are considered uncertain in the design model, can be effectivelyused as additional synthesis parameters to achieve a robust design.

Furthermore, the automatic generation of NDI control laws from the symbolic air-craft model in Modelica as described in the previous chapter is used, considerablyrelieving the derivation, coding and verification tasks.

This chapter is structured as follows. In Section 5.1 Nonlinear Dynamic Inversionis reviewed. In Section 5.2 the aircraft model is described and in Section 5.3 theautomatic generation of control laws from the symbolic model implementationin Modelica is reviewed. In Section 5.4 the tuning of the synthesis parametersfor the lateral-directional dynamics is described. The results of three design can-didates will be discussed and compared in Section 5.5. Flight test results withthe controller are discussed in Section 5.6. Finally, in Section 5.7 conclusions aredrawn.


5.1 Nonlinear Dynamic InversionIt is assumed that the nonlinear system model can be brought into the followingform (assumption 1):

x = f(x,pk,pu) + g(x,pk,pu)uc (5.1)

ycmd = hcmd(x,pk,pu) (5.2)

yo = ho(x,pk,pu) (5.3)

where x ∈ IR nx is the state vector, uc ∈ IR nu c is the input vector1, ycmd ∈ IR ny c m d

is the vector of outputs that are to be controlled, and yo ∈ IR ny o contains anyother outputs. In this chapter it is assumed that the number of controlled inputsand outputs is equal: ny cmd = nuc (assumption 2). In case this assumptiondoes not hold the method is still applicable, see for example Section 4.5. Twotypes of model parameters are distinguished: parameters that are known on-line(pk ∈ IR np k ) and parameters that are uncertain (pu ∈ IR np u ). The vector uc onlycontains inputs that may be used by the controller. Any other system inputs areconsidered as known parameters (∈ pk). Note that (5.1) is affi ne in uc, but itsuffi ces if uc can be explicitly solved from this equation.As a first step, the output equation hcmd is differentiated:

ycmd =∂ hcmd(x,pk,pu)

∂ xT

d x

d t= Lfhcmd(x,pk,pu) + Lghcmd(x,pk,pu)uc (5.4)

where Lfhcmd and Lghcmd are first order Lie-derivatives of hcmd along f and g

respectively:

Lfhcmd =d hcmd

d xT· f, Lghcmd =

d hcmd

d xT· g (5.5)

It is assumed that, in addition to assumption 2, all outputs have relative order 1with respect to at least one of the inputs and that Lghcmd(x,pk,pu) is non-singular(assumption 3). In that case Nonlinear Dynamic Inversion can be applied2. Thederivative Lghcmd(x,pk,pu) may be singular for the following reasons:

1. one or more of the differentiated outputs do not analytically depend on atleast one of the control inputs. In that case, higher order Lie derivatives ofthese outputs need to be derived;

2. two or more output derivatives analytically depend on the same controlinput. In that case, the choice of control variables should be reviewed,since the current combination of control variables can not be controlledindependently;

1The input vector appears in an affine way. This is not necessary for the application of NDI,b ut it m ak es the sub seq uent sym b olic d erivation easier and m ore illum inating .

2NDI is thus a special case of feed b ack linearisation, where only one d iff erentiation of eachcontrolled variab le is req uired to enab le inversion [1 2 1 ].

5.1 Nonlinear Dynamic Inversion 117

3. in case assumption 2 does not hold (i.e. l 6= k). This may be overcome bycomputing a pseudo-inverse, or by implementing control allocation logic (ifk > l) (see Chapter 4 and Refs. [5, 118]).

Even if Lghcmd(x, pk, pu) is non-singular one has to be careful that the physicalrelations are suitable for inversion. An example is given in Section 4.4 for loadfactor control.With assumption 3, inversion of eq. (5.4) results in the following control law:

uc = (Lghcmd(x, pk, p∗u))−1

(y′cmd − Lfhcmd(x, pk, p∗u)) (5.6)

The vector y′cmd ∈ IR nycmd contains the demanded rates of ycmd and serves asthe new control input. Note that the dot-notation () has been replaced with aprime (′), since y′cmd has become an independent control variable. The vectorx contains the states as computed or estimated from available outputs ycmd andyo. It is assumed that all states are observable from ycmd and yo (assumption 4).The vector p∗u contains assumed values for the uncertain parameters in the inversemodel equations. Usually, nominal parameter values are chosen. Substitution ofthe control law (eq. (5.6)) into (5.4) results in:

ycmd = Lfhcmd(x, pk, pu) (5.7)

+ Lghcmd(x, pk, pu) · (Lghcmd(x, pk, p∗u))−1

(y′cmd − Lfhcmd(x, pk, p∗u))

In the ideal case, when pu and x are known exactly (i.e. p∗u = pu and x = x),equation (5.7) reduces to:

ycmd = y′cmd ⇐ ⇒ ycmd(t) =

∫ t

0

y′cmd(τ)dτ (5.8)

The nonlinear system dynamics have been cancelled by the NDI control law, andthe output response to each corresponding input is reduced to an integrator. It isnecessary to carefully check the internal (zero) dynamics of the closed-loop system.For stability it is required that the relation between uc and ycmd is minimum phase(assumption 5) [121]. Sometimes, instability of a slow mode is acceptable. In caseof fighter aircraft it often occurs that the phugoid mode becomes slowly divergent.This however can usually be handled by the (auto-) pilot without problems [123].Using a linear feedback law it is now possible to impose the desired closed-loopdynamics:

y′cmd = K(s)(yre f − ycmd) (5.9)

where yre f is the commanded output vector. Substitution in (5.8) gives:

ycmd(s) = (Ims + K(s))−1

K(s)yre f (s) (5.10)

The resulting closed-loop system is depicted in Figure 5.1. Alternatively, a so-called two degrees-of-freedom linear design may be applied:

y′cmd = K2(s)(yre f − ycmd) + sycmd (5.11)


Statees tim atio n& f ilter in g

A ir c raftd y n am ic s

( p , p )k u

I n v er s ec o n tr o ller( p , p )k u

*

L in earc o n tr o ller

K ( s )

yr e f

uc

x

ycm d

Is~

ycm d'yo

Figure 5.1: Closed-loop system with Dynamic Inversion control laws

where yref is generated from a command filter. The control law K2(s) then has apurely regulatory function, whereas for K(s) this is to be combined with shapingof the command signal. The latter structure has been used in the ground controlapplication in Chapter 4.From the derivation above, it will be clear that Nonlinear Dynamic Inversion hasa number of advantageous features:

1. the method is straight-forward and easy to understand (equation (5.6));

2. the closed-loop system shows excellent nominal performance, since it is fullydecoupled and desired dynamics can be easily imposed (equations (5.8),(5.10));

3. gain-scheduling is avoided, since the control law cancels the nonlinear systemdynamics over its operating envelope (equation (5.10)).

Applicability of NDI is restricted by the assumptions made, but for rigid bodyaircraft these are in most cases fulfilled. Advantage 2 additionally requires that allstates are known accurately (x = x). For aircraft this is a reasonable assumption,since the states can be analytically computed from the available air data andinertial reference sensors.Dynamic inversion also has apparent disadvantages:

1. no direct way to address robustness to modelling errors. From equation (5.7)it can be seen that in case x 6= x and p∗u 6= pu the dynamics are notfully cancelled, still resulting in nonlinear closed-loop behaviour betweenthe commanded and actual command variables;

2. the control laws inherit the complexity of the synthesis model.

The main contribution of this chapter lies in addressing the first disadvantage.Regarding the second disadvantage, it must be noted that for complex models,manual derivation, coding and validation of the control laws may be tedious anderror-prone. However, in Chapter 4 it has been shown that these tasks can beautomated to a large extent.

5.2 The Aircraft Model 119

5.2 The Aircraft ModelThe control laws are designed for DLR’s Advanced Technologies Testing AircraftSystem (ATTAS) [12]. The aircraft model was implemented using the modellinglanguage Modelica and the Flight Dynamics Library. Implementation details canbe found in the application example in Chapter 2. In this section the equationsof motion are briefly reviewed, based on the inputs, outputs, and parameters asgrouped in the previous section.

5.2.1 Nonlinear model equations

The Newton-Euler equations of motion describe force and moment equations withrespect to body-fixed axes with their origin in the centre of gravity:

Ωb = I(pk, pu)−1 [MAx(xa, pk, pu) + MAu

(xa, pk, pu)uc + MT (xa, uo, pk, pu)

− Ωb × I(pk, pu)Ωb] (5.12)

Vb = m−1[

FA(xa, uc, pu) + FT (xa, pu) − Ωb × Vb + Tbe(Θ) [0, 0, g]T]

(5 .1 3 )

w h e re Ω b = [pb, q b, r b]T a re th e b o d y a n g u la r ra te s, Vb = [ub, v b, w b]

T is th ein e rtia l v e lo c ity v e c to r a lo n g th e b o d y a x e s. MT a n d FT a re m o m e n ts a n d fo rc e sin d u c e d b y th ru st, MA a n d FA a re a e ro d y n a m ic m o m e n ts a n d fo rc e s. T h e v e c to rxa d e sc rib e s th e a ir m a ss-re fe re n c e d sta te o f th e a irc ra ft, uc a re a e ro d y n a m icc o n tro ls, c o n sistin g o f a ile ro n s (δA), e le v a to r (δE) a n d ru d d e r (δR), uo c o n ta in sth e th ro ttle se ttin g s fo r b o th e n g in e s. T h e g ra v ity c o n sta n t is g. T h e in e rtiate n so r I(pk, pu) h a s u n c e rta in ty a n d is p ro p o rtio n a l to th e m a ss m ∈ pk. T h em a trix Tbe(Θ) re p re se n ts th e d ire c tio n c o sin e m a trix fro m th e E a rth -re fe re n c e dh o riz o n ta l re fe re n c e fra m e Fe in to b o d y a x e s Fb:

Re = TT

be(Θ) Vb (5 .1 4 )

w h e re Re is th e p o sitio n ra te w ith re sp e c t to th e in e rtia l re fe re n c e fra m e . Inth e A T T A S m o d e l th e E a rth -fi x e d fra m e is a ssu m e d to b e a n in e rtia l re fe re n c e .F in a lly , th e re la tio n b e tw e e n th e a ttitu d e ra te s a n d th e b o d y a n g u la r ra te s isg iv e n b y [1 5 ]:

Θ = TΦb(Θ) Ω b (5 .1 5 )

w h e re Θ = [φ , θ , ψ ]T d e sc rib e s th e a ttitu d e o f th e a irc ra ft w ith re sp e c t to th eh o riz o n ta l N E D fra m e a n d TΦb(Θ) (∈ IR 3×3) d e p e n d s o n th e c u rre n t a ttitu d e .T h e u n c e rta in p a ra m e te rs pu in e q s. (5 .1 2 ) a n d (5 .1 3 ) a re in d u c e d b y to le ra n c e s o nso -c a lle d a e ro d y n a m ic c o e ffi c ie n ts a n d m o m e n ts o f in e rtia , se e T a b le s 2 .5 a n d 2 .4 .F o r e x a m p le , th e d e riv a tiv e o f th e a e ro d y n a m ic ro llin g m o m e n t c o e ffi c ie n t Cl w ithre sp e c t to th e ro ll ra te p is w ritte n a s:

Clp = Clp(1 + ∆ Clp)


where Clp is the nominal coefficient value, ∆Clp is the tolerance parameter, andClp is the perturbed coefficient. B ased on experience, the tolerances are up to10% for longitudinal coefficients and up to 30% for lateral and ground eff ect-related coefficients, see Table 2.5. In the frame of this chapter, lateral uncertainparameters are of prime interest, since the control laws for lateral dynamics willbe focused on. Therefore:

pu =[

∆CY , ∆Clβ , ∆Cnβ, ∆Clp , ∆Cnp

, ∆Clr , ∆Cnr, (5.16 )

∆ClδA, ∆CnδA

, ∆ClδR, ∆CnδR

, ∆Ix x , ∆Ix z , ∆Iz z

]T

The parameters ∆CY . . .∆CnδRare the aforementioned tolerances on aerody-

namic model coefficients. The parameters ∆Ix x , ∆Ix z , ∆Iz z represent toleranceson elements of the inertia tensor.From the model implemented in M odelica, simulation code for the design model isgenerated according to user-specified inputs and outputs. The model outputs con-sist of 23 measurements available to the control system, see Table 2.2. The modelinputs consist of aircraft controls uc, uo and white noise signals for turbulenceand IL S signal disturbance models, see Table 2.4.

5.2.2 Linearised model equationsThe control laws developed in this chapter will undergo worst-case stability analy-sis, requiring the model to be available in linear form. To this end, the simulationmodel may be linearised numerically as described in Section A.4. H owever, foruncertainty analysis it is more useful to perform symbolic linearisation, since thisallows the resulting linear state space matrices to be expressed as a function ofthe uncertain parameters. In case this dependency is rational, the symbolic ma-trices may be further transformed into a L inear Fractional Transformation (L FT)[132, 44]. From its implementation in object-oriented form, symbolic linearisationcan be done automatically, resulting in the following set of equations:

δx = A(pk, pu, x0(pk, pu), u0(pk, pu))δx

+ B(pk, pu, x0(pk, pu), u0(pk, pu))δuc (5.17 )

δy = C(pk, pu, x0(pk, pu), u0(pk, pu))δx

+ D(pk, pu, x0(pk, pu), u0(pk, pu))δuc

with:

δx = [δVb, δΩb, δE , δRe]T

δuc = [δuc, δuo]T (5.18 )

where δ.. indicates small perturbations from the values around the linearisationpoint. Finally, δy represents perturbations from measurement signals availableto the control laws. O nly for simplicity of notation, disturbance models (e.g.turbulence, sensor noise) and actuator models have been left out from (5.17 ).

5.2 The Aircraft Model 121

The aerodynamic state vector xa may therefore be replaced with x. Of course, inthe actual model for design these effects are fully accounted for.

Especially for aircraft, the linear model may strongly depend on the point oflinearisation, which is characterised by the type of equilibrium, the location inthe operating envelope, and the model parameters pk and pu. For this reasonthe state space matrices in (5.17) explicitly depend on the equilibrium state andinput vectors in the linearisation point: x0(pk, pu) and uc0

(pk, pu). Exact symboliclinearisation requires literal expressions for these variables, which are not alwayseasy to be found. A practical, but approximate approach has therefore beenproposed by the author in Reference [132]. From object oriented models x0(pk, pu)and uc0

(pk, pu) can actually be solved for. However, for aircraft the expressionsmay involve long polynomials, mak ing further transformation into for exampleLFT form very challenging.

In the frame of this chapter only lateral control laws and parameters relatedto lateral dynamics of the aircraft will be considered. Assuming the aircraft issymmetric, lateral states and controls in straight and level fl ight are zero:

pb = 0 trim constraint: straight and level fl ightrb = 0 trim constraint: straight and level fl ightφ = 0 trim constraint: wings levelψ = 0 trim constraint: direction of fl ightvb = 0 trim constraint: zero side slip angle (symmetrical aircraft)

(5.19 )

Since the ATTAS model assumes a symmetric aircraft, yawing and rolling mo-ments as well as lateral side force are zero in the trim condition, resulting in zerodefl ection of rudder and ailerons [105]. As an example, the aerodynamic rollingand yawing moments are:

Cn = (1 + ∆Cnβ)Cnβ

β + (1 + ∆CnδR)CnδR

δR + (1 + ∆CnδA)CnδA

δA

+ (1 + ∆Cnp)Cnp

pbb

2Vtas

+ (1 + ∆Cnr)Cnr

rbb

2Vtas

+ (1 + ∆Cnr)Cn

β

βb

2Vtas

Cl = (1 + ∆Clβ )Clββ + (1 + ∆ClδR)ClδR

δR + (1 + ∆ClδA)ClδA

δA

+ (1 + ∆Clp)Clp

pbb

2Vtas

+ (1 + ∆Clr )Clr

rbb

2Vtas

(5.20)

C oefficients lik e Cnβmay in turn depend on landing gear status, ground effect,

angle of attack etc. Due to (5.19 ), the control defl ections δA and δR must be zeroto ensure zero aerodynamic moments.

Since lateral uncertain parameters have no direct effect on the longitudinal dy-namics, the equilibrium may be characterised by x0(pk) and uc0

(pk). For a givenfl ight condition and set of k nown parameters pk = pk0

, the symbolic linear modelmay then be written as follows:

δx = A(pu)δx + B(pu)δuδy = C(pu)δx + D(pu)δu

(5.21)


For example, linearising the moment equation (5.12) results in:

δΩb = I(pu)−1 (5.22)[

∂ (MAx(x, pu) +MAu

(x, pu)uc +MAT(x, uo, pu))

∂xT

∣

∣

∣

∣

x=x0,uc=uc0,uo=uo0

δx

+ MAu(x0, pu)δuc +

∂MAT(x, uo, pu)

∂uTo

∣

∣

∣

∣

x=x0,uo=uo0

δuo

]

Note that the second term could be written in its differentiated form straightaway, since the dependency on uc is simply affine. The inertial term Ωb× I(pu)Ωb

disappears after linearisation around straight and level flight, since Ωb0 is zero.In case of equilibrium in turning flight (i.e. Ωb0 6= 0) the linearised term must beretained.From (5.20) it can be seen that the uncertain parameters appear in simple poly-nomial form (this also holds for the tolerances on the inertia matrix I). For thisreason, the elements of the state space matrices in (5.21) depend on uncertain pa-rameters in a rational way, allowing for symbolic transformation into LFT form[132, 44].

5.3 Dynamic inversion attitude control lawsObjective of the design is to develop attitude control laws that will be part ofof and autoland system, providing command augmentation and stabilisation. Tothis end, the controller structure as depicted in Figure 5.2 is used. This structurebasically consists of two control loops: control of attitude angles (Θ) and controlof body angular rates (Ωb). Alternatively, one may apply the procedure for NDIto the attitude angles. Time differentiation immediately results in eqn. (5.15):

Θ = TΦb(Θ) Ωb (5.23)

Differentiating one more time:

Θ = TΦb(Θ) Ωb + TΦb(Θ) Ωb (5.24)

The term Ωb involves the moment equations (5.12), so that an analytic relation-ship with aerodynamic control surface deflections arises. However, in this designa slightly different procedure is followed. Instead of (5.24), the following equationis inverted:

Θ = TΦb(Θ) Ωbcm d(5.25)

so that:

Ωbcm d= T−1

Φb (Θ) Θ′cm d (5.26)

where Ωbcm dare desired body angular rates and:

Θ′cm d = KE (Θr ef − Θ) (5.27)

5.3 Dynamic inversion attitude control laws 123

hereKE is a diagonal gain matrix and Θref is the vector with commanded attitudeangles. This basically implies that the derivative of the matrix TΦb(Θ) is neglected,which is possible provided that attitude accelerations are small. The reason forthis simplification is that this allows for opening the attitude loop, commandingΩbcmd

or Θ′cmd may be combined directly. This allows the use of the same set ofattitude control laws with different autopilot functions or for manual control. Anexample will be given in the course of this chapter. In addition, neglecting thederivative of TΦb(Θ) reduces computational effort in the Flight Control Computer(FCC).

Statees tim atio n& f ilter in g

A ir c raftd y n am ic s

( p , p )k u

I n v er s ec o n tr o ller( p , p )k u

*

L in earc o n tr o ller

K ( s )

ref

uc

xa^

Is~

cm d'T (E)b

- 1

KE

ref

cm d'

yo

Figure 5.2: Attitude control system

The tracking control laws for Ωbre fwill be designed using the Dynamic Inversion

procedure as discussed in Section 5.1 for y = Ωb. Differentiating Ωb directlyresults in the moment equation (5.12):

Ωb = Lfh+ Lghuc = I(pk, pu)−1 [MAx(xa, pk, pu) +MT (xa, uo, pk, pu)

− Ωb × I(pk, pu)Ωb] + I(pk, pu)−1MAu(xa, pk, pu)uc (5.28)

where Lgh = I(pk, pu)−1MAu(xa, pk, pu). Since the three control inputs in uc

are primarily moment generating devices around the three aircraft body axes,I(pk, pu)−1MAu

(xa, pk, pu) is (in the normal flight regime) non-singular, fulfillingassumption 3 in Section 5.1. Inversion of (5.28) gives:

uc = MAu(xa, pk, p

∗

u)−1 [I(pk, p∗

u)Ω′bcmd−MAx

(xa, pk, p∗

u)

− MT (xa, uo, pk, p∗

u) + Ωb × I(pk, p∗

u)Ωb

]

(5.29)

Note that xa has been replaced with xa and pu with p∗u. In the ideal case theform of (5.8) results:

Ωb = Ω′bcmd


Usually, nominal values for the parameters in p∗u are substituted, which corre-sponds to inverting the nominal aircraft model. In this design elements of p∗u areused as additional synthesis parameters, as will be discussed in Section 5.4.For the linear control law K (s) a P I structure is used (Figure 5.2):

Ω′bcmd= Ki

∫ t

0

(Ωbref− Ωb)d τ +KpΩb (5.30)

The closed-loop system is depicted in Figure 5.2. The integrator on (Ωbref−Ωb) is

required to compensate for steady errors in the moment equations. In equilibriumflight Ωbref

= Ωb = 0 holds. The integral part then allows a moment error(for example, due to unmodelled aircraft asymmetry) to be compensated for vianonzero Ω′bcmd

. W ithout integrator, Ωbrefwould have to be nonzero, resulting in

a steady state error in Θ (eqn. (5.27)). The optimisation of the linear gains willbe discussed in Section 5.4.

5.3.1 Automatic control law generation

The inverse control law of eq. (5.29) is automatically generated from the aircraftmodel in Dymola using the procedure described in Chapter 4, see Figure 4.8.First step in this procedure is to adapt the model for inversion:

1. wind and terrain models are removed;

2. dynamic motivator and engine models are replaced with static versions.

The motivator dynamics are modelled as first order low-pass filters and are con-siderably faster than the aircraft flight dynamics. For this reason, these dynamicsare neglected in the inversion. In addition, the filter approximation is only in-tended to represent a certain amount of actuator bandwidth and to implementrate and position limits. The internal filter state would have to be replaced withmeasurement of the actual control surface deflection. This measurement howeveris not reliably available on most aircraft types.In the simulation model the radio altitude Hra is computed using the terrainmodel. This object is removed and the radio altitude is defined as input instead.Similarly, all variables computed in the atmospheric model are obtained from airdata measurements instead. However, the component has to be retained in orderto compute the Mach number, since this variable is not available from sensors.The resulting aircraft model is left with the twelve flight dynamics states [Vb, Ωb,Θ, Re]

T . A body angular rate controller is selected from the Flight DynamicsLibrary, see Figure 5.3. The inputs of this controller are commanded body an-gular rates (pqrC, right) and outputs are the required control surface deflections(Co n trS u rf ,left). The body angular rates are obtained from the data bus. The in-tegrator left to the gain block integrates desired angular rate commands (Ω′bcmd

).The result is connected to the body angular rates in the data bus as well. W hentranslating the model, this integrator is automatically removed by differentiation,


Figure 5.3: Body angular rate controller from Flight Dynamics Library. Thebus connector below is connected to the aircraft data bus. Input are commandedbody rates, output are required control deflections.

so that in the model Ωb = Ω′bcmd. More details on this process can be found in

Chapter 4.The object is added to the ATTAS model as depicted in Figure 5.4.The modified model, including the linear controller, is depicted in Figure 5.4.The inputs and outputs of the latter have been propagated as model inputs andoutputs respectively. Note that, compared with Figure 2.12, the control surfacesContrSurf have become outputs. Other inputs like thrust and stabiliser settingsare commanded by the pilot or autopilot and have therefore been retained, sincethese effects are necessary for computation of e.g. the pitching moment.The model may now be translated into executable form. In order to make surethe model states are obtained from measurement instead of integration withinthe inverse model, they are automatically turned into inputs by setting a cor-responding option for translation. The Dymola formula manipulator and codegenerator automatically generates the Nonlinear Dynamic Inversion control lawsof eq. (5.29) in c-code, ready for execution in Simulink and in the Flight ControlComputer (FCC)3.

5.3.2 Implementation aspectsDynamic Inversion control laws provide “ full” control deflections, i.e. a quasi-steady component for trimming the aircraft, and a varying component for ma-noeuvring. Due to model deviations, the static part may be slightly wrong. As aconsequence, the closed loop system may show considerable transient behaviour

3After very carefully checking the code, of course.


Figure 5.4: ATTAS model for automatic generation of NDI control laws. Notethat the terrain and runway models have been removed. The FCS component(Figure 5.3) has been connected to the aircraft. Control deflections (ContrSurf)have become outputs, commanded angular rates (pqrC) have been defined asinputs (compare with Figures 2.12 and 5.2).

when the controller is initialised. For this reason, the following algorithm is used:if init

Ω′bcmd= 0 // override input

compute uc from DI // static uc

uc0= uc // store static uc

δuc = 0 // zero control defl.

else

Ω′bcmd= Ω′bcmd

// normal use

compute uc from DI

δuc = uc − uc0// output control defl.

end


By setting Ωbcmd= 0 at the initial computation, only the static control deflections

are obtained. These are subtracted from the control inputs from then onwards,so that the controller only provides control deflection “delta’s” with respect tothe momentary control deflections at initialisation. The above code is added afterthe NDI control laws.The success of any flight control law design stands or falls with the quality ofthe sensor signals. These signals are processed in the State estimation & filtering

block in Figure 5.2. The ATTAS aircraft has a sophisticated sensor system pro-viding inertial and air data measurements, allowing for easy computation of allstates. However, it is very important to be aware that the air data signals pickup atmospheric disturbances that, without further processing, would propagatethrough the control laws and result in high control activity. In addition, the angleof attack (AoA) sensor of ATTAS is on a long nose boom, which due to its flexi-bility adds a 6 Hz disturbance to the signal. For this reason, the signal has to becombined with inertial measurements using complementary filtering techniques:

α =1

τs + 1α+

τs

τ s + 1αi (5.31)

where αi is the inertial angle of attack computed from:

αi = (θ − γa) (5.32)

with γa ≈ −VZ

Vt a s. The time constant is set to 1 s, causing the air data content to

be cut off well below the 6 HZ disturbance signal. An example result is shown inFigure 5.5The side slip angle is available as a measurement on ATTAS, but this is notthe case on most airliners. For this reason, use of the signal was to be avoided.The signal can be estimated as follows. Writing the side force equation in theaerodynamic model in the form:

CY = CYβ(M,α, . . . )β + CYres t

(δR,M, α, . . . )

the side slip angle can be approximated from:

βest =1

CYβ

[

nymg

1/2ρ V 2casS

− CYres t

]

(5.33)

The right hand term can be computed using the coefficients from the model andthe available measurements. To reduce the propagation of noise due to turbulence,the signal is complementarily filtered with:

βi =−nyg + g sinφ cos θ

Vtas

− rb cosα+ pb sinα (5.34)

which is less affected by turbulence at higher frequencies. The filter, depicted inFig. 5.6, is second order, preventing a bias in β in case of longer term cross windshear. The gains K1 and K2 are set to 0.2 and 1 respectively.


0 5 10 15 20 25 302

3

4

5

6

7

8

time (s)

α (d

eg)

measured α

compl. α

Figure 5.5: Comparison between measured and complementarily filtered angle ofattack signal

For larger aircraft also inertial measurements require careful attention. Being at-tached to the airframe, the sensors not only pick up flight path-related, but alsostructural dynamics. This is taken into consideration in placement and attach-ment of sensors, but usually their signals still need to be filtered using low-passor notch filters.

5.4 Robust parameter synthesisThe tuning of the control laws for the lateral and longitudinal aircraft dynamicsis performed separately. In this chapter only the lateral part will be discussed.

5.4.1 Robustness

Robustness is major issue in the design of Dynamic Inversion control laws, andis usually addressed in the synthesis of the linear part of the controller. This canfor example be done using a robust synthesis method such as H∞ or µ-synthesis[5], if a classical PID controller structure is not sufficient.In this contribution, free synthesis parameters are tuned for robustness usingmulti-objective optimisation in combination with a multi-model approach [61].These synthesis parameters are the gains of the control law in equation (5.30), aswell as the parameters p∗u in the inverse model equations (5.29), which are consid-ered uncertain in the synthesis model. In the synthesis, robustness is addressed

5.4 Robust parameter synthesis 129

1

s

1

sK

1

K2

est

i

.

.

^

-+++

+

+

Figure 5.6: Complementary side slip filter

in th ree w ays:

1 . P arameters pk in th e inv erse control law s th at can b e determined on-line,are ob tained as additional measu rement inpu ts. In th is case, th ese are th eairc raft mass and centre of g rav ity location.

2 . R ob u stness to parametric u ncertainties is ach iev ed v ia a mu lti-model ap-proach . A set of models w ith w orst-case parameter comb inations w ith re-spect to one or more desig n criteria is addressed simu ltaneou sly in th e opti-misation, allow ing for trade-off ov er criteria u nder nominal and w orst-caseconditions. In th is w ay, g lob al rob u stness, i.e. ov er larg e parameter rang es,can b e addressed.

3 . R ob u stness to u nspec ified u ncertainties (eg . compu tational time delays,u nmodelled dynamics) is addressed v ia local u nspec ific rob u stness marg insas optimisation criteria. T h ese are listed in T ab le 5 .1 (gmada ... pmasr).

M u lti-model optimisation implic itly assu mes th at th e control law w ill also b erob u st for model cases in b etw een th ose addressed du ring parameter synth esis.H ow ev er, no g u arantee can b e g iv en. A n additional intention of th e local ro-b u stness criteria is to cov er su ch model cases. T h e g eneral idea is sk etch ed inF ig u re 5 .7 for tw o parameters pu = [a, b ]T . T h ree model cases are u sed in opti-misation (p0

u, p1

u, p2

u). In each case, local rob u stness marg ins are considered as

criteria. T h e sizes of th e areas b ou nded b y th e dotted lines refl ect th e ach iev ab leperformance rob u stness arou nd th e th ree model cases for a controller desig nedw ith p∗

u= p0

u. O b v iou sly, th e area arou nd th e nominal case is larg e, th e areas

arou nd th e w orst cases are smaller. T h e local areas tog eth er do not cov er th ecomplete parameter space b ou nded b y [am in , am a x , b m in , b m a x ]. O ptimisationof model parameters in th e inv erse controller w ill resu lt in th e point p∗

uin th e

parameter space, w h ich can b e interpreted as th e b est location for model inv er-sion. A s a resu lt, th e size of th e area (b ou nded b y solid line) arou nd th e nominalpoint decreases, th ose arou nd th e w orst cases b ecome larg er, so th at th e completeparameter space is cov ered.


The approach sketched above is used for synthesis of the N D I controller param-eters. The assumptions can be validated by performing a µ-analysis on theclosed loop system afterwards. This req uires interconnection of the linearisedcontroller with the aircraft L FT representation, obtained from symbolic lineari-sation sketched in S ection 5.2.2. B esides stability robustness, µ-analysis allowsfor finding worst-case damping and stability margins in the uncertain parameterspace, valid for the linearised model.

a

b

bm ax

bm in

bn o m

am axam in an o m

p =u

a

b

n o m in alc as e p u

0

N D I p u

c as e p u

1

c as e p u

2

L in e ar ro bus tn e s s c rite ria (lo c al ro bus tn e s s )

O p tim iz atio n fo r 3 m o d e l c as e s(g lo bal ro bus tn e s s )

P aram e te rs p ac e

Figure 5.7: G eneral idea of addressing global and local robustness (dashedbounds: achievable robustness areas with p∗

u= p0

u, solid bounds: achievable areas

with optimised p∗u)

5.4.2 StabilityAs mentioned in S ection 5.1, one always has to carefully check the internal dy-namics of the closed-loop system. B ut also, sensors, FCS delays, actuators andsignal processing add dynamics to the over-all system. For this reason, duringeach optimisation step the closed-loop system is linearised and all eigenvaluesof the lateral modes are determined. The minimum damping is computed andaddressed as an optimisation criterion (damp in Table 5.1, to be discussed inthe following section). After parameter synthesis, minimum damping is furtherassessed in ex tensive robustness analysis.

5.4.3 Design criteriaFor tuning the controller parameters, design req uirements need to be formulatedin the form of computational criteria, suitable for optimisation. For the lateralsynthesis parameters of the N D I controller, these criteria are computed from threenonlinear simulations and the linearised closed-loop system, see Table 5.1. The


Linear analysis:

nam e d escrip tio n c o m p u tatio n

g m ad a g ain m arg in at δA act. see R em ark s

p m ad a p h ase m arg in at δA act.

g m ad r g ain m arg in at δR act.

p m ad r p h ase m arg in at δR act.

g m asp g ain m arg in at p sens.

p m asp p h ase m arg in at p sens.

g m asr g ain m arg in at r sens.

p m asr p h ase m arg in at r sens.

d am p m inim u m d am p ing m iniζi

S im u latio n 1 : S tep : φr e f = 1 0 d eg , ψr e f =g

Vtas

tanφr e f

p h rt rise tim e φ see R em ark s

p h o s o v er sh o o t φp h st ’settling tim e’ φ m ax t > 15s|φ− φr e f |

p sd d rise tim e ψ

p h d s ’settling tim e’ ψ m ax t > 15s|ψ − ψr e f |

d d a m ax δA m ax t > ts+ 0.2s|δA|S im u latio n 2 : S tep : ψr e f = 5 d eg , φr e f = 0

p srt rise tim e ψ see R em ark s

p so s o v er sh o o t ψp sst ’settling tim e’ ψ m ax t > 15s|ψ − ψr e f |

p h er erro r φ∫

15

0φd t/ 1 5 s

d d r m ax δR m ax t > ts+ 0.2s|δR|S im u latio n 3 : C ro ss w ind step : wy = 1 6 m / s at t = 0 s

p h w e m ax . φ m ax |φ|d rp eak p eak δR m ax t < 5s |δR|

Table 5.1: Design criteria for multi-objective optimisation. Remarks: All com-putations: symmetrical horizontal flight; altitude=10 0 0 ft; nominal aircraft load-ing. Step times: ts = 1 s. Rise time: ∆ t between 10 % and 9 0 % of command.Gain/ phase margins: as in Matlab Control Toolbox [79 ]

controller is designed as a stability and command augmentation system of anautoland system (Chapter 6 ). The intention is to use the same NDI-controller asinner loop for the localiser mode (approach path tracking), and the align mode(aircraft alignment with the runway, shortly before touchdown). During localisertracking, the NDI controller has to track co-ordinated roll angle and heading ratecommands. For this reason, simulation 1 is performed, from which criteria such asrise time and overshoot of a combined roll angle and heading rate step commandare computed. During the align phase, roll and heading angles are controlledindependently. The roll angle is used to track the localiser, the heading angle isused for de-crab in case of cross-wind. Therefore, simulation 2 is performed, fromwhich criteria are computed based on a step command on ψr e f .The criteria phst, phds, psst address settling times, although a different measureis used to actually compute these criteria. Disturbance rejection is assessed insimulation 3 through a heavy cross wind step.


5.4.4 Scaling of criteriaIn multi-objective optimisation, relative importance of criteria is expressed viascaling. E specially in case of conflicting requirements, this gives the designer aneffective means to make trade-offs and to set priorities. Scaled criteria have tobe formulated such, that the objective is to minimise them, and a value less thanone is considered satisfactory.The scalings as applied to the criteria in Table 5.1 are given in Table 5.2. Scaling

crit, bad good demand typeunit low low valuephrt (s) – – 2.5 cphos (-) – – 0.05 cphst (°) – – 0.1 mpsdd (s) – – 10 mphds (°/ s) – – 0.1 mdda (°/ s) – – 20 mpsrt (s) – – 5 cpsos (-) – – 0.05 cpsst (°) – – 0.2 mpher (°) – – 0.1 mddr (°/ s) – – 10 mphwe (°) – – 6.7 mdrpeak (°) – – 25 mgmada (dB ) 4 6 – mpmada (°) 30 60 – cgmadr (dB ) 4 6 – mpmadr (°) 30 60 – cgmsp (dB ) 4 6 – mpmsp (°) 30 60 – cgmsr (dB ) 4 6 – mpmsr (°) 30 60 – cdamp (-) 0.6 0.7 – cLegend:m=minimise, c=ineq uality constraint, °=deg

Table 5.2: Scaling of criteria

can be performed by division of the criterion by its demand value:

ci(T ) = ci(T )/ di

where ci(T ) and di are the computed value and demanded value of criterion irespectively, and T denotes the current set of tuning parameters. Scaling can alsobe done using so-called ’good-bad’ values [50]. Here, a special type of ’good-bad’scaling is used, as will be explained for gmada in Figure 5.8 . The demand is thatthe gain margin is at least 4 dB (’bad-low’). Any value larger than 6 (’good-low’)is considered equally good and therefore scaled to 0. Below 6 dB, the scaled valueincreases linearly, such that a value of 1 is reached for the bad-low value of 4dB. Any value between 4 and 6 dB is acceptable, any value lower than 4 dB isconsidered unacceptable (bad). As an example, if the gain margin is 3 dB, itsscaled value equals 1.5.


1

4 63

1.5

g o o d -lo wb a d -lo w

g a in m a rg in (d B )scale

dcrite

rion

acceptable goodb a d

Figure 5.8: Scaling of gmada w ith good -b ad v alu e s

Som e of th e crite ria in T ab le 5 .2 are treate d as ine q u ality constraints (i.e . ci(T ) ≤1 ). F or e x am p le , a rise tim e of 2 .5 s is d e m and e d . If th is is satisfi e d , th e re is nop oint to fu rth e r m inim ise th is crite rion, since th is w ill u nnece ssarily go at th e costof stab ility and control e ff ort.

5.4.5 Parameter synthesisF or p aram e te r sy nth e sis th e m u lti-ob jectiv e op tim isation tool M O P S (M u lti-O b jectiv e P aram e te r Sy nth e sis [5 0 ]) w as u se d . A th eore tical b ack grou nd of th ism e th od ology is p rov id e d in Section 6 .6 , b u t it is not nece ssary for u nd e rstand -ing th is section. A s a fi rst ste p , only th e gains of th e late ral p art of th e linearcontrolle r (e q . (5 .3 0 )) w e re tu ne d . T h e re su lting v alu e s can b e fou nd in th e fi rstcolu m n of T ab le 5 .3 .U sing th is gain se t, a w orst-case analy sis w as p e rform e d . It w as assu m e d th atw orst-case s occu r in th e corne rs of th e p aram e te r sp ace . F or th is reason, u nce rtainp aram e te rs w e re se t to th e ir m inim u m or m ax im u m v alu e s and for all p aram e te rcom b inations th e crite ria in T ab le 5 .1 w e re e v alu ate d . In ord e r to lim it th e nu m -b e r of sim u lations, tole rance s on m om ents of ine rtia, as w e ll as th e le ss re le v antcoe ffi cients CnδA

and ClδR, w e re le ft ou t, re d u cing th e nu m b e r of com b inations

from 2 14 = 1 6 3 8 4 to 2 9= 5 1 2 . T h re e m od e l case s w e re se lecte d : th e one w ithm ax im u m (scale d ) v alu e for damp (w orst d am p ing), th e one w ith m ax im u m gmsr

(low e st gain m argin at y aw rate sensor, u nstab le ), and th e one w ith m ax im u mpmadr (m inim u m p h ase m argin at ru d d e r actu ator). T h e se crite ria w e re m ostsensitiv e to th e p aram e te r v ariations and consid e re d m ost critical.T h e th re e w orst m od e l case s w e re ad d e d to th e nom inal m od e l in a m u lti-m od e lse t. Som e crite ria scalings for th e se w orst-case s w e re sligh tly re lie v e d com p are dw ith th e v alu e s in T ab le 5 .2 . F or th e m u lti-m od e l se t a new p aram e te r sy nth e sisw as p e rform e d . F irst, only th e linear gains w e re op tim ise d . T h e v alu e s are giv enin th e second colu m n of T ab le 5 .2 . T h e re su lts w e re not satisfactory , as w ill b ed iscu sse d in Section 5 .5 .D u ring th e th ird p aram e te r sy nth e sis, not only th e linear gains, b u t also e igh t offou rte en ae rod y nam ic tole rance s in th e p aram e te r v ector p∗

uin th e N D I control

law s (se e (5 .2 9 )) w e re av ailab le for tu ning b y th e op tim ise r. T h is can b e inte r-p re te d as search ing for th e b e st location for m od e l inv e rsion ov e r all p aram e te r


tuning (1) nom. (2) m.mod. (3) m.mod. unitpar. design design 1 design 2

linear controller gains:Kip 2.9 2.28 2.52 –Kpp

3.47 2.86 2.92 s−1

Kir 2.0 2.56 2.17 –Kpr

1.6 1.95 1.96 s−1

Kφ 0.41 0.41 0.41 s−1

Kψ 0.7 0 0.7 0 0.7 0 s−1

aerodynamic coefficients in p∗u (eq. (5.29))∆CY 0 0 -0.3 –∆Clβ 0 0 -0.3 –∆Cnβ

0 0 -0.23 –∆Clp 0 0 -0.07 –∆Clr 0 0 -0.3 –∆Cnp

0 0 -0.3 –∆Cnr

0 0 -0.15 –∆CnδR

0 0 0.3 –

In Figure 5.2:KE = d ia g ([KΦ, Kθ, Kψ])Ki = d ia g ([Kip , Kiq , Kir ])Kp = d ia g ([Kpp

, Kpq, Kpr

])

Table 5.3: C ontroller synthesis parameters

configurations. For this reason, the aerodynamic parameters were limited to thesame bounds as the assumed maximum tolerances in the simulation model, i.e.30% . The optimisation run involved 14 synthesis parameters and 4 × 22 = 88criteria (4 model cases). The result can be found in the third column of Table 5.3.Note that the optimiser tends to push most of the parameters to their maximumor minimum values. R elieving the bounds on the model parameters could improvethe results further, but this would imply that the inverse model corresponds to adesign model configuration outside its assumed parameter envelope.

5.5 Assessment

5.5.1 Analysis w.r.t. synthesis criteria

The three designs (Table 5.3) will be compared now using so-called parallel co-ordinates [51] (Figure 5.9) and two simulations (simulation 1 and 3 from Table 5.1)for two model cases (Figure 5.10: nominal and Figure 5.11: worst damping).

In parallel co-ordinates, all scaled criterion values are plotted on an individual axisand connected through a line; one graph corresponds to one controller parameter

5.5 Assessment 135

phrt

phos

phst

phds

psdd

dda

psrt

psos

psst

pher

ddr

phw

edr

peak

GM

AD

AP

MA

DA

GM

AD

RP

MA

DR

GM

SP

PM

SP

GM

SR

PM

SR

DA

MP

phrt

phos

phst

phds

psdd

dda

psrt

psos

psst

pher

ddr

phw

edr

peak

GM

AD

AP

MA

DA

GM

AD

RP

MA

DR

GM

SP

PM

SP

GM

SR

PM

SR

DA

MP

Model 1 (nominal) Model 2 (worst damping)

1

Figure 5.9: Criteria in parallel co-ordinates (solid arrow = ineq. constraint,dashed arrow = minimise). In Figs. 5.9, 5.10, 5.11 – controller 1: solid, controller2: dashed, controller 3: fat-solid

set. The horizontal line indicates a value of one. Criteria values below this lineare considered satisfactory. Parallel co-ordinates are normally visualised on-lineduring optimisation with MOPS. They give quick insight in the optimisationprogress, which criteria are hard to satisfy, and criteria that confl ict and thushave to be compromised.

The labelled axes correspond with the criteria in Table 5.1. To the left the criteriafor the nominal model are shown, to the right those of one of the worst-case modelsare depicted. Due to space restrictions, the other model cases are not shown. Thethin solid lines (in all subsequent figures) show the results for controller 1 (i.e.optimised with the nominal model only). The bad damping of model 2 (scaled2.5, actual ζ = 0.25) was reason to include this model case in the multi-modelset. The results from optimisation of the linear controller parameters only, withthe selected multi-model set, are represented by the dashed line. The dampingis increased to a satisfactory level (criterion below 1-line), this goes clearly atthe cost of rudder activity (ddr,drpeak), the roll response due to a side wind stepph w e (see also Figure 5.11), the phase margin at the rudder actuator pmadr,and the damping of the nominal model. The even worse damping in Figure 5.11seems in contradiction with the better damping criterion level, but inspection


0 5 10 15 200

2

4

6

8

10

12Phi

t [s]

Deg

0 5 10 15 200

0.5

1

1.5

2

2.5PsiDot

t [s]

Deg

/s

0 5 10 15 20−10

−5

0

5

10Phi (cross wind step)

t [s]

Deg

0 5 10 15 20−10

−5

0

5

10PsiDot (cross wind step)

t [s]

Deg

/s

φref = 10 deg

(dψ/dt)ref = 1.5 deg/s

φref = 10 deg


Figure 5.10: Roll and wind step responses for model case 1 (for legend, see captionof Figure 5.9)

of the results revealed that the rudder actuator rate saturates due to its higheractivity. In one of the model cases (not shown) even instability occurs. The fatsolid line shows the results for controller 3, for which linear gains, as well as modelparameters in the NDI controller were optimised. From the parallel co-ordinates itis immediately clear that all criteria are satisfied (this also holds for the two modelcases not shown). E specially Figure 5.11 shows considerable improvement. Sincethe third controller no longer inverts the nominal model, it can be expected thatdecoupling in the nominal case deteriorates. This can be verified in Figure 5.9for the criterion pher 1-1 (roll angle error due to a heading angle command).For controllers 1 and 2 (for which p∗u = pu) decoupling is excellent, whereas forcontroller 3 it is considerably worse, but still acceptable.

Of course, with the gain set of controller 3, a new parameter study was performed.All stability related criteria are satisfied for the investigated parameter combina-tions (no figure). In a number of cases the roll angle error 15 s after the rollcommand step (phst) and the roll angle response due to the wind step (phwe)degrade considerably. In no case instability due to rate saturation occurred. Onemore optimisation with an additional model case, (worst phst) was performed,but the results did not improve much.

5.5 Assessment 137

0 5 10 15 200

2

4

6

8

10

12Phi

t [s]

Deg

0 5 10 15 200

0.5

1

1.5

2

2.5Psidot

t [s]

Deg

/s

0 5 10 15 20−10

−5

0

5

10Phi (cross wind step)

t [s]

Deg

0 5 10 15 20−10

−5

0

5

10PsiDot (cross wind step)

t [s]

Deg

/s

φref = 10 deg


φref = 10 deg


Figure 5.11: Roll and wind step responses for model case 2 (for legend, see captionof Figure 5.9)

5.5.2 Robust stability analysis with µ

During parameters synthesis robustness has been addressed by combining a multi-model approach with local robustness measures in the form of criteria. A reliableway to verify the assumptions behind this approach is to perform a µ-analysis[9]. The structured singular value µ is an indicator for the smallest magnitude ofparameter tolerances for which a specific design criterion is violated or the sys-tem becomes unstable. W hat makes µ-analysis particularly useful is that on theone hand the indicated smallest magnitude is guaranteed, and on the other handthat critical parameter combinations are returned, which may be verified andfurther analysed in for example nonlinear simulation. The number of uncertainparameters that can be addressed simultaneously may be considerably larger ascompared with testing parameter combinations in batch computations (gridding).Only testing minimum and maximum parameter values, the number of evalua-tions doubles with each additional parameter. Finally, µ-analysis not only coversboundaries of the parameter space, but will also detect worst-case combinationswithin.

µ-Analysis has two main drawbacks:

• the method requires the system model in the form of a L inear FractionalTransformation (L FT). Obviously, the analysis result stands or falls withvalidity of this model;


• most criteria cannot be translated directly into a form that can be handledby µ-analysis. For example, rise time can only be addressed indirectly,whereas an H∞ norm-based criterion is handled directly.

For the ATTAS model an LFT is very easily generated, since the object-orientedimplementation allows for symbolic linearisation as a function of the uncertain(lateral) parameters, see Section 5.2.2. The transformation from the symbolicstate space model into an LFT is performed using algorithms described in Refs. [63,132, 44]. In Ref. [54] J uliana presents an approach for combined LFT modellingof the system to be controlled and the NDI control law. H owever, in the frameof this work the aircraft LFT model and linearised controller model are obtainedindividually. One reason is that, as will be described subsequently, actuationsystem dynamics and delays in the Flight Control Computer (FCC) are presentbetween the aircraft model and NDI controller.

In order to verify the robust synthesis approach, gain and phase margins areof most interest. These margins are, although indirectly, very well addressed inµ-analysis [120].

ATTASE Q M

Ac tSe n s N D I K ( s )F C C

E s t

m

la t

ref

x

y

L e g e n d :Ac t = a c tu a to r sE Q M = e q u a tio n s o f mo tio nE s t = s ta te e s tima tio n a n d f ilte r in gF C C = f lig h t c o n tr o l c o mp u te rK ( s ) = lin e a r c o n tr o l la wN D I = N D I c o n tr o l la wSe n s = s e n s o r s

zla t wla t

Figure 5.12: Interconnection structure of the closed-loop ATTAS model for µ-analysis

Figure 5.12 depicts a so-called interconnection structure of the ATTAS modelwith the NDI control laws. This structure is a linearised representation of Fig-ure 5.2, with explicit dependence on uncertain parameters. Most blocks areself-explanatory. For example, the block A T T A S E Q M contains the linearisedequations of motion, the F C C block represents computational time delays in theflight control computer, approximated via first-order Pade filters. The signalnames start with “ δ” to indicate that these are perturbations with respect to thetrimmed flight condition. The block ∆m will be discussed shortly.

The block A T T A S E Q M has a lower feedback loop via ∆la t . This block has alllateral uncertain parameters as listed in (5.16) on its diagonal, normalised withrespect to their bounds. This implies that, compared with the gridding-basedworst-case search during parameter synthesis, this time all parameters, including

5.5 Assessment 139

inertia-related ones, CnδAand ClδR

are taken into account4. The structure of∆lat is given by:

∆lat =

diag[

∆IxI4, ∆IzI3, ∆IxzI6, ∆CY I4, ∆Clβ , ∆Cnβ, ∆Clp , ∆Cnp

,

∆Clr , ∆CnrI3, ∆ClδA

, ∆CnδA, ∆ClδR

, ∆CnδR

]

: ∆C.., ∆I.. ∈ IR

(5.35)

Note that for example the coefficient ∆CY is repeated four times on the diagonal.This arises during transformation of the symbolic state space model into LFT form[44]. The ..-sign indicates that the tolerance has been normalised with respect toits bounds. For example, since −0.3 ≤ ∆ClδA

≤ 0.3, ∆ClδA= 0.3∆ClδA

, so that

−1 ≤ ∆ClδA≤ 1.

The interconnection structure is first used to perform robust stability analysis ofthe three controller versions discussed in the previous sections. To this end thestructure is transformed into the form depicted in Figure 5.13. The block ∆lat

has been removed and its original input zlat and output wlat have been promotedto system outputs and inputs respectively. The command reference δΘr e f is setto zero. The block ∆m is not required for now and therefore omitted.

ATTASE Q M


E s t x

y

zla t

wla t

Figure 5.13: Input and outputs of the interconnection structure for µ stab ilityanaly sis

T he transfer function from wlat to zlat is called Mattas (s). T he structured sing ularv alue µ of Mattas (j ω ) w ith respect to ∆ lat ∈ ∆lat is defi ned as [2 7 , 9 ]:

µ∆lat(Mattas (s)) =

1

m in σ(∆ lat) : ∆ lat ∈ ∆lat, det (I − Mattas (s)∆ lat) = 0

(5 .3 6 )

If no ∆ lat ∈ ∆lat m ak es (I−Mattas (s)∆ lat) sing ular, µ∆lat(Mattas (s)) = 0 . E x act

com putation of µ is hardly ev er possib le. H ow ev er, suffi ciently tig ht upper andlow er b ounds can usually b e com puted.

4It must be noted that this LFT model was not available during the design of the controllers.R eason is that at that time no tools were available that allowed an LFT of suffi ciently low orderto be obtained from the p arametric state sp ace model (5 .2 1 ) for the full p arameter vector: onlyan LFT with the reduced p arameter set of S ection 5 .4 was of p ractical use. In the mean time,H eck er [4 4 ] develop ed highly imp roved algorithms for generation of low-order LFTs, allowingthe p resented LFT w.r.t. to the full p arameter set to be generated and used for the p resentanaly sis.


In case s = jω singularity of (I−Mattas(jω)∆lat) implies that the feedback systemof Mattas and ∆lat has a pole jω crossing the imaginary axis. A practical robuststability check is therefore performed by computing µ∆lat

(Mattas(s)) for s = jω

over a dense grid along the imaginary axis. The maximum value of µ over thisgrid indicates that at this point the worst ∆lat that causes an eigenvalue to crossthe imaginary axis is smallest. In case the smallest norm σ(∆lat) is smaller than1 and Mattas(s) is stable, no parameter combination will destabilise the closedloop system in F igure 5.12.

P lots of µ∆lat(Mattas(jω)) for the three controller parameter sets (Table 5.3) are

shown in F igure 5.14. S ince the peak values of the three curves are well be-low one, the three controllers achieve robust stability against the real parametricuncertainties in (5.16). Interestingly, controller 2 req uires a slightly smaller per-turbation (1 / 0.77 = 1.30) for instability, which indicates that the margin to thispoint is somewhat less.

10−2

10−1

100

101

102

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Frequency [rad/s]

µ u

pper

bou

nd

µ Upper bound for controller parameter sets

K1K2K3

Figure 5.14: C omparison of µ∆lat(Mattas(jω)) for three controller parameter sets

of Table 5.3: the lower the maximum value, the better

D uring design work, achieving a worst-case damping of ζ = 0.4 was consideredacceptable. W hether this has been achieved or not can be eff ectively verifiedby a µ computation. To this end, a µ analysis is performed along the ζ = 0.4damping line in the complex plane, instead of the imaginary axis. The results areshown in F igure 5.15. C learly, controller 1 req uires a magnitude of ∆lat of only1 / 2.50 = 0.4 in order to make a pole cross the 0.4-damping line in a pole map.

5.5 Assessment 141

Controller 3 is considerably better, but still requires only σ(∆lat) ≈ 1/1.9 = 0.53:within the uncertain parameter envelope, lower closed-loop damping will occur.The curve for controller 2 is quite interesting: compared with controller 1 thepeak has moved to the right and increased. Since the damping value in Figure 5.9has achieved an acceptable level for model case 2, the problem has shifted to aparameter case that was not included in the multi-model parameter synthesis. Ingridding-based parameter studies with controller 2, a new badly damped case wasactually found and included in the model set for optimisation of controller 3.

10−2

10−1

100

101

102

0

0.5

1

1.5

2

2.5

Frequency [rad/s]

µ u

pper

bou

nd

µ Upper bound for controller parameter sets: 0.4 damping

K1K2K3

Figure 5.15: Comparison of µ∆lat(Mattas(s)) for three controller parameter sets

of Table 5.3 along 0.4 damping line

It is very easy to figure out the worst-damping level over all parameter combi-nations by iterating the damping line along which the µ-analysis is performeduntil a peak value of 1 is found. This indicates that the norm σ(∆)lat = 1, sothat the worst case is on the border of the parameter envelope. The interest-ing result is that each controller has a worst damping level of ∼0.16. Figure 5.16nicely illustrates this for controller 3. From the µ-analysis along the 0.16 dampingline the worst-case combination was retrieved from the lower bound computation.Scaling the corresponding matrix ∆lat = ∆c r it with a factor 0.95, 1.0 and 1.05,a pole can be observed crossing the 0.16 damping line. Apparently, regardingworst damping the three controllers are surprisingly similar. It is also interestingto note that the worst damping for controller 1 is worse (0.16 instead of 0.25)than the one found from gridding over minimum and maximum parameter values


(giving rise to Model 2 in Figure 5.9). This is not surprising, since in the latteranalysis the number of parameters had to be reduced from 14 to 9, in order tomake computation times acceptable.

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1−1

−0.5

0

0.5

1

1.5

2

Real

Imag

Pole crossing the worst−case damping line

0.95 ∆crit

1.0 ∆crit

1.05 ∆crit

Figure 5.16: Worst case damping line for controller 3: 0.16 for σ (∆crit) = 1

Finally, it is interesting to find out worst gain and phase margins for the threecontrollers in order to evaluate robustness to unspecified uncertainties on top ofthe parametric ones (Section 5.4.1). To this end, the interconnection structure inFigure 5.12 is used, this time with the block ∆m involved, see Figure 5.17. In R ef.[120] Shin and B alas show that, for the ith actuator command, the infinity normof the transfer function between wmarg in i

and zmarg in iequals the inverse of the

smallest distance a N yquist plot of the corresponding actuator input gets to the(-1,0) point: the larger this distance, the larger the gain and phase margin at thespecific actuator input. This is very useful, since µ-analysis can be used to assessworst-case performance for infinity norm-based measures [9]. The technicalitiesare beyond the scope of this chapter and not required for interpretation of thefollowing analyses.Figures 5.18 and 5.19 show nominal and worst-case N yquist plots for the closed-loop system, with the actuator loop cut open at the aileron input. The circle radiirepresent the shortest distances of the curves to the (-1,0) point. These radii havebeen computed from the interconnection structure in Figure 5.17 with ∆lat = 0and:

∆m = diag [∆A, 0, 0, 0, · · · ]

5.5 Assessment 143

ATTASE Q M


E s t x

y

zla t wla t

zm a rg in wm a rg in

Figure 5.17: Input and outputs of the interconnection structure for µ worst-margin analysis

where ∆A is a scalar that acts on the first actuator input (this is the aileroncom m and , see F ig ure 2 .3 ). F ig ure 5 .1 8 v alid ates that the circle just touches theN y q uist plot at its closest prox im ity to the (-1 ,0 ) point. A lthoug h this m easured oes not ex actly m atch the d efinitions of g ain and phase m arg ins, the und erly ingid ea is id entical. A pparently , at the aileron input the m arg ins for the three con-trollers are sim ilar in case param eter tolerances are zero. T his is confirm ed b y theparallel co-ord inates in F ig ure 5 .9 : controller 1 has a slig htly b etter phase m arg in(i.e. has a low er scaled criterion v alue for PMADA). F ig ure 5 .1 9 show s the resultsfor the ind iv id ual w orst cases of the three controllers. C learly , controller 2 has asm aller w orst-phase m arg in w ith respect to the uncertain param eters in (5 .1 6 ).

N ote that a fourth curv e has b een includ ed in the w orst-case N y q uist d iag ram(m ark er “ + ” ). T his is a m eans of v alid ation of the result. T he w orst-case m arg in iscom puted using the A T T A S L F T m od el. T he correspond ing uncertain param eterv alues can of course also b e set in the nonlinear A T T A S sim ulation m od el. N u-m erically linearising this m od el and com puting the N y q uist curv e should g iv e thesam e result. T his v alid ation is perform ed for controller 3 in each of the sub seq uentw orst-case N y q uist plots. R eferring to C hapter 2 , this illustrates an im portantb enefit of the ob ject-oriented m od elling approach: since the nonlinear sim ula-tion m od el and L F T hav e b een autom atically g enerated from the sam e A T T A Sm od el im plem entation (F ig ure 2 .1 0 ), the tw o runtim e m od els are autom aticallyconsistent.

F ig ures 5 .2 0 and 5 .2 1 show the sam e N y q uist result for the rud d er input. N otethat the m arg ins for controller 2 are sm aller, w hich ag ain correlates w ith F ig ure 5 .9(PMADR ). T he w orst case m arg in of controller 3 is slig htly b etter than the oneof controller 1 . A lso note that the circles d o not touch the N y q uist curv e b ya sm all m arg in. T his is the result of a g ap b etw een com puted b ound s on µ:the upper b ound results in the sm aller (w orst) circle, the low er b ound s returnsw orst-case param eters: either the upper b ound is too conserv ativ e, or the low erb ound com putation d id not fully conv erg e to the actual w orst-case (w hich is q uitecom m on).


−1.5 −1 −0.5 0 0.5−1

−0.5

0

0.5

1

real

imag

Nyquist plot @ aileron, nominal case

K1K2K3

Figure 5.18: Nominal case Nyquist plots (at aileron) for three controller param-eter sets of Table 5.3

Figures 5.22 and 5.23 show the results for the margins at the roll rate sensor (tothis end, the ∆ m-feedback in Figure 5.12 has been moved from the actuator inputto the sensor output). Note that for controller 3 the worst case margin is slightlybetter, at the cost of the nominal case margin.In case of the yaw rate sensor (Figures 5.24 and 5.25) a clear improvement isvisible for controller 3. Finally, results for the roll angle sensor are shown inFigures 5.26 and 5.27 .From the above analyses it can be concluded that controller 3 has slightly bettermargins than controller 1. Apparently controller 2, although optimised with worstcase damping and margins model cases, shifted robustness problems to othercorners of the uncertain parameter space. H owever, performance of controller 3with respect to step response measures (especially, roll angle rise time, controlactivity) is better as compared with controller 1. U nfortunately, this can onlybe verified by gridding the parameter space, instead of an elegant µ-analysis,since most performance criteria used in optimisation do not directly translateinto infinity norm bounds as the gain and phase margins do.

5.5 Assessment 145

−1.5 −1 −0.5 0 0.5−1

−0.5

0

0.5

1

real

imag

Nyquist plot @ aileron, worst case

K1K2K3K3−ref

Figure 5.19: Worst case Nyquist plots (at aileron) for three controller parametersets of Table 5.3

−1.5 −1 −0.5 0 0.5−1

−0.5

0

0.5

1

real

imag

Nyquist plot @ rudder, nominal case

K1K2K3

Figure 5.2 0 : Nominal case Nyquist plots (at rudder) for three controller param-eter sets of Table 5.3


−1.5 −1 −0.5 0 0.5−1

−0.5

0

0.5

1

real

imag

Nyquist plot @ rudder, worst case

K1K2K3K3−ref

Figure 5.21: Worst case Nyquist plots (at rudder) for three controller parametersets of Table 5.3

−1.5 −1 −0.5 0 0.5−1

−0.5

0

0.5

1

real

imag

Nyquist plot @ roll rate sensor, nominal case

K1K2K3

Figure 5.22: Nominal case Nyquist plots (at roll rate sensor) for three controllerparameter sets of Table 5.3

5.5 Assessment 147

−1.5 −1 −0.5 0 0.5−1

−0.5

0

0.5

1

real

imag

Nyquist plot @ roll rate sensor, worst case

K1K2K3K3−ref

Figure 5.23: Worst case Nyquist plots (at roll rate sensor) for three controllerparameter sets of Table 5.3

−1.5 −1 −0.5 0 0.5−1

−0.5

0

0.5

1

real

imag

Nyquist plot @ yaw rate sensor, nominal case

K1K2K3

Figure 5.24 : Nominal case Nyquist plots (at yaw rate sensor) for three controllerparameter sets of Table 5.3


−1.5 −1 −0.5 0 0.5−1

−0.5

0

0.5

1

real

imag

Nyquist plot @ yaw rate sensor, worst case

K1K2K3K3−ref

Figure 5.25: Worst case Nyquist plots (at yaw rate sensor) for three controllerparameter sets of Table 5.3

−1.5 −1 −0.5 0 0.5−1

−0.5

0

0.5

1

real

imag

Nyquist plot @ roll angle sensor, nominal case

K1K2K3

Figure 5.26 : Nominal case Nyquist plots (at roll angle sensor) for three controllerparameter sets of Table 5.3

5.5 Assessment 149

−1.5 −1 −0.5 0 0.5−1

−0.5

0

0.5

1

real

imag

Nyquist plot @ roll angle sensor, worst case

K1K2K3K3−ref

Figure 5.27: Worst case Nyquist plots (at roll angle sensor) for three controllerparameter sets of Table 5.3


5.6 Flight test results

The control laws presented in this chapter have been flight tested as part of anautomatic landing system, described in the following chapter. Although flightperformance evaluation concentrated on disturbance rejection during glide slopeand localiser tracking and on flare and alignment performance, a number of testswas initiated with lateral flight path, vertical flight path, and approach speedoff sets respectively.

In the following the results of a localiser capture manoeuvre during the secondapproach of the second test flight at Hanover airport will be discussed briefly.

Figure 5.28 shows deviations from the glide slope, localiser, and selected approachspeed respectively. The initial geometric localiser deviation is approximately 275m. The autopilot reduces this deviation to zero by commanding roll and headingrates to the ND I inner loop controller, see Figure 5.29. The plots shows excellenttracking of these commands, initially banking to the right and then to the leftwhen capturing the localiser track. Lateral acceleration is shown in Figure 5.29:its low mean level implies good turn co-ordination behaviour. Commanded andmeasured body angular rates are shown in Figure 5.31. In spite of the noiselevel, good tracking performance can be recognised. Finally, corresponding con-trol deflections can be found in Figure 5.32. Since commanded and measuredvalues match closely, the assumption that actuator dynamics can be neglected(Section 5.3.1) is proved to be valid. Although less of interest here, deviationbetween commanded and actual deflections for the elevator (δE) are caused bythe auto-trim system, which was active in parallel to the autoland control laws.

It is also interesting to have a look at the air data signals. The measured and com-plementary filtered values (Section 5.3.2) are compared in Figure 5.33. Clearly,noise content is reduced tremendously without any bias.

Finally, it is interesting to note that the ND I control laws performed very wellfrom the first flight test: not a single change was required, demonstrating goodrobustness and performance of the system and, within the scope of the designproject, first-time right capability of the applied design approach.

5.7 Conclusions

Nonlinear D ynamic Inversion is a very attractive methodology for flight controllaw design, since nonlinearities of the flight dynamics are compensated, avoidingthe need for gain-scheduling.

In this contribution it was shown how two drawbacks of ND I can be successfullycoped with:

In the first place, global robustness to parametric uncertainties in the synthesismodel was addressed via a multi-model approach, and local robustness to unspe-cific uncertainties was addressed via linear robustness criteria in multi-objectiveoptimisation. It was shown that, in addition to linear controller parameters, phys-ical parameters in the inverse model equations, that are considered uncertain in

5.7 Conclusions 151

0 20 40 60 80 100 120 140 160 180 200

−100

0

100ep

sGLD

(mA

)

0 20 40 60 80 100 120 140 160 180 200−100

0

100

epsL

OC

(mA

)

0 20 40 60 80 100 120 140 160 180 200−5

0

5

Ver

ror (m

/s)

time (s)

allign init

flare init

retard init

Figure 5.28: Tracking of glide slope, localiser and selected approach speed (errorsignals) (approach 2-2)

the design model, are very effective as additional tuning parameters to achieve arobust design.In the second place, it was shown that manual formula manipulation, coding andverification work can be avoided by automatic generation of the inverse controllaws from the same symbolic aircraft model in M odelica as from which the simu-lation model was obtained.Three controller variants were compared regarding performance and robustness.For this µ-analysis was used, based on an LFT model automatically generatedfrom the aircraft model in M odelica. For parameter synthesis, worst model caseswere sought using a grid-based search with a reduced parameter set. µ-Analysisbased on the full parameter set revealed that worst cases had not always beenfound. Unfortunately, the required low-order LFT model was available only wellafter the design work had been finished. The methodology is now actually usedto search for stability-related adverse model cases for inclusion in multi-modeldesign optimisation. E ven µ-analysis as an optimisation criterion by itself can beincluded.The designed control laws were used as inner loops in an autoland system thatwas successfully implemented and flight tested in the DLR Advanced TechnologiesTesting Aircraft System (ATTAS) [12].


0 20 40 60 80 100 120 140 160 180 200−5

0

5

10

15

φ (d

eg)

DI command following: solid = command, dashed= measured

0 20 40 60 80 100 120 140 160 180 200−10

0

10

θ (d

eg)

0 20 40 60 80 100 120 140 160 180 200−5

0

5

time (s)

dψ/d

t (de

g/s)

Align

Align

Flare

Figure 5.29: Attitude command tracking (approach 2-2)

0 20 40 60 80 100 120 140 160 180 200−0.1

0

0.1

time (s)

ny (−

)

Figure 5.30: Lateral acceleration response (approach 2-2)

5.7 Conclusions 153

0 20 40 60 80 100 120 140 160 180 200−10

0

10pb

(deg

/s)

DI command following (2): solid = command, dashed= measured

0 20 40 60 80 100 120 140 160 180 200−5

0

5

qb (d

eg/s

)

0 20 40 60 80 100 120 140 160 180 200−10

0

10

time (s)

rb (d

eg/s

)align

flare

align

Figure 5.31: Body angular rate tracking (approach 2-2)

0 20 40 60 80 100 120 140 160 180 200−5

0

5

δA (d

eg)

Control deflections: solid=command, dotted=measured

align

0 20 40 60 80 100 120 140 160 180 200−5

0

5

10

δE (d

eg) flare

0 20 40 60 80 100 120 140 160 180 200−6−4−2

02

δR (d

eg)

align

0 20 40 60 80 100 120 140 160 180 2000

50

100

time (s)

N1 (%

)

retard

Figure 5.32: Control deflections (approach 2-2)


0 20 40 60 80 100 120 140 160 180 20060

70

80

Vca

s (m/s

)

Compl. filtering: solid=filt., dotted=measured

0 20 40 60 80 100 120 140 160 180 200−5

0

5

10

α (d

eg)

0 20 40 60 80 100 120 140 160 180 200−5

0

5

time (s)

β (d

eg)

Figure 5.33: Complementary filtering of speed, angle of attack and side slip angleestimation (approach 2-2)

Chapter 6

Design of autoland controllerfunctions with multi-objectiveoptimisation

156 Design of autoland controller functions

Abstract

The application of multi-objective optimisation to the design of longitudi-nal automatic landing control law s for a civil aircraft is discussed. Thecontrol law s consist of a stability and command augmentation, a speed /fl ight path tracking, a glide slope guidance, and a fl are function. Multi-objective optimisation is used to synthesise the free parameters in thesecontroller functions. Performance criteria are thereby computed from lin-ear as w ell as nonlinear analy sis. Robustness to uncertain and vary ingparameters is addressed via linear robustness criteria, and via statisti-cal criteria computed from on-line Monte C arlo analy sis. F or each con-troller function an optimisation problem set-up is defi ned. S tarting w iththe inner loops, the synthesis is sequentially expanded w ith each of theseset-ups, eventually leading to simultaneous optimisation of all controllerfunctions. In this w ay , dynamic interactions betw een controller compo-nents are accounted for, and inner loops can be compromised, such thatthese can be used in combination w ith diff erent outer loop functions. Thisreduces controller complex ity w hile providing good over-all control sy stemperformance.

C o n tributio n s

• Proof-of-concept of the fl ight control law design and modellingmethodologies presented in C hapters 2 and 5 , describing the designof an autoland sy stem from modelling up to fl ight test;

• A strategy for tuning of large control sy stems, comprising multipleinteracting loops and functions;

• Direct use of Monte C arlo-based statistical certifi cation criteria inoptimisation of automatic landing control law parameters.

P ublicatio n

G ertjan L ooye and H ans-Dieter J oos: Design of Autoland ControllerFunctions with Multi-O bjective optimisation, J ournal of G uidance, C on-trol, and Dynamics V ol. 2 9 (2 ), American Institute of Aeronautics andAstronautics, 2 0 0 6 , pp. 4 7 5 – 4 8 4 .

Design of autoland controller functions 157

THE development of automatic landing (autoland) control laws for civil air-craft is a demanding task, since high safety standards have to be met before

operational use under most adverse weather and visibility conditions (i.e. Cate-gory III [37]) is allowed. The landing mission, consisting of glide slope trackingand flare/ runway alignment shortly before touch down, is relatively straight for-ward. However, the autoland design task is complicated by the large amount ofvarying and uncertain parameters involved. In the first place, aircraft loading andconfiguration parameters may vary for each landing case. Secondly, environmentparameters, such as runway, approach terrain, ILS system, atmosphere, and windcharacteristics are different for each landing. A number of these parameters isillustrated in Figure 6.1. For certification, autoland performance under varyingaircraft and environment parameters has to be demonstrated via extensive MonteCarlo (MC) analysis, augmented with flight test validation [37]. The control de-signer also has to account for uncertain parameters in the aircraft design modelto cover differences with the actual aircraft, or even with the high-fidelity andflight test-validated model used for final Monte Carlo assessment [74].

Autoland control laws consist of functions for glide slope and localiser hold, flareand runway alignment (in case of cross wind), and inner loops for stability andcommand augmentation. The design of these functions heavily relies on engineer-ing skills and experience of the designer(s) and may involve considerable trial-and-error in order to tune the controller parameters to meet design specifications.Important problems hereby are the large number of design criteria that have tobe addressed in face of the aforementioned varying and uncertain model parame-ters, as well as the criteria themselves. Typical autoland design requirements, forexample regarding touch down performance, do not always translate easily intocomputational criteria that can be handled by commonly used controller synthe-sis methods. As a consequence, the designer has to manually iterate betweenassessing autoland performance and adjusting controller (synthesis) parameters.

In this chapter, the problems sketched above are addressed by the application ofa flight control law design process that is based on multi-objective optimisation[50, 73]. This approach involves optimisation of free design parameters in a pre-defined controller structure (eg. gains, filter time constants), or in a controllersynthesis set-up (eg. Q and R weighting matrices in LQ -synthesis), with respectto a possibly large set of computational criteria and constraints. An importantadvantage is that these criteria and constraints may be directly derived from engi-neering design specifications and computed from linear and/ or nonlinear analysisof the closed loop system. For example, touch down velocity and distance, com-puted from a nonlinear automatic landing simulation, are valid synthesis criteria.Relative importance of criteria is expressed via scaling.

Features of multi-objective optimisation will be exploited to address robustnessto varying landing and aircraft parameters and uncertainty. Stability robustnessto aircraft model uncertainty is addressed via gain and phase margins as designcriteria. As part of the certification effort, performing Monte Carlo analysis isrequired to prove that the probability of exceeding bounds on specific landingparameters (e.g. maximum sink rate at touch down) under varying environment


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)

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atm

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Figure 6.1: Typical parameters and disturbances during a landing

Design of autoland controller functions 159

and aircraft parameters and disturbances is below a certain level. It will be shownthat these stochastic measures can be directly incorporated as optimisation cri-teria, providing an effective means to address robustness to varying aircraft andenvironment parameters. A similar principle for robust multi-objective tuning ofcontrol laws was proposed by Schy and G iesy [117], using a linear approach to com-pute stochastic properties of performance measures. Wang and Stengel [134] useweighted-sum optimisation to minimise the risks of violating design criteria withrespect to stochastic parameters. The risks are computed using on-line MonteCarlo analysis. In this work, on-line Monte Carlo analysis will be performed tocompute risk-based criteria with respect to varying landing parameters, providinga direct link with certification requirements, and allowing additional criteria tobe computed from individual MC simulations.

The actual optimisation problem is formulated to minimise the maximum overall scaled criteria, under the specified inequality or equality constraints [61]. Thisso-called min-max approach allows the designer to steer compromise solutionsbetween conflicting criteria by adjusting their scalings. The applied methodologywas developed by K reisselmeier and Steinhauser and has been implemented in thecomputer-aided control system design tool MOP S (Multi-Objective P arameterSynthesis) [52], developed at the DLR Institute of Robotics and Mechatronics. Asimilar min-max optimisation approach was proposed by Schy et al. [117], and isalso applied in the Control Designer’s Unified Interface (CONDUITR) by Tischleret al. [129].

The optimisation approach makes it possible to simultaneously tune all design pa-rameters in all control law functions. This saves manual iterations to harmoniseperformance of individual functions and allows controller complexity to be re-duced, since one set of inner loop functions may be tuned to work in combinationwith different outer loop functions. However, performing this optimisation in oneshot is impractical. For the designer it becomes diffi cult to keep track of the largeamount of criteria and design parameters, making it hard to decide if and wherethe controller structure needs to be enhanced, to recognise and adjust conflictingcriteria, etc. For this reason, a stepwise tuning strategy is proposed. Inner loopsfunctions are tuned first, then the optimisation task is expanded step-by-step withthe outer loop functions, eventually leading to simultaneous optimisation of alltuning parameters in all functions. The designer can thus concentrate on one func-tion at a time, but eventually the optimisation addresses the integrated system.Since tuning parameter values resulting from one optimisation provide startingvalues for the next, highly effi cient local optimisation algorithms in general workwell.

In this chapter the design of the longitudinal part of an autoland system for asmall passenger aircraft is discussed. More details on the lateral modes can befound in Ref. [73]. An in-depth discussion on robust tuning of the lateral innerloops is presented in Chapter 5 of this thesis.

This chapter is structured as follows. In Section 6.1 the aircraft model is brieflydescribed. In Section 6.2 the applied design process is discussed: the individualsteps in this process are described in Section 6.3 (controller architecture), in Sec-


tion 6.4 (optimisation set-ups for the controller functions), in Sections 6.5 and 6.6(the optimisation strategy and optimisation problem formulation), and in Sec-tion 6.7 (analysis of design results). Some flight test results are briefly discussedin Section 6.8. Finally, conclusions are drawn in Section 6.9.

6.1 The aircraft modelAs in Chapter 5, the aircraft is DLR’s V FW-614 (∼ 30 passengers) ATTAS (Ad-vanced Technologies Testing Aircraft System), which has been configured as afly-by-wire test bed [12].The basic structure, inputs, outputs and parameters of the model are describedin Section 2.5. The aerodynamics are valid for the landing configuration andinclude unsteady effects, ground effect, and interaction with the engine exhaust.Aerodynamic coefficients as well as the moments of inertia have tolerances between10% and 30% , see Table 2.5. Ground effect coefficients and dynamic stabilityderivatives have the highest level of uncertainty. For example, the tolerance onthe derivative of the aerodynamic pitching moment coefficient Cm with respectto the pitch rate q is written as:

Cmq= Cmq

(1 + ∆Cmq) (6.1)

where Cmqand Cmq

are the perturbed and nominal values respectively, and ∆Cmq

is the tolerance: −0.3 ≤ ∆Cmq≤ 0.3. The vector containing all uncertain pa-

rameter tolerances in the aircraft model is pu.The available controls are ailerons δA, elevator δE , rudder δR, and engine throttlesettings δT H 1,2, see Table 2.3. The tail plane angle of incidence δT is used fortrimming. The control surface actuators are linear, but rate and position limited.The turbofan engine dynamics and thrust computation are nonlinear. The fuelcontrol unit shows backlash behaviour, equivalent to several degrees of throttleinput.The wind model includes wind shear as present in the earth’s boundary layer, Dry-den turbulence filters (as specified in Ref. [37]), as well as additional parametrisedwind shear models. For the atmosphere, approach terrain, runway and ILS equip-ment characteristics, parametrised models are included. The environment-relatedparameters are contained in the vector pe (see Figure 6.1).The model outputs are the measurements available to the control system, seeTable 2.2: calibrated airspeed Vc a s , true airspeed Vta s , ground speed Vg, bodyangular rates p,q, r , attitude angles φ , θ , ψ , load factors nx,ny,nz, track andflight path angles χ , γ , angle of attack α, vertical speed VZ , deviations from theILS beam εL O C (localiser), εG S (glide slope, both in mA), radio and barometricaltitude Hr a ,Hb a r o , and the mean fan shaft speed of both engines N1. Thesensor models are linear, but the output signals are quantised. The signals N1,εL O C ,εG S , and α are corrupted with noise. Finally, parameters related to theaircraft configuration (e.g. the mass m, and the centre of gravity location xC G )may be assumed known to the controller and are collected in the vector pk.

6.2 The applied design process 161

6.2 The applied design process

The design process as applied for the automatic landing control laws is depicted inFigure 6.2. As a first step (A), the global architecture is defined. Detailed design

Autoland control lawarch ite cture de v e lop m e nt& functional b re ak down

Controlle r functionde taile d de v e lop m e nt:- arch ite cture (T )- com p utational crite ria

M ode l de -v e lop m e nt

1 a

2

A

B

Autoland controlle rnonline ar as s e s s m e nt

Autoland control lawsre ady for h ardware te s ting

Latest(flig h t test)m o d el d ata

D

3

Controlle rinte g ration

O p tim iz ation ofinte g rate d controlle r

1 b

C

C u rren tm o d el d ata

F u n c tio n alD esig n R eq ./C ertific atio n

sp ec s

Figure 6.2: optimisation-based autoland control laws design process

of functions within this architecture is addressed in step B, involving detailedspecification of the control law structure, and the formulation of function-specificcomputational criteria. The controller structures are integrated into the autolandsystem, which, in combination with the aircraft model, can then be used for closedloop analysis required for criteria computation.

The actual optimisation of the controller functions is performed in step C. Herethe tuning strategy as sketched in the introduction is applied, eventually allowingall controller components to be optimised simultaneously.

Performance and robustness of the resulting autoland system is assessed in stepD of Figure 6.2. The iteration loops 1a and 1b involve adjustments to compu-tational criteria or component architectures, in case optimisation or assessmentresults are not satisfactory, or in order to further improve them. In case of se-vere shortcomings, the overall controller structure may have to be reconsidered(loop 2). Loop 3 may be required in case of major model updates. However, byexplicitly addressing model uncertainty in the process, this loop may possibly beavoided.


6.3 The controller architecture

This section addresses step A (global architecture) and partially step B (detailedfunction architectures) in the design process in Figure 6.2. The selected globalstructure for the autoland controller is depicted in Figure 6.3. It has previouslybeen applied to a large transport aircraft [73]. Three main loops can be identi-

DynamicInversion

inverseth ru st m a p

L a tera l P a thT ra c k ing (P D )

G ST E C S

Va p p

A R

e

Tc

ref ,ref

.

F la re(va r. Ta u )

A lig n

R eta rd

fla re

a lig n

ref

reta rd

g u id a nce sp eed /p a thtra ck ing

sta b . & com m a nda u g m enta tion

rw y

L O Cye

F eed b a ck sig -na l synth esis

M ea su redsig na ls

T

B L -c o m p

pk

(inner loop s)

Figure 6.3: Autoland controller architecture (Remark: for block inputs, onlycommand and error sig nals show n)

fi ed, separated by dashed lines: S tability and C ommand Augmentation (S C A),S peed/ P ath T racking (S P T ), and g uidance. C omplex ity of autopilot control law smay be considerably reduced by implementing one component for each functiononly. T his is an important feature of the T otal E nerg y C ontrol S ystem (T E C S )w hich is used as a S P T function: a sing le speed/ path tracking control law canbe used as the core of a complete set of long itudinal autopilot modes [6 6 , 6 5 ].In this autoland architecture, the same principle is applied to the S tability andC ommand Aug mentation (S C A) function, w hich is used by all S P T functions.In this chapter, only the desig n of the long itudinal controller functions w ill bediscussed. A brief description of each of the functions is g iv en in the follow ing .

Inner loops

T he task of the inner loops is to improv e stability and to achiev e robust tracking ofcommand variables (φref , θref , ψ ′c m d ). T he use of θref as inner loop commandvariable allow s for direct control ov er the pitch attitude dynamics. E speciallyduring fl are, these play an important role in pilot acceptance.

T he inner loops w ere desig ned w ith N onlinear D ynamic Inv ersion (N D I) [3 4 ].Inv erse model eq uations compensate the nonlinear aircraft dynamics, resulting in

6.3 The controller architecture 163

uniform and decoupled command responses, so that (manual) gain scheduling ofthe control laws is avoided. Note that in this way also known aircraft loadingand configuration parameters (pk), provided to the controller, are automaticallycompensated for. A more detailed discussion on the inner loops can be found inChapter 5.

The closed loop dynamics are shaped using a linear outer loop control law:

q′cmd = Kθ(θref − θ) −Kqq (6.2 )

where q′cmd is the pitch acceleration command to the NDI controller, θref is thecommanded pitch attitude angle, and Kθ and Kq are constant gains.

Longitudinal tracking and glide slope mode

F or longitudinal speed and flight path tracking during the approach, the Total En-ergy Control System (TECS) is used [66], see F igure 6.4. The TECS-architecture

KTI

S

KE I

S

KFs peed lo o p(path prio rity )

a

Vg

.

Ve

g

.

Tc

W

ref

KTP

KE P

2 -KF

-

+

+

+

-

++

+

-

+

a

0

p

p

+

+

KV

gVcas

Vcas

+

-

C

+-

-

+

Vc

g

.c

e

e

Figure 6.4: TECS controller structure (subscript e denotes an error from a com-manded value)

off ers pilot-like decoupled tracking of speed and flight path angle commands. Theinput signals are air mass referenced flight path angle γa ≈ −VZ/ Vcas and filtered

acceleration ˆV/ g and the errors from the commanded values. H ere, TECS con-trols pitch attitude θref and thrust (per unit weight, δ T c/ W ). An inverse thrustmap and an engine backlash compensation scheme (B L comp) are used to gener-ate appropriate throttle commands (F igure 6.3). The gain KF allows for shiftingcontrol priority to flight path tracking. The speed loop is opened in case thrustsaturates.

The feedback signal ˆV/ g is obtained from complementarily filtering of the mea-sured calibrated airspeed Vcas and the time derivative of the inertial speed V , withtime constant τV , see F igure 6.5 and Ref. [66]. The acceleration V is computed


1

v-

+

Vc a s

V.

V

.

1

s

Vc a s0

V^

Figure 6.5: Block diagram of the complementary speed filter

from:

V =√

V Tv Vv

d V

d t=

1

V

(

V Tv Vv

)

d V

d t=

1

VV T

v

−Tvb

nx

ny

nz

g −

00g

(6.3)

where Vv is the velocity vector in the vehicle-carried vertical frame Fv. Compu-tation of the acceleration thus requires load factors, Euler attitude angles (for thedirection cosine matrix Tvb ), and the inertial speed components in Fv. Note thatthe load factors have negative sign with respect to the aircraft body axes. Thecomponents of Vv are obtained from:

Vv = V(

cos γ cos χ , cos γ sin χ , − sin γ)T

(6.4)

The acceleration error ¯Ve/g is computed from proportional feedback of the cali-brated airspeed:

¯Ve = KV (Vap p − Vcas) −ˆV (6.5)

where Vap p is the selected approach speed. Note that Vcas instead of the comple-

mentary filtered V is used for feedback. The reason is that during wind shear theaircraft has to accelerate or decelerate in order to maintain airspeed, i.e. V 6= 0.Consequently, the integrator output in Figure 6.5 will have a steady state error[23].The structure of the glide slope mode is depicted in Figure 6.6. The flight pathangle error is input to the TECS controller and is computed as follows: [66]:

γe =1

V

[

Kh

1

τhs + 1∆h−

ˆh

]

(6.6)


1

hs+1

in it

Kh

1

sh.

-1

V

Z.

++

+

+-

in it

V^

x

. e

h.

c o m p lem en ta r y f ilter

hG S

~

S in (-3 /5 7 .3 )

Figure 6.6: Block diagram of the glide slope mode

where V is the complementarily filtered calibrated airspeed signal, and ∆hGS isthe height error estimated from the glide slope signal εGS , which is filtered withtime constant τh in order to remove high frequency signal noise (see Figure 6.6).

The estimated vertical speed with respect to the glide slopeˆh is obtained from

complementarily filtering of ∆hGS and the vertical velocity with respect to theglide slope: V sin(γg ldnom

)+VZ . In th e la tte r e x p re ssio n VZ is th e v e rtic a l v e lo c ityin Fv a n d γgld nom

= −3/1 8 0 ∗ π ra d is th e n o m in a l g lid e slo p e a n g le . T h e e x a c ta n g le m a y b e h a lf a d e g re e m o re o r le ss a n d is n o t k n o w n , b u t n o te th a t in ste a d y

sta te th e c o n trib u tio n toˆh is z e ro . T h e tim e c o n sta n t o f th e c o m p le m e n ta ry fi lte r

is τh.

Flare law

F o r th e fl a re la w , th e so -c a lle d v a ria b le T a u p rin c ip le [6 4 ] w a s ch o se n (F ig u re 6 .7 ).It fe a tu re s c o n sta n t in itia tio n h e ig h t (Hf la r e ) a n d lo w to u ch d o w n d isp e rsio nu n d e r v a ry in g w in d c o n d itio n s, sin c e th e tim e c o n sta n t o f th e fl a re m a n o e u v reis p o sitio n , ra th e r th a n tim e -re fe re n c e d a n d c o m p u te d o n -lin e (h e n c e th e n a m e“ v a ria b le ta u ” ). In th e fi g u re Vg is th e g ro u n d sp e e d a n d Hb ia s is a re fe re n c ea ltitu d e so m e w h a t b e lo w th e ru n w a y su rfa c e , m a k in g su re so m e sin k ra te is le ftw h e n th e a irc ra ft m a in w h e e ls to u ch th e g ro u n d . A fe e d fo rw a rd p a rt is a d d e dc o n sistin g o f a ra m p c o m m a n d to in c re a se p itch (θR a m p ) a s w e ll a s th e m e a n T E C S

a ttitu d e c o m m a n d θ0, m u ltip lie d b y KF W (o b v io u sly , th e sta rtin g v a lu e fo r tu n in g

is 1 ). T h e a n g le θ0 is o b ta in e d b y lo w -p a ss fi lte rin g θr e f c o m m a n d e d b y T E C Sd u rin g th e a p p ro a ch , a n d h o ld in g th e v a lu e fro m fl a re in itia tio n (F ig u re 6 .7 ).

T h e v e rtic a l sp e e dˆh is o b ta in e d b y c o m p le m e n ta ry fi lte rin g th e ra d io a ltitu d e

Hr a a n d VZ, re su ltin g in a ru n w a y re fe re n c e d sig n a l w ith lo w n o ise c o n te n t, se e

F ig u re 6 .8 . T h e d iff e re n c e ∆ hr w y =ˆh − (−VZ) is c a u se d b y a p o ssib le ru n w a y

slo p e , fro m w h ich a n a d d itio n a l fe e d fo rw a rd c o m m a n d is g e n e ra te d v ia KR W .D u rin g fl a re th e th ro ttle s a re re ta rd e d a t a c o n sta n t ra te δT H c

(p ro p o rtio n a l toth e g ro u n d sp e e d a t fl a re in it), su ch th a t th e se re a ch id le p o sitio n a t th e p ro je c te d


!

" ! "

# $

%

%

%

& # ' ( ) *

+

%

, ,

Figure 6.7: Variable Tau-based flare law architecture

Hr a

1

sh f l

.

-1

Z.

++

-

lim ite r-

d e a d z o n e

Hr w y

.

Hr w y

.

-H.

0

fla r e

Hr a r e se t a tf la r e in it

Figure 6.8 : Structure of complementary vertical speed filter


touchdown point.

Hra

tim e

a c tu a l Hra

Hra fro m s e n s o r(e x a g g e ra te d )

Hfla re

0.1 s

H - Hra ,s e n s ra

tim e0.1 s

0

Htr

Htr

e rro r in in itia l Hra d u e to s te p -tra n s ie n ta t u n d e la y e d fla re in it

n o e rro r in in itia l Hra w h e n fla re in it d e la y e d

fla re

fla re in it (d e la y e d )

approach terain

nom inal g lid e s lope

Hfla re

cros s ing : f lare init

Hfla re

R u nw ay T hres hold

Hr a

S ea w all(ex ag g erated )

Htr

aircraft f lig ht path(m ain w heels )

Figure 6.9: Flare initiation due to sea wall

A somewhat tricky situation is sketched in Figure 6.9 (top). At some airports,like Funchal, M adeira, the approach path is over water, resulting in a step-wiseterrain elevation shortly before the runway threshold (also known as “sea wall”).In case the aircraft arrives low from the glide slope, this sea wall may triggerthe flare mode due to the step-wise change in Hra. Due to the (in this case)100 ms time constant of the radio altimeter, its reading may still decrease further(Figure 6.9 , middle). H owever, from flare initiation Hra becomes a feedback signaland is used to initialise the vertical speed filter, see Figure 6.7 and 6.8. Due tothe sensor transient, the flare law will unnecessarily react to the small altitudeerror (Figure 6.9 , below), and the filter will be initialised wrongly. The solutionto this problem is simple: after reaching the flare initiation, activation of the flaremode is delayed with 100 ms.


6.4 Optimisation problem set-ups

This section addresses step B in the design process in Figure 6.2. For each individ-ual controller function, an optimisation problem set-up is defined. Such a set-upincludes properties of the free parameters in the controller structure, computa-tional design criteria (including properties such as scaling, minimise or ineq ualityconstraint, etc.) that apply to the specific function, as well as macros and modelsto compute these criteria. Visualisation of analysis results is configured as well,allowing the designer to q ualitatively monitor design progress.

Stability and Command Augmentation (SCA)The gains Kθ and Kq are tuning parameters for command shaping. If necessary,uncertain model parameters that also appear in the inverse model eq uations (p∗u),can be effectively used as additional tuning parameters to improve robustness, seeChapter 5 . The criteria for the longitudinal part of the N DI inner loop are listedin Table 6.1. Those based on simulation 1 are intended for command shaping,those on simulation 2 for disturbance rejection, and those based on linear analysisare intended to guarantee closed-loop stability (over all eigenvalues, includingzero dynamics) and stability robustness to model uncertainty (eg. time delays,unmodelled dynamics).

name description computationSimulation 1, step: θref = 5 deg, no turb ulence

THrt rise time see remarksTHos over shoot ,,

THcontr δE control activity maxt> 1.3s|δE |Simulation 2 , h eav y turb ulence

THturb disturbance rejection 1

6 0

∫ t= 6 0

t= 0(θref − θ)2d t

L inear analy sis

gmAD gain margin at δE-act. see remarkspmAD phase margin at δE-act. ,,gmST gain margin at θ-sens. ,,pmST phase margin at θ-sens. ,,DAMP di min. damping (lon) minζi (longitudinal)

Table 6.1: Criteria for the SCA function. R emarks: All computations: sym-metrical horizontal flight; altitude=1000 ft; nominal aircraft loading. Step times:ts = 1 s. R ise time: ∆t between 10% and 90% of command. G ain/ phase margins:computed using margin-command [79].

In multi-objective optimisation, relative importance of criteria is expressed viascaling. Especially in case of conflicting req uirements, this gives the designer aneffective means to make trade-offs and to set priorities. Scaled criteria have tobe formulated such, that the objective is to minimise them, and that a value lessthan one is considered satisfactory.

6.4 Optimisation problem set-ups 169

Criteria scaling can be performed by division of each criterion by its demandedvalue:

ck(T ) = ck(T )/dk (6.7)

where ck(T ) and dk are the computed value and demanded value of criterion krespectively, and T denotes the current set of tuning parameters. Scaling can alsobe done using so-called ’good-bad’ values [50], as will be illustrated for gmAD inFigure 6.10. The demand is that the gain margin is at least 4 dB (’bad-low’). Any

Figure 6.10: Scaling of gmAD w ith good -b ad v alu e s

v alu e large r th an 6 (’good -low ’) is consid e re d e q u ally good and th e re fore scale d to0 . B e low 6 d B , th e scale d v alu e increase s linearly , su ch th at a v alu e of 1 is reach e dfor th e b ad -low v alu e of 4 d B . A ny v alu e b e tw e en 4 and 6 d B is acce p tab le , anyv alu e low e r th an 4 d B is consid e re d u nacce p tab le (b ad ). A s an ex am p le , if th egain m argin is 3 d B , its scale d v alu e e q u als 1 .5 . In th e sam e fash ion, ’good -h igh ’and ’b ad -h igh ’ v alu e s can b e sp ecifi e d . It m u st b e note d th at th e corne r b e tw e enth e slop ing line (< 6 d B ) and th e h orizontal line p rojecting crite ria v alu e s to z e rois (sh arp ly ) rou nd e d u sing an ex p onential fu nction, in ord e r to m ak e su re th escale d crite rion v alu e re m ains sm ooth .T h e scalings ap p lie d to th e SC A crite ria are giv en in th e fi rst p art of T ab le 6 .4(T H rt ... DAM P d i). Som e of th e crite ria are treate d as ine q u ality constraints(i.e . ck(T ) ≤ 1 ). F or e x am p le , an ov e rsh oot of le ss th an 5 % is d e m and e d . Ifth is is satisfi e d , th e re is no p oint to fu rth e r m inim ise th is crite rion, since th is m ayu nnece ssarily go at th e cost of rise tim e . T h e crite rion T H tu rb is p assiv e : its v alu eis com p u te d at each ite ration ste p for m onitoring, b u t ignore d b y th e op tim ise r.O f cou rse , p assiv e crite ria m ay b e activ ate d any tim e .

S peed / P a th T ra ckin g (S P T ) co n tro l la w sT h e task of th e T E C S controlle r is d ecou p le d sp e e d and fl igh t p ath angle track ing,w h ile p rov id ing ad e q u ate stab ility m argin. T h e se task s are re fl ecte d b y crite riath at are com p u te d from th re e nonlinear sim u lations and linear analy sis, se e T a-b le 6 .3 . Sim u lations 1 and 2 are intend e d for fl igh t p ath angle and sp e e d ste pre sp onse sh ap ing. Sim u lation 3 is intend e d to asse ss tu rb u lence re jection. T h ecorre sp ond ing scalings are giv en in T ab le 6 .4 . T h ose for d am p ing and stab ilitym argins are as in T ab le 6 .4 .


Criterion Bad Good Good Bad De- Type(u nit) low low h ig h h ig h / m andLongitudinal SCA criteria

TH rt (s) – – – – 2 .5 cTH os (-) – – – – 0 .0 5 cTH contr (deg / s) – – – – 1 2 .0 mTH tu rb (deg 2) – – – – 0 .0 8 pg m A D (-) – – 4 .0 6 .0 1 mpm A D (deg ) – – 4 0 6 0 1 cg m S T (-) – – 4 .0 6 .0 1 mpm S T (deg ) – – 4 0 6 0 1 cDA M P di (-) – – 0 .5 0 .7 1 cLongitudinal SP T criteria

GA rt (s) – – – – 1 6 .0 cGA os (-) – – – – 0 .1 cGA st (deg ) – – – – 0 .2 mTH c m d (deg / s) – – – – 2 .0 mGA V A (m / s) – – – – 0 .5 mV A rt (s) – – – – 2 0 .0 cV A os (-) – – – – 0 .1 0 cV A st (m / s) – – – – 0 .5 mdTH R (deg / s) – – – – 0 .1 5 mV A GA (deg ) – – – – 1 .0 mN Z tu rb (s−2) – – – – 0 .0 0 7 pTH Ctu rb (deg 2) – – – – 0 .1 6 7 pTH R tu rb (deg 2) – – – – 5 pG lide slop e criteria

GS rt (s) – – – – 2 0 cGS os (-) – – – – 0 .1 2 mGS st (m ) – – – – 2 cm ax GS dev (m A ) – – – – 2 0 0 mm ax GS dev 5 0 (m A ) – – – – 2 0 0 mm eanGS dev (m A ) – – – – 1 mm ax TH E dev (deg ) – – – – 3 mm ax V Cdev (m / s) – – – – 8 .0 mF lare criteria

X TDnom (m ) 3 6 0 3 8 0 4 0 0 4 2 0 1 mV Z TDnom (m / s) -3 .0 -2 .8 -2 .4 -2 .2 1 mTH g rad (rad/ s) – – – – 1 cdH g rad (m / s) – – – – 1 cdDE m ax (deg / s) – – – – 1 0 cm eanH TP 6 0 (m ) 8 1 0 1 2 1 5 1 mstdev H TP 6 0 (m ) – – – – 1 .3 mlim H TP 6 0 (m ) 0 5 – – 1 cm eanX TD (m ) 3 0 0 3 5 0 4 0 0 4 5 0 1 mstdev X TD (m ) – – – – 7 5 mlim X TD (m ) – – – – 6 8 0 cm eanV Z TD (m / s) -6 -4 -2 -1 .5 1 mstdev V Z TD (m / s) – – – – 1 .4 mlim V Z TD (m / s) – – – – 3 .1 c

Table 6.2: Scalings of all optimisation criteria. Remark: c=inequality constraint,m=minimise, p=passive


name description computationSimulation 1, step: γc = 3 deg, no turb .

G Art rise time γ see remarks Table 6.1G Aos over shoot γ ,,G Ast ’settling time’ γ maxt> 25s|γc − γ|

TH cmd θ cmd eff ort max|θr e f |G AV A max. speed deviation max|∆Vca s|Simulation 2 , step: ∆Vc = 10m / s , γ = −3 d eg , no turb .

V Art rise time Vca s see remarks Table 6.1V Aos over shoot Vca s ,,V Ast ’settling time’ Vca s maxt> 25s|Vc − ∆Vca s|

dTH R throttle activity max|δT H 1c|

V AG A max. γ deviation max|∆γ|Simulation 3 , h eav y turb ulence:

trimmed on g lid e slope, Vw in d = 15.4m / s

N Z turb load factor variation 1

6 0

∫ t= 6 0

t= 0∆n2

zd t

TH Cturb θ cmd eff ort 1

6 0

∫ t= 6 0

t= 0∆θ2

r e f d t

TH Rturb throttle activity 1

6 0

∫ t= 6 0

t= 0∆δ2

T H c

d t

L inear analy sis

gmAD G M δE-act. see remarks Table 6.1pmAD P M δE-act. ,,gmSG G M at γ-sens. ,,pmSG P M at γ-sens. ,,gmSV G M at Vca s-sens. ,,pmSV P M at Vca s-sens. ,,D AM P min. damping miniζi

Table 6.3: Criteria for the SP T function. Remarks: ∆ denotes deviation fromtrimmed value

Of course, the SP T control laws always work via the SCA system. Thus, ex-cept for the TECS gains, also tuning parameters in the N D I controller aff ect theperformance criteria in Table 6.3.

Glide slope modeFor the glide slope mode again command shaping criteria are applied, see Table 6.4(simulation 1). A more important aspect during glide slope and approach speedtracking is disturbance rejection, whereas pitch attitude dynamics and throttleactivity have to be limited for passenger comfort and pilot acceptance reasons.D esign requirements were based on indicators listed in the second column of Ta-ble 6.4. One possibility is to derive computational criteria from analytical covari-ance analysis. For this work, a diff erent approach was tried, based on nonlinearapproach and landing simulations performed in on-line M onte Carlo analysis thatis used to compute statistical flare law criteria (to be discussed in the next sub-


section), see Table 6.4. Since each landing is performed with different parametervectors pe, pk, their variation is implicitly addressed in optimising for disturbancerejection. Due to the large number of simulations involved, the risk that the op-timiser may anticipate a specific noise signal, is reduced. However, the randomgenerators used in the Monte Carlo analysis are re-set before each run, so that theith individual simulation in different Monte Carlo runs always uses the same setof parameters values and noise signals. This is required in order to prevent noisycriteria due to variations in simulated disturbances. The scalings on the criteriaare given in Table 6.4.

name description computationNonlinear simulation, offset of 50 m above glide slope, aircrafttrimmed parallel to the G S, no turbulence.GSrt rise time see remarks Table 6.1GSos over shoot ,,GSst ’settling time’ maxt>30s|Ho ffset|M onte C arlo simulations, all effects (eg. turbulence,nonlinearities) includedmaxGL Ddev max. abs. vertical maximc

max10s< t< tf l|εG S (t)|

deviation from GSmaxGL Ddev50 abs. vert. dev. from maximc

|εG S (tfli M C)|

GS, at flare init

meanGL Ddev mean deviation∑nmc

i=1

∫ tf l i M C

t=10εG S (t)dt/...

from GS∑nmc

i=1(tfli M C

− 10)maxTHEdev max pitch angle dev.

from mean value maximcmax10s< t< tf l

|θ(t) − θ|maxVCdev max. speed maximc

max10s< t< tf l|Vap p i M C

−deviation Vcasf i l t

− 0.5m/s|

Table 6.4: Criteria for the Glide Slope mode. Remarks: nmc = to tal n u mb er o f

M o n te C arlo simu latio n s, imc = in d ex o f in d iv id u al M C simu latio n , tf li M C= fl are

in itiatio n time fo r simu latio n imc, .. in d icates mean v alu e fo r 1 0 s < t < tf li M C, Vca s f i l t

is Vca s fi ltered w ith 5 s time co n stan t.

Flare modeFor the flare mode deterministic and stochastic criteria are considered. The de-terministic criteria (Table 6.5, 6.4) are computed from a nonlinear landing simu-lation. The stochastic criteria are computed from on-line Monte Carlo analysis.To this end, nM C = 400 nonlinear landing simulations are performed in whichall disturbances are applied. Before each landing simulation iM C , 16 operationalparameters (∈ pk, ∈ pe, see Figure 6.1) are selected randomly, according to pre-scribed statistical properties. After completing the simulations, the mean valuesand standard deviations of so-called risk-parameters are determined. The longi-tudinal risk parameters are: the height of the main gear over the runway at 60


Name Specification Computationdescription

Nonlinear simulation, no disturbances,nominal conditionsX TDnom touchdown xtd(ttd)

point

VZTDnom vert. touch- Hra(ttd)down speed

THgrad θ may not 1 − mint1≤t≤ttdθ(t)

ch a n g e sig n

d H g ra d VZ m a y n o t 1 + m a x t1≤t≤ttdVZ

ch a n g e sig n

d D E m a x e le v . ra te m a x f l≤t≤ttd| δE |

ttd = to u ch d o w n tim e , t1 = fl a re in it tim e + 2 s

Table 6.5: D e te rm in istic fl a re c rite ria

m fro m th e th re sh o ld (H T P 6 0 ), to a sse ss th e risk o f sh o rt la n d in g s, th e ru n w a yto u ch d o w n d ista n c e fro m th re sh o ld (X T D ), to a sse ss th e risk o f lo n g la n d in g s,a n d th e v e rtic a l sp e e d w ith re sp e c t to th e ru n w a y su rfa c e (V Z T D ), to a sse ssh a rd la n d in g s. F ro m th e m e a n v a lu e s a n d sta n d a rd d e v ia tio n s, th e d istrib u tio na n d c u m u la tiv e d istrib u tio n fu n c tio n s c a n b e c o m p u te d , a ssu m in g th a t th e se a reG a u ssia n . B a se d o n E A S A C S -A W O sp e c ifi c a tio n s, e a ch risk p a ra m e te r h a s alim it v a lu e fo r w h ich th e p ro b a b ility o f e x c e e d a n c e m u st b e p ro v e d to b e le ss th a n1 0−6 (a v e ra g e risk a n a ly sis).

F o r e a ch o f th e risk p a ra m e te rs, th e m e a n v a lu e , sta n d a rd d e v ia tio n , a n d p ro b -a b ility o f e x c e e d in g th e lim it v a lu e a re a d d re sse d v ia o p tim isa tio n c rite ria , se eT a b le 6 .4 (b e lo w ). T h e p ro b a b ility c rite ria a re a d d re sse d a s illu stra te d v ia a ne x a m p le . T h e lim it v a lu e fo r XT D is 9 1 5 m . A s o p tim isa tio n c rite rio n , th e a c tu a lv a lu e o f XT D fo r w h ich th e p ro b a b ility o f e x c e e d in g e q u a ls 1 0−6 is ta k e n :

XT D l i m ,6: P (XT D ≥ XT D l i m ,6

) = 1 0−6 (6 .8 )

XT D l i m ,6(fo u n d b y in te rp o la tio n ) is d iv id e d b y its d e m a n d e d v a lu e o f 9 1 5 m

a n d h a n d le d a s a n in e q u a lity c o n stra in t (XT D l i m ,6/9 1 5 < 1 ). T h is is e q u iv a le n t to

d e m a n d in g P (XT D > 9 1 5 ) < 1 0−6. H o w e v e r, in o rd e r to a ch ie v e m o re m a rg in (o r,lo w e r p ro b a b ility o f e x c e e d a n c e ), th e d e m a n d e d v a lu e h a s b e e n se t to 6 8 0 m , o r:XT D l i m ,6

/6 8 0 m < 1 , so th a t P (XT D > 9 1 5 ) << 1 0−6. F o r c e rtifi c a tio n n a m e ly ,a lso lim it risk s m u st b e c o m p u te d . T h is in v o lv e s re p e a te d M o n te C a rlo a n a ly se sw h e re e a ch tim e o n e m o d e l p a ra m e te r is h e ld fi x e d a t o n e o f its e x tre m e v a lu e sw h ile o th e r p a ra m e te rs v a ry a s b e fo re . C o m p a re d w ith th e sta n d a rd a n a ly sis, th ere q u ire m e n ts o n X T D a re h a rd ly re lie v e d (a p ro b a b ility o f e x c e e d in g th e 9 1 5 mm a rk o f < 1 0−5 m u st b e p ro v e d ). F o r th is re a so n in th e o p tim isa tio n a n a d d itio n a lm a rg in is a im e d fo r b y d e m a n d in g a sm a lle r lim it v a lu e .


6.5 Controller optimisation strategyThis section addresses step C in the design process in Figure 6.2. For tuning theautopilot functions, the multi-objective optimisation environment MOPS (Multi-Objective Parameter Synthesis [50]) is used. MOPS allows multiple optimisationsub-tasks (set-ups), as defined in the previous section, to be comfortably combinedinto a single one. This feature allows for tuning controller functions simultane-ously, as will be described shortly.In Section 6.4, for each controller function an optimisation set-up has been defined.Optimising each function independently does not guarantee suffi cient performanceof the complete system, since in spite of time scale separation between sequentialloops, considerable dynamic interaction may be left. This especially holds forthe SCA in combination with the flare and SPT functions. For this reason, theintention is to tune all controller functions simultaneously. However, in order tosteer the optimisation process in a structured way, and to keep an overview overthe large amount of criteria, simultaneous optimisation is not performed in oneshot. Instead, tuning is started with the SCA inner loop, and then sequentiallyexpanded with the problem set-ups for SPT and guidance functions. The tuningprocess is depicted in Figure 6.11.

For each set-up:* selection of m od el cases* set criteria properties

2

A ug m en t con tr. fun ctionoptim iz ation set-ups

1

T un in g & C om prom isin g 3

C

C on troller fun ctionassessm en t

4

In teg rated con troller

augm

entnextcontr

.fu

nction

O ptim iz ation prob lem set-upsfor con troller fun ction s

2

1

FunctionalD e s ig n R e q ./C e rtification

s p e cs

Figure 6.11: optimisation-based autoland control laws design process (step C inFigure 6.2)

For a combined optimisation task (step 1), criteria properties (scaling, type) areadjusted and, if desired, multiple model cases are selected (step 2). The latterallows for compromising performance between nominal and worst-case model pa-rameter combinations (pe, pu, pk) and is therefore an eff ective means to address

6.5 Controller optimisation strategy 175

performance robustness [50]. After tuning and compromising via multi-objectiveoptimisation in step 3 , performance and robustness of the resulting controllerfunctions are assessed (step 4). After optimisation or assessment, the designermay decide to adjust criteria scaling (loop 1,2) in order to influence compromisesolutions. In case of robustness problems, worst model cases may be added to theoptimisation (loop 2). In case the result is satisfactory, the next controller func-tion set-up is added (step 1). Eventually, all controller functions are optimisedsimultaneously.

As already mentioned, the SCA inner loop function is tuned first. The linearcontroller parameters (Kθ, Kq) are used for command shaping. Since dynamicinversion is sensitive to modelling errors, special attention has to be paid to thisissue (Chapter 5). For this reason, performance of the optimisation result isevaluated for all combinations of extreme values of the longitudinal parametersin pu. If necessary, selected worst cases may be included in the optimisation1.

N ext, the optimisation set-up for the TECS-based flight path and speed trackingloop is added. The SCA set-up is retained, but the critical criteria are changedinto inequality constraints (step 2 in Figure 6.11). During optimisation the innerloop gains Kθ, Kq may thus be adjusted to improve TECS performance, but theoptimiser is prevented from distorting the achieved SCA performance by choos-ing gains Kθ, Kq that are only valid with TECS connected. Again, the newlyoptimised parameters are used as a start for the following step.

N ext, the glide slope mode is added to the optimisation. Criteria of the TECSset-up are also set as inequality constraints. At this point, the Monte Carlobased criteria are left out, since their computation is too time consuming for anintermediate optimisation step. During tuning of the glide slope mode it becameclear that tight path tracking could not be sufficiently achieved. Fortunately,in the TECS structure the gain KF (normally 1) can be used to (temporarily)shift priority to flight path tracking. Opening the speed loop (see Figure 6.4) isalso helpful. The parameter KF may be adapted when the glide slope mode isconnected. Evaluation of the criteria for TECS alone (as in previous optimisation)is performed for KF = 1 and the speed loop closed. R egarding tracking, the glideslope mode is most demanding. It is expected that other autopilot modes can beadded later on, without, or with only minor adjustment to TECS gains.

Finally, the flare mode is added, including Monte Carlo based criteria. Thoserelated to the glide slope criteria are activated as well now (Table 6.4). Theoptimisation task now includes all longitudinal autoland functions. All gains maybe adjusted, and criteria from all sub-tasks (see Section 6.4) are active. Those inthe SCA and TECS set-ups are set as inequality constraints. This on one handallows these functions to be adjusted to improve outer loop performance, but onthe other hand prevents distortion of performance of the functions without theflare or GS mode connected.

1Contrary to roll and yaw control, for pitch attitude control it turned out that the use of

uncertain m odel param eters in the inv erse m odel eq uations in the control law was not neces-

sary. M ain reason is that the m ax im um uncertainty lev els of m ost long itudinal aerodynam ic

coeffi cients are lower than those of the lateral coeffi cients, see T ab le 2 .5 .


Augmented design set-up:(set-up to the left is retained)

Active tuner: SCA TECS GS flare unitKθ 1.6 2.0 2.0 2.1 s−2

SC

A

Kq 2.5 3.2 3.2 3.3 s−1

KEI 0.2 0.3 0.2 s−1

KT I 0.4 0.3 0.43 s−1r a d−1

KEP 0.56 0.4 0.33 –KT P 0.6 1.0 1.2 r a d−1

KV 0.12 0.12 0.12 s−1

KF * 1.0 1.0 0.28 –

SP

T(T

EC

S)

τV * 10 * 10 8.8 sKh 0.06 0.06 s−1

τh 2.5 2.6 s

GS

τh 15 3.9 sKFL -0.04 r a d(m/s)−1

Hb ia s 1.4 m

θR a mp 0.18 r a d/sKFW 1 –KR W 0.12 r a d(m/s)−1

Hr eta r d 6.0 m

Fla

re

Hfla r e 12.1 m

Table 6.6: Development of tuner parameter values. ’* ’ = inactive

6.6 Formulation of the basic optimisation problemFor each design step, the problem of tuning and compromising (step 2 in Fig-ure 6.11) is formulated as a weighted min-max optimisation problem, comprisingall active criteria, over all sub-tasks (set-ups as defined in section 6.4), over allselected model parameter cases per sub-task:

minT

maxij k∈Sm

cij k(T, pij )/dij k

cij k(T, pij ) ≤ dij k, ijk ∈ Si (6.9)

cij k(T, pij ) = dij k, ijk ∈ Se

Tmin ,l ≤ Tl ≤ Tma x ,l (6.10)

where: Sm is the set of criteria to be minimised, Si is the set of inequality con-straints, and Se is the set of equality constraints; T is a vector containing thetuning parameters Tl to be optimised, lying between the upper and lower boundsTmin ,l and Tma x ,l, respectively; cij k ∈ Sm is the kth normalised criterion of the jth

model parameter case in the ith optimisation sub-task (e.g. flare, SCA) and dij k isthe corresponding demand value, which serves as a criterion weight (Section 6.4);pij denotes a parameter vector of the ith sub-task, defining the jth model case.

6.7 Controller optimisation results 177

The criteria cijk ∈ Si, Se are used as inequality and equality constraints respec-tively. The affiliation of criteria to one of the groups Sm, Si or Se respectively,can be changed at any time depending on the design progress. This feature is forexample used when adding the TECS set-up to the SCA one: the SCA criteria arechanged from minimisation to inequality constraints, allowing TECS performanceto be improved while the demanded level of performance of the SCA function isallowed to deteriorate to the level of demanded criteria values (dijk). The opti-misation problem (6.9) is formulated automatically by the MOPS environment.Adding optimisation sub-tasks or model cases, and setting properties of criteriais done with the help of a graphical user interface, or via scripts [52].

The min-max optimisation problem is solved by reformulating it as a standardNonlinear Programming (NL P) problem with equality, inequality and simplebound constraints. This reformulation is done fully automatically, after whichthe NL P problem is solved by using one of several available powerful solvers im-plementing local and global search strategies. Besides efficient gradient-basedsolvers, also gradient-free direct search-based solvers (usually more robust, butsomewhat less efficient) are available to address problems with noisy or non-smooth criteria. Solvers based on statistical methods or genetic algorithms areavailable as well. For this work, for the SCA and TECS functions SequentialQ uadratic Programming (SQ P) was used, after augmenting the glide slope andflare set-ups a pattern search method was applied, which turned out to cope betterwith the criteria derived from on-line Monte Carlo analysis.

6.7 Controller optimisation results

In this section the performance of the final controller (step D in Figure 6.2), aswell as intermediate results, will be assessed. Figure 6.12 shows the result of theoptimisation in so-called parallel co-ordinates. All scaled criterion values havebeen plotted on an individual axis and connected through a line (i.e. one graphcorresponds to one tuning parameter set T ). The fat horizontal line indicates avalue of one. Criteria values below this line are considered satisfactory. Parallelco-ordinates are standard graphical output during optimisation with MOPS, giv-ing quick insight in the optimisation progress, in criteria that are hard to satisfy,and in criteria that conflict and thus have to be compromised [50]. The represen-tation will be used here to compare the intermediate optimisation steps. Criteriavectors belonging to the different optimisation set-ups have been separated bythick vertical lines.

The dash-dotted line (marker ’×’) in Figure 6.12 represents the result after opti-misation of the SCA function. The other set-ups have not been involved yet, sothat the line can only be drawn for the SCA criteria. The resulting tuner param-eter values can be found in the first column of Table 6.6. Since all scaled criterionvalues are below 1, the result is regarded satisfactory. The corresponding pitchattitude command response and the Nyquist curve for the loop opened at the ele-vator actuator have been plotted in Figure 6.13 (dash-dotted curves). The phase


1

SCA TECS GS FLARE MONTE CARLO

THrt

THos

THco

ntr

THtu

rbgm

AD

pmA

Dgm

SQ

pmS

QD

AM

Pdi

GA

rtG

Aos

GA

stTH

cmd

GA

VA

VA

rtV

Aos

VA

stdT

HR

VA

GA

NZt

urb

THC

turb

THR

turb

gmA

Dpm

AD

gmS

Gpm

SG

gmS

Vpm

SV

DA

MP

GS

rtG

Sos

GS

stX

TDno

mV

ZTD

nom

THgr

addH

grad

DE

dot

mea

nHTP

60st

devH

TP60

limH

TP60

mea

nXTD

stde

vXTD

limX

TDm

eanV

ZTD

stde

vVZT

Dlim

VZT

Dm

axG

LDde

vm

axG

LDde

v50

mea

nGLD

dev

max

THE

dev

max

VC

dev

Figure 6.12: Scaled criteria in parallel co-ordinates: ’-.’ = SCA optim., ’- -’ =

SCA+ SP T optim., ’– ’=complete optimisa tion . Scaled criteria values below the fathorizontal line satisfy demanded values.

0 2 4 6 8 10 124

6

8

10

12Step response θ

time (s)

θ (d

eg)

−2 −1 0 1 2−2

−1

0

1

2

real

imag

Nyquist @ act.

Figure 6.13 : SCA results (for legend, see caption of Figure 6.12)


margin is 85 deg, which is larger than the good-low value of 60 deg (Table 6.4)demanded for pmAD. As depicted in Figure 6.12 on the pmAD axis, the scaledcriterion value is thus 0 (note that this is the case for all linear criteria).The result of the combined optimisation of the SCA and SPT function is rep-resented by the dashed line. This time, the curve can be drawn for the parallelco-ordinates up to DAM P (dashed line, marker ’o’). The resulting tuning param-eter values can be found in the second column of Table 6.6. All scaled criteriavalues are below one, except for T H cmd and T H R tu rb (set passive). However,exceedance with 20 and 40% was considered acceptable, since these criteria werenot considered critical. Corresponding step responses on γa and Vc as can be foundin Figure 6.14 (dashed lines). Due to further adjusting Kθ and Kq, SCA criteriavalues have changed in Figure 6.12. T H rt increased somewhat, but not beyond 1(i.e. rise time < 2.5s), since the criterion was set as an inequality constraint inthe optimisation. Damping (DAM P d i) and control effort (T H co n tr) have deteri-orated, but are still acceptable (< 1). Intermediate parameter studies with theSCA and TECS functions for longitudinal aerodynamic coefficients in pu revealedthat criteria values did not degrade to unacceptable levels. For this reason it wasdecided to proceed with the gains as found from optimisation with the nominalaircraft model.

0 20 40 60−1

0

1

2

3

4

5

γ

Vcas

time (s)

θ,γ

(deg

), V ca

s (m/s

)

θ

step response γ

0 20 40 60−2

0

2

4

6

8

10

12

γ

Vcas

time (s)

γ (d

eg),

V cas (m

/s)

step response Vcas

Figure 6.14: TECS results (for legend, see caption of Figure 6.12)

Results of the combined optimisation of SCA, SPT, and GS set-ups will not bediscussed (third column of Table 6.6). The final optimisation step involves allset-ups augmented into a single optimisation task. The result is represented by asolid line in Figure 6.12 (marker ’*’). The corresponding tuner parameter valuescan be found in the fourth column of Table 6.6.Regarding the nominal flare manoeuvre (Figure 6.15), all criteria (X T Dn om ...

DE d o t) are satisfactory, except for T H gra d (Table 6.5). It turned out that aslight nose drop during the flare (0.3 deg) had to be tolerated, unless considerableemphasis was put on feedforward. This however made it hard to meet MonteCarlo assessment criteria that will be discussed shortly. At this point, it becameclear that architectural enhancements will be necessary to further improve the


0 1 00 2 00 3 00 4 00 500

0

5

1 0

1 5

D is t. from thre s hold (m)

Hra

(m

)R a dio a ltitu de (m)

0 100 200 300 400 5002

3

4

5

6

Dist. from threshold (m)

θ (d

eg)

Pitch attitude response

Figure 6.15: Flare results

−5000 −4000 −3000 −2000 −1000 0−10

−5

0

5

10

Dist. to threshold (m)

εG

LD (m

A)

Glide slope offset

backlash compensatedbacklash not compensated

−5000 −4000 −3000 −2000 −1000 015

20

25


Thro

ttle

(deg

)Throttle activity

Figure 6.16: G lid e S lo p e results

d esig n .A m ajo r c o n cern fo r g lid e slo p e track in g w as en g in e th ro ttle b ack lash . S in ce T E C Suses th ro ttle fo r fl ig h t p ath track in g , un ac cep tab le o sc illatio n s aro se. T h ese c o uldb e red uced v ia a c o m p en satio n sch em e usin g m easurem en t o f th e m ean fan sh aftsp eed N1 o f b o th en g in es. Fig ure 6 .1 6 sh o w s n o m in al g lid e slo p e track in g p er-fo rm an ce an d th ro ttle activ ity w ith o ut (d o tted ) an d w ith b ack lash c o m p en satio n(so lid ). B o th th ro ttle activ ity an d o sc illatio n s h av e red uced c o n sid erab ly . T h em ax im um o v er-all d ev iatio n (maxGLDdev) an d m ax im um d ev iatio n 5 0 m b efo reth resh o ld (maxGLDdev5 0 ) un d er turb ulen t c o n d itio n s w ere d iffi cult to im p ro v eb ey o n d th e criteria v alues sh o w n in Fig ure 6 .1 2 (scalin g s h ad to b e reliev ed as w ell,see T ab le 6 .4 ). In sp ectio n o f in d iv id ual lan d in g s fro m th e M o n te C arlo an aly sisrev ealed th at th ese m ax im um v alues o c curred d urin g ex trem e w in d sh ears (>9ft/ s), caused b y a c o m b in atio n o f h eav y turb ulen ce an d th e stan d ard w in d p ro fi leas a fun ctio n o f h eig h t [3 7 ]. In furth er tun in g , such cases sh o uld b e elim in ated .T h e T E C S related p aram eters h av e b een c o n sid erab ly m o d ifi ed d urin g th e fi n alo p tim isatio n step . C learly , p itch attitud e c o m m an d s (T H cmd) h av e d ecreased ,


at the cost of throttle activity (dTHR). This is related to minimising pitch at-titude excursions during glide slope tracking (maxTHE dev). H owever, the lattercriterion was most difficult to improve. Again, it turned out that the maximumdeviations occurred during heavy wind shears. Figure 6.17 shows an examplesimulation from Monte Carlo analysis with a θ deviation of ∼5 deg due to a 3sec. wind shear of ∼9 ft/s. R egarding the SCA related criteria, these slightly,but acceptably, deteriorate due to further adjustments of Kθ and Kq. The stepresponse and N yq uist curve (solid) in Figure 6.13 confirm this.

−6000 −4000 −2000 0 20000

2

4

6

8

10Theta (°)

−6000 −4000 −2000 0 200030

40

50

60

70

80

90N1 (%)

−6000 −4000 −2000 0 2000−30

−25

−20

−15

−10

−5


Wx (m/s)

−6000 −4000 −2000 0 200058

60

62

64

66

68

70

72


Vc (m/s)

Figure 6.17: W orst case simulation from Monte Carlo analysis

The most important criteria for certification are based on risk analysis from MonteCarlo assessment. The optimised result can be found in Figure 6.18 . The left halfof the figure shows distribution of the risk parameters H TP 60, X TD , and V Z TDcomputed from the mean and standard deviations over 2000 landings (duringoptimisation, only 400 were used). To the right the resulting cumulative distribu-tions can be found. As an example for interpretation, the probability of landingat a sink rate (V Z TD ) higher than 2 m/s is 10−2.4, as indicated in Figure 6.18 .The graph should stay outside the shaded area, so that the probability of landingharder than 3 m/s (for ATTAS) is less than 10−6 (risk to be demonstrated). Thishas clearly been achieved by the optimiser. Incorporating these statistical criteriain the optimisation was found extremely useful, since EASA CS-AW O robustnesscriteria could be addressed (and fulfilled) directly.


5 10 15 200

0.5

1HTP60

200 400 6000

0.5

1XTD

−1 0 1 2 30

0.5

1VZTD

0 5 10

−5

0Log[p(HTP60 < x)] (m)

0 500 1000

−5

0log[p(XTD > x)] (m)

0 1 2 3 4

−5

0log[p(VZTD > x)] (m/s)

−2.4

Figure 6.18: Monte Carlo simulation results

6.8 Flight test results

In this section some relevant flight test results will be briefly reviewed. Moredetails have been published in Ref. [12]. For the glide slope mode the same landingas in Section 5.6 will be discussed. Figure 6.19 depicts commanded and measuredcalibrated airspeed, air mass-referenced flight path angles, and accelerations. Theapproach started parallel to, but somewhat below the glide path. The glide slopemode therefore initially commands a flight path angle of zero degrees in orderto capture it (see Figure 5.28). After capture, the flight path angle returns toapproximately -3 deg. Figure 6.20 shows throttle commands from TECS. Clearly,a slight thrust increase is commanded in order to increase the flight path anglefor glide slope capture. Also note the eff ect of backlash compensation (dashedline) on the commanded throttle setting.

Figure 6.21 shows the vertical and lateral speed errors with respect to the glideslope and localiser respectively (solid lines). Although the latter is less of interesthere, both signals have been estimated in a similar way, using complementary fil-ters. The lower frequency content is obtained from diff erentiating filtered verticaland lateral path errors respectively. The higher frequency content is the speederror computed with respect to the nominal flight path angle of γgld = −3 degand the rounded runway heading (set by the pilot) respectively (dashed lines).For vertical speed, the filter is depicted in Figure 6.6. Comparing the solid anddashed lines, the glide slope is apparently slightly less steep than -3 deg and theactual runway heading is slightly rotated counter clock-wise with respect to the

6.8 Flight test results 183

reference value entered by the pilot (see also lower plot in Figure 6.23).

0 20 40 60 80 100 120 140 160 180 20060

70

80

Vca

s (m/s

)

TECS cmd following: solid=command, dotted=measured

0 20 40 60 80 100 120 140 160 180 200−10

−5

0

5

γa (d

eg)

flare

0 20 40 60 80 100 120 140 160 180 200−2

0

2

time (s)

dV/d

t (m

/s2 ) retard

Figure 6.19: Tracking of TECS-related variables (approach 2-2)

The flare manoeuvre for the third landing during the fourth flight test is depictedin Figure 6.22 (top left). U nfortunately, the touch down point from the runwaythreshold could only be estimated optically, but was about 400 m. The pitchangle is depicted left below and clearly shows a slight (but unacceptable) nosedrop before touch down. The same holds for the sink rate (top right). The engineresponse (N1, mean value for both engines) is depicted bottom right and decreasesdue to throttle retard.

The feedforward part of the flare law (Figure 6.7) is based on a moving average of

the pitch angle commanded by TECS during the approach: θ0. The developmentof this value is shown in Figure 6.23 (upper plot). Finally, it is interesting to lookat the complementary vertical speed filter within the flare mode (Figure 6.8).After flare initialisation this filter estimates the vertical speed with respect to therunway surface, as well as the component due to a possible runway slope. Thetime responses are depicted in Figure 6.24. Apparently, the runway goes slightlyup-hill in flight direction, since the sink rate with respect to the runway surfaceis slightly larger (more negative). During design it quickly turned out that evena small runway slope (1 % ) can make the difference between a too soft landingand an unacceptably hard one.


0 20 40 60 80 100 120 140 160 180 200−20

−10

0

10

20

30

time (s)

δTH

R (d

eg)

Backlash comp: solid=before, dashed=after comp.

Retard

Figure 6.20: Throttle inputs commanded by TECS (approach 2-2)

0 20 40 60 80 100 120 140 160 180 200−4

−3

−2

−1

0

1

2

Hdo

t erro

r (m)

dotted=compl. filter solid=computed with γgld

=−3

0 20 40 60 80 100 120 140 160 180 200−4

−2

0

2

4

6

time (s)

Ydo

t erro

r (m)

solid=compl. filter dashed=computed with ψrwy

Figure 6.21: Estimated vertical and lateral speed errors (approach 2-2)

6.9 Conclusions 185

122 124 126 128 130 132−5

0

5

10

15Radio altitude

Hra

[m]

122 124 126 128 130 132−1

0

1

2

3

4

touch down

Sink rate

VZ [m

/s]

122 124 126 128 130 132−1

0

1

2

3

4

5

touch down

Pitch attitude

θ [d

eg]

Time [s]122 124 126 128 130 132

35

40

45

50

55

60

65Engine response

N1

[−]

Time [s]

Figure 6.22: Flare manoeuvre during landing 4-3

6.9 ConclusionsThe application of an optimisation-based design process for automatic landingcontrol laws has been discussed.Multi-objective optimisation has proved to be very powerful to handle the largenumber of design criteria. Design requirements regarding touch down performanceand glide slope tracking could be directly translated into numerical design crite-ria. For the Stability and Control Augmentation (SCA) function and Speed andPath Tracking (SPT) function standard step response criteria (computed fromnonlinear simulations), as well as stability-related criteria like damping and gainand phase margins have been successfully used.It has further been demonstrated that robustness to varying (aircraft and envi-ronment) parameters can be successfully addressed with the help of stochasticcriteria computed from on-line Monte Carlo analysis. In the case of autoland,this implies that part of the certification criteria have been directly addressed inthe optimisation.Final optimisation of the system has been successfully performed for all func-tions simultaneously. This has been used to tune a single SCA function to workwith the SPT and glide slope functions, with the flare law, as well as on its own.optimisation of the integrated system may thus help to reduce control law com-


0 20 40 60 80 100 120 140 160 180 200

−2

0

2

4

time (s)

θc (d

eg)

0 20 40 60 80 100 120 140 160 180 20080

85

90

95

100

time (s)

ψrw

y (deg

)flare

θc from TECS

χ (deg) align

filtered χ (τ=20s) = estimated rwy heading

selected runway heading

filtered (τ=30s)

Figure 6.23: Computation of moving average pitch attitude and runway headingangles (approach 2-2)

plexity. However, in order to keep complexity of the optimisation problem for thenumerical algorithms and for the designer in hand, a stepwise tuning strategy hasbeen proposed. After optimising the SCA function, the outer loop functions aresequentially added, allowing the designer to concentrate on one function at time,but eventually resulting in optimisation of all design parameters in the integratedsystem.The design process resulted in good performance of the autoland system and,according to the Monte Carlo analysis results, is well able to cope with varyingaircraft and environment parameters. The flare law turned out to be close to itsperformance limits (eg. a slight undesirable nose drop had to be accepted). Forfurther improvement of performance and robustness, the structure may have tobe enhanced.The presented controller structure used in this design was successfully flight testedin September 2000 on DL R’s ATTAS (Advanced Technologies Testing AircraftSystem). Some of the results have been briefly discussed.

6.9 Conclusions 187

120 122 124 126 128 130−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

Time [s]

VZ, H

dot rw

y [m/s

]

Complementary altitude rate filter

flare

Hdotrwy

−VZ

Figure 6.24: Complementary filtering of radio altitude and sink rate (approach4-3)


AcknowledgementsThe presented design is based on the autoland system that was developed by DLRwithin the project Robust and Efficient Autopilot control Laws design (REAL),sponsored by the Commission of the European Community (contract nr. BRPR-CT-98-0627). The DLR design team consisted of Hans-Dieter J oos and Gert-jan Looye (from the Institute of Robotics and Mechatronics, Oberpfaffenhofen),and Wulf Monnich and Dehlia Willemsen (from the Institute of Flight Systems,Braunschweig). The Matlab-based Monte Carlo assessment tool SIMPALE wasdeveloped by project partner ONERA, France. Finally, the authors would like tothank the reviewers for their valuable comments.

Chapter 7

Conclusions

190 Conclusions

Abstract

The main result of this thesis is a new, highly automated process structure

for design of complex fl ight control laws. P urpose of this chapter is to

describe this process, and how the other contributions of this thesis fi t

in. The result is validated against the objectives set in the introduction,

namely to accommodate multi-disciplinary control law design, as well as

preliminary control design methods for early design stages of the aircraft.

A ll contributions have been validated on industrial applications. B ased on

this, some lessons learnt will be reviewed. P art of these lessons give rise

to recommendations for future work.

C o n tributio n s

• A n integrated and automated process structure for design of complex

control laws.

7.1 An integrated fl ight control law design process 191

THE objective of this thesis is to propose a design process and design method-ologies that inherently facilitate a multi-disciplinary approach to the design

of flight control laws. The methodologies presented in the individual chapters ofthis thesis provide the building blocks for such a newly structured process. Thebasis for this process has been laid in Chapter 6, focusing on automatic landingdesign. In this final chapter the generalised structure and its components will bediscussed.

7.1 An integrated fl ight control law design process

Figure 7.1 depicts the proposed design process structure. It covers the off-linedesign phase as indicated in Figure 1.3. Since the structure is a generalisedversion of the one proposed and applied in Chapter 6 (Figure 6.2), the principalsteps will be described only briefly.

The first step is the definition of the over-all control law architecture. As also indi-cated in Figure 1.3, the resulting structure will heavily depend on the philosophyand preferences of the aircraft manufacturer. Typical examples are the selectionof command variables for manual control, the integration of autopilot and man-ual control functions, the implementation of protections against exceedance ofenvelope parameters, etc. From developing the architecture a break-down intosub-systems and controller functions usually arises naturally.

As a new element, a possibility for rapid-prototyping is accommodated in thedesign process between architecture development and detailed design of the func-tions: Preliminary FCL design. As discussed in Chapter 4, this is very useful incase new types of functions are introduced for which key design choices like com-mand variables, control allocation, etc. are not yet obvious to the design team.Chapter 4 describes an example for control laws for drive-by-wire control of air-craft on the ground. Compared with Figure 1.3, the rapid-prototyping processadds a fast design loop, indicated by “ 0” in Figure 7.1.

Even more important from a multi-disciplinary point of view, rapid-prototypingallows for early generation of standard control functions for analysis purposes inthe aircraft (preliminary) design process. This is depicted in Figure 7.2. Sincethe aircraft geometry may change rapidly in the pre-design phase, the FCL designloop has to be fast as well. This is enabled by the object-oriented modellingmethodology, allowing for fast model updates and for automatic generation ofnonlinear control laws, based on inverse control methods like Nonlinear DynamicInversion. In Figure 7.2 the block Prel. design of functions represents the rapid-prototyping process discussed in Chapter 4, see Figure 4.8.

Although the rapid prototyping process will only be applied to a number of keyfunctions of the eventual control system, the methodology gives the flight con-trols team the possibility to already become a design (rather than specification)contributor to the most preliminary aircraft configuration. The methodology forexample allows for accurate preliminary closed-loop flying quality analysis forthe current aircraft configuration, deriving “ just right” specifications for sizing of

192 Conclusions

Flight control lawarchite ctu re d e v e lop m e nt& fu nctional b re ak d own

Multi-disciplinaryaircraft m od e l

Inte gration

FCLinte gration

FCL fu nction d e taile d d e s ign:- archite ctu re (T )- com p u tational crite ria* p e rform ance , e tc.* m u lti-d is cip linary

L ib rarie s :- archite ctu re com p one nts (T )- as s ociate d op tim is ation s e t-u p s- algorithm s for crite ria- ...

P re lim inaryFCL d e s ign

M u lti-ob je ctiv e op tim is ationof inte grate d FCL

FCL m u lti-d is cip linaryas s e s s m e nt

0

C

D

B

A

Com p one ntlib rarie s

m od e l d ata s ou rce s- cu rre nt AC d e s ign s tatu s- late s t e x p e rim e ntal re s u lts

R ob u s t FCL s re ad y for

b y s p e cialis t d e p artm e nts

- hard ware te s ting- load s analy s is- ae ros e rv oe las tic analy s is- ...

F unctio nal de sig nre q uire m e nts /C e rtificatio n spe cs

De s ignP hilos op hy

1

Figure 7.1: New integrated FCL design process structure

control surfaces and actuators, and for flight loads analysis.

The functional breakdown in Figure 7.1 and rapid prototyping (if exercised) leadto the following main step, namely FCL function detailed design. As explained inChapter 6, this involves detailed development of the control function structure,as well as formulation of numerical design criteria for tuning of the free design pa-rameters. The process structure in principle allows a control law structure basedon any synthesis method: it does not matter whether the design parameters arecontroller variables or weighting function parameters. This means that struc-tures from previous design programs can be incorporated, or, as demonstratedin Chapters 5 and 6, promising architectures like Nonlinear Dynamic Inversionand the Total Energy Control System (TECS) can be selected and exploited totheir full extent. This fulfils the constraint posed in Chapter 1 that it must be


Fast design

cycle

Baseline FCLarc h itec tu re

A irc raft m o d elinteg ratio n

P rel. d esig no f fu nc tio ns

O p en-lo o p and c lo sed -lo o pflig h t d y nam ic s analy sis

Aircraft preliminary /d etailed d es ig n

Rapid prototyping designprocess Chapter 4

Selection of command variables ( )ycm d

A llocation of controls ( )uc

A dd linear controller /command filters (from library )

Inversion of A /C model

B asic decisions

A u tomatic

Working design

P reliminary analy sis(validate/comp are basic decisions)

okno

y e s

B atch / real-timesimu lation

C onstru ct aircraft model

.

.

S p e cifica tio n s ,A ircr a ft- le v e la n a lys is r e s ults

D esig nP h ilosop h y

A Cfreez e

no

y e s

d e ta ile dd e s ig n

0

Figure 7.2: Rapid-prototyping preliminary design process structure

possib le to incorporate ex isting k now -h ow , ex perience, and lessons learnt. F ora giv en control law function most of th e w ork in th is design step needs to b edone only once. A rch itectures, associated computational scripts for criteria com-putation (“ optimisation set-ups” ), criteria, etc. may b e stored in and retriev edfrom a repository, sh ow n to th e left in F igure 7 .1 . E x amples of set-ups h av e b eendiscussed in C h apters 5 and 6 .A fter detailed design of F C L functions, tuning of free design parameters can b eperformed. T h is is done automatically w ith th e h elp of multi-ob jectiv e optimi-sation, b ased on computed numerical criteria and th eir scalings as formulated inth e prev ious design step. In case of multiple interacting functions, it is importantto apply an optimisation strategy in order to allow th e designer to steer and k eeptrack of th e tuning process. S uch a strategy h as b een dev eloped in C h apter 6 , seeF igure 6 .1 1 . A s compared w ith manual tuning, multi-ob jectiv e optimisation h asv ery attractiv e adv antages:

• T h e tuning process is repeatab le and easy to document. T h is allow s inter-mediate design results to b e easily reproduced afterw ards;

• A much larger set of design criteria can b e h andled simultaneously. T h isis th e k ey to multi-disciplinary control law design, since criteria related tooth er disciplines can b e included in th e optimisation to search for trade-off s.

194 Conclusions

The second advantage allows design iterations along loop 2 (Figure 1.3) to be elim-inated. Although not of multi-disciplinary nature, Chapter 6 has clearly validatedthis: M onte Carlo analysis of autoland control laws usually results in iterationsalong loop 2. B y directly addressing this analysis in the tuning process, controllaws resulted that automatically fulfi lled the underlying safety req uirements.

After tuning and compromising of the design parameters, the fl ight control lawsare extensively tested in order to validate if (multi-disciplinary) design req uire-ments have been met in all fl ight conditions, for all aircraft confi gurations, etc.(lower block in Figure 7.1). In Chapter 5 µ-analysis has been applied to assess sta-bility robustness of three versions of N D I control laws. The use of object-orientedmodelling allowed for automatic generation of the req uired LFT representationof the closed loop system. D ue to guaranteed bounds on the structured singularvalue µ it could be demonstrated that, in face of the uncertain parameters, thecontrollers are robustly stable, have a minimum damping number of ζ = 0 .2, andhave stability margin to spare against other unspecifi ed uncertainties in the formof worst-case gain and phase margins.

B esides q uantitative assessment, q ualitative analysis of the control laws in theform of interactive real-time desktop simulation is a valuable tool as well (Ap-pendix A). It for example allows for easy testing of closed-loop responses to adverseinput signals. This q uickly reveals issues like singularities, undesired transient be-haviour in case of switching, etc. that remained undetected in batch simulations.In this way, interactive desktop simulation can be a strong help to improve ma-turity of control laws released from the off-line design phase.

O nce the assessment step has been passed, control laws may be released for hard-ware and fl ight testing, as well as for detailed aeroservoelastic and fl ight loadsanalysis by specialists in those fi elds. The latter is req uired for certifi cation, butshould, compared with today’s process, no longer result in new control designiterations along the aforementioned loop 2 (Figure 1.3).

To the right in Figure 7.1 various design iteration loops (A,B ,C,D ) have beendrawn. These loops have been described in Chapter 6 and belong to loop 1 inFigure 1.3.

O ne of the most important elements of the new design process is multi-disciplinaryaircraft modelling. In Figure 7.1 this aspect is indicated by Multi-disciplinary

aircraft m ode l integratio n. The development of multi-disciplinary aircraft modelsis discussed in Chapters 2 and 3. O bject-oriented modelling, available in the formof the free language specifi cation M odelica, is highly suitable for this purpose:

• From a m ode l co nstructio n point of view, object-oriented modelling takesplace at the level of physical eq uations. This allows various componentsto be implemented in their discipline-specifi c form (multi-body system, hy-draulic scheme, electronic circuit). At the same time, the various compo-nents can be combined in a single model. P hysical eq uation-based modellingalso allows for one-to-one implementation of physical objects and phenom-ena. These features result in intuitive and understandable model structures,even if the level of detail becomes high. In Chapter 2 a new structure for


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Figure 7.3: Modelling process for control law design and simulation

aircraft flight dynamics models has been introduced, combined with a dedi-cated flight dynamics library with reusable components. This library allowsfor model implementation of rigid and flexible aircraft in slow-low up tofast-high flight regimes;

• An object-oriented model does not require causality to be fixed before imple-mentation. This offers great flexibility for the flight control design process,since different types of runtime models as used for different control analy-sis methods can be automatically obtained from one and the same aircraftmodel implementation. This has been exploited to its full extent in thisthesis. For example, the rapid-prototyping process is based on automaticgeneration of inverse model-based control laws. Since the inverse modelequations originate from the same source as the simulation model, a workingdesign (for nominal uncertain parameter settings) is obtained straight-away.In Chapter 5 this feature was used to generate NDI control laws, but also toobtain a Linear Fractional Transformation form of the model. W orst-caseanalyses were performed, which could be validated on the nonlinear modelgenerated for simulation analysis.

196 Conclusions

Model construction and automatic code generation for runtime models has beensummarised in Figure 7.3: model construction (a.o. with the help of compo-nent libraries) is performed above, various applications for control law design andsimulation are depicted below. Each runtime model form may be automaticallygenerated from the same model, depending on inputs, outputs, parameters and,in some cases, states selected by the user.

One of the most notorious disciplinary interactions in flight control law design isthe influence on flight loads and aeroelasticity, and vice versa. In order to includethese aspects in the design model, overlaps in data between aeroelastic and flightmechanics models need to be addressed before implementation. This has beensubject of Chapter 3. In this chapter the Residualised Model (RM) method hasbeen extended, allowing for correct integration of rigid and aeroelastic aircraftmodel data. Although not exercised in this thesis, these models may be used formulti-disciplinary flight control law optimisation, see Ref. [60] for an example. Inthe mean time, for integration and computation of model data a dedicated processhas been developed, called DAMIP (Dynamic Aircraft Model Integration Process).Details are provided in Ref. [59 ]. DAMIP largely automates the integration ofmodel data from various sources, as well as their implementation in the object-oriented aircraft model.

7.2 Recent application examples

Each of the contributions in this thesis has found industrial application, has beenflown, or at least, has been validated on industry-relevant applications.

The process structure as depicted in Figure 7.1 has been applied in the EU -fundedproject REAL (Robust and Effi cient Autopilot control Laws design). WithinREAL a first design was made for a generic transport aircraft. The applicationto ATTAS (described in Chapters 5 and 6) was intended to prove effi ciency of theproposed design process by posing a severe time limit and by requiring the designto be flight ready. The ATTAS design took five weeks and resulted in six out ofsix successful automatic landings, see Figure 7.4 and Ref. [12]. Within the scopeof the REAL project, a first shot design could be achieved: flight test results didnot require any re-tuning of the control laws.

With the help of the Flight Dynamics Library as presented in Chapter 2 an object-oriented integrated aeroelastic aircraft model for real time simulation and loadsanalysis was implemented. The model has 8 56 continuous and discrete states(flight mechanics, structural dynamics, actuators, sensors, unsteady aerodynam-ics, landing gears) and 118 9 2 scalar variables and model equations. The effi ciencyof the generated simulation code allows for real-time interactive simulation, incombination with similar control laws as developed in Chapters 5 and 6. Thesimulation has been combined with 3-D stereo visualisation, allowing for photo-realistic visualisation of the deforming aircraft in flight and on the ground, seeFigure 7.5.

The Residualised Model method (Chapter 3) has found industrial application for

7.2 Recent application examples 197

Autopilot modes:- IL S track in g- flare / alig n

R ob ust D y n amicIn v . in n er loopcon troller

ua

c

T h rustsen sors

Figure 7.4: ATTAS during an automatic landing

Figure 7.5 : Interactive simulation and 3-D Stereo visualisation of the flexibletransport aircraft model at the ILA airshow in Berlin, 2006. Equipment set-up(left) and large public interest (middle). The blue arrows on the aircraft representdistributed aerodynamic loads, computed on-line in the simulation model (right)

flight loads computation for civil aircraft. Manoeuvre loads analyses are based onrigid flight dynamics simulation models (including quasi-static aeroelastic effectsonly). The Residualised Model method has allowed aeroelastic models from gustresponse analysis to be integrated in the simulation of loads design manoeuvres,in order to assess the effect of structural dynamics on peak and fatigue loads.

In the mean time, the RM method has been extended to rigid aerodynamic modelsdescribing distributed air loads, instead of total aerodynamic forces and momentsthat act on the aerodynamic reference point only [107].

The methodology of rapid-prototyping (Chapter 4) has been used in the projectV ECTOR (V ectoring, Extremely short take-off and landing, Control, Tailless Op-erations Research) [40] to evaluate reduced vertical tail configurations on thethrust-vectored X -31A experimental combat aircraft (see cover of this thesis).This evaluation involved ground-based simulation of combat and post-stall ma-noeuvres by test and fleet pilots (Figure 7.6) at the Patuxent River Naval Air

198 Conclusions

Station in the US. Changes to the vertical tail severely impact lateral-directionaldynamics of the aircraft, requiring adaptation of the control laws. The proposedrapid-prototyping design process allowed for updating the X-31A model with newaerodynamic databases and re-generation of nonlinear control laws on-site withindays. The design and evaluation of the control laws is discussed in more detail inRef. [123].In the VECTOR application the results have been used to make recommendationsfor controllability of the aircraft with smaller tail configurations. The rapid-prototyping process herewith offers a methodology to more accurately assess theairframe configuration in interaction with the control laws and achievable handlingperformance. Supporting of such far reaching airframe design decisions, the rapid-prototyping design methodology is a relevant contribution to the flight control aswell as the over-all aircraft design process.

Figure 7.6: Evaluation of control laws for the X-31A with reduced vertical tail inthe ground based simulator at the Naval air station at Patuxent River, MD. Testand fleet pilots rated the control laws for standard flight and combat handlingand post-stall manoeuvres as level 1 on the Cooper-H arper scale.

7.3 Lessons learnt

Probably one of the most valuable recent documents on flight control law designis the report that was produced by the RTO working group 23: Flight Control

D esign – B est P ractises [111]. As the title suggests, this report summarises bestpractises resulting from lessons learnt in various military flight control projects.During the development work and applications of the concepts described in thisthesis, a number of lessons was learnt as well, sometimes intuitively, sometimesthe hard way. These lessons will be briefly shared in the following. Althoughsome seem obvious, they are easily forgotten in practice. Some of the lessons give

7.3 Lessons learnt 199

rise to recommendations for future research.

• When provided with tolerances on model parameters to be taken into ac-count during control law design, do not be comfortable in case analysis withrespect to these tolerances indicates robustness to spare (e.g. a µ value forrobust stability of 0.5). Modelling errors usually arise from other sources aswell, which may completely spoil achieved performance, or even stability.Always check worst-case margins to unspecified uncertainties, for exampleusing (multi-variable) gain and phase margins.

• Tolerances on model data are very easily specified. However, the probabilitythat all parameters are at the extreme end of their tolerances is very low. Forthis reason, just providing parameter tolerances is not sufficient: probabilityof parameter combinations should be kept in mind. This gives rise to arecommendation for future work in the next section.

• Robustness analysis by parameter gridding usually is the only practical ap-proach to investigate most simulation-based design criteria. Critical casesmay be easily missed (see Chapter 5). Use more reliable methods, like µ-analysis, in case the criterion can be translated into the method-specificform (e.g. an H∞-norm). This lesson also gives rise to a recommendation.

• Consider radical structural changes in control functions, if necessary. Some-times a control law structure just is not sufficient to meet the spectrumof performance and robustness requirements. The proposed design processfully supports a structure change, since the optimisation set-ups can befurther used.

• Thoroughly understand the control law structure. For example, know whatis happening with controller states in steady flight conditions. In the ATTASdesign a lack of an integrator in the lateral inner loops (Chapter 5) alloweda slight aircraft asymmetry to propagate into a static lateral flight pathdeviation. This was detected during first ground simulator tests. Also,methods like Nonlinear Dynamic Inversion are very straight forward, butmay result in surprises when applied without physical understanding (seeChapter 4).

• Most aerodynamicists do not speak the language of coefficient tables andapplication rules, as used most aircraft simulation models. In order to de-velop reliable models from computational aerodynamic methods, make suremutual understanding of model data and their interpretation exists.

• Do not underestimate turbulence. Extensively test control law behaviourat maximum turbulence levels, especially check control activity for rate andposition saturations.

• Use interactive desktop simulation and visualisation to validate control laws.Especially in case of manual control, this allows for basic analysis of flying

200 Conclusions

qualities, for visual verification of command variable decoupling, for qualita-tive analysis of behaviour of nonlinear functions, switches, protections, etc.In this way, problem areas that otherwise tend not to show up until pilotedflight simulator evaluation can be addressed early-on at the engineer’s workstation, saving considerable simulator time and cost [123].

• Do not use desktop simulation and visualisation to interactively design flightcontrol laws. As also stated in [111], qualitative analysis provides importantfeedback in the design process, but tuning should be based on relevantquantitative design criteria.

7.4 Recommendations for future researchIn Chapter 5 it was demonstrated that parametric linearisation of the aircraftdesign model is extremely useful for robustness analysis. It allowed for automaticgeneration of a Linear Fractional Transformation of the ATTAS model for robust-ness analysis. Since the LFT stemmed from the same object-oriented model as theone used for nonlinear simulation analysis, worst-cases found from the analysiscould be exactly verified. In the ATTAS case some luck was involved, since theequilibrium point did not depend on the parameters of interest. This was a differ-ent case for robustness analysis of autopilot designs for the Research Civil Aircraft(RCAM), detailed in Refs. [71, 131]. In these references a pragmatic approachwas followed to include influence of parameters on the trim condition. Combiningsymbolic trimming and symbolic linearisation is required to make methods likeµ-analysis a mature tool for use in the flight control design process. The effort toderive low-order LFTs will be high, but a generally applicable successful outcomewill be invaluable.An important aspect in control law design that has not been covered in this thesisis fault tolerance. Currently, this aspect is manually addressed in the controllerarchitecture, in order to anticipate failure scenarios that may be expected to occur(e.g. loss of air data). One approach to automatically address failures of actuatorsor damage to the airframe is to combine on-line identification techniques withNonlinear Dynamic Inversion [130, 100]. It is recommended that methodologiesare integrated in the design process that allow fault tolerance to be addressed asinherently as robustness in the synthesis of FCL parameters. This also impliesthat detailed system models including failure scenarios need to be integrated inthe aircraft flight dynamics model.An important (implicit) lesson from the design work in Chapters 5 and 6 is thatsimple bounds on uncertain model parameters, even though these are handled wellby robust design methods, may result in overly conservative, even insufficientlyperforming control laws. Worst-cases tend to be in corners of the parameter spacethat usually are very unlikely to occur in practice. In the case the designer hasto handle multiple uncertain parameters, information on probability or combinedvariations therefore should be an inherent part of the uncertainty specification.In this thesis, Monte Carlo analysis was performed in the parameter optimisation

7.4 Recommendations for future research 201

in order to address autoland performance under stochastically varying landingconditions. Uncertainty in the aircraft model was addressed in a deterministicway. Synthesis of control laws in face of stochastic uncertainty has been addressedby Schy and G iesy [117] and Wang and Stengel [134]. As a start, investigatingthe integration of these approaches into the proposed design process is stronglyrecommended.

202 Conclusions

Appendices

Appendix A

Model building for control lawdesign

206 Model building for control law design

Abstract

This appendix describes the preparation of (object-oriented) aircraft mod-els for automatic generation of various types of analysis models for usein the fl ight control law design process. A nalysis models for nonlinear(real time) simulation, linear analysis, L FT-based robustness analysis,and trimming are discussed.

C o n tributio n s

• A G U I-based procedure for defi ning trimming problems for nonlinearaircraft models.

A.1 Simulation models for design analysis 207

THE design of flight control laws requires the availability of models that ac-curately describe the aircraft flight dynamics and allow for evaluation of all

design criteria of interest.This appendix is concerned with the application of object-oriented aircraft modelsin the flight control law design process. These models are needed in various forms.For example, linear synthesis techniques require locally linearised state space mod-els around flight conditions that have been selected as design points. Modernrobustness analysis methods, like µ-analysis, require models in the form of a lin-ear fractional transformation (LFT), whereby uncertain parameters are explicitlypresent and pulled out of the model into a feedback structure. Optimisation-based design, as discussed in the introduction, can directly work on a nonlinearmodel, but as for the examples above, this model should provide all necessaryoutput to compute design criteria of interest and should allow for external accessto model parameters. Some control design techniques, like Nonlinear DynamicInversion (NDI), even directly incorporate model equations as an integral part ofthe control laws.In this appendix it will be shown how

• simulation models,

• inverse models for computation of equilibrium conditions, and

• linear (parametric) models

can be generated by appropriate specification of input, output, parameter, andstate vectors. Generation of inverse models for various control law structures isdescribed in Chapter 4 of this thesis.

A.1 Simulation models for design analysisMost design work as described in this thesis has been performed using Mat-lab/ Simulink [81]. In combination with the computational engine of Matlab thisblock diagram-based environment provides a useful tool for experimentation withand tuning of control laws. However, each block is limited to the form of acausal input-output structure, as illustrated in the feedback interconnection inFigure A.1. Consequently, for control design purposes the object-oriented aircraftmodel needs to be made available with interfaces as depicted.Throughout this appendix, the inputs to an aircraft model will be divided intothree groups:

1. Control inputs (uc(t) ∈ IR nuc): These are any inputs that are available tothe control law s;

2 . Noises (n(t) ∈ IR nn): These are inputs for (w hite) noise sig nals, e.g . forg eneration of rand om turbulence, sensor d isturbances, etc .;

3 . O th er in p u ts (uo(t) ∈ IR nu o ): These are any other inputs that are not usedby the control law s, lik e the horizontal stabiliser setting , or fl ap setting s.


Flight DynamicsM o d e l

yS im ( o u tp u ts f o r simu latio nanalysis)

yM e a s (me asu r e me ns fo rf e e d b ack )

p (co nstant mo d e l p arame te rs)

uo ( o the r inp u ts)

uc (co ntr o l inp u ts)

n (no ise s)

Flight C o ntr o lL aw s

P ilo t co mmand s /se ttings

Figure A.1: Feedback structure with aircraft model and flight control laws

N ote that the total number of inputs eq uals: nu = nuc + nuo + nn. Two types ofoutputs are distinguished:

1 . Measurements for feedback (yM e a s(t) ∈ IR nym): These are sensor signalsthat are available to the control laws;

2. Outputs for analy sis (yS im(t) ∈ IR nys ): These are outputs that provide ad-ditional signals useful for q ualitative and q uantitative analysis of the closed-loop system.

The total number of outputs eq uals: ny = nyc + nys. Finally, the model parame-ters p are set at initialisation and to not change during a simulation. C ommonlyused parameters are aircraft weight and balance (mass, C oG location), off sets onuncertain aerodynamic coeffi cients, etc.A fter model translation an algorithm results that computes state derivatives andoutputs in the form:

x(t) = f(x(t),uc(t),uo(t),n,p,t)yM e a s(t) = hm(x(t),uc(t),uo(t),n,p,t)yS im(t) = hs(x(t),uc(t),uo(t),n,p,t)

(A .1 )

In E x ample A .1 the translation process will be demonstrated for the A TTA Sex ample from S ection 2.5 .

Example A.1: Generating a simulation model for ATTAS

Figure A.2 once more shows the top level of the object-oriented aircraft model example

of AT T AS . T he connector at the top of the actual aircraft object provides external

A.1 Simulation models for design analysis 209

access to the databus as shown in Figure 2.10. This databus basically represents anavionics bus in a contemporary civil or military aircraft. In Figure A.2 two objectscalled Avionics and C ontrols have been added. This block is intended to create aninput / output interface of the model for implementation in a simulation program. Inthis example case, the external inputs U c (control inputs), U o (other inputs) and n

(noises) comply with the division as in Figure A.1. The same holds for the outputsy M e a s and y S im . N ote that the noise connects directly to the aircraft object and isdistributed internally over turbulence, sensor noise models, etc. The reason is thatit was found physically inappropriate to have noise passed on via the databus. TheAvionics object extracts the desired outputs from the databus, whereas the C ontrols

object sends the inputs (besides noise) to the databus.

Figure A.2: Specification of inputs and outputs at the hierarchical top level ofthe ATTAS model for implementation in the feedback structure in FigureA.1

After automatic translation an input-output model as depicted in Figure A.1 is ob-tained. The specifi c inputs, outputs and parameters are listed in detail in Tables 2.3 ,2.2, and 2.4 .


A.2 Interactive desktop simulationFor control law design simulation analysis is initially performed off-line in orderto quantitatively assess control law performance. H owever, interactive real timesimulation and visualisation can be a valuable tool in the off-line design phase aswell. W ith the help of instrument displays and visualisation of the aircraft andenvironment the engineer can “ fly” the control laws at his desktop. Especially incase of manual control, this allows for basic analysis of flying qualities, for visualverification of command variable decoupling, for qualitative analysis of behaviourof nonlinear functions, switches, protections, etc. In this way, problem areas thatotherwise tend not to show up until piloted flight simulator evaluation can beaddressed early-on at the engineer’s work station, saving considerable simulatortime and cost [123].In 19 9 7 Rasmussen brought the Aviator V isual D esign Simulator (AV D S) to mar-ket, providing open interfaces for integrating own (closed-loop) aircraft flight dy-namics models [10 3]. A more recent example is V isEngine, developed by AeroL absAG [1]. This tool allows for addition of stylistic visualisation elements, like vec-tors to indicate flight direction, loads, etc. As a special development for the D L RInstitute of Robotics and M echatronics, on-line visualisation of aircraft deforma-tion was integrated. Cheap alternatives are FlightGear [2] and X -P lane [3]. Thesepackages are primarily games, but allow a limited number of flight variables to bemodified from an external simulation tool, e.g. via U D P .The Flight D ynamics L ibrary provides a standard interface for interactive desktopsimulation tools like V isEngine, containing variables like W GS8 4 co-ordinates,attitude angles, flap and gear settings, etc.

A.3 Computation of initial conditionsFor numerical evaluation of control law performance the accurate determination ofinitial conditions before simulation is crucial. Simulation-based evaluation criteriaare usually based on specific manoeuvres, starting from a strictly specified initialflight state (e.g. straight and level flight, co-ordinated turn, 1.6 6 7 g pull-up).This allows performance at various controller or model parameter settings tobe compared consistently, which obviously is particularly important when usingoptimisation-based parameter tuning. Also in case the model is to be linearisedaround a flight condition it is important to make sure this is done around anaccurately defined equilibrium condition.Starting from equation A.1 the most trivial initial condition is specified by settingthe state derivatives to zero:

x0 = 0 = f(x0, uc0, uo0

, n0, p, t0)yMeas0

= hm(x0, uc0, uo0

, n0, p, t0)ySim0

= hs(x0, uc0, uo0

, n0, p, t0)(A.2)

where t0 is the initialisation time, x0 = x(t = t0), etc. In this particular casethe unknowns are x0, yMeas0

and ySim0, instead of x(t), yMeas(t) and ySim(t)

A.3 Computation of initial conditions 211

as in (A.1). Solving the above equations is also called trimming and may forexample be done using a Newton-Raphson-based algorithm (as for example inthe MINPACK library [87]). In practice, not all elements of x0 will be fixedat zero and some of the elements of x0 will be set. More importantly, specificoutput values in ySim0

and yMeas0are fixed at values that characterise the flight

condition, requiring appropriate entries in uc0and uo0

to be determined. A simpleprocedure to specify such equilibrium conditions will be discussed by means ofExample A.2 below.

From equation A.2 it can be seen that model initialisation may be done in twoways:

1. solve x0, yMeas0and ySim0

from (A.2) using a nonlinear equation solver oroptimisation algorithm;

2. specify x, yMeas and ySim in the object model as unknowns, set x = 0 andtranslate the model. This results in the equations:

x0 = f inv (x0, uc0, uo0

, n0, p, t0)yMeas0

= hm(x0, uc0, uo0

, n0, p, t0)ySim0

= hs(x0, uc0, uo0

, n0, p, t0)(A.3)

where in this case the inverse of f with respect to x0 results. Note that x

is no longer a state vector, but computed from x = x0 instead.

The first variant is more flexible, since the type of initial condition can be setby the user without having to newly translate the model. The second variantresults in model equations for a specific type of initial condition, but may beconsiderably more efficient since the unknowns are algebraically solved for. Theresult may even be exact when no internal variables are solved iteratively in thetranslated model.

A.3.1 Trim computation using the model ODE

In the case of flight dynamics, solving (A.2) usually does not make sense. Forexample, forcing time derivatives of the aircraft CoG position with respect to theinertial frame (Re = [xe, ye, ze]

T to be zero does not allow the aircraft to haveany velocity. Furthermore, equilibrium for given (sub-sets of) state values ratherthan given inputs uc, uo, n are of interest. Equation (A.2) however is a goodstarting point for a simple procedure to formulate a physically sensible trimmingproblem:

1. Make sure that all constraint variables that define the trim condition soughtas well as free variables that may be used to maintain this condition, arevisible and accessible through ySim,c, uo,c. n, and/ or p in (A.1). In caseof redundant controls, add co-ordination rules to the model as far as thesecannot be specified as constraints on uo,c;


2. After generation of a simulation model in the form of (A.1), configure thetrimming problem as in (A.2). In this situation the number of equationsand unknowns are namely balanced and equal nx + ny;

3. Identify and set constraints that apply to the equilibrium condition thatis sought. Some flight mechanical insight is required in order to select theminimum set of variables that characterises the equilibrium condition;

4. For each single additional constraint defined in step 2, release one variablethat was originally assumed fixed (i.e. in x0, u0, p0). Again, some modelinsight and experience may be required to release variables that make sensefor eventually solving the system of equations.

Starting from (A.2), the procedure aims to maintain the amount of nx + ny

variables to be solved from the fixed set of nx + ny equations. In the followingexample it will be shown how this procedure can be implemented in the form ofa graphical user interface.

Example A.2: Definition of a trimming problem for ATTAS

As an example, a trimming problem for the ATTAS model is defined. Objective is tofind an eq uilibrium in a level turn at given altitude, roll angle, and velocity.The model has the following fl ight dynamics states:

x = (ub, v b, w b, xe, ye, ze, pb, q b, r b, φ, θ , ψ )T

The outputs and inputs are listed in Tables 2.2 and 2.3 respectively. Actuator andengine models are left out for this example: the commanded control surface defl ectionseq ual the actual defl ection angles and the engine commands eq ual the actual relativefan shaft speed (relative to the maximum value). The trimming problem is formulatedin the four steps proposed in the main text:

Step 1:The trim condition for a co-ordinated level turn can be defined by:

• a constant velocity, e.g. Vcas = V0 = 80 m / s (obviously, only one out ofvelocities like Vtas, Vcas, Vg should be specified);

• a level fl ight path: γ = γ0 = 0 r a d ;

• a roll angle of, say φ = φ0 = 60 d e g = π / 3 r a d ;

• ... in a co-ordinated turn, i.e. ny = 0.

Additionally, the turn is to start:


• at zero heading: χ = χ0 = 0 rad;

• in an altitude of: h = h0 = 304 m;

The model inputs and outputs can be found in Tables 2.2 and 2.3: all variables neededare readily at hand. The aircraft further has two engines and it is desired that theseare set at the same thrust values. This can be achieved by using a single throttlecommand in the model that applies to both engines, or the thrust diff erence betweenboth engines, made available as an additional output, may be required to be zero.Since the thrust diff erence is available (output ySim(4)), the latter variant is used.

Step 2:E quation A.2 is represented in the following table, containing the states and only aselection of the inputs and outputs that are relevant for the trimming problem andfor evaluating the final result (the remaining inputs are assumed to be fixed at zero):

value c x value c x value c y value c u

– ub 0 x ub – Vcas 0 x δA– vb 0 x vb – γ 0 x δE– wb 0 x wb – χ 0 x δR– xe 0 x xe – nx 0 x δT– ye 0 x ye – ny 0 x N11

– ze 0 x ze – nz 0 x N12

– pb 0 x pb – β– qb 0 x qb – dF– rb 0 x rb– φ 0 x φ

– θ 0 x θ

– ψ 0 x ψ# unknowns - # equations: 0

For each variable that is known, an “ x” -marker is set in the constraint column (C ).D efault values for the state derivatives are zero, as in (A.2).

Step 3 :In the first place, the rates of the position co-ordinates are allowed to vary freely,whereas the initial position is fixed at the desired location. In this case this is at analtitude h0.



– ub 0 x ub – Vcas 0 x δA– vb 0 x vb – γ 0 x δE– wb 0 x wb – χ 0 x δR0 x xe 0 - xe – nx 0 x δT0 x ye 0 - ye – ny 0 x N11

-304 x ze 0 - ze – nz 0 x N12

– pb 0 x pb – β– qb 0 x qb – dF– rb 0 x rb– φ 0 x φ

– θ 0 x θ

– ψ 0 x ψ# unknowns - # equations: 0

Now the constraints as formulated in step 1 may be applied:


8 0 ub 0 x ub 8 0 x Vcas 0 x δA– vb 0 x vb 0 x γ 0 x δE– wb 0 x wb 0 x χ 0 x δR0 x xe 0 xe – nx 0 x δT0 x ye 0 ye 0 x ny 0 x N11

-304 x ze 0 ze – nz 0 x N12

– pb 0 x pb – β– qb 0 x qb 0 x dF– rb 0 x rbπ/3 x φ 0 x φ

– θ 0 x θ

– ψ 0 x ψ# unknowns - # equations: -6

Note that the initial value of ub has been set to V0 as well: this provides a goodstarting value and prevents the first model evaluation during trimming to start atzero velocity, causing singularity problems in most aerodynamic models.

Step 4 :Note that in the previous table six additional constraint variables have been specified.Six other variables now have to be released. In order to maintain speed and flightpath angle, thrust settings (via N11 and N12) and tail plane incidence angle (δT ) aremost obviously used. L ateral load factor ny and roll angle φ are primarily maintained

with the help of ailerons δA and rudder δR. Of course, yaw rate ψ can not be zero inturning flight, which is released for the initial track angle χ. The following settingsnow result:



80 ub 0 x ub 80 x Vcas 0 - δA– vb 0 x vb 0 x γ 0 x δE– wb 0 x wb 0 x χ 0 - δR0 x xe 0 xe – nx 0 - δT0 x ye 0 ye 0 x ny 40 - N11

-304 x ze 0 ze – nz 40 - N12

– pb 0 x pb – β– qb 0 x qb 0 x dF– rb 0 x rbπ/3 x φ 0 x φ

– θ 0 x θ

– ψ 0 - ψ# unknowns - # equations: 0

U sually, appropriate starting values for inputs and states may have to be found. Thisagain requires some experience. In this case for example, N1 is set to 40 % , sincethrust may be a considerably nonlinear function of the fan shaft speed around idle.Furthermore, the trimmed solution is very unlikely to be found at zero thrust.

The described procedure is very easily realised in a G raphical U ser Interface (G U I),as for example depicted in Figure A.3. Finally, after solving the nonlinear system ofequations, the following values result:


8 0.3 8 ub 0 x ub 8 0 x Vc a s 1 .9 1 e-2 δA

-0.1 8 vb 0 x vb 8 1 .1 7 x γ 0 x δE

1 1 .1 3 wb 0 x wb 0 x χ -4 .5 0e-2 δR

0 x xe 8 1 .1 7 xe -2 .7 7 e-1 nx -5 .8 2 e-2 δT

0 x ye 0 ye 0 x ny 8 9 .0 N11

-3 04 .0 x ze 0 ze 1 .9 7 nz 8 9 .0 N12

-1 .4 1 e-2 pb 0 x pb -2 .2 2 e-3 β1 .7 9 e-1 qb 0 x qb 0 x d F

1 .03 e-1 rb 0 x rb

π / 3 x φ 0 x φ

6 .8 3 e-2 θ 0 x θ

1 .2 2 e-2 ψ 2 .07 e-1 ψ

# un k n o w n s - # eq uatio n s: 0

Note the the total load factor√

(n2

x+ n

2

z) = 1 .9 9 , i.e. dou b le the n orm al g rav ity for

w hich 6 0 deg co-ordin ated tu rn s are w ell k n ow n . A lso n ote that the side slip an g le is

n on zero: althou g h the aircraft m odel is sy m m etric, the n on zero b ody an g u lar rate rb

cau ses a sm all side force that is com p en sated v ia side slip an g le an d ru dder defl ection .

A ll zero variab les are zero u p to m achin e accu racy.


Figure A.3: Example GUI for defining trim conditions for ATTAS. Activating/deactivating

th e ch eck b ox b etw een th e variab le name and valu e (C -colu mn) sets/releases a constraint vari-

ab le. Th e in-b alance b etw een u nk now ns and th e nu mb er of eq u ations is indicated b y th e nu mb er

in th e top-left corner (zero, in th is case). Th e “ Trim” b u tton on top actu ally ru ns th e trim al-

gorith m (nonlinear eq u ation solver). Th e “ L in.” performs a linearisation arou nd th e trimmed

fl igh t condition. Th e “ Script” b u tton au tomatically generates a script for trimming and lin-

earising th e aircraft model. H ereb y th e settings in th e GUI are applied in a w ay th at is easily

adapted b y th e u ser. F inally , in case of a constrained variab le in th e GUI, an additional rou nd

ch eck -b ox appears in th e “ M -colu mn” . Activating th is b ox cau ses th e generated script to per-

form trim compu tations in a loop for valu es of th e selected variab les. Th is graph ical interface is

au tomatically set u p u sing M atlab fu nctions b ased on data provided w ith a translated M odelica

model.


A.3.2 Trim computation by model inversion

The same symbolic algorithms that are used for generating the model ODE canof course just as w ell be ap p lied to directly solv e the set of model eq uations forthe free v ariables from the trimming p roblem. A gain, this may be done in fi v esimp le step s, starting from an object-oriented model imp lementation as discussedin C hap ter 2 :

1. M ak e sure that the v ariables (ex cep t for states and state deriv ativ es) thatcharacterise the eq uilibrium condition are av ailable as model outp uts, andthat controls av ailable to maintain this condition are av ailable as inp uts. A tthis stage, the model w ould still translate into an ODE;

2 . “ Decoup le” state deriv ativ es from the corresp onding states and set them tozero. A s in case of S ection A .3 .1, the confi guration as in eq uation A .2 hasbeen achiev ed;

3 . N ow fi x v ariables that characterise the eq uilibrium condition to their ap -p rop riate v alues. This is best done by defi ning those as model inp uts orp arameters, allow ing them to be set for diff erent fl ight conditions after-w ards;

4 . F or each new ly fi x ed v ariable release one that w as p rev iously fi x ed. This isbest done by defi ning it as an outp ut, so that after translation the v alue isk now n outside the model.

5 . Translate the model.

N ote that this p rocedure is similar to the one p rop osed in S ection A .3 .1.S ince the model does not contain states anymore, only algebraic eq uations in theform of an comp uter algorithm result, w ith desired trim state v alues as inp uts andreq uired eq uilibrium variables for states and control inp uts as outp ut arguments.S ince the symbolic translation algorithm w ill try to reduce the dimensions ofsystems of eq uations to be solv ed internally to a minimum (e.g. using the tearingalgorithm as sk etched in Ex amp le A .1), the trim comp utation w ill be considerablyfaster as comp ared w ith solv ing the ODE as one big system of eq uations. In thecase of A TTA S the diff erence is in the order of magnitude 5 to 10 (during multip lecalls the time lap sed for the inv erse model v ersion w as mostly too small to bemeasured w ith the C P U clock ).

Example A.3: Model inversion for trim computation for ATTAS

Figure A.4 once more shows the top level hierarchy of the ATTAS model. As in

E x ample A.1 the inputs and outputs are inserted into and ex tracted from the aircraft

object with the help of an Avionics block. This version is confi gured to have the


inputs and outputs as in Example A.2. The known outputs are collected in Ytrim, the

unknown ones in Ya . The unknown inputs are in U trim, the known ones in U o . The

O I and IO objects now revert the Ytrim outputs into inputs and U trim inputs into

outputs respectively. The re-declaration of states and state derivatives is done via

modifier statements added to the A irc ra ft object. This is done at the depicted level,

so that this object is not changed by itself. The known states and state derivatives

may be adjusted from outside with the help of model parameters.

Figure A.4: Specification of inputs and outputs at the hierarchical top level ofthe ATTAS model for trim computation

A.4 Linearisation of the aircraft modelMost controller synthesis methods are linear by nature and require the aircraftmodel to be available in linear form. L inearisation may be performed by derivingthe J acobians of f(...) and h(...) in (A.1) at given initial conditions, resulting inthe well-known state space A, B, C, and D matrices:

δx = Aδx + Bδu (A.4)

δy = Cδx + Dδu (A.5)

(A.6 )

A.5 Parametric models for robustness analysis 219

where δ implies a small deviation from initial values, e.g. δx = x − x0. And:

A = ∂f(x,u,p,t)∂xT

∣

∣

∣

x=x0,u=u0,p=p0,t=t0B = ∂f(x,u,p,t)

∂uT

∣

∣

∣

x=x0,u=u0,p=p0,t=t0

C = ∂h (x,u,p,t)∂xT

∣

∣

∣

x=x0,u=u0,p=p0,t=t0D = ∂h (x,u,p,t)

∂uT

∣

∣

∣

x=x0,u=u0,p=p0,t=t0

(A.7 )

Most simulation platforms perform linearisation of (A.1) instead. For example,the A-matrix is approximated by [15]:

Aj =f(x0 + ∆xj , u0, p0, t0) − f(x0 − ∆xj , u0, p0, t0)

2δxj

(A.8 )

where j is the column index and ∆xj = [0, · · · , 0, δxj , 0, · · · , 0]T .W ith the help of the algorithms as used for translation of object-oriented models,linearisation of the latter may, under some circumstances, be performed symboli-cally. To this end, it is required that (A.1) can be symbolically solved from full setof model equations. An interesting application is briefly presented in the followingsection.

A.5 Parametric models for robustness analysisOver the last decades control law design and analysis methods have been devel-oped that are able to explicitly take into account uncertainty in model parameters.Most of these methods require the model to be cast in the form of a so-called Lin-ear Fractional Transformation (LFT). An LFT can best be interpreted pictorially.In Figure A.5 the uncertain parameters have been separated from the model M

and placed on the diagonal of the matrix ∆. This matrix interacts with M in afeedback way, allowing the application of dedicated analysis algorithms. A modelin the form of an LFT is obtained in two steps:

1. Obtain a symbolic linear state space model in the form:

δx = A(p)δx + B(p)δu

δy = C(p)δx + D(p)δu (A.9)

(A.10)

where p is the vector with model parameters;

2. Transform (A.10) into an LFT, either manually, or with the help of dedicatedsymbolic tools [44, 63]. This transformation is possible, provided that theelements of the state space model depend on the elements of p in a rationaland polynomial way [27 ].


M

u

z w

y

Figure A.5: An (upper) linear fractional transformation (LFT)

Since for the second step sophisticated algorithms are available that allow forgeneration of LFTs of low order, the first step is usually considerably more com-plicated, especially when the model is only available in the form (A.1). It is thenusually required to go back to the model equations and derive the elements of thestate space matrices by hand, see for example R ef. [39]. As already mentionedin the previous section, object-oriented modelling allows the nonlinear ordinarydifferential equations to be derived symbolically. As a result, linearisation maybe performed symbolically as well, see eq. (A.7). B y skipping the substitutionof p0 for p, the resulting state space model explicitly depends on this vector asin (A.10) [132]. An important problem is that p may influence the equilibriumcondition of the system. For example, if the aircraft mass is a model parameter,in order to trim the aircraft in straight and level flight at a given airspeed VTAS0

,the required angle of attack to maintain vertical force equilibrium will increaseas a higher mass is selected. As a consequence, the vectors x0 and u0 need tobe determined as a function of p as well. Unfortunately, even for aircraft thismay result in too a complicated model for transformation into an LFT. For thisreason, two alternatives are at hand:

1. Substitution of nominally values p0 in computation of x0 and u0 (see Sec-tion A.3) and substituting these when symbolically linearising the model.The resulting LFT is only valid for small parameter offsets from p0. There-fore, “local” LFTs may be derived at a grid of operating points [132];

2. Identify the elements of (A.10) that depend on the equilibrium condition,isolate the dependent terms with the help of text-book approximations as forexample provided in appendix A of [15]. Then find a polynomial dependencyon p by performing multiple linearisations over a grid of flight conditions.This procedure is followed in [132] and described in more detail in [131] and[71].

Appendix B

Equations of motion of afl exible aircraft

222 Equations of motion of a fl exible aircraft

Abstract

In this appendix the equations of motion of a flexible aircraft are derived.It is assumed that the airframe may be represented by a collection of elas-tically interconnected grid points w ith lumped masses rig idly attached tothem. A s a fi rst step, basics of dynamic analy sis of such a sy stem arereview ed. T he derivation of the equations of motion w ill be based on L a-grangian mechanics. Inertial coupling betw een over-all motion and elas-tic deformation is minimised by the choice of mean axes as body referenceframe. R emaining coupling terms may be neg lected after making a numberof explicit assumptions. C onsequently , interaction betw een deformationand over-all motion are only induced by external (aerodynamic) loads.

B.1 Review of structural dynamics 223

PURPOSE of this appendix is the derivation of the nonlinear equations ofmotion of a deformable aircraft . Linear structural dynamics models as used

for flutter and loads analysis assume that the airframe may be considered as acollection of lumped masses and inertias attached to grid points that are inter-connected elastically. After determining eigenfrequencies, the linear equations ofmotion of these models are obtained in modal form. A number of properties ofthese modal equations of motion leads to significant simplifications in the over-allnonlinear equations of motion of the flexible aircraft. Therefore, basic principlesof structural dynamics will be reviewed first. Then the equations of motion willbe derived using Lagrangian dynamics.

B.1 Review of structural dynamicsThe structural equations of motion of a lumped-mass system in physical co-ordinates of condensed grid points are given by (see for example [101]):

Mggxg + Kggxg = Fg (B.1)

The subscript g refers to a collection of structural grid of points with one or morelumped masses attached to them. For now, it is assumed that each grid point hassix degrees of freedom, with respect to some local reference system. The degreesof freedom of all grid points are contained in the (quite large) vector xg:

xg = [. . . , δxj−1, δyj−1, δz j−1, δϕ j−1, δθ j−1, δψ j−1,

δxj , δyj , δz j , δϕ j , δθ j , δψ j , . . .]T

where δx, δy, δz indicate translational degrees of freedom, δϕ , δθ , δψ are rota-tional degrees of freedom, and j is an index: j = 2, 3, · · · , ng. The total numberof grid points is thus ng, the total length of xg is nd o f = 6ng.The matrices Mgg and Kgg are (symmetric, band-diagonal) mass and stiffnessmatrices respectively. The vector Fg describes external forces and moments actingon the grid points in the directions of the degrees of freedom as described by xg.The mass matrix not only includes mass and mass distribution of the airframestructure, but also of payload, fuel and other consumables. Mass distribution isfurther affected by the aircraft configuration, like gear and flap positions. As aconsequence, it is important to keep in mind that the matrix Mgg, all equations itoccurs in and results it affects, are only valid for the specific aircraft configurationand loading.

B.1.1 Structural eigenvalue problem

Substitution of xg = xg0ejω t into (B.1) and setting Fg = 0 gives rise to the

following eigenvalue problem (modal analysis):

Kgxg0= ω2Mgxg0

(B.2)


where ω is an eigenfrequency. Solution of this eigenvalue problem results in the(squared) eigenfrequencies ω1, ω2, · · · , ωndof

and eigenvectors Φgh1, Φgh2

, · · · ,Φghndof

. These eigenvectors represent physical translation and rotation of the grid

points associated with each eigenfrequency and are referred to as the mode shapesof the structure. Usually, a considerably lower number of eigenvalues / modes isactually computed than the total number of degrees of freedom: nh < < ndof .

In case none of the grid points has been constrained (as it is assumed here),the first six eigenvalues are zero and the associated eigenvectors involve rigidtranslation and rotation (i.e. without deformation) with respect to the referenceaxes. These eigenvectors will be referred to as ΦgR1

, ΦgR2, · · · , ΦgR6

and are theso-called rigid-body modes of the structure. The elastic modes will be referredto as ΦgE1 , ΦgE2 , · · · , ΦgEnh−6 . Besides a fixed rotation (or directional signs)and origin of axes, the rigid body modes correspond to the degrees of freedom indirection of, or around aircraft body axes in flight mechanics models.

It is noted once more that the results of the modal analysis must always be ac-companied with the aircraft weight and loading configuration. Each new loading,in principle, requires a new modal analysis.

B.1.2 The half-generalised equations of motion

As a next step, the degrees of freedom in xg are written as a linear combinationof the obtained mode shapes. For now it is assumed that all modes are available(i.e. nh = ndof ):

xg = Φghη (B.3)

=[

ΦgR1, ΦgR2

, · · · , ΦgR6, ΦgE1

, ΦgE2, · · · , ΦgE(nh−6)

]

η

= [ΦgR, ΦgE ]

[

ηR

ηE

]

where η is a vector of mode shape multipliers, referred to as generalised co-ordinates, and Φgh is the modal matrix. Note that the mode shapes and gen-eralised co-ordinates have been divided into rigid (subscript R) and flexible (sub-script E) ones. Substitution into equation (B.1) gives:

MggΦghη +KggΦghη = Mghη +Kghη = Fg (B.4)

At this point structural damping may be introduced artificially, e.g. by assigninga damping ratio to each mode:

Mghη +Bghη +Kghη = Fg (B.5)

The matrices with subscript gh are also called “half-generalised”, since the rowsrepresent physical degrees of freedom, the columns apply to generalised co-ordinates.


The half-generalised mass, damping and stiffness matrices can be partitionedaccording to the mode shapes (B.3):

[MgR, MgE ]

[

ηR

ηE

]

+ [BgR, BgE ]

[

ηR

ηE

]

+ [KgR, KgE ]

[

ηR

ηE

]

= Fg (B.6)

where R and E refer to rigid and elastic modal degrees of freedom respectively.The matrices KgR and BgR will be zero zero, since rigid motion of the structuredoes not induce internal stresses.

B.1.3 The generalised equations of motionFinally, equation (B.5) is left-multiplied with ΦT

gh:

ΦTgh (Mghη +Bghη +Kghη) = ΦT

ghFg ⇐⇒

Mhhη +Bhhη +Khhη = ΦTghFg

This is the generalised equation of motion, providing a second order differentialequation that may be used to solve for η. As in the half-generalised case thematrices may be partitioned into rigid, flexible and coupled sub-matrices. Leftmultiplication of (B.6) with ΦT

gh = [ΦgR, ΦgE ]T gives:[

MRR 00 MEE

] [

ηR

ηE

]

+

[

0 00 BEE

] [

ηR

ηE

]

+

[

0 00 KEE

] [

ηR

ηE

]

= [ΦgR, ΦgE ]TFg (B.7)

H ereby use has been made of the fact that KgR = BgR = 0 as well as the factthat mode shape eigenvectors are orthogonal with respect to the mass matrix:

ΦTghiMggΦghj

=

Mhhiif j = i

0 if j 6= i(B.8)

see for example [101] for a proof.The short-hand form of (B.7) and with aerodynamic loads applied to the right-hand side is known as the flutter equation:

Mhhη +Bhhη +Khhη = qQhha e r oη (B.9)

where q is the dynamic pressure and Qhha e r oare generalised aerodynamic loads,

its ith column representing generalised forces induced by mode shape φi. Theflutter equation is the basic model representation in aeroelasticity, serving as astarting point for loads and flutter analyses.The generalised mass matrix Mhh, the stiffness matrix Khh (due to B.2), and thegeneralised damping matrix Bhh (as a function of Khh and Mhh) are diagonal, sothat the differential equations are decoupled. For the elastic degrees of freedom:

MEEiηi +BEEi

ηi +KEEiηi =

MEEi(ηi + 2ζiωiηi + ω2

i ηi) = ΦTgEi

Fg


where i < nh refers to the mode i. Here BEE is assumed to have been selectedsuch that in the second order differential equations the damping number of modei is ζi.As already mentioned, in most engineering applications the number of modes thatis actually computed is considerably lower than the total number of degrees offreedom: nh << ndof . Since ndof may be in the order of 104, massive computationtime is saved by applying the modal transformation xg ≈ Φghη in the equations ofmotion, whereby in the case of aircraft nh is usually in the order of magnitude 102.Accuracy is suffi ciently preserved in case the nth

h eigenfrequency is well beyondthe frequency range of interest in analysis [96]. In case Fg does not depend on η,truncation of modes beyond the frequency range of interest still results in exactmodal response of the retained modes of vibration, otherwise, the ≈-sign is to beused in the modal equations.Due to mutual orthogonality rigid body modes usually do not represent puretranslation along, or rotations about, the reference system that is used for modalanalysis, but linear combinations thereof. These mode shapes are therefore re-placed with purely translational and rotational ones in the direction of respectivelyabout the mean body axes, which will be defined more precisely later on. As aresult, MRR will mostly no longer be diagonal [113]. Fortunately, orthogonalitybetween rigid and flexible modes is preserved, since the new rigid modes can bewritten as a linear combination of the original ones. This will be of great impor-tance in developing nonlinear equations of motion for the unrestrained aircraft.The mode shapes Φghi

may be scaled with a constant factor. This is frequentlyused to normalise the generalised mass matrix. Here it is assumed that the cor-rected rigid body modes represent unit translations and rotations, so that MRR

contains total mass and moments of inertia with respect to the mean body axes.

B.1.4 Recovery of physical degrees of freedomFinally, after solving the structural equations of motion for η, the physical degreesof freedom can be recovered using:

xg ≈ Φghη (B.10)

the ≈-sign has been introduced due to truncation of higher-order modes (nh <<

ndof ) and since modal response is not exact in case Fg depends on xg. The firstand second time derivatives give deformation rate and acceleration respectively atthe structural grid points. For loads analysis the internal stresses in the structureare of particular interest. Having solved (B.7) for η, equation (B.1) may be usedto determine resultant loads Lg acting on the lumped mass elements:

Lg = Kggxg ≈ Kghη (B.11)

Since Lg is computed directly from modal displacement, this method of computa-tion is called “mode displacement” method [14]. The method requires relativelylittle computational effort, but is notorious for slow convergence as a function of


the number of modes nh. As a consequence, a large number needs to be consid-ered in order to obtain sufficiently accurate results. Alternatively, the so-calledforce summation method may be used:

Lg ≈ Fg −Mghη −Bghη (B.12)

where loads are computed from summation of external, mass and damping forces.In case η = η = 0 then Lg = Fg. This implies that neglected modes are at leaststatically taken into account [14, 101]. In order to compute total loads at stationsof interest on the airframe (e.g. wing root, fin root, aft fuselage, etc.), the loadsacting on the involved grid points are summed and interpolated.Combining the equations (B.11) and (B.12), a more accurate estimation of xg

may be obtained:

xg ≈ K−1gg Lg

This is known as the mode acceleration method [14, 101].

B.1.5 Orthogonality of modesAn important characteristic of free vibration modes resulting from modal analysiswithout constraints is that flexible modes are orthogonal to rigid modes and toeach other, with respect to the mass matrix Mgg (equation B.8). Now supposethat a linear combination of rigid modes results in the following rigid motion atthe grid points i:

[d0]g =

d0 + φ0 × r1φ0

d0 + φ0 × r2φ0

...d0 + φ0 × rng

φ0

(B.13)

note that the only difference per grid point is ri, its undeformed location in theairframe body axes. Also, ng is the number of grid points. Remember that eachgrid point is assumed to have six degrees of freedom, so that: ndof = 6ng. Alsosuppose that a linear combination of elastic modes results in the following flexibledeflections at all grid points (collected in a single vector):

[δd]g =

d1

δφ1

d2

δφ2

...dng

δφng

(B.14)


Orthogonality of all flexible modes with respect to rigid ones now requires that:

[d0]Tg Mgg[δd]g = 0 (B.15)

In case the centre of gravity of the lumped mass does not match the location ofthe associated grid point, the total mass matrix Mgg (B.1) is block diagonal. Byderiving the equations of motion of lumped mass i with respect to grid point i,the contribution of each grid point consists of the 6× 6 block Mggi

(see also [97],page 91):

Mggi=

[

miI3 −mis k ew (li)mis k ew (li) Ii

]

(B.16)

where Ii is the inertia tensor, li the vector from the grid point to the centre ofgravity (body axes), and mi the mass of lumped mass i.

Equation B.15 may then be rewritten as follows:

∑

i

[d0]Ti Mggi

[δd]i = 0 (B.17)

⇐⇒∑

i

[

d0 + φ0 × riφ0

]T [

miI3 −mis k ew (li)mis k ew (li) Ii

] [

di

δφi

]

= 0

⇐⇒ dT0

∑

i

mi(di + δφi × li) +

φT0

∑

i

[miri × (di + δφi × li) +mili × di + Iiδφi] = 0

Since φ0 (rigid rotation) and d0 (rigid translation) are independent, orthogonalityimplies that the individual terms are zero:

∑

i

mi(di + δφi × li) = 0 (B.18)

∑

i

[miri × (di + δφi × li) +mili × di + Iiδφi] = 0 (B.19)

Note that di and δφi arise from a linear combination of mode shapes and arenot independent. The two equations above will be extremely useful in derivingnonlinear equations of motion for the unrestrained aircraft.

Finally, the rigid body modes are considered:

ΦgiR =

[

I3 −s k ew (ri)O3 I3

]

(B.20)

where −s k ew (ri) arises from (B.14). The first three modes are translational (inmean axes x, y, z-direction), the latter three are rotational (around the mean x,

B.2 The equations of motion of a fl exible aircraft 229

y, z-axes). Left and right multiplication with the mass matrix gives:

∑

i

[

I3 −skew(ri)O3 I3

]T [

miI3 −miskew(li)miskew(li) Ii

] [

I3 −s k e w (ri)O3 I3

]

=

[

mI3 −

∑

imis k e w (ri + li)

∑

imis k e w (ri + li)

∑

i(Ii − mi[s k e w (ri + li)

2− s k e w (li)

2])

]

B.2 The equations of motion of a fl exible aircraft

F ro m a p u re ly m e ch a n ic a l p o in t o f v ie w , th e e q u a tio n s o f m o tio n fo r a fl e x ib lea irc ra ft in v o lv e d e sc rib in g th e d y n a m ic s o f a d e fo rm a b le b o d y th a t a rb itra rilytra n sla te s a n d ro ta te s w ith re sp e c t to a n in e rtia l re fe re n c e fra m e . A c o n sid e ra b lea m o u n t o f re fe re n c e s is a v a ila b le a d d re ssin g th e d e riv a tio n o f th e se e q u a tio n s. A l-re a d y in o n e o f th e fi rst sta n d a rd b o o k s o n a e ro e la stic ity , B isp lin g h o ff a n d A sh le ya d d re ss th e d y n a m ic s o f th e ’u n re stra in e d v e h ic le ’ [1 4 ]. M iln e [8 6 ] d e riv e d th ee q u a tio n s fro m fi rst p rin c ip le s a n d u se d th e se to p e rfo rm in g e q u ilib riu m a n d sta -b ility a n a ly sis fo r th e d e fo rm a b le a irc ra ft. A lso E tk in in [3 5 ] a d d re sse s fl e x ib lea irc ra ft fl ig h t d y n a m ic s. W a sz a k a n d S ch m id t [1 3 6 , 1 3 7 ] d e riv e th e e q u a tio n s o fm o tio n fo r th e fl ig h t d y n a m ic s o f ’a e ro e la stic v e h ic le s’ u sin g L a g ra n g e ’s e q u a tio n s.T h e n ic e a sp e c t o f th is p a p e r is th a t a ll sim p lifi c a tio n s a n d a ssu m p tio n s a re e x -p la in e d ste p b y ste p in a c le a r a n d c o n siste n t w a y . A lso a n u m e ric a l e x a m p le isp ro v id e d , w h ich p ro b a b ly is th e o n ly in te g ra te d fl ig h t d y n a m ic s a n d a e ro e la stica irc ra ft m o d e l p u b lic ly a v a ila b le .

T h e re fe re n c e s liste d a b o v e h a v e a n im p o rta n t d ra w b a ck . T h e e q u a tio n s d e riv e da re v a lid fo r c o n tin u o u s e la stic b o d ie s. H o w e v e r, stru c tu ra l d y n a m ic s m o d e ls fo ra irc ra ft a re u su a lly o b ta in e d fro m F in ite E le m e n t m o d e ls in th e fo rm o f rig idlu m p e d m a sse s a n d in e rtia s a tta ch e d to g rid p o in ts th a t a re in te rc o n n e c te d b ye la stic e le m e n ts (S e c tio n B .1 ). T h e e q u a tio n s o f m o tio n fo r u n re stra in e d lu m p e dm a ss sy ste m s h a v e b e e n d e riv e d b y B u ttrill e t a l. [2 1 ]. U n lik e e a rlie r w o rk b ya m o n g o th e rs C a v in [2 4 ] a n d C e rra [2 5 ], th is re fe re n c e a d d re sse s in e rtia l c o u p lin gb e tw e e n fl e x ib le rig id m o tio n a n d in te n tio n a lly re ta in s sm a ll se c o n d -o rd e r e ff e c ts.B u ttrill in tu rn re fe rs to w o rk b y M o rin o [9 4 ]. Z e ile r e t a l. [1 4 3 ] a lso a d d re ssn o n lin e a r e la stic ity e ff e c ts. A n o th e r d e riv a tio n c a n b e fo u n d in th e (slig h tly in -a c c e ssib le ) re fe re n c e b y Y o u sse f [1 4 2 ]. T h e w o rk o f B u ttrill w a s d e v e lo p e d w ith inth e N A S A L a n g le y F IT te a m (F u n c tio n a l In te g ra tio n T e ch n o lo g y ). T h is te a mh a d b e e n fo rm e d in 1 9 8 5 in re sp o n se to a g o a l se t b y N A S A L a n g le y R e se a rchC e n te r (L a R C ) to “ D e v e lo p m u lti-d isc ip lin a ry In te g ra tio n M e th o d s to Im p ro v eA e ro sp a c e S y ste m s” . T h is te a m g e n e ra te d a n im p re ssiv e a m o u n t o f k n o w le d g e o nm u lti-d isc ip lin a ry a irc ra ft m o d e l in te g ra tio n [8 , 1 4 3 , 1 8 , 2 0 , 1 2 8 ]. A lso in sp a c e -c ra ft lite ra tu re e q u a tio n s o f m o tio n o f u n re stra in e d lu m p e d -m a ss sy ste m s h a v eb e e n d e riv e d , se e fo r e x a m p le [6 9 ]. B e y o n d a e ro sp a c e a p p lic a tio n s, S ch w e rta sse k(a n d m a n y o th e rs) a d d re sse s fl e x ib le b o d ie s in m u lti-b o d y sy ste m s, se e fo r e x a m -p le [1 1 6 , 1 1 5 ].


A complete derivation of equations of motion of lumped mass systems, compatibleto Finite Element models is very lengthy, especially due to inertia tensors of masselements, and the fact that grid points of the FE model not necessarily match thecentres of gravity of lumped masses. Buttrill for this reason omits the rotationaldegrees of freedom from his published derivation. He does not explicitly foreseean offset between a grid point in the FE model and the corresponding lumpedmass centre of gravity. Hanel does address this offset as well as rotational inertiaof lumped mass elements [43]. However, the derivation is not pulled throughcompletely and simplifications and linearisation are performed early on in theprocess.

For the development of integrated equations of motion for the airframe as to beimplemented in the object-oriented model structure in Figure 2.5, the followingis required:

1. differential equations for dynamics of the unrestrained flexible airframe ofappropriate complexity;

2. any simplification must be associated with one or more explicit assumptions;

3. aiming at integration of available model components, availability of modalanalysis results should be exploited.

The derivation is performed in the following subsections.

B.2.1 Approach for derivation

For derivation of the equations of motion for the unrestrained flexible airframeLagrange’s equations will be used. This in the first place requires formulation ofkinetic and potential energy terms.

As already indicated, the following basic assumption is made:

• Assumption 1: The aircraft airframe may be described as a collection oflumped rigid mass elements with an associated mass mi and moment ofinertia Ii.

Such a lumped-mass model is manually and/ or automatically generated froman FE-model: local masses are condensed in assigned grid points (nodes) withspecified degrees of freedom [144]. Figure B.1 shows the position and orientationof the airframe reference system b with respect to an (assumed) inertial system e.

In addition, throughout the derivation the following assumptions are made:

• Assumption 2 : Translational and rotational deformations with respect tothe reference shape of the airframe are small;

• Assumption 3 : Linear elastic theory applies.


xb

yb

zb

xe

ye

ze

i

inertial reference frame

Re

0

Re

i

pi

d = lo c a l defo r m a tio nr = u n defo r m ed lo c a tio np = r + d

i

i i i

i

C o G

ub

vb

wb

pb

qb

rb

s i

dVdi

i

Figure B.1: Flexible aircraft in free flight

This reference shape may either be the jig shape or the flight shape in some trimcondition. Small rotational deformations imply that orientation of a lumped masselement may be described using a vector product rather than a cosine matrix .An infinitesimal mass element dm within the lumped mass i is considered, seeFigure B.1. The inertial position of this element with respect to the body axesframe is given by:

Rδi = Rb + ri + di + (I + skew(δ φ i))sδi (B.21)

where:

• Rb: the location of the origin of the body reference frame;

• ri: the location of a reference point on the mass element i with respect tothe body axes. This reference point corresponds with a node or grid pointin the condensed FE model. The position vector may represent the locationwithin the undeformed airframe (i.e. jig shape), or within the staticallydeformed airframe in a given trim condition (i.e. flight shape);


• di: is the (dynamic) translation of element i due to deformation of theairframe;

• δφi is the (dynamic) rotation of element i due to deformation of the airframe;

• sδi is the position vector to the mass element, measured with respect tothe point ri, in a local reference frame that rotates with the lumped masselement. For convenience, it is assumed that this reference system is parallelto the body axes in jig or flight shape condition;

• (I + skew(δφi)) is a linearised version (assumption 2!) of the cosine matrixrotation the vector sδi from the element reference frame into body axes.The term skew(δφi) is a matrix notation for the vector product of δφi withanother vector, in this case sδi.

The velocity of the mass element is given by:

Rδi = Vb + (di + δφi × sδi)︸︷︷︸

˙di

+Ωb × (ri + sδi︸︷︷︸

ri

+ (di + δφi × sδi)︸︷︷︸

di

) (B.22)

where:

• Vb: the inertial velocity of the origin of the body frame, resolved in bodyaxes;

• Ωb: the inertial rotational velocity of the body frame with respect to theinertial frame, resolved in body axes;

•˙di: the velocity of a mass element in lumped mass i, with respect to thebody reference system;

• ri: the undeformed position of a mass element in lumped mass i, withrespect to the body reference system;

• di: the translation of position of a mass element in lumped mass i due todeformation, with respect to the body reference system;

In the next sub-section the following very useful relations will be used frequently:


a × b = skew(a)b a, b ∈ R3

skew(a) =

0 −a3 a2

a3 0 −a1

−a2 a1 0

a × b = −b × a = skew(−b)a

skew(a)T = −skew(a)

skew(a)skew(b) = abT− aT bI

∑

i

∫

Vi

=∫

V

(B.23)

B.2.2 Kinetic energy

Based on the kinematics of all elementary mass elements the total kinetic energyis computed as follows:

Tk in =1

2

∫

V

RT

δiRδiρdV (B.24)

where ρ is the local material density, dV is an infinitesimal volume element, andV is the over-all volume of the aircraft. The mass of the element is dm = ρdV .Expanding the terms gives:

Tk in =1

2

∫

V

(

Vb + ˙di + Ωb × (ri + di))T (

Vb + ˙di + Ωb × (ri + di))

ρdV

=1

2V T

b Vbm +1

2ΩT

b

∫

V

(ri + di)T (ri + di)I − (ri + di)(ri + di)

T ρdV Ωb

+1

2

∫

V

˙dT

i

˙diρdV + V T

b

∫

V

˙diρdV + V T

b (Ωb ×

∫

V

(ri + di)ρdV )

+

∫

V

(Ωb × (ri + di))T ˙diρdV (B.25)

Note that the first equation is exactly the form as used in [137], and, replacingthe integral by the summation symbol, also as in [21]. The second element in the


second equation contains the momentary inertia tensor, as can be seen as follows:∫

V

(Ωb × (ri + di))T (Ωb × (ri + di))ρdV

=

∫

V

((ri + di) × Ωb)T ((ri + di) × Ωb)ρdV

=

∫

V

(skew(ri + di)Ωb)T (skew(ri + di)Ωb)ρdV

= ΩTb

∫

V

skew(ri + di)T skew(ri + di)ρdV Ωb

= ΩTb

∫

V

((ri + di)T (ri + di)I − (ri + di)(ri + di)

T )ρdV Ωb

= ΩTb IΩb

Now the following two assumptions are made:

• Assumption 4: V ariation of the inertia tensor as a function of airframedeformation may be neglected, i.e. I = constant;

• Assumption 5 : Deformation and deformation rates are collinear (i.e. are in

the same direction), so that ˙di × di = 0;

It must be stressed that none of these assumptions are necessary, but they areconsidered reasonable and do considerably reduce length of the derivation andcomplexity of the outcome [135, 19]. The last term of (B.25) then becomes:

∫

V

(Ωb × (ri + di))T ˙diρdV = ΩT

b

∫

V

(ri + di) ×˙diρdV ≈ ΩT

b

∫

V

ri ×˙diρdV

The origin of the body reference frame is positioned in the centre of gravity ofthe airframe:

rc g =

∫

VriρdV

∫

VρdV

= 0 (B.26)

The kinetic energy term reduces to:

Tkin =1

2V T

b Vbm +1

2ΩT

b IΩb +1

2

∫

V

˙dTi

˙diρdV

+ V Tb

∫

V

˙diρdV + V Tb (Ωb ×

∫

V

diρdV ) + ΩTb

∫

V

ri ×˙diρdV

The third term of this expression is expanded as follows:

1

2

∫

V

˙dTi

˙diρdV =1

2

∫

V

(di + δφi × sδi)T (di + δφi × sδi)ρdV (B.27)

=1

2

∫

V

dTi di + 2dT

i (δφi × sδi) + (δφi × sδi)T (δφi × sδi)ρdV


Dealing with a lumped mass system, the mass and inertia tensor of an element i

are defined as follows:

mi =

∫

Vi

ρdV (B.28)

Ii =

∫

Vi

skew(sδi)T skew(sδi)ρdV =

∫

Vi

sTδisδiI − sδis

TδiρdV (B.29)

li =

∫

Vi

sδiρdV =

∫

Vi

sδiρdV∫

Vi

ρdVmi (B.30)

where Vi indicates integration over the volume of the lumped mass i. The vectorli is the location of its centre of gravity with respect to the local origin in ri.Further note that Ii is the inertia tensor with respect to a local reference systemwith its origin in ri and parallel to the body axes in undeformed state.

Integration now over the total airframe volume involves summation over all lumpedmasses with volume Vi:

1

2

∫

V

˙dTi

˙diρdV =1

2

∑

i

dTi dimi +

1

2

∑

i

δφTi Iiδφi +

∑

i

δφTi li × dimi

B.2.3 Floating reference frames

The last three terms of (B.25) involve coupling between rigid and elastic motion.The importance of these terms can be influenced by a suitable choice of bodyreference system. This is most fundamental step in derivation of the equationsof motion of an unrestrained airframe. Already in 1891, Tisserand proposed afloating reference system (i.e. not physically attached to some location of thesystem, i.e. airframe), with respect to which the total relative momentum andangular momentum equal zero. This provides six constraints locating the referenceframe. The total relative momentum equals:

Hre l =∂

∂t

∫

V

(ri + di)ρdV =

∫

V

˙diρdV = 0 (B.31)

where ˙ri = 0 and the density has been assumed to be constant. This implies that:

∫

V

(ri + di)ρdV = constant (B.32)

The location of the centre of gravity with respect to the reference system is thusinvariant. Starting from the undeformed position (compare with equation B.26):

∫

V

riρdV = constant = rcgm (B.33)


where rcg is the location of the centre of gravity. Since rcg was set to zero in(B.26):

∫

V

riρdV =

∫

V

(ri + di)ρdV = rcgm = 0 (B.34)

so that∫

V

diρdV = 0 (B.35)

Further expansion finally gives:

∫

V

diρdV =

∫

V

(di + δφi × sδi)ρdV =

∑

i

mi(di + δφi × li) = 0 (B.36)

The angular momentum is slightly more complicated:

Hrelr o t=

∫

V

(ri + di) × (∂

∂t(ri + di))ρdV = 0 (B.37)

This expression is nonlinear in the deformation and therefore diffi cult to satisfy inpractice. Therefore a ’practical’ or linearised version is used (again, with ˙ri = 0):

Hrelr o t=

∫

V

ri ×˙diρdV = 0 (B.38)

A reference frame satisfying this linearised constraint, in combination with (B.31)is also called ’Buckens’ frame. In [137] the constraints are called ’practical’ meanaxis conditions. See reference [22] for a more thorough treatment on different typesof floating reference systems. Finally, further expanding the terms of (B.38) gives:

∫

V

ri ×˙diρdV =

∫

V

(ri + sδi) × (di + δφi × sδi

)ρdV (B.39)

=∑

i

[miri × (di + δφi × li) + mili × di + Iiδφi] = 0

=∂

∂t

∑

i

[miri × (di + δφi × li) + mili × di + Iiδφi] = 0

Comparing with equation B.18, the conditions (B.36) and (B.39) are satisfiedright away in case di and δφi result from a linear combination of orthogonal modeshapes from free-free vibration analysis. This implies that the orientation of themean axes simply complies with the direction of the rigid translational modeshapes (along axes of reference system used for modal analysis), with its originin the momentary centre of gravity. It will be clear that mean axes are a verysensible choice as body reference system in case mode shapes are available fromfree-free vibration analysis:


Vibration modes that result from modal analysis of the unrestrained system auto-

m atically fulfi l the linearised T isserand constraints due to orthogonality of thesemodes w ith respect to rig id (i.e. w ithout internal stresses) translational and rota-tional motion of the airframe.Note that at this point the following assumption has been implicitly made:

• Assumption 6 : A set of orthogonal modes for the airframe is available, re-sulting from free-free modal vibration analysis. Deformation of the airframemay be written as a linear combination of these mode shapes.

Thus, from now on di and δφi are no longer arbitrary, but related as follows:[

di

δφi

]

= ΦgiEηE =

[ΦgiEt

ΦgiEr

]

ηE (B.40)

where ΦgiE are the rows of the modal matrix for three translational (ΦgiEt) and

three rotational (ΦgiEr) degrees of freedom of grid point i. The subscript E,

as in (B.3), indicates that only elastic modes have been included. The columndimension of the matrices is the number of flexible modes nhE

that is taken intoaccount.The kinetic energy is now written as:

Tkin =1

2V T

b Vbm +1

2ΩT

b IΩb (B.41)

+1

2

∑

i

dTi dimi +

1

2

∑

i

δφTi Iiδφi +

∑

i

δφTi li × dimi

=1

2V T

b Vbm +1

2ΩT

b IΩb

+1

2

∑

i

(dTi dimi + δφT

i Iiδφi + δφTi li × dimi − dT

i li × δφimi)

=1

2V T

b Vbm +1

2ΩT

b IΩb

+ ηTE

1

2

∑

i

(ΦTgiEt

miI3ΦgiEt+ ΦT

giErIiΦgiEr

+ ΦTgiEr

miskew(li)ΦgiEt− ΦT

giEtmiskew(li)ΦgiEr

)ηE

=1

2V T

b Vbm +1

2ΩT

b IΩb

+ ηTE

1

2

∑

i

[ΦT

giEtΦT

giEr

][

miI3 −miskew(li)miskew(li) Ii

] [ΦgiEt

ΦgiEr

]

ηE

And in its final short-hand form (see also (B.16)):

Tkin =1

2V T

b Vbm +1

2ΩT

b IΩb +1

2ηT

EΦTgEMggΦgE ηE

=1

2V T

b Vbm +1

2ΩT

b IΩb +1

2ηT

EMEE ηE (B.42)


B.2.4 Potential energyPotential energy of the airframe consists of a gravitational and a strain component.At this point as an additional assumption is made:

• Assumption 7: G ravity is constant over the airframe.

The gravitational potential energy is:

Ug = −

∫

V

(TebRδi)T GeρdV (B.43)

= −

∫

V

RTδiρdV TbeGe

= −

∫

V

(Rb + ri + di)T ρdV Tb0G0

= −mRTb TbeGe

where Ge is the direction of gravity acceleration in earth-fixed axes:

Ge = [0, 0, g ]T

if gravity is assumed to be perpendicular to the earth surface. Remember thatthe origin of the body mean axes is always in the momentary centre of gravity,so that the integral for all terms except for Rb becomes zero. The minus sign isinduced by the downward direction of ze.The strain energy for undamped free vibration modes is [14]:

Us =1

2ηT

EKhhηE =1

2ηT

EMEEΩ2ηE (B.44)

where

Ω2 = diag(ω2

1, ω2

2, · · · , ω2

nh)

G eometric nonlinearities are not considered here: for transport aircraft these aregenerally considered to be of secondary importance. Zeiler and Buttrill do addressthese effects in Ref. [143].

B.2.5 Application of Lagrange’s equationsLagrange’s equations are given by [78, 21]:

∂

∂t

∂(T − U)

∂Vb

+ Ωb ×

∂(T − U)

∂Vb

−∂(T − U)

∂∫

Vbdt= F (B.45)

∂

∂t

∂(T − U)

∂Ωb

+ Ωb ×

∂(T − U)

∂Ωb

−∂(T − U)

∂∫

Ωbdt= M (B.46)

d

dt

∂(T − U)

∂ηE

−∂(T − U)

∂ηE

= Q (B.47)


B

yV

yB

zV

xB

xV

k3

k2

k1

x , zvB

y , zB B

x , yV V

z

R

θ

θ

ψ

ψ

φ

φ

xE

yE

zE

CoG

Figure B.2: Transformation from vehicle-carried vertical into body axes

Note that since Vb and Ωb are resolved in mean body axes, differentiation withrespect to inertial axes is applied using:

d

dt=

∂

∂t+ Ωb×

This does not apply to the latter equation, since ηE is not a position vector relyingon a specific reference system. With

∫Vbdt = Rb, application of (B.45) is straight

forward and results in the following equations of motion:

m[

Vb + Ωb × Vb − TbeG0

]

= F (B.48)[

IΩb + Ωb × IΩb

]

= M (B.49)

MEE

[ηE + Ω2ηE

]= QE (B.50)

B.2.6 Kinematic equations

In equations (B.48) the gravity acceleration is transformed into body axes viathe cosine matrix Tb0. This matrix depends on the attitude of the mean axessystem with respect to earth-fixed axes, expressed as a function of the Eulerangles φ, θ, ψ. The attitude angles are indicated in Figure B.2. First the earthaxes are translated to the aircraft centre of mass, resulting in the vehicle-carriedvertical frame (subscript V ). Next this frame is rotated about the zv (ψ), the k2

(θ), and xb (φ) axis respectively into the body (mean axes) frame.


The transformation from earth-fixed into body axes is described by [15]:

Tbe =

1 0 00 cosφ sinφ0 − sinφ cosφ

cos θ 0 − sin θ0 1 0

sin θ 0 cos θ

cosψ sinψ 0− sinψ cosψ 0

0 0 1

The position of the aircraft centre of gravity with respect to the earth-fixed axesis given by:

Vv =

xyz

v

= TTbeVb = TebVb (B.51)

The relation between the body angular rates and the Euler attitude angular ratesis described by [15]:

Θ = Tφ bΩb = Tφ b

pb

qbrb

(B.52)

with the transformation matrix:

Tφ b =

1 sinφ tan θ cosφ tan θ0 cosφ − sinφ0 sinφ/ cos θ cosφ/ cos θ

(B.53)

Accelerations at locations of interest on the airframe are derived as follows (seeFigure B.1).The position of location i on the airframe with respect to earth-fixed axes is:

Riie

= R0e+ Teb(ri + di) (B.54)

Differentiation results in the local velocity:

Rei= Re0

+ Tebdi + TebΩb × (ri + di) (B.55)

or:

Rei= Teb

(

Vb + di + Ωb × (ri + di))

(B.56)

Differentiating one more time gives:

Rei= Teb

(

Vb + Ωb × Vb + Ωb × di + di + Ωb × di + Ωb × (ri + di)

+ Ωb × Ωb × (ri + di))

Finally, the velocity and acceleration components in body co-ordinates are:

Rbi= Vb + di + Ωb × (ri + di) (B.57)

Rbi= Vb + Ωb × Vb

︸︷︷︸

1

+ 2Ωb × di︸︷︷︸

2

+ di︸︷︷︸

3

+ Ωb × (ri + di)︸︷︷︸

4

+ Ωb × Ωb × (ri + di)︸︷︷︸

5

with the following individual terms:


1. acceleration of origin of body axes (usually aircraft centre of gravity);

2. Coriolis acceleration;

3. local modal accelerations;

4. local acceleration due to offset from origin;

5. centrifugal acceleration.

The local deformation di and its rates are given by (mode displacement):

di = ΦgE ηE , di = ΦgE ηE , di = ΦgE ηE

This implies that velocities and accelerations at grid points (represented by rows inΦgE) may be computed directly. For airframe locations in between, interpolationis required.

B.2.7 Application of local forces and momentsIn section as a first step an arbitrary force and moment at grid point i is applied:

Fgi=

[Tbi 00 Tbi

]

Xi

Yi

Zi

Li

Mi

Ni

(B.58)

A local reference system i is considered attached to the grid point and undergoesthe same translation and rotation. In undeformed condition, the reference systemis parallel to the mean axes. In deformed condition, the rotation matrix from thelocal into body axes is given by Tbi.The contribution of a local force to the generalised force F in (B.48) is:

Fi = Tbi

Xi

Yi

Zi

(B.59)

the contribution to the generalised moment M in (B.49) is given by:

Mi = Tbi

Li

Mi

Ni

+ (ri + di) × Tbi

Xi

Yi

Zi

(B.60)

where (ri + di) is the deformed location of grid point i with respect to the centreof gravity.


Finally, the contribution to QE in (B.50) is:

QE,i = ΦTgiE

Xi

Yi

Zi

Li

Mi

Ni

(B.61)

where ΦgiE are the rows of the modal matrix ΦgH that correspond to the sixdegrees of freedom of grid point i.In the frame of this work, local forces and moments are generated by the powerplants or attached inertias. Assuming engine attachment points are among thegrid points, the application to the equations of motion as in the equations abovemay be done directly.

B.2.8 Summary of resultIn this appendix the equations of motion for an unrestrained aircraft have beenderived. The equations and underlying assumptions are summarised in the fol-lowing:

m[

Vb + Ωb × Vb − Tb0G0

]

= F (B.48)[

IΩb + Ωb × IΩb

]

= M (B.49)

Mhh

[ηE + Ω2ηE

]= Qh (B.50)

(B.62)

The underlying assumptions are:1 The aircraft airframe may be described as a collection of lumped

rigid mass elements with an associated mass mi and moment ofinertia Ii.

2 Translational and rotational deformations with respect to the ref-erence shape of the airframe are small.

3 Linear elastic theory applies (i.e. also geometric stiffening is ne-glected).

4 A set of orthogonal modes for the airframe is available, result-ing from free-free modal vibration analysis. Deformation of theairframe may be written as a linear combination of these modeshapes.

5 Gravity is constant over the airframe.6 Variation of the inertia tensor as a function of airframe deforma-

tion may be neglected, i.e. I = constant.7 Deformation and deformation rates are collinear (i.e. are in the

same direction), so that ˙di × di = 0.

K inematic relations are given by:

B.3 Concluding remarks 243

Attitude of mean axes and position of its origin:

Θ = TφbΩb (B.63)

R0 = TebVb (B.64)

Velocity and acceleration at grid point:

Rbi= Vb + di + Ωb × (ri + di) (B.65)

Rbi= Vb + Ωb × Vb + 2Ωb × di + di + Ωb × (ri + di)

+ Ωb × Ωb × (ri + di)

where:

di = ΦgE ηE , di = ΦgE ηE , di = ΦgE ηE (B.66)

B.3 Concluding remarksThe form of the equations (B.62) are the same as those in [21] after making thelisted assumptions. However, in the derivation in this section it has been shownthat for

1. lumped mass of which the centre of gravity does not match the location ofthe corresponding grid point;

2. lumped masses with inertia tensor

the equations are still valid under the same assumptions. This has great practicalvalue, since this allows available modal data (i.e. the generalised mass matrixMhh

and the in-vacuo free vibration eigenfrequencies in Ω, as obtained from modalanalysis using a finite element model of the airframe) to be directly integrated inthe airframe equations of motion. Correlating the last equation of (B.62) with(B.7), it appears that linear equations of motion as resulting from FE-analysismay be incorporated directly, only leaving out the already separated rigid bodymodes (ηR). The rigid body dynamics are covered by the first two equations in(B.62), which are exactly the same as the commonly used Newton-Euler equationsin flight mechanics simulation.The listed assumptions are in general valid for transport aircraft. Interestingly,especially due to the assumptions 4, 6, and 7, no inertial coupling between flex-ible and over-all (rigid) motion exists. Coupling does arise via the (generalised)aerodynamic forces and motion, which depend on rigid as well as flexible dynam-ics of the airframe. Note that this simplification only implies that no structuralmodes are exited. However, when studying local effects, like loads at engine py-lon attachments, the inertial effects may no longer be negligible, see for exampleRef. [107]. Also, for military aircraft with stores manoeuvring at high roll rates,loading of elastic modes due to centrifugal forces, Coriolis forces, and body angu-lar accelerations may be significant [21]. In [107, 21] the equations of motion are


derived without making the assumptions 6 and 7. The resulting inertial couplingterms are worked out and their influence is investigated.

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Nomenclature

Symbols (Roman)c M e a n a e ro d y n a m ic ch o rd le n g th

q D y n a m ic p re ssu re

Ib T o ta l a irc ra ft in e rtia te n so r w .r.t. Fb

N1 M e a n o f fa n sh a ft sp e e d s o f a ll e n g in e s

A L in e a r sta te sp a c e m o d e l – sta te m a trix

Aj C o lu m n j o f sta te sp a c e A -m a trix

A.. P a rtitio n o f m a trix A

A0..

,A1..

,A2..

R F A a e ro d y n a m ic stiff n e ss, d a m p in g , a n d m a ss m a tric e s

B L in e a r sta te sp a c e m o d e l – sta te in p u t m a trix

Bj C o lu m n j o f sta te sp a c e B -m a trix

B.. P a rtitio n o f m a trix B

Bg h H a lf-g e n e ra lise d stru c tu ra l d a m p in g m a trix

Bh h G e n e ra lise d stru c tu ra l d a m p in g m a trix

b lp l B a ck la sh o f p o w e r le v e r

C L in e a r sta te sp a c e m o d e l – o u tp u t sta te m a trix

Cj C o lu m n j o f sta te sp a c e C -m a trix

Cl T o ta l a e ro d y n a m ic m o m e n t c o e ffi c ie n ts a ro u n d x -a x e s o f Fa

Cm T o ta l a e ro d y n a m ic m o m e n t c o e ffi c ie n ts a ro u n d y -a x e s o f Fa

Cn T o ta l a e ro d y n a m ic m o m e n t c o e ffi c ie n ts a ro u n d z -a x e s o f Fa

CX T o ta l a e ro d y n a m ic fo rc e c o e ffi c ie n ts a lo n g x -a x e s o f Fa

CY T o ta l a e ro d y n a m ic fo rc e c o e ffi c ie n ts a lo n g y -a x e s o f Fa

CZ T o ta l a e ro d y n a m ic fo rc e c o e ffi c ie n ts a lo n g z -a x e s o f Fa

D L in e a r sta te sp a c e m o d e l – o u tp u t in p u t

D O u tp u t m a trix o f R F A la g sta te sp a c e e q u a tio n s

Dj C o lu m n j o f sta te sp a c e D -m a trix

260 Nomenclature

Drwy x-Distance from A/C REF to runway threshold

d T Thrust difference b etween left and right engines

E Input matrix of RFA lag state space equations

F Force vector

F (..) S ystem of equations, depending on variab les ..

f(..) S tate space model – nonlinear state derivative function

f(..) (N onlinear) function of ...

Fg Force vector in directions of xg

Fx Force in x-direction of Fb

Fy Force in y-direction of Fb

Fz Force in z-direction of Fb

FAbAerodynamic force vector, w.r.t. Fb

g Gravity acceleration

Gb Gravity acceleration vector in Fb

Gv Gravity acceleration vector in Fv

Gtyre Cornering stiffness of tyre

h Altitude in W GS 8 4 co-ordinates

h(..) S tate space model – nonlinear output function

hm S tate space model – nonlinear output function for yM e as

ho(...) S tate space model – nonlinear output function for other outputs than command vari-ab les

hs S tate space model – nonlinear output function for yS im

Hbaro Barometric altitude signal

hggld Glide slope reference height

Hlg Height of landing gears ab ove ground

Hra Radio altitude signal

Hrwy Height of runway threshold ab ove sea level

ht e rrain Local terrain elevation over W GS 8 4 ellipsoid

Ht r Terrain height step b efore threshold

HW S S tarting height of wind shear ab ove runway

I< n> Identity matrix with dimension n

j Complex numb er j =√−1

k Reduced frequency k = ωc/ V a

K.. Controller gain

K..(s) Controller transfer function

Kgg S tructural stiffness matrix

Nomenclature 261

Kgh Half-generalised stiffness matrix

Khh Generalised stiffness matrix

Lf h(..) First-order Lie-derivative of function h(..) along vector fi eld f

lH Distance between. wing and stabiliser AC’s

Lrwy Length of runway

M Nominal matrix part in static LFT model

M Moment vector, Mach number

m Total aircraft mass

M(s) Nominal system part in dynamic LFT model

Mx Moment around x-direction of Fb

My Moment around y-direction of Fb

Mz Moment around z-direction of Fb

MAbAerodynamic moment vector, w.r.t. Fb

Mgg Structural mass matrix

Mgh Half-generalised mass matrix

Mhh Generalised mass matrix

n V ector with noise inputs

N1i Fan shaft speed of engine i

N1dist White noise for engine fan shaft speed

N1 Engine fan shaft speed (low pressure spool) [% ]

nE Number of elastic modes (sub-set of h-set)

ng Number of DOF of all structural grid points (g-set)

nh Number of structural modes (h-set)

nn Dimension of noise input vector

np Dimension of parameter vector p

nR Number of rigid modes (sub-set of h-set)

nu Dimension of input vector

nw Dimension of the vector w(t)

nw Dimension of vector w(t)

nX Dimension of vector X(t)

nx Dimension of state vector

nx Load factor in body negative x-direction

nX Number of control and gust inputs

ny Dimension of output vector

ny Load factor in body negative y-direction

nz Load factor in body negative z-direction

262 Nomenclature

nz Vertical load factor in dir. of body z-axis

nuc Dimension of uc

nuo Dimension of uo

nycmd Dimension of ycmd

nym Dimension of yMeas

nys Dimension of ySim

p Vector with model parameters

P (..) Probability of ..

pb Angular velocity of aircraft around body x-axis

pe Vector of (unknown) environment parameters

pk Vector containing parameters that may be considered to be known to the control laws

pu Vector containing parameters that are unknown

qb Angular velocity of aircraft around body y-axis

Q.. (Half-)generalised aerodynamic loads

R Diagonal matrix with RFA lag state poles

rb Angular velocity of aircraft around body z-axis

Re Aircraft CoG position vector in geodetic axes

Re Position vector of aircraft CoG in Fe

ri Relative degree of model output i

rarp Position aerodynamic reference point with respect to the CoG in Fb

Rgear Position of landing gear (bottom of tyre) in FEC I

Rterrain Projected position vector of gear-ground contact point in FEC I

s Laplace variable

S Wing reference area

s Laplace variable

T Vector with design parameters

t Time

T Total thrust

T0 Temperature at mean sea level (MSL)

Tφ b Cosine matrix relating body and Euler angular rates

Tbv Trafo matrix from Fv into Fb

TWSxTime constant for specific wind shear

u Vector with model inputs

U Control inputs

ub Velocity of aircraft CoG in body x-direction

uc Vector with control inputs

Nomenclature 263

uo Vector with other model inputs

V Inertial speed (norm)

Vb Inertial velocity vector along Fb

vb Velocity of aircraft CoG in body y-direction

Vv Inertial velocity vector along Fv

VW Wind velocity in Fb

VZ Vertical speed (positive down) in Fv

Vapp Selected calibrated approach speed

Vcas Calibrated airspeed

Veas Equivalent airspeed

Vg Ground speed

Vtas True airspeed

wb Velocity of aircraft CoG in body z-direction

WSxSpecific wind shear in Fe x-direction

WX33 Wind (3 3 ft above ground) in Fe x-direction

WY 33 Wind (3 3 ft above ground) in Fe y-direction

wngld White noise for glide slope disturbance

wnloc White noise for localiser disturbance

wnTurbxWhite noise for x-turbulence in Fe

wnTurbyWhite noise for y-turbulence in Fe

wnTurbzWhite noise for z-turbulence in Fe

x Vector with model states

X(t) Vector containing potential state variables in set of model equations

xc Reference state vector in model-following control scheme

xe Aircraft CoG x-Position in geodetic axes

xg Struct. grid point DOF in Fs (g-set)

xL Lag state vector

xR Vector with rigid aircraft state variables

xCG x-Position of CoG in Fac

xLEVector with aerodynamic lag states, driven by airframe deformation

xLRVector with aerodynamic lag states, driven by rigid motion

xLXVector with aerodynamic lag states, driven by control defl ections and wind

xMLG x-Position of main gear in Fb

xN W x-Position of nose gear in Fb

xRbFD body states xT

Rb= [V T

b, ΩT

b]

264 Nomenclature

xRbRigid body states: velocity and angular rates w.r.t. Fb

xReFD position/attitude states: xT

Re= [RT

e , ΘT ]

xReRigid body states: position and attitude w.r.t. Fe

Xtr x-Position of height step w.r.t. threshold

xwheel x-Position of wheel in Fb

xP O Srwy Inertial x-position of runway threshold

y Vector with model outputs

ye Aircraft CoG y-Position in geodetic axes

yo Vector containing other outputs than command variables

ycmd Vector containing command variable outputs

yMeas Vector containing sensor measurements

yrwy y-Offset of aircraft CoG from runway centre line

ySim Vector containing model outputs for simulation analysis

ywheel y-Position of wheel in Fb

ze Aircraft CoG z-Position in geodetic axes

Symbols (Greek)α Aerodynamic angle of attack

αdist (Sinusoidal) alpha sensor disturbance

β Aerodynamic angle of side slip

βtyre Slip angle of tyre w.r.t. ground surface

χ Flight path heading angle

∆CD U ncertainty level of CD

∆Cl0 U ncertainty level of Clβ, Clβ,l g

, and Clβ,g r

∆ClδaU ncertainty level of Clδa

and Clδa,α

∆CLδehU ncertainty level of CLδeh

∆ClδrU ncertainty level of Clδr

∆CLg rU ncertainty level of CLαg r

and CLF , g rh

∆CLhU ncertainty level of CLh

∆Clp U ncertainty level of Clp and Clp,M a

∆Clr U ncertainty level of Clr and Clr,α

∆Cm0U ncertainty level of Cm0

, Cml g, CmM a

, Cm0, g r , and Cmβ

∆Cmq U ncertainty level of Cmqw band Cmq, l g

∆Cn0U ncertainty level of Cnβ

, Cnβ,l g, and Cnβ,g r

∆CnδaU ncertainty level of Cnδa

Nomenclature 265

∆CnδrUncertainty level of Cnδr

∆Cnp Uncertainty level of Cnp and Cnp,α

∆Cnr Uncertainty level of Cnr , Cnr,lg, and Cn

β

∆CY Uncertainty level of CY

∆Ixz Uncertainty level of Ixz

∆Ix Uncertainty level of Ix

∆Iy Uncertainty level of Iy

∆Iz Uncertainty level of Iz

∆ Structured uncertainty matrix in LFT model

δ Vector of control surface deflections

∆εloc Displacement of localiser signal

δA Aileron deflection angle

δE Elevator deflection angle

δR Rudder deflection angle

δT Tailplane rotation angle

δAm axMaximum aileron deflection

δEm axMaximum elevator deflection

δEm i nMinimum elevator deflection

δRm axMaximum rudder deflection

δThr1cThrottle command for engine 1

δThr2cThrottle command for engine 2

δAm axMaximum aileron rate

δEm axMaximum elevator rate

δRm axMaximum rudder rate

η Generalised mode shape co-ordinates

ηE Generalised elastic mode shape co-ordinates

ηR Generalised rigid mode shape co-ordinates

ηX Control surface deflections (from trim)

γ Flight path climb angle

γa Air mass-referenced flight path climb angle

γgld ILS glide slope angle

γrwy Slope of runway

γtr Slope of terrain before runway

µ Aerodynamic bank angle

µ Structured singular value

266 Nomenclature

ω Frequency

Ωb Inertial body angular velocity vector along Fb

φ Aircraft body axes Euler angle: roll

ΦgE Modal matrix: elastic modes

Φgh Modal matrix

ΦgR Modal matrix: rigid modes

ψ Aircraft body axes Euler angle: yaw

ψrwy Heading of runway

τ Time constant of first order filter

θ Aircraft body axes Euler angle: pitch

Θ Vector with Euler angles [φ, θ, ψ]T

θNW Commanded steering angle of nose wheel

θsteer Commanded steering angle of wheel

εgld Glide slope displacement signal

εloc Localiser displacement signal

Fa Aerodynamic reference system origin in ARP, x in dir. of airspeed, z down

Fb Body-fixed reference system origin in CoG, x to AC nose, z down

Fe Earth-fixed reference system (inertial) origin on earth surface, orientation NED

Ff Flight path-oriented reference system origin in CoG, x in direction of V, z down

Fs Structural reference system (inertial) origin e.g. at AC nose, x to AC tail, z upw.

Fv Vehicle-carried vertical reference system origin in CoG, orientation NED

Fac Aircraft-fixed reference system, parallel to Fb, origin in agreed aircraft-specific referencepoint

FECEF Earth-centred Earth-fixed reference system

FECI Earth-centred Inertial reference system

∂ ε∂ α grunc

Uncertainty level of ∂ ε∂ α gr

∂ ε∂ α unc

Uncertainty level of ∂ ε∂ α

Subscripts0 Initial value

A Aerodynamic, aileron

a Aerodynamic, w.r.t. Fa

b Body, w.r.t. Fb

c Commanded value

cas Calibrated airspeed

cmd Commanded

Nomenclature 267

e Error value

E Elevator, elastic

eas Equivalent airspeed

ext external

f Flight path, w.r.t. Ff

FD Flight dynamics

g Struct. grid point DOF in (g-set)

gE Right-generalised, only elastic deformation

gh Right-generalised

gR Right-generalised, only rigid motion

H Horizontal stabiliser

h W.r.t. generalised co-ordinates (h-set)

hh Generalised

inv Inverse

max Maximum

min Minimum

MLG Main landing gear

NW Nose wheel

o pt Optimum, optimal

R Rudder, rigid

r Rotational

T Thrust

t Translational

tas True airspeed

tyre From or of tyre

v W.r.t. Fv

W Wind

WB Wing-body combination

X Controls and wind

dist Disturbance

EE Elastic - Elastic matrix partitions

ER Elastic - Rigid matrix partitions

lag Lag term in RFA equation

other Non-aerodyn. forces/moments (e.g. thrust)

RE Rigid - Elastic matrix partitions

RR Rigid - Rigid matrix partitions

268 Nomenclature

unst Unsteady

Otherχ Second time derivative of χ

δχ Deviation from initial value: χ− χ0

δχ Small perturbation of variable χ

χ Time derivative of χ

χ χ Augmented with aerodyn. effects

χ Time domain representation of χ(s)

χ−1 Inverse of χ

SuperscriptsT Transpose

λ Longitude angle in WGS84 co-ordinates

φ Latitude angle in WGS84 co-ordinates

χ Actual value of χ (includes uncertainty)

χ Mean value of χ (e.g. applies to state vector feedback)

χ′ Dummy-derivative of χ

χri rthi time derivative

χ Estimated, or complementary filtered value of χ

AbbreviationsAC Aerodynamic CentreAC AircraftAE Aeroelasticity, AeroelasticAFCS Automatic Flight Control SystemAIAA American Institute of Aeronautics and AstronauticsAIC Aerodynamic Influence CoefficientALC Active Loads ControlAP AutopilotARP Aerodynamic Reference PointAT AutomatisierungstechnikATHR Auto throttleATTAS Advanced Technologies Testing Aircraft SystemAVDS Aviator Visual Design SimulatorAWO All Weather OperationsBSCU Braking and Steering Control UnitCat. CategoryCFD Computational Fluid DynamicsCFRP Carbon Fibre Reinforced PlasticCIT Comfort In TurbulenceCG Centre of Gravity

Nomenclature 269

CoG Centre of GravityCONDUIT Control Designer’s Unified InterfaceCS Certification SpecificationsCV Command VariableDAE Differential Algebraic EquationDAMIP Dynamic Aircraft Model Integration ProcessDGLR Deutsche Gesellschaft fur Luft- und RaumfahrtDLM Doublet-Lattice MethodDLR Deutsches Z entrum fur Luft und Raumfahrt,

German Aerospace CentreDME Distance Measuring EquipmentDOF Degree(s) Of FreedomDymola Dynamic Modelling Laboratory

Dynamic Modelling LanguageEASA European Aviation Safety AgencyECI Earth-Centred InertialECEF Earth-Centred Earth-FixedEGM9 6 Earth Gravitation Model 19 9 6EPR Engine Pressure RatioEQ M Equation(s) of MotionFADEC Full Authority Digital Engine ControllerFBW Fly-By-WireFC Flight ControlFCC Flight Control ComputerFCS Flight Control SystemFCL Flight Control LawFD Flight DynamicsFDL Flight Dynamics LibraryFE Finite ElementFEM Finite Element MethodFIT Functional Integration Technology (NASA program)FG Flight GuidanceFG& C Flight Guidance and ControlFL Feedback LinearisationFM Flight ManagementFM Flight MechanicsFMS Flight Management SystemFSA First Shot ApproachGARTEUR Group for Aeronautical Research and Technology in EURopeGPS Global Positioning SystemGS Glide slopeIEEE Institute of Electrical and Electronics EngineersIFASD International Forum on Aeroelasticity and Structural DynamicsIFFC Inverse FeedForward CompensationILA Internationale Luftfahrt AusstellungILS Instrument Landing SystemISA International Standard AtmosphereISV Servo Valve Control CurrentJ AA J oint Aviation AuthoritiesJ AR J oint Aviation RegulationsK S K onig-Schuler methodLaRC Langley Research CenterLFT Linear Fraction TransformationLQ Linear Q uadraticMC Monte CarloMCDU Mode Control and Display Unit

270 Nomenclature

MD Multi-DisciplinaryMFC Model Following ControlMOPS Multi-Objective Parameter SynthesisMSL Mean Sea LevelNASA National Aeronautics and Space AdministrationNATO North Atlantic Treaty OrganisationNDI Nonlinear Dynamic InversionNED North-East-DownNGA National Geospatial-Intelligence Agency (NGA)NLP Nonlinear ProgrammingODE Ordinary Differential EquationOOM Object-Oriented ModellingRFA Rational Function ApproximationRM Residualised Model methodRM Institute for Robotics and MechatronicsRCAM Research Civil Aircraft ModelREAL Robust and Efficient Autopilot control Law designRealCAM REAL Civil Aircraft ModelRef. ReferenceRTO Research & Technology OrganisationSC Structural ControlSCA Stability and Command AugmentationSLC Structural and Loads ControlSPT Speed / Path TrackingSQP Sequential Quadratic ProgrammingTECS Total Energy Control SystemTHCS Total Heading Control SystemTP Technical PublicationTR Technical ReportTU Technische Univeritat, Technische Univeriteit, Technical UniversityUDP User Datagram ProtocolVECTOR Vectoring, Extremely short take-off and landing, Control, Tailless

Operations ResearchVFW Vereinigte Flugtechnische WerkeVHF Very High FrequencyVLM Vortex-Lattice MethodVOR VHF Omni-directional Radio RangeWGS84 World Geodetic System 1984WMM-2xxx World Magnetic Model valid from year 2xxxX WB eX tra Wide Body

Index

active loads control system, 3Airbus, 4Airbus A340, 4aircraft

pre-design phase, 83aircraft control

direct law, 6mechanical, 3

aircraft control functionsprimary, 4secondary, 4

aircraft-on-groundautomatic inversion, 104object oriented implementation, 87,

101ATTAS, 40automatic flight control system, 3automatic landing (autoland), 9, 10, 185,

196certification, 31, 43, 87control laws, 31, 40, 157, 162, 194functions, 175performance, 157, 201requirements, 157system, 3, 99, 115, 150, 156, 159,

161, 186, 188

behavioural approach, 23bicycle model approximation, 85, 92, 93,

97, 100

Carbon Fibre Reinforced Plastic, 11command filter, 94, 96, 100command variables, 83

selection, 98control allocation, 98, 103

damping matrix, 224deformable aircraft, 223design process

aircraft, 55, 191, 198flight control law, 7, 8, 10–15, 17,

55, 99, 157, 159, 161, 162, 168,174, 185, 186, 191, 194–196,198–200

rapid prototyping, 97, 101, 193structure, 15, 191, 192

desktop simulator, 105differential algebraic equation, 92differential algebraic index, 92dummy derivative, 94Dymola, 87Dynamic Aircraft Model Integration Pro-

cess (DAMIP), 196dynamics

internal, 97zero, 97

Earth Gravitational Model 1996, 30Earth-Centred Earth-Fixed frame, 87Earth-Centred Inertial frame, 87envelope protection, 3equation

nonlinear solver, 90equations of motion

generalised, 225half-generalised, 224

feedback linearisation, 83, 84, 96, 97,115, 116

feedback signal synthesis, 99filter

complementary, 127, 128, 150, 163–165, 182

272 INDEX

notch, 128First-Shot Approach in flight control law

design, 13flight control, 3, 4, 7, 16, 25, 47, 101,

198computer, 4, 125, 138department, 13functions, 16law design, 6, 13–15, 27, 40, 83, 97,

115, 127, 150, 156, 196model, 16process, 7, 99, 157, 195, 198, 200,

206, 207laws, 5, 7, 9–13, 21, 27, 28, 55, 74,

82, 115, 194, 200, 207system, 3–5, 9, 11, 55

flight guidance and control system, 3flight management system, 3, 4flutter

analysis, 223, 225department, 55equation, 225margin, 9, 16, 55

fly-by-wire, 4landing system, 114system, 4test bed, 40, 160

force summation method, 227free vibration modes, 227full authority digital engine control sys-

tem, 5

gain scheduling, 83GARTEUR, 7, 28generalised co-ordinates, 224

handling commonality, 4

inline integration, 104Instrument Landing System (ILS), 31,

36, 41, 43, 87, 120, 157, 160Inverse Feedforward Compensation, 83inversion

automatic, 83iron bird, 10

Lie derivatives, 95Linear Fractional Transformation, 25loads (flight loads), 4, 21, 55, 196, 197,

210, 227, 243active control, 4analysis, 9, 12, 21, 25, 55, 60, 76,

192, 194, 196, 197, 223, 225,226

department, 16, 55, 83design manoeuvre, 197discipline, 13fatigue, 4, 197gust, 9, 55manoeuvre, 55, 197model, 65unsteady aerodynamic, 61

Lockheed 1011 Tristar, 4loop capability, 25

mass matrix, 223Mean Sea Level, 30modal

analysis, 223matrix, 224response, 226

modeacceleration method, 227displacement method, 226elastic, 224rigid body, 224shapes, 224

modelaircraft

flexible, 89rigid-body, 89

causality, 85compiler, 83differential equations, 96inverse

stability, 96inversion, 83, 95

automatic, 93, 105object-oriented

translation of, 90Model Following Control, 83

INDEX 273

Modelica, 20, 27, 28, 37, 43, 47, 65,86, 89, 114, 115, 119, 120, 151,194, 216

Association, 27environment, 38Flight Dynamics Library, 76, 87translator, 27

modellingblock diagram, 84object-oriented, 83, 84signal flow, 84

Monte Carlo analysis, 9, 156, 157, 159,171–173, 177, 179–181

criteria based on, 175, 185MOPS (Multi-Objective Parameter Syn-

thesis), 13, 133, 135, 159, 174,177

multi-body system, 89multi-disciplinary

aircraft model, 2, 20, 21, 194aircraft model integration, 194, 229aircraft modelling, 194design analysis, 13design requirements, 194flight control law design, 12, 14, 190,

193, 194, 196model, 27, 47, 83model implementation, 27model integration, 14, 87modelling, 25, 85simulation model, 21system model, 82

multi-purpose control and display unit,4

Nonlinear Dynamic Inversion, 115, 116,118, 123

nonlinear dynamic inversion, 16, 17, 83,97, 125, 128, 150, 162, 175,191, 192, 199, 200, 207

nonlinear equations of motion, 223

optimisationalgorithm, 159criterion, 130, 159, 173

min-max, 159, 176, 177multi-model, 129multi-objective, 13, 14, 17, 105, 128,

131–133, 150, 151, 157, 168,175, 185, 193

problem, 177problem set-up, 168, 174, 175, 193,

199process, 174

ordinary differential equation, 92orthogonality of modes, 225, 227

rapid prototyping, 83REAL project, 40relative degree, 94, 96

simulationbatch, 99desktop, 105, 194pilot-in-the-loop, 83, 99real-time, 105

stiffness matrix, 223structural

damping, 224degrees of freedom, 223eigenfrequencies, 224eigenvalue problem, 223

structural control system, 3structured singular value µ, 137, 140

analysis, 130, 137, 138, 140–142,144, 151, 194, 199, 200, 207

bounds, 139, 143robust stability, 199synthesis, 128

Total Energy Control System (TECS),162–165, 169, 171, 175, 177,180, 182, 183, 192

trim computation, 101

World Geodetic System 1984, 30World Magnetic Model, 30

274 INDEX

Samenvatting

REGELWETTEN in boordcomputers van moderne verkeers- en gevechtsvlieg-tuigen maken het verschil uit tussen de vliegeigenschappen van het vliegtuig

op zichzelf en de vliegeigenschappen zoals deze door de piloot en passagier er-varen worden. Deze regelwetten bestaan uit terug- en voorwaartskoppelingenvan gemeten sensoren commandosignalen die in geschikte stuuruitslagen wor-den omgezet. Dit gebeurt zo, dat zowel gewenst stuurgedrag als ook aangenaamvliegcomfort voor de passagier bereikt wordt. Bovendien zijn het de regelwettendie automatische besturingstaken waarnemen.Het zal duidelijk zijn dat regelwetten voor vliegtuigbesturing aan een groot aantalprestatie- en veiligheidseisen moeten voldoen, zodat ze betrouwbaar in alle vlieg-toestanden, onder slechte weersomstandigheden, bij uitval van hardware compo-nenten, enz. hun taak vervullen. De terugkoppeling van sensorsignalen beın-vloedt en wordt beınvloed door vliegmechanika, door het dynamische gedrag vande structuur, besturingshardware, sensoren, motoren, landingsgestel, enzovoorts.Door toedoen van regelwetten ontstaan bovendien dynamische interacties tussendeze aspecten, hetgeen het ontwerpen van regelwetten voor vliegtuigen tot eenmulti-disciplinaire taak bij uitstek maakt.Juist het multi-disciplinaire aspect in het ontwerp wordt van steeds grotere beteke-nis. Ten einde de efficientie van vliegtuigen verder te verbeteren, gaat men fysiektot het (veilige) uiterste om een zo hoog mogelijk prestatieniveau uit het vliegtuigals geheel te verkrijgen. Dit leidt bijvoorbeeld tot lichte en daarmee vaak behoor-lijk flexibele constructies en tot het gebruik van actuatoren die zeer maatgesnedenzijn. De mogelijkheid tot het implementeren van regelwetten wordt bovendiensteeds meer uitgebuit, bijvoorbeeld om belastingen in de vlucht actief te reduc-eren, het vliegtuig als geheel te stabiliseren (vooral bij gevechtsvliegtuigen) en omvibraties in de constructie actief te dempen. Het huidige ontwerpproces voor vlieg-tuig regelwetten is helaas niet goed geschikt voor een inherent multi-disciplinaireaanpak. Hoofdreden is dat interacties met andere ontwerpdisciplines voor hetgrootste deel pas na het eigenlijke ontwerpwerk door gespecialiseerde afdelingenvan de vliegtuigproducent geanalyseerd worden. Dit leidt doorgaans tot aanvra-gen de regelwetten aan te passen, hetgeen resulteert in nieuwe ontwerpslagen.Het doel van dit proefschrift is de ontwikkeling van concepten en methoden vooreen nieuw ontwerpproces van regelwetten dat een inherent multi-disciplinaire aan-pak omvat. De mogelijkheid daartoe staat of valt met de beschikbaarheid van dy-

276 Samenvatting

namische vliegtuigmodellen die zich niet, zoals gebruikelijk, tot de vliegmechanikabeperken, maar tevens structuurdynamica, belastingen, systeemdynamica, enz.voldoende nauwkeurig beschrijven. Zo een geıntegreerd model laat het bijvoor-beeld toe de bandbreedte (reactiesnelheid op stuursignalen) van het vliegtuig teverhogen, terwijl de daarmee toenemende belasting van de constructie tegelijk-ertijd nauwkeurig in ogenschouw kan worden genomen. Dit bespaart naderhandlangdurige ontwerpiteraties met de afdeling die voor belastingsberekeningen ver-antwoordelijk is.

Het maken van geıntegreerde vliegtuigmodellen vereist een geschikte generiekemodelstructuur en het juiste gebruik van beschikbare databronnen, vooral in-dien daarin overlappingen bestaan (daarover gelijk meer). In dit proefschriftwordt een modelstructuur gedefinieerd die gebaseerd is op de techniek van object-georienteerd modelleren. Deze techniek staat een directe implementatie van fysiekevergelijkingen toe, in plaats van de gangbare implementatie van differentiaal-vergelijkingen (die door de gebruiker eerst handmatig uit de fysieke vergelijkingenmoeten worden afgeleid). Op deze manier kunnen modelcomponenten aan de enekant door het direct programmeren van –in de specifieke vakdiscipline gangbare envertrouwde– fysieke vergelijkingen geımplementeerd worden. Aan de andere kantkunnen componenten uit verschillende disciplines nog steeds makkelijk samen-gevoegd worden, omdat de onderliggende object-georienteerde modelleringstaaleen een dezelfde is. De eerste bijdrage van dit proefschrift is een nieuwe algemenestructuur voor multi-disciplinaire vliegtuigmodellen en de implementatie ervan inde modelleringstaal Modelica.

Zelfs in het geval van een inherent multi-disciplinaire ontwerpaanpak worden deregelwetten naderhand door gespecialiseerde vakafdelingen geanalyseerd met be-trekking tot flutter, structuurbelastingen en interacties met systeemhardware.Het is daarom van enorm belang dat voor geıntegreerde vliegtuigmodellen model-databases (of op zijn minst, berekeningsmethoden) uit de gespecialiseerde afdelin-gen worden gebruikt en deze dus niet nog eens zelf ontwikkeld worden. Anderskunnen alleen al door incompatibiliteiten tussen modellen vermijdbare ontwerpit-eraties ontstaan. Echter, vooral in het geval van aeroelasticiteit ontstaat hier eenprobleem. Tussen aeroelastische en vliegmechanische modellen bestaan namelijkbehoorlijke overlappingen. De tweede bijdrage van dit proefschrift is een proce-dure de bewegings- and aerodynamische vergelijkingen achter deze modellen opeen geschikte manier te integreren.

Voor numerieke simulatie is het nodig het vliegtuigmodel in de vorm van dif-ferentiaal- of differentiaal-algebraısche vergelijkingen ter beschikking te hebben.Het mooie aan object-georienteerde modelimplementatie is dat deze vereiste vormvolledig automatisch uit de geımplementeerde fysieke modelvergelijkingen wordengegenereerd. Daartoe staan betrouwbare symbolische algoritmes ter beschikking.Behalve gangbare simulatiemodellen kunnen met deze algoritmes evengoed inversemodellen gegenereerd worden. Nu bestaan er verschillende ontwerpmethoden,zoals “ Nonlinear Dynamic Inversion” (NDI), die door middel van het inverterenvan modelvergelijkingen in regelwetten resulteren die voor praktisch alle vliegom-standigheden geldig zijn. De derde bijdrage van dit proefschrift is het idee deze

Samenvatting 277

ontwerpmethode en de mogelijkheid automatisch inverse modellen te genereren, tecombineren in een “rapid-prototyping” ontwerpproces voor regelwetten. Dit steltde ontwerper ertoe in staat in hele korte ontwerpcycli te experimenteren met stu-urvariabelen, te gebruiken stuurvlakken, de dimensionering van deze stuurvlakken(of eisen daartoe), enz. Dit is een nieuw hulpmiddel om over ver–strekkendeontwerpkeuzes in een vroeg stadium beter gefundeerde beslissingen te kunnennemen. Tevens stelt deze snelle procedure de ontwerpafdeling in staat al in eenvroeg stadium van het vliegtuigontwerp ook aan andere vakafdelingen eerste func-tionele regelwetten te leveren, zodat ook zij vroegtijdig eerste ontwerpanalysenkunnen beginnen of voorbereiden. In dit proefschrift wordt de ontwerpproceduregedemonstreerd aan de hand van een regelsysteem voor het manoeuvreren vaneen vliegtuig op de grond.

Indien het “rapid prototyping” ontwerp veelbelovend is, wordt overgegaan totgedetailleerd ontwerp van de regelwetten. In dit proefschrift wordt dit gedaanvoor regelwetten voor automatisch landen, toegepast op een klein passagiersvlieg-tuig. Hierbij wordt in het bijzonder op praktische aspecten zoals synthese vanterug te koppelen sensorsignalen en op verschillen tussen de voor inversie ge-bruikte modellen en de eigenschappen van het eigenlijke vliegtuig. Bij NDI ishet gebruikelijk het laatste probleem door middel van robuuste regelwetten ineen lus buitenom aan te gaan. Als een vierde bijdrage wordt in dit proefschrifteen andere manier voorgesteld, namelijk het met betrekking tot robuustheid opti-maliseren van onzekere modelparameters in het inverse model. Deze bieden extragraden van vrijheid die voor een robuuster ontwerp uitgebuit kunnen worden.

In het huidige ontwerpproces van regelwetten voor vliegtuigen worden de ontwerp-parameters doorgaans met de hand ingesteld. Echter, multi-disciplinair ontwerpenbetekent dat een veel groter aantal ontwerpcriteria uit verschillende vakrichtingentegelijkertijd in ogenschouw moet worden genomen. Daarom wordt het gebruikvan zogenaamde “multi-objective” optimalisatie aangeraden. De onderliggende“min-max” formulering van het optimalisatiecriterium staat het toe een zeer grootaantal individuele multi-disciplinaire ontwerpcriteria en -beperkingen in rekeningte brengen en zo geschikte compromis-oplossingen te vinden. Het relatieve belangvan deze individuele criteria wordt door middel van schalingen uitgedrukt dieeenvoudig fysisch interpreteerbaar zijn. Het optimalisatieprobleem is natuurlijkniet convex. Het gaat er echter om automatisch geschikte ontwerpoplossingente vinden. De modellen die voor het berekenen van de criteria nodig zijn, wor-den uit de eerder voorgestelde object-georienteerde modellen gegenereerd. Alseen vijfde bijdrage wordt in dit proefschrift een ontwerpprocedure voorgesteld diede optimalisatiemethode uitbreidt met het oog op grote complexe regelwetstruc-turen met vele onderling afhankelijke functies. Er wordt een ontwerpstrategietoegepast waarbij nieuwe functies sequentieel aan de synthese van de regelwettenworden toegevoegd, hetgeen stapsgewijs leidt tot optimalisatie van het gehele sys-teem. Deze strategie staat bijvoorbeeld toe functies in binnenste regellussen zoin te stellen dat ze met verschillende functies in lussen daarbuiten samen kunnenwerken. Daar op deze manier voor iedere taak nog slechts een functie nodig is,leidt dit tot eenvoudiger regelwetstructuren.

278 Samenvatting

De hierboven beschreven bijdragen zijn gecombineerd in een nieuw, eenvoudigermulti-disciplinair ontwerpproces voor regelwetten voor vliegtuigbesturing. Ditproces is in zijn geheel toegepast op het ontwerp van een stuurautomaat die heteen klein passagiersvliegtuig mogelijk maakt automatisch te landen. De hierbovenbeschreven ontwerpstrategie staat het toe dezelfde regelwetten in de binnenste lusin combinatie met alle functies in regellussen daaromheen (voor het afvangen, hetvolgen van het aanvliegpad, snelheidsregeling, enz.) te gebruiken. Alvorens eenautomatisch landingssysteem in testvluchten beproefd kan worden, worden navoltooiıng van het ontwerp uitgebreide Monte Carlo analysen uitgevoerd, hetgeenvaak tot nieuwe ontwerpiteraties voort. In het beschreven ontwerp wordt dezeanalyse direct bij het instellen van de ontwerpparameters als ontwerpcriteriumopgenomen. Op deze manier is aangetoond dat het voorgestelde ontwerpprocesdaadwerkelijk in staat is zulke ontwerp iteraties naderhand te vermijden. Deresulterende regelwetten zijn met succes in zes automatische landingen beproefd.Hierbij bleek dat de ontwerpparameters reeds tijdens het ontwerpproces goedwaren ingesteld en dat geen noodzaak tot aanpassingen bestond.

Acknowledgements

THIS thesis is a spin-off result of several projects I conducted or have beeninvolved in during some ten years as a research engineer at the German

Aerospace Center (DLR) in Oberpfaffenhofen, Germany. For the opportunity tocompile results of scientific significance into a PhD thesis, I first of all would liketo thank Prof.dr.ir. J.A. (Bob) Mulder of the faculty of Aerospace Engineeringat the Delft University of Technology, and Dr. J. Bals, my department head atDLR.Bob strongly encouraged me to write this thesis. I would like to thank him for hiscontinuous enthusiastic support and for his patience during writing it, and aboveall, for willing to be my promotor. I am happy that the start of my career asa junior researcher in his Control and Simulation group, although via a detour,eventually led to this thesis.I would like to thank Dr. J. Bals, head of the Dynamics and Control departmentat the DLR Institute of Robotics and Mechatronics, for offering a most pleasantworking environment, for getting me involved in very interesting and challengingprojects, and for his strong support and encouragement to write this thesis.I am very grateful to Dr. Q.P. (Ping) Chu for his strong encouragement during thewriting of this thesis, and for very interesting discussions on Nonlinear DynamicInversion control. I also would like thank him for dealing with administrative workat the university in preparation of the promotion, relieving me from travellingshortly after the birth of my son. I am very proud that Ping has been willing tobe co-promotor.I owe special thanks to the person who first came up with the idea of pursuinga PhD degree, Samir Bennani. Samir supervised my MSc project at the DelftUniversity of Technology, introduced me to emerging robust and nonlinear con-trol theory, got me involved in the (in hindsight) legendary GARTEUR projectFM(AG08) on Robust Flight Control, and gave me the opportunity to presentour results on µ-based robust flight control and Nonlinear Dynamic Inversion atseveral Benelux meetings and other conferences.I further would like to thank Tony Lambregts. I think I was very lucky thatmy graduation work coincided with his sabbatical at the Faculty of AerospaceEngineering. Discussions with Tony have been real eye openers and taught methe indispensable practical aspects of flight control. His written material has beenof enormous help for years afterwards.

280 Acknowledgements

My first project (freelance) at DLR, back in 1996, was set up by Samir and Prof.G. Grubel, at that time department head of the Control Design Engineering (nowDynamics and Control) department at the Institute of Robotics and Mechatronicsat DLR. I would like to thank Prof. Grubel offering this opportunity at DLR,for offering the position in his department one year later, and for his strongencouragement to write this thesis.I would like to thank many people at the Institute of Robotics and Mechatronicsat DLR. Hereby I especially mention those involved in flight dynamics, control,loads and aeroelasticity: Dieter Moormann, Lester Faleiro (my former office col-league), Hans-Dieter Joos (with whom I intensively co-operated in a.o. the REALproject on autoland control), Andras Varga, Reinhold Steinhauser (my current of-fice colleague), Christian Ballauf, Simon Hecker, Thiemo Kier, Marion Reijerkerk,Moriz Scharpenberg, and Christian Reschke. Two former students, Philipp Nageland Andre Rudolph provided valuable contributions to the GARTEUR FM(AG-17) project (Chapter 4) and the flexible aircraft simulator presented at the ILA2006 respectively. I further would like to thank Frau Jaschinski (our group’s for-mer secretary), and Monika Kohler and Christine Traurig (our group’s currentsecretaries) for all their help in organisational, bureaucratic and travel matters.In Delft, I further would like to thank all members and former members of the Con-trol and Simulation group, especially Jan-Willem van Staveren, Bertine Markus,Peter Kraan, and Henk Lindenburg.Finally, I especially would like to thank my parents, who enabled and uncondi-tionally supported my studies in Delft and strongly encouraged me to write thisthesis. I am very grateful to my sister Lieneke, and brothers Ernstjan, and RobertJan (who made the painting depicted on the cover), for their continuous strongsupport, even though I have been living quite far away. I am most grateful to mywife Claudia, who has given me enormous support and love during the writingof this thesis. I am amazed and grateful I could finish it the same year we gotmarried, bought and renovated a house, and saw the birth of our son, Hendrikjan.

This thesis is dedicated to Claudia.

Mundraching, GermanyDecember 2007

Gertjan Looye

Curriculum Vitae

Personal data:Full name: Gertjan Hendrik Nicolaas LOOYEDate of birth: May 28, 1972Place of birth: Maasland, The NetherlandsNationality: Dutch

Education:1984 – 1990 Christelijke Scholengemeenschap “Westland-Zuid”,

Vlaardingen. Diploma: VWO1990 – 1996 Delft University of Technology, Aerospace Engineering

Diploma: M.Sc. (with honours)Graduation thesis: Design of a Flight Controller forthe Research Civil Aircraft Model using µ-Synthesis.Prepared within Stability & Control Group(Prof.dr.ir. J.A. Mulder)

Professional career:1996 – 1997 Research assistant in Stability & Control Group,

Faculty of Aerospace Engineering,Delft University of Technology, The Netherlands

1997 – Research Engineer at the Institute of Robotics andMechatronics, German Aerospace Center,DLR-Oberpfaffenhofen, Germany.Principal fields of work:- Multi-disciplinary aircraft model integration- Flight control law design and design methods

STELLINGEN

behorende bij het proefschrift

An Integrated Approach to Aircraft Modelling andFlight Control Law Design

Gertjan H.N. Looye, 16 januari 2008

1. D e in d e v lieg tuig m od ellering v eel g eb ruik te uitd ruk k ing “ fl ex ib el v lieg tuig ”is een p leonasm e.

2. In d e p rak tijk v an h et ontw erp en v an reg elaars is niet “ rob uuste stab iliteit” ,m aar “ rob uuste stab iliteitsm arg e” d e b elang rijk ste sp ec ifi catie.

3 . M onte C arlo analyse zoals d eze w ord t toeg ep ast op reg elw etten v oor au-tom atisch land en, is v anuit h et oog p unt v an m od elonzek erh eid op te v attenals een nom inale p restatieanalyse. [D it p roefsch rift, h oofd stuk 6]

4 . V anuit p rak tisch e ov erw eg ing en zou h et z inv ol z ijn m eer th eoretisch ond er-zoek te rich ten op h et in rek ening b reng en v an onzek erh ed en m et statistisch eeig ensch ap p en en m et ond erling e afh ank elijk h ed en.

5 . Het feit d at een M atlab sk rip t d oorloop t zond er “ W arnig s” en “ E rrors”b etek ent niet d at h et resultaat ook zond er “ W arning ” te interp reteren is, ofg een “ E rrors” b ev at.

6. D e synth ese m eth od e “ Nonlinear D ynam ic Inv ersion” m ag alleen in com b i-natie m et fysisch inz ich t toeg ep ast w ord en.

7 . D e d uim stok d ank t z ijn p rak tisch e b ruik b aarh eid aan h et feit d at d e cosinusv an k leine h oek en ong ev eer g elijk aan een is.

8. B ij re-im p lem entatie en v alid atie v an een m od el k an een relatief v ersch il ind e ord e v an g rootte v an 10−6 (b ijv oorb eeld in een tijd srep onsie) m oeilijk aanslech ts num eriek e eff ecten w ord en toeg esch rev en. E en d erg elijk e afw ijk ingd uid t eerd er of op een nog aanw ez ig e fout of een aanp assing in d e nieuw eim p lem entatie.

9 . V oor iem and v an b uiten h et v ak is th e uitd ruk k ing “ rob ust control” eenonjuiste form ulering (m isnom er).

10. Het fund am entele v ersch il tussen tech niek en als M od el-F ollow ing C ontrol,Nonlinear D ynam ic Inv ersion en Inv erse F eed -F orw ard C ontrol z it h em ind e h erk om st v an d e v ariab elen d ie in d e inv erse m od elv erg elijk ing en w ord eng esub stitueerd . [D it p roefsch rift, h oofd stuk 4 ]

11. K arak teristiek e k enm erk en v an D uitse snelw eg en z ijn d e afw ez ig h eid v an eeng enerale snelh eid slim iet en h et feit d at een inv oeg strook zeer v aak d oor eensch erp e b och t w ord t ing eleid . Het laatste k enm erk b ep erk t d e m og elijk h eidsnelh eid te m ak en b ij h et inv oeg en, h etg een v ak er tot g ev aarlijk e situatiesleid t d an h et eerste k enm erk .

D eze stelling en w ord en v erd ed ig b aar g each t en z ijn als zod anig g oed g ek eurd d oord e p rom otor, P rof.d r.ir. J .A . M uld er.

PROPOSITIONS

belonging to the thesis

An Integrated Approach to Aircraft Modelling andFlight Control Law Design

Gertjan H.N. Looye, January 16th, 2008

1. T he phrase “flexible aircraft”, which is commonly used in aircraft modelling,is a pleonasm.

2. In flight control law design practice, “robust stability margin” rather than“robust stability” is the most important specification.

3. From a model uncertainty point of view, Monte Carlo analysis as applied inautoland control law design should be considered as a nominal performanceanalysis. [T his thesis, chapter 6]

4. From a practical point of view, it would be expedient to put more theoreticalresearch effort into handling of uncertainties with statistical properties andinter dependencies.

5. A Matlab script that executes without “warnings and errors” does not implythat its results may be interpreted without warning, or do not contain anyerrors.

6. T he synthesis method “Nonlinear Dynamic Inversion” may only be appliedin combination with physical insight.

7. T he carpenter’s rule owes its practical use to the fact that the cosine ofsmall angles approximately eq uals one.

8. When re-implementing and validating a model it is hard to still justify a10−6 order-of-magnitude relative difference (e.g. in a time response) as a“numerical effect”. S uch a difference rather indicates an error or adaptationin the new implementation.

9. T o someone not familiar with the subject, the phrase “robust control” is amisnomer.

10. T he fundamental difference between techniq ues like Model-Following Con-trol, Nonlinear Dynamic Inversion, and Inverse Feed-Forward Control is inthe origin of the variables that are substituted into the inverse model eq ua-tions. [T his thesis, chapter 4]

11. Characteristic features of German highways are firstly the absence of a gen-eral speed limit and secondly the fact that the slip road is often precededby a sharp turn. T he latter reduces the possibility of gaining speed beforeentering the highway, more freq uently resulting in dangerous situations thanthe first feature.

T hese propositions are considered defendable and as such have been approved bythe supervisor, Prof.dr.ir. J.A. Mulder.

An Integrated Approach to Aircraft Modelling and Flight ...

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