Advances in Dynamics & Control Sivasundaram

NONLINEAR SYSTEMS IN AVIATION, AEROSPACE,AERONAUTICS, AND ASTRONAUTICS

ADVANCES INDYNAMICS AND CONTROL

Nonlinear Systems in Aviation, Aerospace,Aeronautics, and Astronautics

A series edited by:S. SivasundaramEmbry-Riddle Aeronautical University, Daytona Beach, FL USA

Volume 1Stability DomainsL. Gruyitch, J.-P. Richard, P. Borne, and J.-C. Gentina

Volume 2Advances in Dynamics and ControlS. Sivasundaram

Volume 3Optimal Control of TurbulenceS.S. Sritharan

ADVANCES INDYNAMICS AND CONTROL

S. SIVASUNDARAM

CHAPMAN & HALL/CRCA CRC Press Company

Boca Raton London New York Washington, D.C.

NONLINEAR SYSTEMS IN AVIATION, AEROSPACE,AERONAUTICS, AND ASTRONAUTICS

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Advances in dynamics and control / edited by S. Sivasundaram.p. cm. -- (Nonlinear systems in aviation, aerospace, aeronautics, and astronautics ; 2)

Includes bibliographical references and index.ISBN 0-415-30852-6 (alk. paper)1. Flight control. 2. Aerodynamics. I. Sivasundaram, S. II. Series.

TL589.4.A33 2004629.135--dc22 2003070019

CONTENTS

List of Contributors viiPreface ix

1 Global Spacecraft Attitude Control Using Magnetic ActuatorsMarco Lovera and Alessandro Astolf 1

2 Adaptive Learning Control for Spacecraft Formation FlyingHong Wong, Haizhou Pan, Marcio S. de Queiroz, and Vikram Kapila 15

3 Spectral Properties of the Generalized Resolvent Operator foran Aircraft Wing Model in Subsonic AirflowMarianna A. Shubov 29

4 Bifurcation Analysis for the Inertial Coupling Problem of a Reentry VehicleNorihiro Goto and Takashi Kawakita 45

5 Missile Autopilot Design Using Dynamic Fuzzy Gain-SchedulingTechniqueChun-Liang Lin, Rei-Min Lai, and Sen-Wei Huang 57

6 Model Predictive Control of Nonlinear Rotorcraft Dynamics withApplication to the XV-15 Tilt RotorRaman K. Mehra and Ravi K. Prasanth 73

7 New Development of Vector Lyapunov Functions and Airplane ControlSynthesisLyubomir T. Gruyitch 89

8 Stabilization of Unstable Aircraft Dynamics under Control ConstraintsM.G. Goman and M.N. Demenkov 103

9 QUEST Algorithms Case Study: GPS-Based Attitude Determinationof Gyrostat SatelliteJinlu Kuang and Soonhie Tan 117

10 Asymptotic Controllability in Terms of Two Measures of HybridDynamic SystemsV. Lakshmikantham and S. Sivasundaram 137

11 Exact Boundary Controllability of a Hybrid PDE System Arisingin Structural Acoustic ModelingGeorge Avalos and Irena Lasiecka 155

12 Flow Field, Temperature, and Dopant Concentration Evolutionin a BridgmanStockbarger Crystal Growth System in a StrictlyZero-Gravity and a Low-Gravity EnvironmentSt. Balint and A.M. Balint 175

13 Identification of Stiffness Matrices of Structural and Mechanical Systemsfrom Modal DataFirdaus E. Udwadia 189

14 A Survey of Applications in StructuralAcoustic Optimization for PassiveNoise Control with a Focus on Optimal Shell ContourSteffen Marburg 205

15 Intelligent Control of Aircraft Dynamic Systems with a New HybridNeuro-Fuzzy-Fractal ApproachPatricia Melin and Oscar Castillo 221

16 Closed-Form Solution of Three-Dimensional Ideal Proportional NavigationChi-Ching Yang and Hsin-Yuan Chen 231

17 Guidance Design with Nonlinear H2/H ControlHsin-Yuan Chen 247

18 A Control Algorithm for Nonlinear Multiconnected ObjectsV.Yu. Rutkovsky, S.D. Zemlyakov, V.M. Sukhanov, and V.M. Glumov 261

19 Optimal Control and Differential Riccati Equations under SingularEstimates for eAtB in the Absence of AnalyticityIrena Lasiecka and Roberto Triggiani 271

20 Nonlinear Problems of Spacecraft Fault-Tolerant Control SystemsV.M. Matrosov and Ye.I. Somov 309

Subject Index 333

CONTRIBUTORS

Alessandro Astolfi, Department of Electrical and Electronic Engineering, Imperial Col-lege, Exhibition Road, SW7 2BT London, EnglandGeorge Avalos, Department of Mathematics and Statistics, University of NebraskaLin-coln, Lincoln, NE 68588-0323, USAA.M. Balint, Department of Physics, University of West Timisoara, Blv. V. Parvan No. 4,1900 Timisoara, RomaniaSt. Balint, Department of Mathematics, University of West Timisoara, Blv. V. Parvan No.4, 1900 Timisoara, RomaniaOscar Castillo, Department of Computer Science, Tijuana Institute of Technology, P.O.Box 4207, Chula Vista CA 91909, USAHsin-Yuan Chen, Department of Automatic Control Engineering, Feng Chia University,Taichung, TaiwanM.N. Demenkov, Department of Computing Sciences and Control, Bauman Moscow StateTechnical University, Moscow, RussiaV.M. Glumov, Trapeznikov Institute of Control Sciences, Russian Academy of Sciences,Moscow, RussiaM.G. Goman, De Montfort University, Faculty of Computing Sciences and Engineering,Hawthorn Building, Leicester LE1 9BH, UKNorihiro Goto, Kyushu University, Fukuoka 812-8581, JapanLyubomir T. Gruyitch, University of Technology BelfortMontbeliard, Site Belfort,90010 Belfort Cedex, FranceSen-Wei Huang, Department of Applied Mathematics, National Chung Hsing University,Taichung 402, TaiwanVikram Kapila, Department of Mechanical, Aerospace, and Manufacturing Engineering,Polytechnic University, Brooklyn, NY 11201, USATakashi Kawakita, Kyushu University, Fukuoka 812-8581, JapanJinlu Kuang, Satellite Engineering Center, The School of Electrical and Electronic Engi-neering, Nanyang Technological University, 639798, SingaporeRei-Min Lai, Institute of Automatic Control Engineering, Feng Chia University, Taichung40724, TaiwanV. Lakshmikantham, Florida Institute of Technology, Department of Mathematical Sci-ences, Melbourne, FL 32901, USAIrena Lasiecka, Department of Mathematics, Kerchof Hall, University of Virginia, Char-lottesville, VA 22904, USAChun-Liang Lin, Department of Electrical Engineering, National Chung Hsing University,Taichung 402, Taiwan

Marco Lovera, Dipartimento di Elettronica e Informazione, Politecnico di Milano, PiazzaLeonardo da Vinci 32, 20133 Milano, ItalySteffen Marburg, Institut fur Festkorpermechanik, Technische Universitat, 01062 Dres-den, GermanyV.M. Matrosov, Stability and Nonlinear Dynamics Research Center, Mechanical Engineer-ing Research Institute, Russian Academy of Sciences, 5 Dmitry Ulyanov Street, Moscow,117333 Russia; Moscow Aviation Institute (State Technical University), 4 VolokolamskoyeAvenue, Moscow, 125871, RussiaRaman K. Mehra, Scientific Systems Company Inc., 500 West Cummings Park, Suite3000, Woburn, MA 01801, USAPatricia Melin, Department of Computer Science, Tijuana Institute of Technology, P.O.Box 4207, Chula Vista, CA 91909, USAHaizhou Pan, Department of Mechanical, Aerospace, and Manufacturing Engineering,Polytechnic University, Brooklyn, NY 11201, USARavi K. Prasanth, Scientific Systems Company Inc., 500 West Cummings Park, Suite3000, Woburn, MA 01801, USAMarcio S. de Queiroz, Department of Mechanical Engineering, Louisiana State Univer-sity, Baton Rouge, LA 70803, USAV.Yu. Rutkovsky, Trapeznikov Institute of Control Sciences, Russian Academy of Sci-ences, Moscow, RussiaMarianna A. Shubov, Department of Mathematics and Statistics, Texas Tech University,Lubbock, TX 79409, USAS. Sivasundaram, Embry-Riddle Aeronautical University, Department of Mathematics,Daytona Beach, FL 32114, USAYe.I. Somov, Stability and Nonlinear Dynamics Research Center, Mechanical Engineer-ing Research Institute, Russian Academy of Sciences, 5 Dmitry Ulyanov Street, Moscow,117333 Russia; Research Institute of Mechanical Systems Reliability, Samara State Tech-nical University, 244 Molodogvardyeskaya Street, Samara, 443100 RussiaV.M. Sukhanov, Trapeznikov Institute of Control Sciences, Russian Academy of Sciences,Moscow, RussiaSoonhie Tan, Satellite Engineering Center, The School of Electrical and Electronic Engi-neering, Nanyang Technological University, 639798, SingaporeRoberto Triggiani, Department of Mathematics, Kerchof Hall, University of Virginia,Charlottesville, VA 22904, USAFirdaus E. Udwadia, Aerospace and Mechanical Engineering, Civil Engineering, Math-ematics, and Information and Operations Management, 430K Olin Hall, University ofSouthern California, Los Angeles, CA 90089-1453, USAHong Wong, Department of Mechanical, Aerospace, and Manufacturing Engineering,Polytechnic University, Brooklyn, NY 11201, USAChi-Ching Yang, Department of Electrical Engineering, Hsiuping Institute of Technology,Taichung, TaiwanS.D. Zemlyakov, Trapeznikov Institute of Control Sciences, Russian Academy of Sciences,Moscow, Russia

PREFACE

Nonlinear phenomena in aviation and aerospace have stimulated cooperation among en-gineers and scientists from a range of disciplines. Developments in computer technologyhave allowed for solutions of nonlinear problems, while industrial recognition of the useand applications of nonlinear mathematical models in solving technological problems isincreasing.

Advances in Dynamics and Control comprises research papers contributed by expertauthors in dynamics and control and is dedicated to Professor A.V. Balakrishnan, Univer-sity of California, Los Angeles, USA in recognition of his significant contribution to thisfield of research.

Professor A.V. Balakrishnan earned his Ph.D. in mathematics from the Universityof Southern California in 1954. Professor Balakrishnan has been with the University ofCalifornia, Los Angeles as a professor of engineering since 1962 and a professor of math-ematics since 1965. He was chair of the Department of System Science in the School ofEngineering from 19691975. Since 1985, he has served as the director of the NASAUCLA Flight Systems Research Center. He was chairman of the Technical Committee onSystem Modelling and Optimization, International Federation of Information Processing,19701980, and is currently the president of the Com Con Conference Board. He has re-ceived honors and awards including: Fellow IEEE (1966), Silver Core IFIP (1977), theGuillemin Prize (1980), the NASA Public Service Medal (1996), and the AACC RichardE. Bellman Control Heritage Award (2001).

The work of Professor A.V. Balakrishnan has been an inspiration for generations ofengineers and applied mathematicians. During more than 40 years of his outstanding sci-entific career, he has made significant contributions to the analysis and design of controlsystems. His contributions range from the theory of optimal control (in the 1960s, he de-veloped a celebrated method the epsilon technique for the computation of optimalcontrols for distributed parameter systems) to filtering and identification theory, to a numberof difficult engineering applications that include the control of aircraft under wing turbu-lence, the control of flexible structures in space, and the aeroelastic modeling of aircraftwings. Professor Balakrishnan has demonstrated the ability to master difficult engineeringproblems in a rigorous mathematical way and has produced effective engineering solutions.

Advances in Dynamics and Control explores many new ideas, results, and directionsin the rapidly growing field of aviation and aerospace. It encompasses a wide range of top-ics including rotorcraft dynamics, stabilization of unstable aircraft, spacecraft and satellitedynamics and control, missile auto-pilot and guidance design, hybrid systems dynamicsand control, and structural and acoustic modeling.

The contributors to Advances in Dynamics and Control are to be highly commendedfor their excellent chapters. The volume will provide a useful source of reference for grad-uate and postgraduate students, and researchers working in areas of applied engineeringand applied mathematics, in university and in industry.

S. Sivasundaram

1Global Spacecraft Attitude ControlUsing Magnetic Actuators

Marco Lovera and Alessandro AstolfiDipartimento di Elettronica e Informazione, Politecnico di Milano, Milano, Italy

Department of Electrical and Electronic Engineering, Imperial College, London, England

The problem of inertial pointing for a spacecraft with magnetic actuators is addressed. Itis shown that a global solution to the problem can be obtained by means of (static) attitudeand rate feedback and (dynamic) attitude feedback. Simulation results demonstrate thefeasibility of the proposed approach.

1 INTRODUCTION

The problem of (global) regulation of the attitude of rigid spacecraft, i.e., spacecraft mod-eled by the Eulers equations and by a suitable parameterization of the attitude, has beenwidely studied in the recent years.

If the spacecraft is equipped with three independent actuators a complete solution tothe set point and tracking control problems is available. In [5, 11] these problems have beensolved by means of PD-like control laws, i.e., control laws that make use of the angularvelocities and of the attitude, whereas [1], building on the general results developed in [3],has solved the same problems using dynamic output feedback control laws. It is worthnoting that, if only two independent actuators are available, as discussed in detail in [4], theproblem of attitude regulation is not solvable by means of continuous (static or dynamic)time-invariant control laws, whereas a time-varying control law, achieving local asymptotic(nonexponential) stability, has been proposed in [10].

The above results, however, are not directly applicable if the spacecraft is equippedwith magnetic coils as attitude actuators. As a matter of fact, it is not possible by meansof magnetic actuators to provide three independent torques at each time instant, yet asthe control mechanism hinges on the variations of the Earths magnetic field along thespacecraft orbit, on average the system possesses strong controllability properties. In [2, 7,8, 13] the regulation problem has been addressed exploiting the (almost) periodic behaviorof the system, hence resorting to classical tools of linear periodic systems, if local resultsare sought after, or to standard passivity arguments, if (global) asymptotic stabilization ofopen-loop stable equilibria is considered.

However, several problems remain open. In particular, if only inertial pointing isconsidered, the global stabilization problems by means of full- (or partial-) state feedbackis still theoretically unsolved. Note, however, that from a practical point of view these

0-415-30852-6/04/$0.00+$1.50c2004 by Chapman & Hall/CRC

2 M. Lovera and A. Astolf

problems have an engineering solution, as demonstrated by the increasing number of ap-plications of this approach to attitude control.

The aim of this chapter is to show how control laws achieving global1 inertial pointingfor magnetically actuated spacecraft can be designed by means of arguments similar tothose in [1, 11], provided that time-varying feedback laws are used and that the controlgains satisfy certain scaling properties. In particular, while previous work ([9]) dealt withthe case of state feedback control for a magnetically actuated, isoinertial spacecraft, thischapter deals with the more general problems of full (attitude and rate) and partial (attitudeonly) state feedback for a generic magnetically actuated satellite.

The chapter is organized as follows. In Section 2 the model of the system is presented,while in Section 3 the model of the geomagnetic field used in this study is described. In Sec-tion 4 a general result on the stabilization of magnetically actuated spacecraft is presented.Namely, using the theory of generalized averaging, it is shown how stabilizing control lawsdesigned for spacecraft with three independent control torques have to be modified to con-struct stabilizing laws in the presence of magnetic actuators. In Section 5 the general theoryis used to design control laws using only attitude information, so avoiding the need for ratemeasurements in the control system. Finally, Sections 6 and 7 present some simulationresults and concluding remarks.

2 THE MODEL

The model of a rigid spacecraft with magnetic actuation can be described in various refer-ence frames [12]. For the purpose of the present analysis, the following reference systemsare adopted.

Earth Centered Inertial reference axes (ECI). The origin of these axes is in the Earthscenter. The X-axis is parallel to the line of nodes, that is the intersection between theEarths equatorial plane and the plane of the ecliptic, and is positive in the Vernalequinox direction (Aries point). The Z-axis is defined as being parallel to the Earthsgeographic northsouth axis and pointing north. The Y-axis completes the right-handed orthogonal triad.

PitchRollYaw axes. The origin of these axes is in the satellite center of mass. TheX-axis is defined as being parallel to the vector joining the actual satellite center ofgravity to the Earths center and positive in the same direction. The Y-axis points inthe direction of the orbital velocity vector. The Z-axis is normal to the satellite orbitplane and completes the right-handed orthogonal triad.

Satellite body axes. The origin of these axes is in the satellite center of mass; theaxes are assumed to coincide with the bodys principal inertia axes.

The attitude dynamics can be expressed by the well-known Eulers equations: [12]:

I = S()I + Tcoils + Tdist, (1)1To be precise, the control laws guarantee that almost all trajectories of the closed-loop system converge to the

desired equilibrium.

Global Spacecraft Attitude Control Using Magnetic Actuators 3

where R3 is the vector of spacecraft angular rates, expressed in body frame, I R33is the inertia matrix, S() is given by

S() =

0 z yz 0 xy x 0

, (2)

Tcoils R3 is the vector of external torques induced by the magnetic coils and Tdist R3 isthe vector of external disturbance torques, which will be neglected in what follows.

In turn, the attitude kinematics can be described by means of a number of possibleparameterizations (see, e.g., [12]). The most common parameterization is given by thefour Euler parameters (or quaternions), which lead to the following representation for theattitude kinematics:

q = W ()q, (3)where q =

[q1 q2 q3 q4

]T = [qTr q4]T is the vector of unit norm Euler parametersand

W () =12

0 z y xz 0 x yy x 0 zx y z 0

. (4)

It is useful to point out that Eq. (3) can be equivalently written as

q = W (q), (5)

where

W (q) =12

q4 q3 q2q3 q4 q1q2 q1 q4q1 q2 q3

. (6)

Note that the attitude of inertially pointing spacecraft is usually referred to the ECI refer-ence frame.

The magnetic attitude control torques are generated by a set of three magnetic coils,aligned with the spacecraft principal inertia axes, which generate torques according to thelaw:

Tcoils = mcoils b(t),where mcoils R3 is the vector of magnetic dipoles for the three coils (which representthe actual control variables for the coils) and b(t) R3 is the vector formed with thecomponents of the Earths magnetic field in the body frame of reference. Note that thevector b(t) can be expressed in terms of the attitude matrix A(q) (see [12] for details) andof the magnetic field vector expressed in the ECI coordinates, namely b0(t), as

b(t) = A(q)b0(t).

The dynamics of the magnetic coils reduce to a very short electrical transient and can beneglected. The cross-product in the above equation can be expressed more simply as amatrix-vector product as

Tcoils = B(b(t))mcoils, (7)


where

B(b(t)) =

0 bz(t) by(t)bz(t) 0 bx(t)by(t) bx(t) 0

(8)

is a skew symmetric matrix, the elements of which are constituted by instantaneous meas-urements of the magnetic field vector.

As a result, the overall dynamics, after application of the preliminary feedback,

mcoils = BT (b(t))u,

can be written asq = W (q)I = S()I + (t)u,

(9)

where (t) = B(b(t))BT (b(t)) 0.

3 MAGNETIC FIELD MODEL

A time history of the International Geomagnetic Reference Field (IGRF) model for theEarths magnetic field [12] along five orbits in PitchRollYaw coordinates for a near-polarorbit (87 inclination) is shown in Fig. 1.

As can be seen, bx(t), by(t) have a very regular and almost periodic behavior, whilethe bz(t) component is much less regular. This behavior can be easily interpreted by notic-ing that the x and y axes of the Pitch-Roll-Yaw coordinate frame lie in the orbit plane whilethe z axis is normal to it. As a consequence, the x and y components of b(t) are affectedonly by the variation of the magnetic field due to the orbital motion of the coordinate frame(period equal to the orbit period) while the z component is affected by the variation of b(t)due to the rotation of the Earth (period of 24 h).

When one deals with the problem of inertial pointing, however, it is more appropriateto consider a representation of the magnetic field vector in Earth centered inertial coordi-nates to be more convenient, as shown in Fig. 2.

4 STATE FEEDBACK STABILIZATION

In this section, a general stabilization result for a spacecraft with magnetic actuators isgiven, in the case of full-state feedback (attitude and rate). For, let q = [0 0 0 1]Tand consider the system

q = W (q)I = S()I +

(10)

and the control law = kpqr kv. (11)

In the light of Theorem 1 in [11], the control law (11) guarantees that qr 0 and 0as t for the closed-loop system (10) and (11). Also, an analysis of the Lyapunovfunction used in the same reference shows that the equilibrium (q, 0) of the closed-loopsystem (10) and (11) is asymptotically stable, while the other possible equilibrium (q, 0)is unstable.


Figure 1 Geomagnetic field in PitchRollYaw coordinates, 87 inclination orbit, 450 km altitude.

Proposition 1 Consider the system (9) and the control law

u = 2kpqr kv. (12)Then, there exists > 0 such that for any 0 < < the control law (12) ensuresthat (q, 0) is a locally exponentially stable equilibrium of the closed-loop system (912).Moreover, almost all trajectories of (912) converge to (q, 0).

Proof. In order to prove the first claim, introduce the coordinates transformation

z1 = q z2 =

. (13)

In the new coordinates, the system (9) is described by the equations:z1 = W (z1)z2Iz2 = S(z2)Iz2 + (t)(kpz1r kvz2). (14)

System (14) satisfies all the hypotheses for the application of generalized averaging theory([6, Theorem 7.5]). Moreover, using the Lyapunov function of Theorem 1 in [11] one canconclude that the system obtained applying the generalized averaging procedure has (q, 0)as locally asymptotically stable equilibrium provided that

= limT

1T

T

0

B(t)BT (t)dt > 0.


Figure 2 Geomagnetic field in Earth-centered inertial coordinates, 87 inclination orbit.

To conclude the proof of the second claim it is necessary to prove that the matrix isgenerically positive definite. For, note that the matrix is obtained by integration of a three-by-three square (symmetric) matrix of rank two, namely the matrix B(t)BT (t). However,the kernel of the matrix B(t)BT (t) is not generically a constant vector, which implies > 0 generically. The set of bad trajectories, i.e., the trajectories for which the matrix is singular, is described by the simple relation

K = Ker(B(t)) = Im(b),for some constant vector b. However, by a trivial property of the vector product one has

K = Im(b(t)) = Im(A(q)b0(t)),hence all bad trajectories are such that, for all t,

A(q)b0(t) = (t)b,

for some scalar function (t), which is obviously a nongeneric condition.

5 STABILIZATION WITHOUT RATE FEEDBACK

The ability of ensuring attitude tracking without rate feedback is of great importance froma practical point of view. The problem of attitude stabilization without rate feedback has


been recently given an interesting solution in [1] for the case of a fully actuated spacecraft.In this section a similar approach is followed in the development of a dynamic control lawthat solves the problem for a magnetically actuated satellite. First, notice that the system(10) and the control law (which is similar in spirit to the one proposed in [1]):

z = q z = kpqr WT (q)(q z), (15)

(where > 0 and > 0) give rise to a closed-loop system having (q, 0, q/) as a locallyasymptotically stable equilibrium and qr 0 and 0 as t . On the basis of thisconsideration, which can be proved by means of the Lyapunov function

V = kp[(q4 1)2 + qTr qr] +12T I +

12(q z)T(q z), (16)

it is possible to give a solution to the magnetic attitude control problem without rate feed-back.

Proposition 2 Consider the system (9) and the control law

z = q zu = 2(kpqr + WT (q)(q z)). (17)

Then there exists > 0 such that for any 0 < < the control law renders the equilib-rium (q, 0, q/) of the closed-loop system (917) locally asymptotically stable. Moreover,almost all trajectories of the closed-loop system converge to this equilibrium.

Proof. As in the case of the state feedback control law we now introduce the coordinatestransformation

1 = q 2 =

3 = z . (18)

In the new coordinates, the system (9) is described by the equations:

1 = W (1)2I2 = S(2)I2 (t)(kp1r + WT (1)(1 3))3 = (1 3).

(19)

Again, system (19) satisfies all the hypotheses for the application of [6, Theorem 7.5] andusing the Lyapunov function given in Eq. (16) one can conclude that the system obtainedapplying the generalized averaging procedure has (q, 0, q/) as a locally asymptoticallystable equilibrium provided that

= limT

1T

T

0

B(t)BT (t)dt > 0,

and this holds nongenerically as demonstrated in the proof of Proposition 1.


Figure 3 Quaternion and angular rates for the attitude acquisition: state feedback controller.


Figure 4 Quaternion and angular rates for the attitude acquisition: output feedback controller.


Figure 5 Quaternion and angular rates for the attitude maneuver: state feedback controller.


Figure 6 Quaternion and angular rates for the attitude maneuver: output feedback controller.


6 SIMULATION RESULTS

To assess the performance of the proposed control laws the following simulation test casehas been analyzed. The considered spacecraft has an inertia matrix given by I = diag[27,17, 25] kg m2 and it is operating in a near-polar (87 inclination) orbit with an altitude of450 km and a corresponding orbit period of about 5600 s. For such a spacecraft, two sets ofsimulations have been carried out; the first is related to the acquisition of the target attitudeq =

[0 0 0 1

]T from an initial condition characterized by a high initial angular rate;the second is related to a point-to-point attitude maneuver from the initial attitude given byq0 =

[0 0 0 1

]Tto the target attitude q = 1

2

[1 0 0 1

]T.

In all cases, both the full-state feedback control law and the control law without ratefeedback have been applied.

The results of the simulations are displayed in Figs. 3 and 4 for the attitude acquisi-tion, and Figs. 5 and 6 for the attitude maneuver, from which the good performance of theproposed control laws can be seen.

7 CONCLUDING REMARKS

The problem of inertial attitude regulation for a small spacecraft using only magnetic coilsas actuators has been analyzed and it has been shown that a nonlinear low-gain PD-likecontrol law yields (almost) global asymptotic attitude regulation even in the absence ofadditional active or passive attitude control actuators such as momentum wheels or gravitygradient booms.

Acknowledgments

The work for this chapter was partially supported by the European network Nonlinear andAdaptive Control and by the MURST project, Identification and Control of IndustrialSystems.

REFERENCES

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3. S. Battilotti. Global output regulation and disturbance attenuation with global stabilityvia measurement feedback for a class of nonlinear systems. IEEE Transactions onAutomatic Control, 41(3):315327, 1996.

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8. M. Lovera. Optimal magnetic momentum control for inertially pointing spacecraft.European Journal of Control, 7(1):3039, 2001.

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10. P. Morin, C. Samson, J.-B. Pomet, and Z.-P. Jiang. Time-varying feedback stabilizationof the attitude of a rigid spacecraft with two controls. Systems and Control Letters,25:375385, 1995.

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2Adaptive Learning Control forSpacecraft Formation Flying

Hong Wong, Haizhou Pan, Marcio S. de Queiroz, and Vikram KapilaDepartment of Mechanical, Aerospace, and Manufacturing Engineering,

Polytechnic University, Brooklyn, NYDepartment of Mechanical Engineering, Louisiana State University, Baton Rouge, LA

This chapter considers the problem of spacecraft formation flying in the presence of peri-odic disturbances. In particular, the nonlinear position dynamics of a follower spacecraftrelative to a leader spacecraft are utilized to develop a learning controller that accounts forthe periodic disturbances entering the system model. Using a Lyapunov-based approach, afull-state feedback control law, a parameter update algorithm, and a disturbance estimaterule are designed that facilitate the tracking of given reference trajectories in the presenceof unknown spacecraft masses. Illustrative simulations are included to demonstrate theefficacy of the proposed controller.

1 INTRODUCTION

Spacecraft formation flying is an enabling technology that distributes mission tasksamong many spacecraft. Practical applications of spacecraft formation flying include sur-veillance, passive radiometry, terrain mapping, navigation, and communication, where us-ing groups of spacecraft permits reconfiguration and optimization of formation geometriesfor single or multiple missions.

Distributed spacecraft performing space-based sensing, imaging, and communicationprovide larger aperture areas at the cost of maintaining a meaningful formation geome-try with minimal error. The ability to enlarge aperture areas beyond conventional singlespacecraft capabilities improves slow target detection for interferometric radar and allowsfor enhanced resolution in terms of geolocation [1]. Current spacecraft formation controlmethodologies provide good tracking of relative position trajectories between a leader andfollower spacecraft pair. However, in the presence of disruptive disturbances, designing acontrol law that compensates for unknown, time-varying, disturbance forces is a challeng-ing problem.

The current spacecraft formation flying literature largely addresses the problem ofcontrol of spacecraft relative positions using linear and nonlinear controllers. For exam-ple, linear and nonlinear formation dynamic models have been developed for formationmaintenance and a variety of control designs have been proposed to guarantee the desired


16 H. Wong et al.

Figure 1 Schematic representation of the spacecraft formation flight system.

formation performance [210]. Specifically, a linear formation dynamic model known asHills equation is given in [8, 11], which constitutes the foundation for the application ofvarious linear control techniques to the distributed spacecraft formation maintenance prob-lem [3, 4, 6, 8, 10]. Model-based and adaptive nonlinear controllers for the leader-followerspacecraft configuration are given in [2, 7, 9]. However, control design to track desiredtrajectories under the influence of exogenous disturbances within the formation dynamicmodel has not been fully explored.

In this chapter, we consider the problem of spacecraft formation flying in the pres-ence of periodic disturbances. In particular, the nonlinear position dynamics of a followerspacecraft relative to a leader spacecraft are utilized to develop a learning controller [12],which accounts for the periodic disturbances entering the system model. First, in Section 2,the EulerLagrange method is used to develop a model for the spacecraft formation flyingsystem. Next, a trajectory tracking control problem is formulated in Section 3. In Sec-tion 4, using a Lyapunov-based approach, a full-state feedback control law, a parameterupdate algorithm, and a disturbance estimate rule are designed that facilitate the trackingof given reference trajectories in the presence of unknown spacecraft masses. Illustrativesimulations are included in Section 5 to demonstrate the efficacy of the proposed controller.Finally, some concluding remarks are given in Section 6.

2 SYSTEM MODEL

In this section, referring to Fig. 1, we develop the nonlinear model characterizing theposition dynamics of follower spacecraft relative to a leader spacecraft using the EulerLagrange methodology. We assume that the leader spacecraft exhibits planar dynamics ina closed elliptical orbit with the Earth at its prime focus. In addition, we consider thatthe inertial coordinate system {X,Y, Z} is attached to the center of the Earth. Next, let(t) R3 denote the position vector from the origin of the inertial coordinate frame to

Adaptive Learning Control for Spacecraft Formation Flying 17

the leader spacecraft. Furthermore, we assume a right-hand coordinate frame {x, y, z}is attached to the leader spacecraft with the y-axis pointing along the direction of the vec-tor , the z-axis pointing along the orbital angular momentum of the leader spacecraft,and x-axis being mutually perpendicular to the y-axis and z-axis, and pointing in thedirection that completes the right-handed coordinate frame.

Modeling the dynamics of a spacecraft relative to the Earth utilizes the fact that theenergy of the spacecraft moving under the gravitational influence is conserved. Next, letthe Lagrangian function L be defined as the difference between the specific-kinetic energyT and the specific-potential energy V , i.e., L = T V . Then, the Lagrangian function forthe leader spacecraft L(t) R is given by

L= 0.5v

2 +

|||| , (2.1)

where a =aTa, for an arbitrary n-dimensional vector a, v(t) = ||v(t)||, with v(t)

as defined in (2.4) below, is the Earth gravity constant, and |||| is the specific-potentialenergy of the leader spacecraft. Similarly, the Lagrangian function for the follower space-craft Lf (t) R is given by

Lf= 0.5v

2f +

||+ qf || , (2.2)

where qf (t) R3 is the position vector of the follower spacecraft relative to the leaderspacecraft, expressed in the coordinate frame {x, y, z} as qf =xf + yf + zf k andvf (t) = ||vf (t)||, with vf (t) as defined in (2.3) below.

Next, let v(t) and vf (t) denote the absolute velocities (i.e., velocity relative to theinertial coordinate system) of the leader spacecraft and the follower spacecraft, respectively,expressed in the moving coordinate frame {x, y, z}. In addition, let vrelf (t) denotethe velocity of the follower spacecraft relative to the leader spacecraft, measured in thecoordinate frame {x, y, z}. Then, it follows that

vf = v + vrelf + k qf , (2.3)

where

v = r + r, (2.4)vrelf = xf + yf + zf k, (2.5)

r(t) R denotes the instantaneous distance of the leader spacecraft from the center of theearth and (t) R is the orbital angular speed of the leader spacecraft. In addition, (t)in the moving coordinate frame {x, y, z} can be expressed as = r.

To obtain the leader spacecraft dynamics relative to the Earth, we use the conservativeform of the Lagranges equations [13] on the leader spacecraft given by

d

dt

(L

) L

= 0, (2.6)

where {r, }, which denotes the set of polar orbital elements describing the motionof the leader spacecraft. This yields a set of differential equations of motion for the leader

18 H. Wong et al.

spacecraft in the chosen coordinate system. Specifically, substituting the magnitude of thevelocity vector of (2.4) into (2.1) yields

L = 0.5r22 + 0.5r2 +

r. (2.7)

For L given by (2.7), an application of (2.6) results in a set of planar dynamics describingthe elliptical motion of the leader spacecraft given as

r r2 + r

= 0, (2.8)r2 = 0. (2.9)

Taking the time derivative of (2.9) produces a dynamic relationship coupling and r asr + 2r = 0. (2.10)

To obtain the follower spacecraft dynamics relative to the Earth, we utilize the samestructure of (2.6) in the form

d

dt

(Lff

) Lf

f= 0, (2.11)

where f {xf , yf , zf}, which denotes the set of cartesian elements describing the mo-tion of the follower spacecraft relative to the leader spacecraft. Substituting the magnitudeof the velocity vector of (2.3) into (2.2) yields

Lf = 0.5(xf r yf )2 + 0.5(r + yf + xf )2 + 0.5z2f +

+ qf . (2.12)

For Lf given by (2.12), an application of (2.11) results in

(r + 2r) + xf 2yf (yf + 2xf

)+ xf+qf3 = 0,(

r r2)+ yf + 2xf + (xf 2yf)+ (yf+r)+qf3 = 0,zf +

zf+qf3 = 0.

Substituting the leader planar dynamics of (2.8) and (2.10) into the above homogeneousdynamics results in the thrust free dynamics of the follower spacecraft relative to the leaderspacecraft, expressed in the coordinate frame {x, y, z}, given by

xf 2yf yf 2xf + xf+qf3 = 0,yf + 2xf + xf 2yf + (yf+r)+qf3

r3 = 0,

zf +zf

+qf3 = 0.(2.13)

After premultiplying the nonlinear dynamics of (2.13) with the follower spacecraftmass mf and using the method of virtual work for the insertion of external forcing terms[13], e.g., disturbance and thrusting forces for the leader spacecraft and the follower space-craft, the nonlinear position dynamics of the follower spacecraft relative to the leader space-craft can be arranged in the following form [14, 15]:

mf qf + C()qf +N(qf , , , r, u) + Fd = uf , (2.14)


where C is a Coriolis-like matrix defined as

C = 2mf

0 1 01 0 0

0 0 0

, (2.15)

N is a nonlinear term consisting of gravitational effects and inertial forces

N = mf

2xf yf + xf||+qf ||3 +

ux

m

2yf + xf + (yf+r)||+qf ||3 r||||3 +

uy

mzf

||+qf ||3 +uz

m

, (2.16)

Fd(t) R3 is a composite disturbance force vector given by

Fd= Fdf

mfm

Fd , (2.17)

where u(t) R3 and uf (t) R3 are the control inputs to the leader spacecraft and thefollower spacecraft, respectively, m is the mass of the leader spacecraft, ux , uy , uz arecomponents of the vector u, and Fdf (t) R3 and Fd(t) R3 are disturbance vectors forthe follower spacecraft and the leader spacecraft, respectively. In this chapter, we assumethat Fd is a periodic disturbance with a known period > 0 such that Fd(t+ ) = Fd(t),t 0. Note that periodic disturbances in formation dynamics may arise due to, e.g., solarpressure disturbance and gravitational disturbance, among others.

The following remarks further facilitate the subsequent control methodology and sta-bility analysis.

Remark 1 The Coriolis matrix C of (2.15) satisfies the skew symmetric property ofxTCx = 0, x R3. (2.18)

Remark 2 The left-hand side of (2.14) produces an affine parameterizationmf + Cqf +N = Y (, qf , qf , , , r, u), (2.19)

where (t) R3 is a dummy variable with components x, y , and z , Y R32 is aregression matrix, composed of known functions, defined as

Y =

x 2yf yf 2xf + xf+qf3 uxy + 2xf 2yf + xf + (yf+r)+qf3

r3 uy

z +zf

+qf3 uz

, (2.20)

R2 is the unknown, constant system parameter vector defined as

=[mf

mfm

]T. (2.21)

Remark 3 In this chapter, we consider that the composite disturbance vector in the positiondynamics of the follower spacecraft relative to the leader spacecraft can be upper boundedas follows

Fd(t) , t 0, (2.22)where is a positive constant and denotes the usual infinity norm.

20 H. Wong et al.

3 PROBLEM FORMULATION

In this section, we formulate a control design problem such that the relative position qftracks a desired relative trajectory qdf (t) R3, i.e., lim

tqf (t) = qdf (t). The effectivenessof this control objective is quantified through the definition of a relative position errore(t) R3 as

e = qdf qf . (3.1)

The control design methodology is to construct a control algorithm that obtains the afore-mentioned tracking result in the presence of the unknown composite disturbance vectordefined in (2.17) and the unknown constant parameter vector of (2.21). We assume thatthe relative position and velocity measurements (i.e., qf and qf ) of the follower spacecraftrelative to the leader spacecraft are available for feedback.

To facilitate the control development, we assume that the desired trajectory qdf andits first two time derivatives are bounded functions of time. Next, we define the parameterestimation error (t) R2 as the difference between the actual parameter vector and theparameter estimate (t) R2, i.e.,

= . (3.2)

In addition, we define the composite disturbance error Fd(t) R3 as the difference be-tween the composite disturbance vector Fd and the disturbance estimate Fd(t) R3, i.e.,

Fd= Fd Fd. (3.3)

Finally, we define the components of a saturation function sat() R3 as

sat(s)i=

{si for |si|

sgn(si) for |si| > s R3 , (3.4)

where i {x, y, z} and sx, sy, sz are the components of the vector s.To facilitate the subsequent stability analysis, we define the saturated disturbance

estimation error variable R3 as () = sat(Fd()

) sat(Fd()).Remark 4 The saturation function (3.4) satisfies the following useful property:(sat(a) sat(b)

)T (sat(a) sat(b)

) (a b)T (a b), a, b R3. (3.5)

4 ADAPTIVE LEARNING CONTROLLER

In this section we develop an adaptive learning controller based on the system model of(2.14) such that the tracking error variable e exhibits asymptotic stability. Before we beginthe control design, we define an auxiliary filter tracking error variable (t) R3 as

= e+ e, (4.1)

where R33 is a constant, diagonal, positive definite, control gain matrix.


4.1 Controller Design

To initiate the control design, we take the time derivative of (4.1) and premultiply theresulting equation by mf to obtain

mf = mf (qdf qf ) +mfe, (4.2)where the second time derivative of (3.1) has been used. Substituting (2.14) into (4.2)results in

mf = mf (qdf + e) + C qf +N + Fd uf . (4.3)Simplifying (4.3) into a more advantageous form, we obtain

mf = Y (qdf + e, qf , qf , , , r, u)+ Fd uf , (4.4)where (2.19) has been used with = qdf + e in the definition of (2.20). Equation (4.4)characterizes the open-loop dynamics of and is used as the foundation for the synthesisof the adaptive learning controller.

Based on the form of the open-loop dynamics of (4.4), we design the control law ufas

uf = Y +K + Fd, (4.5)where K R33 is a constant, diagonal, positive definite, control gain matrix. Guidedby the subsequent Lyapunov stability analysis, the parameter update law for in (4.5) isselected as

= Y T , (4.6)

where R22 is a constant, diagonal, positive definite, adaptation gain matrix. Finally,the disturbance estimate vector Fd is updated according to

Fd(t) = sat(Fd(t )

)+ kL(t), (4.7)

where kL R is a constant, positive, learning gain.Using the control law of (4.5) in the open-loop error dynamics of (4.4) results in the

following closed-loop dynamics for :

mf = Y K + Fd. (4.8)In addition, computing the time derivative of (3.2), using the fact that the parameter vector is constant, and the parameter update law of (4.6), the closed-loop dynamics for theparameter estimation error is given by

= Y T . (4.9)

4.2 Stability Analysis

The proposed control law of (4.5)(4.7) provides a stability result for the position andvelocity tracking errors as illustrated by the following theorem. In order to state the mainresult of this section, we define

= min

(K +

12kLI3

), (4.10)

22 H. Wong et al.

where I3 is the 3 3 identity matrix and min() denotes the smallest eigenvalue of amatrix.

The adaptive learning control law described by (4.5)(4.7) ensures asymptotic con-vergence of the position and velocity tracking errors as delineated by

limt e(t), e(t) = 0. (4.11)

Proof. We define a nonnegative function as follows:

V (t) =12mf

T +12T1+

12kL

tt

()T()d. (4.12)

Taking the time derivative of (4.12) along the closed-loop dynamics of (4.8) and (4.9)results in

V (t) = T(Fd(t)K

)+

12kL

T (t)(t) 12kL

T (t )(t ). (4.13)

Substituting the composite disturbance estimate of (4.7), noting that sat(Fd(t )

)=

Fd(t) due to (2.22), and utilizing the definition of (3.3) into (4.13) produces

V (t) = T(Fd(t)K

)+ 12kL

T (t)(t)

12kL(Fd(t) + kL

)T (Fd(t) + kL

). (4.14)

Expanding (4.14) and combining like terms produces

V (t) = T(K +

12kLI3

) 1

2kL

(FTd (t)Fd(t) T (t)(t)

). (4.15)

Utilizing the inequality of (3.5) and definitions of Fd and , we can upper-bound (4.15) asfollows:

V (t) T(K +

12kLI3

). (4.16)

Taking the norm of the right-hand side of (4.16) and using the definition of (4.10) results inV (t) 2. (4.17)

Since V is a nonnegative function and V is a negative semidefinite function, V is a non-increasing function. Thus V (t) L as described by

V ((t), (t), (t)) V ((0), (0), (0)), t 0. (4.18)Using standard signal chasing arguments, all signals in the closed-loop system can now beshown to be bounded. Using (4.8) along with the boundedness of all signals in the closed-loop system, we now conclude that (t) L. Solving the differential inequality of (4.17)results in

V (0) V ()

0

(t)2dt. (4.19)


Figure 2 Actual trajectory of follower spacecraft relative to leader spacecraft.

Since V (t) is bounded, t 0, we conclude that (t) LL2, t 0. Finally, using

Barbalats Lemma [16, 17], we conclude that

limt (t) = 0. (4.20)

Using the definition of in (4.1), the limit statement of (4.20), and Lemma 1.6 of [16],yield the result of (4.11).

5 SIMULATION RESULTS

The illustrative numerical example considered here utilizes orbital elements to propagatethe leader spacecraft in a low-altitude orbit similar to the TechSat 21 mission specifications[1]. The adaptive learning control law described in (4.5) is simulated for the dynamics ofthe follower spacecraft relative to the leader spacecraft. The leader spacecraft is assumedto have the following orbital parameters a = 7200 km, e = 0.001, (0) = 0 rad, T = = 1.6889 hr, where a is the semimajor axis of the elliptical orbit of the leader spacecraft,e is the orbital eccentricity of the leader spacecraft, (t) R is the time-varying trueanomaly of the planar dynamics of the leader spacecraft, and T is the orbital period of theleader spacecraft. The relative trajectory between the follower spacecraft and the leaderspacecraft was generated by numerically solving the thrust-free dynamics given by (2.13).The initial conditions for the relative position and velocity between the follower spacecraftand the leader spacecraft were obtained in the same manner as in [15] and are given by

qdf (0) = [0 20 1]m, qdf (0) = [149.1374 0 0] mhr . (5.1)

24 H. Wong et al.

Figure 3 Tracking error of follower spacecraft relative to leader spacecraft.

Additional parameters used for simulation within the spacecraft formation flying sys-tem are as follows = 5.165862481021 m3hr2 , m = 100 kg, mf = 100 kg, u = [0 0 0]

T

N, Fd = [1.9106 1.906 1.517]T sin(2 t) 105N. Finally, in the following sim-

ulation, the parameter and disturbance estimates were all initialized to zero.The control, adaptation, and learning gains, in the control law of (4.5)(4.7), are

obtained through trial and error in order to obtain good performance for the tracking errorresponse. The following resulting gains were used in this simulation K = diag (3, 3, 3) 103, = diag (1, 1, 1), = diag (1, 1) 102, and kL = 1000. The actual trajectory qf ,shown in Fig. 2, is initialized to be the same as the desired trajectory in (5.1). Figures 3and 4 show the tracking error e and velocity tracking error e, respectively. The controlinput uf and the disturbance estimate Fd are shown in Figs. 5 and 6, respectively. Finally,since we assume the leader is in a thrust free orbit about the earth, the second componentin the parameter estimate is neglected while the first component of the parameter estimateis shown in Fig. 7.

6 CONCLUSION

In this paper, we designed an adaptive learning control algorithm for the position dynamicsof the follower spacecraft relative to the leader spacecraft. A Lyapunov-type design wasused to construct a full-state feedback control law and parameter and disturbance estimatesthat facilitate the tracking of reference trajectories with global asymptotic convergence.Simulation results were given to illustrate the efficacy of the control design in the presenceof unknown periodic disturbance forces.


Figure 4 Velocity tracking error of follower spacecraft relative to leader spacecraft.

Figure 5 Control effort for follower spacecraft.

Acknowledgments

Research was supported in part by the National Aeronautics and Space AdministrationGoddard Space Flight Center under Grant NAG5-11365; AFRL/VACA, WPAFB, OH; andthe New York Space Grant Consortium under Grant 39555-6519.

26 H. Wong et al.

Figure 6 Disturbance estimate for follower spacecraft.

Figure 7 Parameter estimate for follower spacecraft.

H.W. and V.K. are grateful to the Air Force Research Laboratory/VACA, Wright-Patterson Air Force Base, OH, for their hospitality during the summer of 2001.


REFERENCES

1. http://www.vs.afrl.af.mil/factsheets/TechSat21.html.2. F. Y. Hadaegh, W. M. Lu, and P. C. Wang, Adaptive control of formation flying space-

craft for interferometry, IFAC Conference on Large Scale Systems, pp. 97102, 1998.3. V. Kapila, A. G. Sparks, J. Buffington, and Q. Yan, Spacecraft formation flying: Dy-

namics and control, AIAA J. GCD., vol. 23, pp. 561564, 2000.4. C. Sabol, R. Burns, and C. McLaughlin, Formation flying design and evolution, Proc.

of the AAS/AIAA Space Flight Mechanics Meeting, Paper No. AAS99121, 1999.5. H. Schaub, S. R. Vadali, J. L. Junkins, and K. T. Alfriend, Spacecraft formation flying

control using mean orbit elements, AAS G. Contr. Conf., pp. 97102, 1999.6. R. J. Sedwick, E. M. C. Kong, and D. W. Miller, Exploiting orbital dynamics and mi-

cropropulsion for aperture synthesis using distributed satellite systems: Applicationsto Techsat 21, D.C.P. Conf., AIAA Paper No. 98-5289, 1998.

7. M. S. de Queiroz, V. Kapila, and Q. Yan, Adaptive nonlinear control of multiplespacecraft formation flying, AIAA J. GCD., vol. 23, pp. 385390, 2000.

8. R. H. Vassar and R. B. Sherwood, Formation keeping for a pair of satellites in acircular orbit, AIAA J. GCD., vol. 8, pp. 235242, 1985.

9. H. Wong, V. Kapila, and A. G. Sparks, Adaptive output feedback tracking control ofmultiple spacecraft, Proc. ACC, 2001.

10. H.-H. Yeh and A. G. Sparks, Geometry and control of satellite formations, Proc.ACC., pp. 384388, 2000.

11. R. R. Bate, D. D. Mueller, and J. E. White, Fundamentals of Astrodynamics. NewYork: Dover, 1971.

12. B. Costic, M. de Queiroz, and D. Dawson, A new learning control approach to themagnetic bearing benchmark system, Proc. ACC, pp. 26392643, 2000.

13. H. L. Langhaar, Energy Methods in Applied Mechanics. New York: John Wiley andSons, 1962.

14. M. de Queiroz, V. Kapila, , and Q. Yan, Adaptive nonlinear control of multiple space-craft formation flying, J. GCD., vol. 23, pp. 385390, 2000.

15. Q. Yan, G. Yang, V. Kapila, and M. S. de Queiroz, Nonlinear dynamics, trajectorygeneration, and adaptive control of multiple spacecraft in periodic relative orbits, AASG. Contr. Conf., Paper No. 00-013, 2000.

16. D. M. Dawson, J. Hu, and T. C. Burg, Nonlinear Control of Electric Machinery. NewYork: Marcel Dekker, 1998.

17. J.-J. E. Slotine and W. Li, Applied Nonlinear Control. Englewood Cliffs, NJ: Prentice-Hall, 1991.

3Spectral Properties of the GeneralizedResolvent Operator for an Aircraft

Wing Model in Subsonic AirflowMarianna A. Shubov

Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX

In this chapter we announce a series of results on the asymptotic and spectral analysis ofan aircraft wing model in a subsonic air flow. This model has been developed in the FlightSystems Research Center of UCLA and is presented in the works by A.V. Balakrishnan.The model is governed by a system of two coupled integro-differential equations and atwo-parameter family of boundary conditions modeling action of the self-straining actu-ators. The unknown functions (the bending and torsion angle) depend on time and onespatial variable. The differential parts of the above equations form a coupled linear hyper-bolic system; the integral parts are of the convolution type. The system of equations ofmotion is equivalent to a single operator evolution-convolution equation in the state spaceof the system equipped with the energy metric. The Laplace transform of the solution ofthis equation can be represented in terms of the so-called generalized resolvent operator.The generalized resolvent operator is a finite-meromorphic function on the complex planehaving the branch cut along the negative real semi-axis. The poles of the generalized resol-vent are precisely the aeroelastic modes and the residues at these poles are the projectorson the generalized eigenspaces. In this paper, we present the following results: (i) spectralproperties of an analytic operator-valued function associated with the generalized resolventoperator; (ii) asymptotic distribution of the aeroelastic modes; (iii) the Riesz basis propertyof the system of mode shapes in the energy space.

1 INTRODUCTION

In this chapter, we formulate a series of results on the asymptotic and spectral analysis ofan aircraft wing model. The model has been developed in the Flight Systems ResearchCenter of the University of California at Los Angeles. The mathematical formulation ofthe problem can be found in the works by A.V. Balakrishnan [24]. This model has beendesigned to find an approach to control the flutter phenomenon in an aircraft wing in asurrounding airflow by using the so-called self-straining actuators.

The model, which is used in [24], is the 2-D strip model that applies to bare wings ofhigh-aspect ratio [12, 15]. The structure is modeled by a uniform cantilever beam that bends


30 M. A. Shubov

and twists. The aerodynamics is assumed to be subsonic, incompressible, and inviscid. Inaddition, the author of [24] has added self-straining actuators using a currently acceptedmodel (e.g., [410, 13, 18, 20, 23, 25, 26]).

Flutter, which is known as a very dangerous aeroelastic development, is the onset, be-yond some speedaltitude combinations, of unstable and destructive vibrations of a liftingsurface in an airstream. Flutter is most commonly encountered on bodies subjected to largelateral aerodynamic loads of the lift type, such as aircraft wings, tails, and control surfaces.The only air forces necessary to produce it are those due to the deflection of the elasticstructures from the undeformed state. The flutter or critical speed uf and frequency fare defined as the lowest airspeed and corresponding circular frequency at which a givenstructure flying at given atmospheric density and temperature will exhibit sustained, simpleharmonic oscillations. Flight at uf represents a borderline condition or neutral stabilityboundary, because all small motions must be stable at speeds below uf , whereas divergentoscillations can ordinarily occur in a range of speeds (or at all speeds) above uf .

Probably, the most dangerous type of aircraft flutter results from coupling between thebending and torsional motions of a relatively large aspectratio wing and tail. The modelpresented in [24] has been designed to treat flutter caused by this type of coupling. Wealso mention paper [14] by P. Dierolf et al., in which the authors treat the flutter problemas a perturbation problem for semigroups.

Our main objective is to find the timedomain solution of the initialboundary valueproblem from [24]. This objective requires very detailed mathematical analysis of theproperties of the system. Now, we describe the content of this chapter. In Section 2, wegive a precise mathematical formulation of the problem. The model is governed by a sys-tem of two coupled partial integro-differential equations subject to a two-parameter familyof boundary conditions. The parameters are introduced in order to model the action of self-straining actuators as known in current engineering and mathematical literature. In Section3, we reformulate the problem and set it into an operator format. We also show that thedynamics are defined by two matrix operators in the energy space. One of the aforemen-tioned operators is a matrix differential operator, and the second one is a matrix integralconvolution-type operator. The aeroelastic modes (or the discrete spectrum of the prob-lem) are closely related to the discrete spectrum of the matrix differential operator whilethe continuous spectrum is completely defined by the matrix integral operator. We notethat, if the speed of an airstream u = 0, then the integral operators vanish and the appropri-ate purely structural problem will have only a discrete spectrum. In Section 4, we formulatethe asymptotic and spectral results related to the matrix differential operator. In Section 5,we present asymptotics for aeroelastic modes and formulate and discuss the importance ofBalakrishnans Theorem (see Theorem 5.2). In Section 6, we discuss the properties of theLaplace transform of the matrix integral operator. In Section 7, we formulate the resultsconcerning the properties of the adjoint operator-valued function, the properties of whichare important for the spectral decomposition of the resolvent operator generated by theaircraft wing model.

In the conclusion of the Introduction, we describe what kind of a control problem willbe considered in connection with the flutter suppression. In the specific wing model con-sidered in the current chapter, both the matrix differential operator and the matrix integraloperator contain entries depending on the speed u of the surrounding air flow. Therefore,the aeroelastic modes are functions of u : k = k(u), (k Z). The wing is stable ifRe k(u) < 0 for all k. However, if u is increasing, some of the modes move to the right

Mathematical Analysis of Wing Model 31

half-plane. The flutter speed ufk for the k-th mode is defined by the relationRek(ufk) = 0.

To understand the flutter phenomenon, it is not sufficient to trace the motion of aeroelasticmodes as functions of a speed of airflow. It is necessary to have efficient representationsfor the solutions of our boundary-value problem, containing the contributions from boththe discrete and continuous parts of the spectrum. Such a representation will provide aprecise description of the solution behavior. It is known that flutter cannot be eliminatedcompletely. To successfully suppress flutte , one should design self-straining actuators (i.e.,in the mathematical language, to select parameters in the boundary conditions that are thecontrol gains and in formulas (2.10) and (2.11) of Section 2) in such a way that fluttedoes not occur in the desired speed range. This is a highly nontrivial boundary controlproblem.

2 STATEMENT OF THE PROBLEM

In this section, we give a precise formulation of the initial-boundary value problem.Following [24], let us introduce the dynamical variables

X(x, t) =(

h(x, t)(x, t)

), L x 0, t 0, (2.1)

where h(x, t) is the bending and (x, t) the torsion angle. The model can be described bya linear system

(Ms Ma)X(x, t) + (Ds uDa)X(x, t) + (Ks u2Ka)X =[f1(x, t)f2(x, t)

]. (2.2)

We will use the notation . (dot) to denote differentiation with respect to t. We use thesubscripts s and a to distinguish the structural and aerodynamical parameters, respec-tively. All 2 2 matrices in Eq. (2.2) are given by the following formulae:

Ms =[m SS I

], Ma = ()

[1 aa (a2 + 1/8)

], (2.3)

where m is the density of the flexible structure (mass per unit length), S is the mass mo-ment, I is the moment of inertia, is the density of air, and a is linear parameter of thestructure (1 a 1).

Ds =[

0 00 0

]Da = ()

[0 1

1 0], (2.4)

Ks =

[E

4

x4 00 G 2x2

]Ka = ()

[0 00 1

], (2.5)

where E is the bending stiffness, and G is the torsion stiffness. The parameter u in Eq.(2.2) denotes the stream velocity. The right-hand side of system (2.2) can be representedas the following system of two convolution-type integral operations:

f1(x, t) = 2t

0

[uC2(t ) C3(t )

]g(x, )d, (2.6)

32 M. A. Shubov

f2(x, t) = 2t

0

[1/2C1(t ) auC2(t ) + aC3(t )

+uC4(t ) + 1/2C5(t )]g(x, )d, (2.7)

g(x, t) = u(x, t) + h(x, t) + (1/2 a)(x, t). (2.8)The aerodynamical functions Ci, i = 1 . . . 5, are defined in the following ways [11]:

C1() =

0

etC1(t)dt =u

e/u

K0(/u) +K1(/u), Re > 0,

C2(t) =

t

0

C1()d, C4(t) = C2(t) + C3(t),

C3(t) =

t

0

C1(t )(u u22 + 2u)d,

C5(t) =

t

0

C1(t )((1 + u)u22 + 2u (1 + u)2)d, (2.9)

where K0 and K1 are the modified Bessel functions of the zero and first orders [1, 19]. Theself-straining control actuator action can be modeled by the following boundary conditions:

Eh(0, t) + h(0, t) = 0, h(0, t) = 0, (2.10)

G(0, t) + (0, t) = 0, , C+ {}, (2.11)where C+ is the closed right half-plane. The boundary conditions at x = L are

h(L, t) = h(L, t) = (L, t) = 0. (2.12)

Let the initial state of the system be given as follows:

h(x, 0) = h0(x), h(x, 0) = h1(x), (x, 0) = 0(x), (x, 0) = 1(x). (2.13)

We will consider the solution of the problem given by Eq. (2.2) and conditions (2.10)(2.13) in the energy space H. To introduce the metric of H, we assume that the parameterssatisfy the following two conditions:

det

[m SS I

]> 0, 0 < u

2G

L

. (2.14)

Let {Ci}2i=1 be the kernels in the convolution operations in (2.6), (2.7), i.e.,

C1(t) = 2(uC2(t) C3(t)),C2(t) = 2 (1/2C1(t) auC2(t) + aC3(t) + uC4(t) + 1/2C5(t)), (2.15)


and let M,D,K be the following matrices:

M = Ms Ma, D = Ds uDa, K = Ks u2Ka . (2.16)Then Eq. (2.2) can be written in the form

MX(x, t) +DX(x, t) +KX(x, t) = (FX)(x, t)), t 0, (2.17)where the matrix integral operator F can be given by the formula

F = t0

C1(t )(dd )d

t0

C1(t )[u +(1/2 a)(dd )]d

t0

C2(t )(dd )d

t0

C2(t )[u +(1/2 a)(dd )]d

=

C1 (

dd ) C2 (u +(1/2 a) dd )

C2 ( dd ) C1 (u +(1/2 a) dd )

, (2.18)

where the notation has been used for the convolution.Remark 2.1. The model described by Eq. (2.17) in the case u = 0 occurs actually in

aeroelastic problems (see classic textbooks [12, 15]) if one ignores aeroelastic forces. How-ever, the boundary conditions in [12, 15] that complemented the system of Eq. (2.17) aretotally different. To the best of our knowledge, the whole problem consisting of Eq. (2.17)(u = 0) and boundary conditions (2.10) and (2.11) has been considered only in papers [5,9].

3 OPERATOR REFORMULATION OF THE PROBLEM

We start with the state space of the system (the energy space). Let H be the set of4-component vector-valued functions = (h, h, , )T (0, 1, 2, 3)T (the super-script T means the transposition) obtained as a closure of smooth functions satisfying theconditions

0(L) = 0(L) = 2(L) = 0 (3.1)in the following energy norm:

||||2H = 1/20

L

[E|0 (x)|2 +G|2(x)|2 + m|1(x)|2 + I|3(x)|2

+S(3(x)1(x) + 3(x)1(x)) u2|2(x)|2]dx, (3.2)

wherem = m+ , S = S a, I = I + (a2 + 1/8). (3.3)

Note that under the first condition in (2.14), formula (3.2) defines a positively definitivemetric. Our goal is to rewrite Eq. (2.17) as the first order in time evolutionconvolutionequation in the energy space. As the first step, we will represent Eq. (2.17) in the form

X +M1DX +M1KX = M1FX. (3.4)

34 M. A. Shubov

Note that, due to the first condition in (2.14), M1 exists.Using formula (3.1), one can easily see that the initial-boundary value problem de-

fined by Eq. (3.4) and conditions (2.10)(2.13) can be represented in the form = iL+ F, = (0, 12, 3)T , |t=0 = 0. (3.5)

L is the following matrix differential operator in H:

L = i

0 1 0 0

EI

d4

dx4uS

S

(G d

2

dx2+ u2

)uI

0 0 0 1ES

d4

dx4um

m

(G d

2

dx2+ u2

)uS

(3.6)

defined on the domain

D(L) = { H : 0 H4(L, 0), 1 H2(L, 0), 2 H4(L, 0),3 H1(L, 0); 1(L) = 1(L) = 3(L) = 0; 0 (0) = 0;

E0 (0) + 1(0) = 0, G

2(0) + 3(0) = 0}, (3.7)

where Hi, i = 1, 2, 4, are the standard Sobolev spaces. F is a linear integral operator in Hgiven by the formula

F=

1 0 0 00 [I(C1) S(C2)] 0 00 0 1 00 0 0 [S(C1) + m(C2)]

0 0 0 00 1 u (1/2 a)0 0 0 00 1 u (1/2 a)

.

(3.8)It turns out that the spectral properties of both the differential operator L and the integraloperator F are of crucial importance for the representation of the solution.

Important Remark. At this moment, we would like to emphasize the difference be-tween the aircraft wing model considered in this chapter and models related to other flexiblestructures that have been studied by the author in recent years. Each of the aforementionedmodels (a spatially nonhomogeneous damped string, a 3-dimensional damped-wave equa-tion with spatially nonhomogeneous coefficients, Timoshenko beam model, and coupleEulerBernoulli and Timoshenko beam model) can be described by an abstract evolutionequation in a Hilbert space H:

(t) = iA(t), (t) H, t 0. (3.9)The dynamics generator A is an unbounded nonself-adjoint operator in H. This operatoris our main object of interest in the aforementioned works. In each of the above examples,A is a specific differential or matrix differential operator in H. The domain of A is definedby the differential expression and the corresponding boundary conditions. A has a com-pact resolvent and, therefore, has a purely discrete spectrum. The main results concerningthe dynamics generator A established in the authors works can be split into the followingthree parts: (i) the explicit asymptotic formulae for the spectrum; (ii) asymptotic represen-tations for the generalized eigenvectors; (iii) the Riesz basis property of the generalizedeigenvectors in H.


In contrast with the evolution equation (3.9), the aircraft wing model is described byan evolutionconvolution equations of the form

(t) = i A(t) +

t

0

F (t )()d. (3.10)

Here () H the energy space of the system, is a 4-component vector function,A (A = L) is a matrix differential operator, and F (t) is a matrix-valued function.

Equation (3.10) does not define an evolution semigroup and does not have a dynamicsgenerator. So it is necessary to explain what is understood as the spectral analysis of Eq.(3.10).

Let us take the Laplace transformation of both parts of Eq. (3.5). Formal solution inthe Laplace representation can be given by the formula

() =(I iA F ()

)1 (I F ()

)0, (3.11)

where 0 is the initial state, i.e., (0) = 0, and the symbol is used to denote theLaplace transform. In order to have representation of the solution in the spacetime do-main, we have to calculate the inverse Laplace transform of Eq. (3.11). To this end, it isnecessary to investigate the generalized resolvent operator

R() =(I iA F ()

)1. (3.12)

In the case of the 1-dimensional wing model, R() is an operator-valued meromorphicfunction on the complex plane with a branch cut along the negative real semi-axis. Thepoles of R() are called the eigenvalues, or the aeroelastic modes. The residues of R()at the poles are precisely the projectors on the corresponding generalized eigenspaces. Thebranch cut corresponds to the continuous spectrum.

Definition 3.1. Those values of for which the equation

[I iL F()] = 0 (3.13)has nontrivial solutions are called aeroelasitc modes. The corresponding solutions arecalled mode shapes.

4 ASYMPTOTIC AND SPECTRAL PROPERTIES OF THE MATRIXDIFFERENTIAL OPERATOR L

In this section, we describe the main properties of the matrix differential operator L[2023].

Theorem 4.1. a) L is a closed linear operator in H whose resolvent is compact,and therefore, the spectrum is discrete [16, 17].b) Operator L is nonselfadjoint unless = = 0. If 0 and 0,then this operator is dissipative, i.e., (L,) 0 for D(L) [16, 17]. Theadjoint operatorL is given by the same matrix differential expression (3.6) on the domainobtained from (3.7) by replacing the parameters and with () and (), respectively.

36 M. A. Shubov

c) When L is dissipative, then it is maximal, i.e., it does not admit dissipative extensions[24].

In the next theorem, we provide asymptotics of the spectrum for the operator L . Itis convenient now to recall the notion of an associate vector [16].

Definition 4.1. A vector in a Hilbert space H is an associate vector of a nonself-adjoint operator A of order m corresponding to an eigenvalue if = 0 and

(A I)m = 0 and (A I)m+1 = 0. (4.1)If m = 0, then is an eigenvector. The set of all associate vectors and eigenvectorstogether will be called the set of root vectors. The following results from [21, 22] hold.

Theorem 4.2. a) The operator L has a countable set of complex eigenvalues. If

=GI, (4.2)

then the set of eigenvalues is located in a strip parallel to the real axis.b) The entire set of eigenvalues asymptotically splits into two different subsets. We callthem the -branch and the -branch and denote these branches by {n}nZ and {n}nZ,respectively. If 0 and > 0, then each branch is asymptotically close to its ownhorizontal line in the closed upper half-plane. If > 0 and = 0, then both horizontallines coincide with the real axis. If = = 0, then the operator L is self-adjointand, thus, its spectrum is real. The entire set of eigenvalues may have only two points ofaccumulation: + and in the sense that ()n and | ()n | const asn (see formulae (4.3) and (4.4) below).c) The following asymptotics are valid for the -branch of the spectrum as |n| :

n = (sgn n)(2/L2)

EI/ (n 1/4)2 + n(), = ||1 + ||1, (4.3)

with being define by = mI S2. A complex-valued sequence {n} is boundedabove in the following sense: supnZ{|n()|} = C(), C() 0 as 0.d) The following asymptotics are valid for the -branch of the spectrum:

n =n

LI/G

+i

2LI/G

ln +

GI

GI

+O(|n|1/2), |n| . (4.4)

In (4.4), ln means the principal value of the logarithm. If and stay away from zero,i.e., || 0 > 0 and || 0 > 0, then the estimate O(|n|1/2) in (4.4) is uniform withrespect to both parameters. There may be only a finit number of multiple eigenvalues of afinit multiplicity each. Therefore, only a finit number of the associate vectors may exist.

The next result is concerned with the properties of the root vectors of the operatorL . Before we formulate this result, we recall the definition of the biorthogonal vectors[16].

Definition 4.2. Two sequences of vectors {n} and {n} in a Hilbert space H aresaid to be biorthogonal if for every m and n,we have

(m, n)H = mn. (4.5)In the case when an operator has a simple spectrum (i.e., there are no associate vectors), thebiorthogonal set consists of the eigenvectors of the adjoint operator. However, in general,


the assumption that the spectrum is simple is quite artificial. Numerical simulations showthat minor changes of the parameters of the problem often result in appearance of numerousmultiple eigenvalues. So, one cannot disregard the existence of associate vectors. In thatcase the relationship between the root vectors of the operator and its adjoint becomes lessobvious. (For the corresponding results, see [22].) It is also shown in [22] that the set ofthe root vectors of the operator L is complete in the state space. (Recall the set of vectorsis complete in a Hilbert space if finite linear combinations of the vectors from the set aredense in this space [16, 17].)

Definition 4.3. A basis in a Hilbert space is a Riesz basis if it is linearly isomorphicto an orthonormal basis, that is, if it is obtained from an orthonormal basis by means of abounded and boundedly invertible operator.

The next result from [22] describes a very important property of the root vectors ofthe differential operator L .

Theorem 4.3. The set of root vectors of the operator L forms a Riesz basis in theenergy space H.

The Riesz basis property of the root vectors has been proven in [22] based on the func-tional model for nonself-adjoint operators by Sz. NagyC. Foias [24], the main elementsof which have been reproduced in Section 7 of [22].

5 ASYMPTOTICAL DISTRIBUTION OF AEROELASTIC MODES

Our first result in this section is the following statement.Theorem 5.1. a) The set of all aeroelastic modes (which are the poles of the gen-

eralized resolvent operator) is countable and does not have accumulation points on thecomplex plane C. There might be only a finit number of multiple poles each of a finitmultiplicity. There exists a sufficientl large R > 0 such that all aeroelastic modes, whosedistance from the origin is greater than R, are simple poles of the generalized resolvent.The value of R depends on the speed u of an airstream, i.e., R = R(u).b) The set of the aeroelastic modes splits asymptotically into two series, which we call the-branch and the -branch. Asymptotical distribution of the and the -branches of theaeroelastic modes can be obtained from asymptotical distribution of the spectrum of the op-erator L . Namely if {n}nZ is the -branch of the aeroelastic modes, then n = inand the asymptotics of the set {n}nZ are given by the right-hand side of formula (4.3).Similarly, if {n = in}nZ is the -branch of the aeroelastic modes, then the asymptoticaldistribution of the set {n}nZ is given by the right-hand side of formula (4.4).

The next result is of particular importance for us. It does not follow from the asymp-totic representations (4.3) and (4.4). This result has been formulated in paper [4]. Theauthor of [4] has suggested the proof using the general ideas of the Semigroup Theory. Weare able to prove the same fact by using a totally different approach. We divide our proofinto two parts, and the first part is formulated as Theorem 6.3 below.

Theorem 5.2. (A.V. Balakrishnan, [4]). For any u > 0, there might exist only afinit number of the aeroelastic modes having nonnegative real parts.

Now we provide an explanation why Theorem 5.2 cannot be considered as a corollaryof Theorem 5.1. We discuss each branch individually. First we note that the -branch ofthe aeroelastic modes may have only a finite number of modes with nonnegative real parts.Indeed, since the points n are asymptotically close to the eigenvalues {n}nZ, and the

38 M. A. Shubov

latter set has a horizontal asymptote in the upper half-plane (see (4.4)), the -branch of theaeroelastic modes {n = in}nZ has a vertical asymptote in the open left half-plane.This means that for |n| N >> 1, all modes {n}|n|N are located in the left half-plane.However, this is not the case for the -branch of the aeroelastic modes. Indeed, the set{n}nZ is asymptotically close to the set {n}nZ, and the latter set is close to the realaxis. Therefore, the -branch of the aeroelastic modes {n = in}nZ is close to theimaginary axis. Moreover, we know that n > 0 (i.e., (in) < 0) for all n Zdue to the fact that the operator L is dissipative. However, the points {n}nZ are notthe eigenvalues of any operator, let alone a dissipative operator. So we cannot be sure that n > 0 (i.e., n < 0) for any value of n. Thus, though the fact that there may be onlya finite number of modes of the -branch with n > 0 is valid, it requires quite involvedproof.

6 STRUCTURE AND PROPERTIES OF MATRIX INTEGRAL OPERATOR

In this section, we describe the properties of the Laplace transform of the convolution-typematrix integral operator given in (3.8).

Lemma 6.1. Let F be the Laplace transform of the kernel of matrix integral operatorfrom (3.8). The following formula is valid for F:

F() =

0 0 0 0

0 L() uL (1/2 a)L()

0 0 0 0

0 N () uN () (1/2 a)N ()

,

where

L() = 2

u

{ S2+[I + (a+ 1/2) S

]T (/u)

}, (6.1)

N () = 2

u

{m

2[S + (a+ 1/2) m

]T (/u)

}, (6.2)

and T is the Theodorsen function define by the formula

T (z) =K1(z)

K0(z) +K1(z), (6.3)

K0 and K1 are the modifie Bessel functions [1, 19].We note that the Theodorsen function is defined as a single-valued analytic function onthe complex plane having a branch cut along the negative real semi-axis. The followingasymptotic approximation is valid:

T (z) = 1/2 + V (z), V (z) = 1/(16z) +O(z2), |z| . (6.4)Taking into account representation (6.5) and the fact that z = /u, we can write F() asthe following sum:

F() = M + N(). (6.5)


The matrix M in (6.6) is defined by the formula

M =

0 0 0 00 A uA (1/2 a)A0 0 0 00 B uB (1/2 a)B

, (6.6)

where according to (6.2) and (6.3), we have for A and BA = u1[I + (a 1/2)S], B = u1[S + (a 1/2)m]. (6.7)

The matrix-valued function N() is defined by the formula

N() =

0 0 0 00 A1() uA1() (1/2 a)A1()0 0 0 00 B1() uB1() (1/2 a)B1()

, (6.8)

where, according to (6.2) and (6.3), we have for A1() and B1()A1() = 2u1V (z)[I + (a+ 1/2)S] 2u1V (z) d1,B1() = 2u1V (z)[S + (a+ 1/2)m] 2u1V (z) d2, z = /u. (6.9)

Therefore, the generalized resolvent (see (3.12) and (3.13)) can be written in the formR() = (I iL MN())1 , (6.10)

where the matrix M and the matrix-valued function N() are defined by (6.7)(6.10).Theorem 6.1. M is a bounded linear operator in H. The operator K introduced

by the formulaK = L iM + iM I, (6.11)

where I is the identity operator, is an unbounded dissipative operator in H.Remark 6.1. The fact that K is dissipative is very important for the proof that the

root vectors of this operator form a Riesz basis in H.Theorem 6.2. N() is an analytic matrix-valued function on the complex plane with

the branch-cut along the negative real semiaxis. For each , N() is a bounded operatorin H with the following estimate for its norm:

NH C(1 + ||)1, (6.12)where C is an absolute constant the precise value of which is immaterial for us.

We recall that Balakrishnans Theorem means that for a given speed u, there maybe only a finite number of unstable mode shapes. In other words, only a finite number ofmodes can be subject to flutter. Let us return to Eq. (3.13) for the aeroelastic modes. Weknow that F() is a sum of the 44 matrix M with constant entries and the matrix-valuedfunction N(), whose norm goes to zero as || . We show that the result similar toBalakrishnams Theorem is valid for the equation obtained from Eq. (3.13) in which theterm containing N() has been omitted, i.e., that an addition of N() cannot destroy theresult obtained for the equation

iL+ M = . (6.13)Theorem 6.3. For a given value of u, there may exist only a finit number of eigen-

values of the operator iL + M having nonnegative real parts.

40 M. A. Shubov

7 PROPERTIES OF THE ADJOINT OPERATORS AND OPERATOR-VALUEDFUNCTIONS

To write the spectral decomposition for the generalized resolvent operator, we need detailedinformation about the set of functions that is biorthogonal to the set of the mode shapes. Itturns out that this biorthogonal set is closely related to the set of nontrivial solutions of thefollowing equation:

S() [iL + M + N()] = . (7.1)

The set of complex points for which Eq. (7.1) has nontrivial solutions will be called theset of the adjoint aeroelastic modes and the set of corresponding solutions will be calledthe set of adjoint mode shapes.

Now we describe the structure of the adjoint analytic operator-valued functionS(). We already know that L is given by the same matrix differential expression(3.6) and the only difference is in the description of the domain of L , i.e., the parameters and from (3.7) should be replaced with () and () respectively.

Theorem 7.1. The operator-valued function M + N() can be represented in thefollowing form:M + N() =

u

0 0 0 0

0 (I bS

) (1 + 2V (/u)

)/ 0

(I bS

)(b+ 2bV (/u)

)/

0 u (1 + 2V (/u)) 0LK(x, ) d 0 u(1 + 2V (/u)

) 0LK(x, ) d

0(S bm

) (1 + 2V (/u)

)/ 0

(S bm

)(b+ 2bV (/u)

)/

,

where b = 1/2 a, b = 1/2 + a. The kernel of the integral operator in (7.2) is given bythe following formulae:

K(x, ) = 1W ()

{sin(x+ L) cos, x ,sin( + L) cosx, x , (7.2)

and W () = cosL, = u/G. If u satisfie the second condition from (2.14),

then W () = 0.Theorem 7.2. The set of adjoint aeroelastic modes is countable and does not have

accumulation points on the complex place C. This set splits asymptotically into two series,which we call the -branch and the -branch. Asymptotical distribution of the andthe -branches of the adjoint aeroelastic modes can be obtained from the asymptoticaldistribution of the spectrum of the operator L . Namely, if {n }n Z is the -branchof the adjoint aeroelastic modes, then n = i

n , and the asymptotics of the set {n }n

Z are given by the right-hand side of formula (4.3). Similarly, if {n = in }n Z isthe -branch of the adjoint aeroelastic modes, then the asymptotical distribution of the set{n } can be obtained from the right-hand side of the formula (4.4) by replacing the sign+ with the sign before the logarithmic term.


Note that due to the latter theorem, one can see that the set of the adjoint aeroelasticmodes is closely related to the spectrum of the operator L .

Finally, we formulate an important result about the operator

K = L + iM. (7.3)

Let us denote by {n}nZ and {n}nZ the and -branches of root vectors of the oper-ator K , respectively.

Theorem 7.3. a) The entire set of the root vectors {n}nZ{n}nZ of the oper-

ator K is complete in H. b) This set forms a Riesz basis in H. The properties (a) and (b)imply that the operator K , is a Riesz spectral operator in the sense of Dunford [22]. c)The operator K , which is adjoint to the operator K , is also Riesz spectral.

To prove Theorem 7.3, we have used the same approach as for the case of the oper-ator L , i.e., the Sz. NagyC. Foias functional model for nonself-adjoint operators. Toformulate the result on the mode shapes, let us use the following notations: {F n }nZ arethe -branch mode shapes, {F n}nZ are the -branch mode shapes.

Theorem 7.4. a) The entire set of the mode shapes {F n }nZ{F n}nZ is complete

in the energy space H. b) The set of mode shapes is quadratically close to the set of rootvectors of the operator K , i.e.,

nZ

||n F n ||2H +nZ

||n F n||2H

42 M. A. Shubov

5. Balakrishnan A.V., Shubov M.A., Peterson C.A., Spectral analysis of EulerBernoulliand Timoshenko beam model. Submitted to: Integral and

Advances in Dynamics & Control Sivasundaram

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