Applicationsofadvancedcontrolmethodsinspacecrafts ......progress,challenges,andfutureprospects∗ Yong-chun XIE †‡ 1,2 , Huang HUANG † ,YongHU 1,2 , Guo-qi ZHANG ( 1 Science

Xie et al. / Front Inform Technol Electron Eng 2016 17(9):841-861 841

Frontiers of Information Technology & Electronic Engineering

www.zju.edu.cn/jzus; engineering.cae.cn; www.springerlink.com

ISSN 2095-9184 (print); ISSN 2095-9230 (online)

E-mail: [email protected]

Review:

Applications of advanced controlmethods in spacecrafts:

progress, challenges, and future prospects∗

Yong-chun XIE†‡1,2, Huang HUANG†1,2, Yong HU1,2, Guo-qi ZHANG1,2

(1Science and Technology on Space Intelligent Control Laboratory, Beijing 100190, China)

(2Beijing Institute of Control Engineering, Beijing 100190, China)†E-mail: [email protected]; [email protected]

Received Mar. 7, 2016; Revision accepted June 24, 2016; Crosschecked Aug. 16, 2016

Abstract: We aim at examining the current status of advanced control methods in spacecrafts from an engineer’sperspective. Instead of reviewing all the fancy theoretical results in advanced control for aerospace vehicles, thefocus is on the advanced control methods that have been practically applied to spacecrafts during flight tests, orhave been tested in real time on ground facilities and general testbeds/simulators built with actual flight data. Theaim is to provide engineers with all the possible control laws that are readily available rather than those that aretested only in the laboratory at the moment. It turns out that despite the blooming developments of modern controltheories, most of them have various limitations, which stop them from being practically applied to spacecrafts. Thereare a limited number of spacecrafts that are controlled by advanced control methods, among which H2/H∞ robustcontrol is the most popular method to deal with flexible structures, adaptive control is commonly used to deal withmodel/parameter uncertainty, and the linear quadratic regulator (LQR) is the most frequently used method in caseof optimal control. It is hoped that this review paper will enlighten aerospace engineers who hold an open mindabout advanced control methods, as well as scholars who are enthusiastic about engineering-oriented problems.

Key words: Spacecraft control, Robust control, Adaptive control, Optimal controlhttp://dx.doi.org/10.1631/FITEE.1601063 CLC number: V448.22; TP273

1 Introduction

In aerospace engineering, reliability probablyhas the highest priority over many other criteria.System engineers usually are risk-adverse and adoptmethods which have already been verified practi-cally. Despite the blooming development of mod-ern control theories, it is indisputable that the clas-sical proportional-integral-derivative (PID) controlstill plays the dominant role in aerospace engineer-ing. Since 1957, nearly 7800 spacecrafts have beenlaunched, among which more than 99% used PID as

‡ Corresponding author* Project supported by the National Natural Science Foundationof China (Nos. 61203075, 61333008, and 61304027) and ChinaMinistry of Science and Technology (No. 2013CB733100)

ORCID: Yong-chun XIE, http://orcid.org/0000-0003-1412-0495c©Zhejiang University and Springer-Verlag Berlin Heidelberg 2016

the baseline controller. Indeed, the already launchedsatellites or spacecrafts have testified that PID con-trol can meet most of the fundamental mission re-quirements. Upon the baseline PID controller, vari-ous techniques, such as structure-bending filters andgain scheduling, have been designed to compensatefor the lack of robustness and adaptability of thebaseline controller. The control systems thereforebecome so complex that they lack flexibility andportability. On the other hand, with the increasingrequirements of space exploration, the spacecraftsnot only tend to exhibit different features, such ashigh flexibility, high-frequency oscillations, or un-known dynamics, but also face ultimate high-levelperformance requirements. This trend challengesthe baseline PID controller and pushes engineers toadvanced control methods. According to Hanson

842 Xie et al. / Front Inform Technol Electron Eng 2016 17(9):841-861

(2002), from 1991 to 2001, among all the launchedvehicles by U.S., Europe, Japan, and Russia (thatare involved with U.S. companies), 41% of the launchvehicle failures could be avoided by some advancedguidance and control where the control system wouldreact quickly to failures and adjust its control param-eters autonomously for different scenarios, instead ofbeing tuned from the ground.

On the other hand, members of the control com-munity are quite enthusiastic about developing noveland fancy control methods, most of which are mo-tivated by the requirements on better robustness,more powerful adaptability, and higher-level systemperformance. There are millions of peer-reviewed re-ports and simulations demonstrating quite satisfac-tory outputs of those advanced control methods. Inchemical engineering, apart from PID control, modelpredictive control (MPC) has become universal andhas already been successfully applied to quite a fewproduct lines. In aerospace engineering, there arerobust control that originated in the 1970s and adap-tive control that showed its potential as early as inthe 1960s. However, in the 21st century, more than99% launched spacecrafts are still using PID. Whatis stopping us?

We believe that there are probably threereasons:

1. Lack of a control-oriented modelFor spacecrafts, the Newton theory, the Euler

theory, and the Kepler theory together ideally illus-trate the kinetic and kinematic dynamics in a preciseway. The corresponding mathematical models arecharacterized by multiple variables, high nonlinear-ities, and strong couplings. The unknown parame-ters determined by the mass, inertial, structure, andworking status are difficult to measure because of thelack of high fidelity ground tests. Moreover, externaldisturbance and structure uncertainties are usuallygenerated from experienced data, which cover onlya limited number of working conditions. Therefore,controller design has to deal with unknown or chang-ing parameters. Meanwhile, even when the dynami-cal model is precise and rigorous, it is usually highlynonlinear, highly ordered, and strongly coupled, andthus has to be simplified by techniques, such as lin-earization around trim conditions or model order re-duction, so as to facilitate controller design.

2. Lack of perspective from engineersScholars are enthusiastic about getting motiva-

tions from practical systems. However, when solv-ing the mathematical problems generated from thosemotivations, various assumptions or constraints haveto be made for a rigorous proof, e.g., the tuning ofinitial values in adaptive control and a known param-eter bound in robust control. Although theoreticallysound, those assumptions are usually impossible tomeet in practice. In other words, theoretical resultsare usually more or less far from practice due to thevarious assumptions and constraints, and thus aredifficult to accept and implement by engineers.

Moreover, taking adaptive control as an exam-ple, it can modify its parameters online so as to fitthe changing environment and unmodeled dynamics.This nature makes adaptive control an ideal methodin aerospace engineering where many uncertaintiesexist. However, adaptive control was never popularin the aerospace industry. The main reason is prob-ably that system engineers without adaptive controlbackground do not know where to start and how tostart, while, on the other hand, an engineer can tunethe parameters of a PID controller by following a listof specific guidelines without too much mathemati-cal knowledge. Besides, engineers are used to judginga system according to analysis in the frequency do-main. Once adaptive control is introduced, engineerswill not be able to quantify the system performancein the frequency domain even if the design is carriedout in this domain.

3. Expense of computationOne has to admit that PID is the simplest con-

trol algorithm requiring the least number of lines ofcodes and computation time. A complex algorithmnot only increases the lines of codes but also bringshuge troubles to system reliability. With the rapiddevelopment of hardware, the implementation of acomplex algorithm may no longer be an issue, andhence the reliability.

To be more specific, in modern aerospace en-gineering, in both academia and industry, the mainfocuses in attitude control are high precision and fastmaneuverability. With these increasing demandsand the new generations of satellites, the main diffi-culties for spacecraft attitude control are as follows:

1. Flexible structureThe flexible mode of a spacecraft comes from its

solar panels, antennas, flexible body, and sloshingeffects in tanks during orbital maneuvers. It is diffi-cult to build accurate dynamic models for these large


flexible space structures on the ground due to the 1ggravity and atmospheric effects, their low stiffnesscharacteristics, etc., which lead to considerable mod-eling uncertainties established on the ground com-pared to the 0g space environment. On-orbit systemidentification is a promising way that helps buildan accurate model with identified parameters. Thisin turn makes the high-performance control designmore feasible.

Even with an accurate model, controller designof the flexible structure is not easy. When the flex-ible structure is modeled as a distributed parame-ter system, currently the design of controller worksonly on simple sticks or boards and is not ready forengineers. By modeling the flexible structure as acentralized parameter model, various control meth-ods including the classical one, the modern controltheory, and the intelligent control theory have beenstudied by researchers.

2. Unknown parametersThe movement of spacecrafts in their orbit fol-

lows the basic physical rules, and most of them canbe described by mathematical expressions. However,the space is full of mysteries, and it is almost impos-sible to anticipate all disturbance and uncertaintiesfrom ground experiments or simulations. Therefore,stability margin is one of the key indexes in PID con-troller design. When the true values are far from theones used for simulation, the PID controller may failto provide an adequate stability margin. As a con-troller with fixed parameters, robust control is themost commonly used method to deal with the un-known parameters. When the parameter uncertain-ties exceed the capability of robust control, adap-tive control should be considered. The main issueis the implementation of those control methods inengineering practice. Engineers have to deal withthe tuning in adaptive control and the calculation ofparameter bounds through the entire flight regime.

3. Changing parametersFor the reentry of spacecrafts, the changes of at-

mosphere in terms of density, temperature, humid-ity, and ionosphere affect the internal and externalmodel parameters. The changes of parameters usu-ally extend to a level beyond the robustness of aPID controller. Robust control is ideal for such sit-uations in which the parameters vary (fast or slow)within a limited domain. Adaptive control with on-line identification can deal with slowly time-varying

parameters over a large range. When it comes tofast-changing parameters over large scales, currentlythere is no effective identification method. Multiplemodel adaptive control which switches between sub-models was believed to be a promising way but wastestified only through a limited number of numericalexamples (Narendra and Han, 2011).

4. High-level requirementsHigh pointing accuracy, agile maneuverability,

and minimum time/energy consumption are thehigh-level requirements in modern satellites. The-oretically, those requirements could be illustratedmathematically with mature solutions. However,for spacecrafts, various uncertainties and all kindsof constraints such as the allowed executing time,energy constraints, and actuator saturations wouldbring huge troubles during optimization. Meanwhile,computation complexity is another reason that stopsmost of the optimization methods from being appliedin practice.

Despite all the obstacles that prevent advancedcontrol methods from being applied in aerospace en-gineering, the increasing demands on modern space-crafts have pushed engineers to embrace advancedcontrol methods. NASA has long been interestedin advanced control theories that can be applied inaerospace. Among those well-known modern con-trol theories, adaptive control, neural network, androbust control are NASA’s favorites; e.g., model-reference adaptive control has been proposed forhighly accurate attitude control of satellites (Scar-ritt, 2008) and the L1 adaptive feedback controlhas been presented for flexible wing (Cao and Ho-vakimyan, 2008; Kharisov et al., 2008).

In this review, we investigate a wide rangeof spacecrafts including satellites, the InternationalSpace Station (ISS), and reentry vehicles. Thefocus is on the advanced control laws that havebeen applied to spacecrafts with flight tests, orhave been tested on ground facilities and generaltestbeds/simulators built with actual flight data.Meanwhile, to understand the effects of zero gravity,several middeck experiments have been conductedonboard the ISS to investigate the robustness andadaptability of different control methods, which arealso included in this paper. The aim is to provide en-gineers with all the possible control algorithms thatare readily available rather than those fancy controlmethods that are tested only within laboratory at


the moment. We hope this material will enlightenaerospace engineers who hold an open mind aboutadvanced control methods and scholars who are en-thusiastic about engineering-oriented problems.

The rest of the paper is organized as follows.Section 2 summarizes the main advanced controllaws that have been applied on-orbit or throughground tests with flight data. Those advanced con-trol laws include optimal control, adaptive control,and robust control. In Section 3, four well-knownsatellites that use advanced control methods to sup-press the flexible modes are introduced. Three ofthem are validated on-orbit and one of them withflight data. Section 4 focuses on the situation withunknown parameters, where identification and adap-tive control algorithms are developed for two satel-lites and one onboard experiment. During the reen-try of spacecrafts, the unknown and fast-changingenvironment poses a huge challenge to the controlsystem. Much effort has been made to deal withthe parameter changes, as introduced in Section 5.Nowadays, high-performance requirements are re-quired for spacecrafts, including ultra-high pointingaccuracy, agile maneuver, and minimum reorienta-tion time. Optimal control methods are introducedto meet those requirements, as shown in Section 6.Finally, conclusions are drawn in Section 7, as wellas future outlook.

2 Advanced control methods

2.1 Linear quadratic regulator

Linear quadratic regulator (LQR) is one of theoptimal control methods that take the states of thedynamical system and control input into account(Antsaklis and Michel, 2007). By optimizing somecriterion, closed-loop gains can be obtained. In gen-eral, the linear state-space equation can be obtainedby linearizing the nonlinear system models aroundthe equilibrium point:

ẋ = Ax+Bu,

where x ∈ Rn is the state vector, u ∈ Rm is theinput vector, and A and B are matrices with ap-propriate dimensions. The optimal state feedbackcontrol u = Kx is derived by minimizing the follow-ing cost function:

J =

∫ ((x(t))TQx(t) + (u(t))TRu(t)

)dt,

where Q and R are positive definite matrices thatgive the compromise between the state transient en-ergy and control input energy. The LQR gain matrixK is given by

K = R−1BTP ,

where P is the unique positive definite solution tothe following algebraic Riccati equation:

ATP + PA− PBR−1BTP +Q = 0.

The typical use of the LQR problem is to determinethe optimal control law K from a given set of weightmatrices Q and R.

2.2 Inverse optimal control

The direct method of designing a control lawwith good performance is to optimize an appropri-ate cost function. For most engineering applications,it is necessary to solve a Hamilton-Jacobi-Bellman(HJB) partial differential equation. The inverse op-timal method offers a feasible approach to obtainan optimal feedback law (Freeman and Kokotovic,1995; 1996). It is based on the fact that the solutionto the appropriate HJB equation can be taken as aLyapunov function that guarantees global stability.

The inverse optimal method avoids the monu-mental task of solving an HJB equation numerically.It consists of two basic stages. The first stage is toconstruct a stabilizing feedback controller called thebenchmark controller. The controller is based on acontrol Lyapunov function. That is, there exists aproper and positive definite function V for the fol-lowing system:

ẋ = f(x) + g(x)u

such that

infu

(Lf(V (x)) + Lg(V (x))u

)< 0,

where x ∈ Rn is the state vector, u ∈ Rm is thecontrol input, f(·), g(·) are continuous functions, andLf(V (x)), Lg(V (x)) are the Lie derivatives of V withrespect to f and g, respectively.

The second stage is to solve a nonlinear pro-gramming problem:

{min J(u) = uTus.t. Lf(V (x)) + Lg(V (x))u ≤ −σ(x).


The solution is

u∗ = −λ2[Lg(V (x))]

T,

where

λ =

⎧⎨⎩

2(Lf(V (x)) + σ(x))

Lg(V (x))[Lg(V (x))]T, Lf (V (x)) > −σ(x),

0, Lf (V (x)) ≤ −σ(x).

Then a minimum norm controller can be obtained(Bharadwaj et al., 1998).

2.3 Pseudo spectral optimal control

In recent years, pseudo spectral (PS) methodshave been used to solve many nonlinear optimal con-trol problems as introduced in Elnagar et al. (1995)and Ross and Fahroo (2004). The PS optimal con-trol theory proposed by Ross and Karpenko (2012) isfounded on the fact that any continuous function canbe approximated to arbitrary precision by a polyno-mial, which is a direct consequence of the Stone-Weierstrass approximation theorem (Rudin, 1975).A crucial question in implementation is how to se-lect an appropriate polynomial basis and a com-putational grid. The most reliable computationalgrids used for spacecraft maneuvers are based on theGauss-Lobatto (GL) points (Ross and Gong, 2010).

According to different polynomial bases andgrids, various PS-based optimal control methodshave been put forward to solve many practical prob-lems in experimental demonstrations and flight oper-ations (Ross and Karpenko, 2012). Usually, a prac-tical optimal control problem requires full consider-ation of the nonlinearity and constraints, and can bedefined as

B :

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

min J(x,u) = E(x(−1),x(1))+∫ 1−1 F (x(t),u(t))dt

s.t. ẋ(t) = f(x(t),u(t)),

e(x(−1),x(1)) = 0,h(x(t),u(t)) ≤ 0,

where F : RNx ×RNu −→ R, E : RNx ×RNu −→ R,f : RNx × RNu −→ RNx , e : RNx × RNx −→ RNe ,and h : RNx × RNx −→ RNh , with Nx, Nu, Ne, Nhthe corresponding dimensions.

Choose an arbitrary grid between the boundarypoints t0 = −1 and tN = 1, where −1 < t1 < t2 <. . . < tN−1 < 1. Using the GL points, the state

function x(t) and control function u(t) can be ap-proximated. By the differentiation and integration ofthe state and control functions, the path constraintsare enforced only at the GL points, and the prob-lem is then transformed to guarantee the values ofthe state and control at the GL points. Therefore,the problem B can be transformed into a relativefinite-dimensional problem BN by PS discretization:

BN :

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

min J(x,u) = E(x(0),x(N))

+N∑i=0

F (x(i),u(i))ωi

s.t.N∑j=0

Dijx(j) = f(x(i),u(i)),

e(x(0),x(N)) = 0,

h(x(i),u(i)) ≤ 0,i = 0, 1, . . . , N,

where D is a square differentiation matrix and ωi isthe weight satisfying

Dij = φ̇j(ti), ωi =

∫ 1−1

φi(t)dt,

φi(t) =gN (t)

g′N(ti)(t− ti), gN(t) =

N∏i=0

(t− ti).

The optimal control problem BN is a nonlin-ear programming (NLP) problem. For smooth sit-uations, as the number of GL points increases, theinterpolation error decreases faster than the polyno-mial rates. PS methods are useful in practice be-cause the optimality verification and validation forthe solution can be quite readily done by the covec-tor mapping theorem (Ross, 2005a; 2005b), and thefeasibility and convergence can also be guaranteed(Gong et al., 2008). A MATLAB implementation ofthe PS method is DIDO, a PS-based optimal controlsolver package (Ross, 2007). The software acts muchlike a ‘black-box’, which allows a user to formulatean optimal control problem in m-code format. AFortran implementation is available under OTIS byNASA (Paris et al., 2006).

2.4 Model reference adaptive control

Model reference adaptive control (MRAC), de-veloped in the 1950s (Åström and Wittenmark,2008), is a rigorous and systematic method in adap-tive control. The basic principle is illustrated inFig. 1. The main idea of MRAC is to make the


output y of the plant track the output yr of a ref-erence model defined beforehand by adjusting thecontroller parameters θ̂. The reference model speci-fies the system performance and tells how to respondto the command signal r. The adaptive law in theparameter adjustment block is used to update theparameters of the controller. The schematic dia-gram of the MRAC system basically consists of twoloops: the first loop is normal feedback control, andthe second loop is parameter adjustment. The mainapproaches to the analysis and design of the MRACinclude the gradient approach, Lyapunov functions,and passivity theory.

Controller

Referencemodel

Parameter adjustment

Plantr

yr

yuθ^

Fig. 1 Block diagram of a model reference adaptivecontrol system

Consider the following nonlinear plant:

ẋ(t) = Ax(t) +B(u(t) + f(x(t))),

where x(t) is a state vector, u(t) is a control vec-tor, A, B are known and controllable, f(x(t)) =θTΦ(x(t)) + ε(x(t)) is an uncertain term which canbe linearly approximated by a set of continuous,differentiable, and bounded basis functions Φ(x(t)),and ε(x(t)) is an approximation error. The referencemodel can be written as

ẋm(t) = Amxm(t) +Bmr(t),

where xm(t) is a reference state vector, r(t) is abounded piecewise continuous command vector, andAm (Hurwitz) and Bm are matrices with appropri-ate dimensions. The aim is to design an adaptivecontroller to ensure x(t) to track xm(t):

u(t) = K1x(t) +K2r(t)−(θ̂(t)

)TΦ(x(t)),

where K1 and K2 are constant matrices and the lastterm is a direct adaptive signal. Note that θ̂ is theestimated value of θ. Assume that there exist K1and K2 such that Am = A+BK1 and Bm = BK2.Then the tracking error equation is as follows:

ė(t) = Ame(t) +B((θ̂ − θ)Φ(x(t)) − ε(x(t))).

The parameter update law which minimizes ‖e(t)‖can be designed as

˙̂θ=−ΓΦ(x(t))((e(t))TP−v(Φ(x(t)))Tθ̂BTPA−1m )B,where Γ is a positive definite matrix, v is a weightingconstant, and P is obtained by solving the followingRiccati equation:

PAm +ATmP = −Q,

where Q is a positive definite matrix. The uniformlyultimately bounded tracking error can be obtainedby choosing a Lyapunov candidate function

V (t) = (e(t))TPe(t) + tr((θ̂ − θ)TΓ (θ̂ − θ)),

where ‘tr’ denotes the trace operation. The detailedproof can be found in Burken et al. (2010).

2.5 Characteristic model-based golden-section adaptive control

Self-tuning is a classical adaptive controlmethod that helps deal with time delay and distur-bance. A self-tuning controller identifies the param-eters according to the disturbed input and output,and generates control signals online.

As a special kind of self-tuning adaptive con-trol, the characteristic model-based adaptive controlmethod was first proposed by Wu (1990). It has re-ceived great development in theory and engineeringapplications over the last decade (Wu et al., 2009).The characteristic model was developed to use a low-order discrete time-varying system to deal with ahigh-order linear or nonlinear system, based on thedynamic characteristics of the plant and the requiredcontrol performance. Rather than dropping informa-tion as in the reduced-order modeling, it compressesall the information of the high-order model into sev-eral characteristic parameters. The characteristicmodel is an online adaptive one so as to fit into thechanging environment. Consider the general trans-fer function for a single-input single-output (SISO)linear time-invariant (LTI) system:

G(s) =bms

m + bm−1sm−1 + . . .+ b1s+ b0sn + an−1sn−1 + . . .+ a1s+ a0

, (1)

where ai (i = 0, 1, . . . , n − 1) and bi (i =0, 1, . . . , m) are constant parameters.

When the control requirement is keeping ortracking a position, its characteristic model can be


described by

y(k + 1) =f1(k)y(k) + f2(k)y(k − 1) + g0(k)u(k)+ g1(k)u(k − 1). (2)

When the LTI system (1) is stable or contains inte-gral components, we can see that (Wu et al., 2001):(1) the coefficients are slowly time-varying; (2) therange of the coefficients can be determined before-hand (Wu et al., 2009); (3) the output of the char-acteristic model becomes arbitrarily closer to that ofthe plant as the sampling period decreases; and (4)the sum of the coefficients at the steady state is equalto 1 if the static gain is 1, i.e.,

f1(∞) + f2(∞) + g0(∞) + g1(∞) = 1.

In practice, we have g0 ∈ [0.003, 0.3] and |g1(k)| ≤g0(k). For a stable plant, if T/Tmin ∈ [1/10, 1/3],where Tmin is the minimum equivalent time constantof the plant, the values of the characteristic parame-ters f1(k) and f2(k) belong to the following set:

DS :

⎧⎪⎨⎪⎩

f1 ∈ [1.4331, 1.9974],f2 ∈ [−0.9999,−0.5134],f1 + f2 ∈ [0.9196, 0.9999].

On the other hand, for an unstable plant, if T/Tmin ∈[1/10, 1/4], the values of the characteristic parame-ters f1(k) and f2(k) belong to the following set:

DN :

⎧⎪⎨⎪⎩

f1 ∈ [1.9844, 2.2663],f2 ∈ [−1.2840,−1],f1 + f2 ∈ [0.9646, 1].

The characteristic parameters can be updatedby the projected gradient algorithm as follows:⎧⎪⎪⎪⎨⎪⎪⎪⎩

θ̂u(k) = θ̂(k − 1)+γφ(k − 1)(y(k)− (φ(k − 1))Tθ̂(k − 1))

δ + (φ(k − 1))Tφ(k − 1) ,θ̂(k) = π

[θ̂u(k)

],

where⎧⎪⎪⎨⎪⎪⎩

φ(k) = [y(k), y(k − 1), u(k), u(k − 1)]T ,θ(k) = [f1(k), f2(k), g0(k), g1(k)]

T,

θ̂(k) =[f̂1(k), f̂2(k), ĝ0(k), ĝ1(k)

]T,

δ > 0 , 0 < γ < 2 are constants, and π[·] is theorthogonal projector.

The estimated parameters are constrainedwithin the convex domain DS or DN.

The characteristic model-based golden-sectionadaptive control u(k) is formulated as (Wu et al.,2009)

u(k) =1

ĝ0(k) + λ

(l1f̂1(k)e(k) + l2f̂2(k)e(k − 1)

− ĝ1(k)u(k − 1)),

where e(k) = yr(k) − y(k), l1 = 0.382, and l2 =0.618. The robust stability of the golden-sectionadaptive control law was proved in Xie and Wu(1992) and Huang (2015).

The golden-section controller is simple and easyto apply in practice. Over the past 20 years, thiscontrol scheme has been applied to more than 400systems belonging to 10 kinds of engineering plantsin the fields of astronautics (such as in the success-ful rendezvous and docking of Shenzhou-8 space-craft (Hu et al., 2011) and reentry adaptive controlof Shenzhou spacecraft (Hu, 1998)) and industry inChina (Wu et al., 2007).

2.6 H∞ and H2 control

Consider a generalized system (Fig. 2), whereG and K are real, rational, and proper. Assumethat the state-space representations of G and K arecontrollable and observable. The exogenous inputw could be disturbance, sensor noise, or commands,z is the controlled output, and v is the measuredoutput. The closed-loop transfer function from w toz is denoted by Twz whose H∞ norm is

‖Twz‖∞ = supw

σ̄(Tzw(jw)).

The H∞ optimal control is to find a controller Ksuch that the H∞ norm of Twz is minimized (Zhouet al., 1996). A smaller H∞ norm indicates that theexogenous input w has less disturbance on the out-put z, meaning the closed-loop system has strongerrobustness to disturbance.

G

K

wz

uv

Fig. 2 Generalized plant and controller configuration


In the more generalized case, we are interestedin the suboptimal problem by finding K such that‖Twz‖∞ < γ.

Once the suboptimal H∞ controller K is found,the system in Fig. 3 is stable for all admissible un-certainty ‖Δ‖∞ < 1/γ.

z w

uv

K

G

Δ

Fig. 3 Generalized plant and controller configurationwith disturbance

The H2 optimal and suboptimal problems aresimilar to the H∞ problem, with the only differencethat the norm of the transfer function is calculatedaccording to

‖Twz‖2 = tr(

1

2π

∫ ∞−∞

T (jw)T ∗(jw)dw)1/2

,

where ‘∗’ represents the conjugate transpose.The synthesis of the H∞ controller K is well-

posed. The original way is to use the Nevanlinna-Pick interpolating method or the operator method,which met with many problems when dealingwith the multi-input multi-output (MIMO) system.Therefore, modernH∞ control theory was built uponthe state-space method (Doyle, 1984; Glover, 1984;Francis, 1987). In the 1980s and 1990s, H∞ controltheory experienced blooming development theoret-ically and practically. Readers are referred to Yu(2002) for the linear matrix inequality (LMI) basedsynthesis of those controllers. The LMI toolbox re-leased by MATLAB in 1995 further provided a handyway to synthesize H∞/H2 controllers.

2.7 μ synthesis

The definition of μ is motivated by finding thesmallest destabilizing matrix Δ to a given matrixM . The solution is

Δd =1

σ̄(M)v1u

∗1, (3)

where M = σ̄(M)u1v∗1 + σ2u2v∗2 + . . . is a singularvalue decomposition.

Suppose Δ is structured by

Δ̄ ={diag(δ1Ir1 , δ2Ir2 , . . . , δSIrS , Δ̄1, Δ̄2, . . . ,Δ̄F ) : δi ∈ C, Δ̄j ∈ Cmj×mj},

where S is the number of repeated scalar blocks andF is the number of full blocks of the block diagonalmatrix Δ. When Δ is block diagonalized by Δ̄, thesmallest perturbation matrix that destabilizes M isEq. (3) with

μΔ̄(M) :=1

min{σ̄(Δ) : Δ ∈ Δ̄, det(I −MΔ) = 0} .

If no Δ ∈ Δ̄ makes I − MΔ singular, thenμΔ̄(M) := 0.

The μ stability of the system in Fig. 4 is givenby the following lemma (Zhou and Doyle, 1999):Lemma 1 The system consisting of M and Δ iswell-posed, internally stable, and ‖Twz‖∞ ≤ β forall Δ(s) ∈ M(Δ̄) with ‖Δ̄‖∞ < 1/β if and only if

supw∈R

μΔ̄P (G(jw)) ≤ β,

where M(Δ̄) := {Δ(·) ∈ RH∞ : Δ(s) ∈ Δ̄, ∀s ∈C̄+}.

z wM

Δ

Fig. 4 System framework

3 Orbital spacecraft with flexiblestructure

Flexibility is probably the biggest challenge inspacecraft control. Flexibility comes from large solarpanels, antennas, and the sloshing effects in tanks. Aflexible structure may produce large structure vibra-tions, which are modeled by a high-order equationwith an infinite number of modes acting over a widefrequency range. Because of the lack of experimentaldata at zero gravity, a flexible structure also bringsparameter uncertainties. Meanwhile, the solar pan-els are usually light-weight with small damping ratiosand can be excited easily.


The modeling of a spacecraft with a flexiblestructure aims to build the coupling equation con-cerning the movements of the spacecraft and struc-ture vibrations. In most cases, the flexible structuresare modeled by the finite element analysis method,and then mixed together into the overall model basedon the Lagrange equations. The order of the modelhas to be reduced so as to facilitate controller design.

In many applications, the traditional PID con-trol has been proved to be short-handed. In thissection, we find as many as three kinds of spacecraftthat used advanced control methods to deal with theflexibility, all of which were tested on-orbit. The well-known Hubble Space Telescope is also introduced inthe end. Its advanced control experiment was carriedout on the ground but with actual flight data.

3.1 Spacebus 4000 telecommunication satel-lite

Thales Alenia Space-France (TAS-F) has devel-oped a geosynchronous telecommunication platformnamed Spacebus 4000, which has been applied to 15telecommunication satellites with satisfactory per-formance. Telecommunication satellites are charac-terized by distinctive flexibilities due to their mov-ing appendages including solar arrays, antennas, andsloshing effects in tanks. According to reports, thoseflexible modes are badly damped at 0.001 with un-certain frequencies at ±30%. To deal with the flex-ibility, a conventional way is to design some filterswith carefully tuned parameters, together with thebaseline PID controller, so as to attenuate its res-onance. For example, after the deployment of thesolar array on the ISS, unexpected flexibility prob-lems were observed, and two filters were added so asto accomplish the reorientation maneuver on-orbit.

TAS-F sponsored research on H∞ controllersynthesis to deal with the flexibility. The controlsystem structure on Spacebus 4000 was presented byCharbonnel (2010).

The H∞ synthesis is to find a controller K(s)to the following optimization problem:

min

∥∥∥∥G([

r

d

]→

[e1e2

])∥∥∥∥∞

, (4)

where G(x → y) is the transfer function from signalx to signal y, r is the reference, d is the disturbance,and ei (i = 1, 2) is the output. A low-pass filterand a high-pass filter are introduced to the outputs,

and a disturbance rejection filter is designed for d.The stability is guaranteed by the H∞ controller,and thus the tuning of the above filters is much re-laxed. One of the key technologies is to solve the op-timization problem. Because of numerical issues andconservative solutions, the Riccati equation basedGlover-Doyle algorithm is used instead of the LMImethod. Meanwhile, the design of the controller is aworst case model with flexible mode frequencies, be-ing close to the control bandwidth so as to improvethe robustness. The delays are treated as first-orderapproximations during the controller design.

The H∞ controller was compared with the clas-sical PID controller on the TAS-F high-fidelity simu-lator AOCS (attitude and orbit control system). TheH∞ controller showed improved stability margins,stronger robustness, better dynamic performance,and less fuel consumption than the PID controllerduring orbit correction maneuvers.

The H∞ approach is the core control algorithmof the Spacebus 4000 platform, which is now theTAS-F industrial baseline. Since 2003, it has beenserving 15 telecommunication satellites with excel-lent pointing accuracy and stability. It is worth men-tioning that the entire design process is impressivelyengineer-friendly. Half a day’s training session wouldallow any engineer to design the controller.

3.2 Engineering Test Satellite-VI/VIII

The Engineering Test Satellite-VI (ETS-VI) wasinitiated by the National Space Development Agencyof Japan (NASDA). It is a three-axis stabilizedgeosynchronous spacecraft with a pair of large light-weight solar panels. The mission of this satellite is toexecute advanced communication experiments. It isa challenge to control such a large flexible spacecraftwith high accuracy due to its structure vibration andcontrol-structure interactions.

ETS-VI was launched in Aug. 1994. After thelaunch, NASDA carried out on-orbit system identi-fication experiments from Dec. 1994 to Mar. 1995 tocheck the validity of the model for ETS-VI obtainedfrom ground experiments. In the system identifi-cation experiments, the attitude angle and rate ofthe satellite’s main body, the thruster drive signals,and the precise accelerometer (PACC) signals wereused as the measurements for two system identifi-cation methods, which are the traditional methodsbased on the polynomial black-box models and a


subspace-based method. Two forms of excitationswere used as the inputs to the spacecraft: impulseexcitations and random excitations. The resultsshowed that both methods constructed an accuratemathematical model of the satellite (Adachi et al.,1999).

During controller design, the full-order modelwas reduced to three low-order ones according todifferent vibration modes. Two types of continu-ous time linear controllers, which were discretized atthe sampling rate of 4 Hz during implementation,were developed to achieve robust stability againstresidual modes and modal parameter errors. Thefirst type is a frequency-dependent LQR with a stateestimator. This type of controller has robustnessagainst the residual modes. The second type is anH∞ controller. The original H∞ control approachcould be ill-conditioned when dealing with the lightlydamped vibration modes and the undamped rigidmodes simultaneously. Therefore, two kinds of meth-ods, namely the robust stability degree assignmentmethod and the direct velocity feedback method,were employed during the H∞ design so as to dealwith the zero poles in the reduced model. To evaluatethe identified model and the controllers, on-orbit ex-periments were carried out after the launch in 1995.The designed controllers held the panel rotation atthe angle of 270◦ or 180◦ according to the experi-mental data. Step response and impulse response ofthe attitude control to the disturbance torque weretested to evaluate the controller performance. Theresults confirmed the validity of the LQR and theH∞design methods based on the reduced-order model,and the ability of these control methods with high-frequency residual modes and parameter uncertain-ties. Besides, online identification was carried outby vibrating the spacecraft with the pseudo-randomand the impulse signals (Kida et al., 1997).

The ETS-VIII, launched in Dec. 2006, is thelargest satellite developed by Japan to date, witha size of 40 m × 37 m and a mass of 3000 kg. Thespacecraft has two large deployable flexible reflectorsand two flexible solar panels that rotate around thepitch axis, which makes dynamic coupling betweenthe three axes, and the system parameters change by25% at the maximum according to the paddle rota-tion angle. To guarantee robust stability of the atti-tude control system for the MIMO linear-parameter-varying (LPV) system against the higher vibration

modes and model parameter uncertainties, four newcontrol methods were designed and examined on-orbit, namely the μ-synthesis linear time-invariantcontroller (Ohtani et al., 2009), the interpolation-based gain scheduling controller (Hamada et al.,2011), the 2-DOF (degree-of-freedom) static outputfeedback controller, and the 2-DOF dynamic out-put feedback controller (Nagashio et al., 2014). Abaseline PD controller for the flight test was usedto be compared with these control methods. FromJune 2009 to Mar. 2010, Japan carried out several or-bital experiments to testify the dynamic performancewith step response, impulse disturbance, square dis-turbance, and random disturbance. Compared to thetraditional PD controller, the advanced controllerscan achieve superior performance.

3.3 Hubble Space Telescope

In Apr. 1990, the United States launched theHubble Space Telescope (HST) to a 611-km circularorbit. Two solar arrays and two high-gain antennaswere deployed on the telescope. When analyzing theflight data, the engineers noticed that, due to day-night changes, the thermally induced deformationsof the solar arrays caused unexpectedly large pertur-bations in the pointing control system. The baselinePID controller of the pointing control system wasunable to deal with such perturbations. Much ef-fort was made from both modeling and control per-spectives. First, several parameters were collectedfrom two on-orbit identification experiments to iden-tify the disturbance frequencies and the three-axissystem transfer functions. Frequency identificationtechniques were used in the two experiments includ-ing the fast Fourier transforms (FFTs) and spec-tral analysis of the measurements on the reactionwheel assemblies (RWAs) and the rate gyro assembly(RGA). Through the identifications, more accuratetransfer functions were obtained (Anthony and An-dersen, 1995). Second, the engineers designed twosixth-order filters to reshape the disturbance rejec-tion transfer function such that the disturbance wasattenuated when it entered the dominant solar arrayfrequencies. Although the modified pointing con-trol system provided superb pointing accuracy, thetuning of the filters was quite tricky, and dependedheavily on the flight data, which was an afterwardremedy and could not be replicated (Nurre et al.,1995).


To avoid similar problems in the later missions,several advanced modern control methodologies weredeveloped and tested using actual flight data col-lected from the HST. The research was sponsoredby the Marshall Space Flight Center. Five moderncontrol strategies were conducted by five different re-search groups. The University of Colorado in Boul-der studied the problem based on the disturbance-accommodating control (DAC) with a reduced-ordermodel (ROM) controller, a disturbance estimator,and a residual mode filter. The Harris Corporationteam solved the problem with the linear quadraticGaussian (LQG) approach. The Ohio University de-sign team proposed an H∞ design method. ThePurdue University team designed two variations ofcovariance controllers. The University of Alabamain Huntsville (UAH) research group designed to-tal isolation (TI) and array damping (AD) strate-gies to deal with solar array disturbance (Bukley,1995). These five research groups testified their con-trol strategies in different ways. The ROM-basedDAC controller showed promising results in the lin-ear MIMO model with quite satisfactory pointing er-rors. The tracking performance of the LQG methodoutperformed that of the PID controller being de-ployed onboard. An 82nd-order H∞ controller de-sign was obtained using the MIMO modal plantmodel at 90◦. However, this controller worked onlyfor this specific model. Therefore, an improved H∞design method was further developed that resultedin improved performance in peak and attitude errors.The covariance controller was evaluated on the 83rd-order actual model. One improvement of this controllaw was the control energy, which was significantlyless than that of the original controller onboard. Fi-nally, UAH developed a planar simulation that in-cluded all the main body and solar array interfacedynamics of the HST, and testified its TI and ADcontrollers in this simulation. HST with the TI con-troller can maintain the pointing stability despite thesolar array vibrations. However, the stability marginand robustness to parameter uncertainty of the ADcontroller were unsatisfactory.

4 Spacecraft with unknown parameters

When parameter variation exceeds the robust-ness of the PID controller, advanced control methodshave to be considered. In this section, the Shenzhou

reentry spacecraft from China and the Data RelayTest Satellite (DRTS) from Japan are introduced.The United States has also conducted a middeck ex-periment onboard the Space Shuttle, which is pre-sented in the end.

4.1 Shenzhou spacecraft

The research on rendezvous and docking (RVD)technology in China began in the late part of thelast century. Automatic control and manual controlwere developed at the same time. China’s first RVDoperation was achieved in Nov. 2011 between space-crafts Shenzhou-8 and Tiangong-1 (Hu et al., 2011).In the next year, the Chinese first manual RVD taskwas also successfully completed by Shenzhou-9 andTiangong-1 in June 2012 (Xie et al., 2013).

Shenzhou spacecraft is composed of an orbitalmodule, a reentry module, and a propulsion module.There are two large solar panels on the propulsionmodule. During RVD, attitude maneuvers and orbitcontrol are frequently performed by firing the appro-priate pairs of thrusters. This excites the flexible vi-bration of the solar panels. Meanwhile, the thrusters’plume, which acts on the solar panels, leads to dis-turbance. Therefore, it is a great challenge to designa controller with high stability, accuracy, and adapt-ability for the RVD mission of Shenzhou spacecrafts.

Because of the infinite-order feature of the flex-ible structure, a characteristic model is introducedthat uses a second-order linear discrete-time modelto describe the dynamics of the flexible structure.Information concerning the high-order feature andnonlinearity is compressed into those characteristicparameters, which are identified online within a con-vex domain. The detailed steps of characteristicmodeling and controller design were given in Wuet al. (2001). Moreover, other difficulties such asserious disturbance due to thrusters’ plume, cross-coupling between attitude and orbit control, andlarge time delay were taken into account in the pro-cess of RVD. The flight data indicate that the highcontrol accuracy of Shenzhou spacecrafts in RVDtasks has reached a high level worldwide (Xie et al.,2013).

4.2 Data Relay Test Satellite

The Data Relay Test Satellite (DRTS), whichwas launched in Sept. 2002 by NASDA, is a research


communications satellite that demonstrates data re-lay experiments on the geostationary orbit (Fujiwaraet al., 2003). The communication antenna couldcause disturbance to the attitude system. Therefore,parameters corresponding to the mass property wereestimated online. A feed-forward control was de-signed based on the estimates (Yamada et al., 2003).The on-orbit flight test showed that the self-tuningadaptive attitude control system has almost the sameperformance as the one predicted on the ground.

4.3 Middeck active control experiment

The middeck active control experiment(MACE) was developed by the Airforce ResearchLaboratory (AFRL), USA, aimed at the modelingand high-accuracy pointing control issues of flexiblestructures in the unknown space environment. Theexperiment was carried out on a test article of 1.7 mlength and 39 kg weight. It consists of a flexiblebus with 17 sensors and 9 actuators including thereaction wheel. There were as many as 50 modesto be controlled, and according to the previousflight experiments, the structural dynamics couldchange significantly at different modes, which poseda huge challenge to controller design from groundso as to meet the stringent pointing accuracy.This test article was delivered to the orbit by theSpace Shuttle flight STS-67 in Mar. 1995. Themicro-gravity on-orbit experiment was carried outonboard the Space Shuttle’s Middeck to evaluate theidentification and controller design. There were twomodels available: the finite element model and themeasurement model. Each of these models contains130−150 states. The original LQG controller failedto provide satisfactory robustness to parameteruncertainty; therefore, five other modern controllerswere developed including the sensitivity-weightedLQG (How et al., 1996), the maximum entropy (Howet al., 1996), the multiple model (How et al., 1996),the Popov controller (How et al., 1994), and the H∞synthesis (Woods-Vedeler and Horta, 1996). Thefirst four controllers were testified on measurementmodels built using open-loop data, and the lastone was validated on-orbit. In the H∞ controllerdesign, the high-order model was reduced using thebalanced model reduction technique (Zhou et al.,1996). Multiplicative uncertainty at the output andadditive plant uncertainty caused by unmodeleddynamics were considered during the H∞ controller

design. The suboptimal H∞ controller design wasapplied by setting γ = 1 (Section 2.6). Accordingto Woods-Vedeler and Horta (1996), over 50 H∞controllers were tested on-orbit. The robustnessof quite a few closed-loop systems with respect todisturbance was maintained within a satisfactorylevel. The overall system was stable during theentire flight experiment. Meanwhile, controllersdeveloped upon the reduced-order model showedidentical performance to that of the one developedon the full-order model. A ground experimenton Popov control based on the absolute stabilitytheory and H2 analysis was introduced by Howet al. (1994). It was demonstrated that, comparedto the optimal LQG design, the Popov controllerguaranteed superior robust performance.

MACE II is a hands-on experiment aboard theISS. The ultimate goal is to implement adaptivestructural control technology in spacecrafts. Theadaptive controller would facilitate controller designon the ground and deal with the change of dynamiccharacteristics and sensor/actuator failures. MACEII was sent to ISS in 2000 aboard the SPACEHABmodule and, after 347 days, it returned aboard STS-105. The adaptive structural control methods wereable to ‘adapt’ whenever they sensed changes in vi-bration or the loss of a sensor or an actuator. Theseadaptive algorithms provided a decrease in vibration,even when a primary actuator experienced failure(Grocott et al., 1994).

One of the most significant achievements of theMACE program was that it showed the limitationsof the traditional fixed-gain control approach andpushed the application of modern control methodsin space engineering.

5 Reentry spacecraft with changingparameters

The reentry of spacecrafts is another challeng-ing task because of the fast time-varying dynamics,model nonlinearities, and large flight envelopes. Inthe transonic regime, there would be large aerody-namic coefficient uncertainties. The unstable andpartially unknown atmosphere further brings diffi-culties to the modeling and controller design. Mean-while, to achieve fast maneuverability, the reentryspacecrafts sometimes contain unstable modes, be-coming the most fragile part of the system.


In this section, several reentry spacecrafts thatimplemented advanced control methods are intro-duced. Towards the changing parameters within alarge scale, although currently there is no theoreticalbreakthrough concerning parameter estimation, en-gineers have proposed several techniques that havebeen proven effective on some specific spacecrafts.For example, during the reentry of Shenzhou space-crafts, input-output transformation was developedsuch that the estimation was carried out within asmall scale instead of a large one.

It is worth noting that reentry spacecrafts usu-ally have military backgrounds, and therefore thereferences are usually limited. Those famous reen-try spacecrafts, such as X-37B, are not included herebecause we could not find published literature withdetailed control system design.

5.1 Italian Unmanned Space Vehicle

The Italian Aerospace Research Center con-ducted an aerospace national research programcalled Unmanned Space Vehicles (USV) in 2003.This program aimed to develop and test new tech-nologies for aerodynamics, guidance, navigation, andcontrol to support the future reusable launch ve-hicles and aerospace planes. The USV programhad been divided into several phases. In the firstphase, the focus was on the subsonic, transonic, andlow supersonic regimes. Two transonic flight mis-sions, DTFT1 and DTFT2, were scheduled in win-ter 2007 and spring 2010, respectively. The test ve-hicle was released from a scientific balloon, whichwas 20 km from the ground for DTFT1 and 24 kmfrom the ground for DTFT2. It then experiencedthe gliding phase and terminal aero energy manage-ment phase subsequentially before landing using arecovery parachute. The DTFT1 flew in a tran-sonic regime with only longitudinal maneuvers. Inthis relatively mild condition, the conventional con-trol augmentation system with the PID controllerwas capable of maintaining the angle of an attackat a constant value. However, the DTFT2 was amore complex mission. After releasing from the bal-loon, it was scheduled to pitch up to reach a pre-defined angle of an attack and accelerated up toMach 1.2 at about 15 km. After that, the vehiclehad to pull down to keep a constant Mach number.Meanwhile, the vehicle had to select an appropriatelanding position out of four preloaded ones online

due to the uncontrolled drop position by the bal-loon. Thus, online trajectory generation and adap-tive tracking would be a huge challenge. The highMach number, rapid maneuvers, and online adaptionmade the traditional flight control system for DTFT1short-handed.

A probabilistic robust control synthesis was cho-sen after being compared to the μ-controller withfuzzy logic gain-scheduling and the direct adaptivemodel-following controller (Corraro et al., 2011a;2011b). According to the post-flight data analysis,this robust control law met with great success, andshould be capable of controlling the terminal area en-ergy management (TAEM) phase of a reentry vehicle(Corraro et al., 2011b; Nebula and Ariola, 2013).

5.2 Space Launch System

For future convenient and reliable access tospace including the ISS in low-Earth orbit (LEO),the Moon, Mars, and near-Earth asteroids, NASAhas initiated the Space Launch System (SLS) con-sisting of various exploration-class launch vehicles.The commonly used test vehicle is the F/A-18, thefull-scale advanced systems testbed. The featuresof aerospace vehicles including interactions amongcontrol surfaces, control−structure interaction, en-gine performance, sensor characteristics, and atmo-spheric behavior are treated specifically. Since 2009,the nonlinear dynamic inversion control and a fewadaptive control algorithms have been testified inthis Space Launch System, and the flight control sys-tem has been validated in many flight tests.

Among the various adaptive control laws, adap-tive augmenting control (AAC), which is a specialkind of model-reference adaptive control, is prob-ably the most successful one. AAC consists of aclassically designed linear controller as the baselinecontroller (mostly PID) and an adaptive total loopgain. Normally, the adaptive gain stays at the min-imum value when the leakage term is not activated.Once the tracking error is too large, the adaptivegain increases immediately to compensate for theperformance loss. Most importantly, when an un-desirable frequency is detected in the control path,the adaptive gain would decrease until the parasiticdynamics is mitigated. The mathematical represen-tation of AAC is (Orr and VanZwieten, 2012)

k̇a =

(kmax − ka

kmax

)ae2r − αkays − β(kT − 1),


where kT = k0 + ka, ka and k0 are the adaptive gainand the minimum gain respectively, kmax is the limitof ka, a is the adaptive error gain, α is the spectraldamper gain, β is the leakage gain, er is the trackingerror, and ys is the output of the filter.

This improved model-reference adaptive controlfinds a balance between the tracking performanceand robustness. It was considered a great successwhen dealing with large pilot inputs, which causedthe loss of X-15-3 in 1967 (Thompson and Hunley,2000).

Both simulations and flight tests were con-ducted. In the flight test, the F/A-18 flew the SLS-like trajectory and in nominal, off-nominal, and fail-ure scenarios. According to the report of NASA inJan. 2015, the F/A-18 had flown more than 100 testcases, a few of which lasted more than 60 min. Ac-cording to the flight results, with AAC on, the systemresponse remained bounded and was capable of deal-ing with the unstable dynamics (Wall et al., 2015).

Besides, to prevent high-frequency oscillationscaused by the high adaptive gain, optimization uponthe L2-norm of the tracking error was studied byBurken et al. (2010). Neural networks were designedto modify the output of the PI controller toward theoptimal criterion. This control system has been sim-ulated in the simulation testbed provided by NASAto show its potential benefits.

5.3 Shenzhou reentry module

The Chinese Shenzhou program was authorizedin 1992. Since then, Shenzhou 1−10 spacecrafts havebeen launched in succession. A total of 10 astronautshave been sent into space and returned to groundsuccessfully. The control and guidance technology ofthe atmospheric entry is a crucial technique in theguidance, navigation, and control (GNC) system of amanned spacecraft. There are several challenges (Wuet al., 2009): (1) The reentry module is a small lift-ing body. This limits the modification capacity of itsmotion trajectory by changing the lift direction. (2)Environmental parameters such as atmospheric den-sity and wind velocity intensively vary with respectto the altitude of the module. Then the aerodynamicparameters such as drag coefficients and lift-to-dragratios of the module are changing in a large scale.This will lead to a large landing error if not consid-ered. (3) Some state constraints should be satisfiedsuch that the acceleration during the reentry must

remain within a safe limit, the landing site should bein the scheduled area, etc.

For the reentry of Shenzhou’s reentry module, astandard reentry trajectory guidance method withan estimated lift-to-drag ratio was proposed (Hu,1998). It shares the advantages of both the land-ing point prediction guidance method and standardtrajectory guidance method. The execution steps areas follows: In the transition phase before the reentry,the calculation of landing point prediction and guid-ance is performed, and a standard reentry trajectoryfor control is obtained. In the reentry phase, the re-sulting reentry trajectory is regarded as the guidancegoal. A characteristic model-based adaptive methodis adopted to estimate the lift-to-drag ratio online,which is used for lift control of the reentry module.To deal with the parameters that vary within a largescale, input-output transformation is developed suchthat the estimation is then carried out within a smallscale instead.

Flight data showed that the guidance and con-trol method could deal with large initial errors, re-duce the fuel consumption, and enhance the stabilityof the reentry. The guidance and control technologyfor the reentry of Shenzhou spacecrafts was quitemature. The dispersions of the spacecrafts’ landingpoints were all controlled within 13 km.

5.4 Chang’e 5 test spacecraft

As the third phase of China’s lunar exploration,the Chang’e 5 mission is scheduled to launch in 2017and is arranged to take samples from the Moon andreturn to the Earth automatically. To guaranteethe success of the mission, China decided to initiatea Chang’e 5 test program to validate the guidanceand control of the return capsule from the Moon.Chang’e 5 test vehicle was launched in Oct. 2014 us-ing a CZ-3C rocket. After traveling along the Earth-Moon transfer orbit for nearly 8 days, it returnedto the Earth on Nov. 1 and landed safely in InnerMongolia.

After returning from the Moon, the reentry ve-locity of Chang’e 5 at the 120 km near-Earth or-bit reached the second cosmic velocity. For accuratelanding at a low speed, the semi-ballistic skip reentrytechnique was chosen. This kind of reentry strategyis quite different from that of the Shenzhou reen-try module from the near-Earth orbit and has neverbeen used in any of the previous returned capsules


in China.For the guidance and control of this semi-

ballistic skip trajectory, a large number of simula-tions showed that the traditional guidance and con-trol methods cannot provide satisfactory accuracyand reliability, or the capsule may jump back to spaceand never come back. There is a strong need foran advanced guidance and control technique. En-gineers from the Beijing Institute of Control Engi-neering, China Academy of Space Technology, pro-posed a first-order all-coefficient adaptive prediction-correction scheme (Hu, 2014). This is a simplerform of the commonly used second-order charac-teristic model-based adaptive control introduced inSection 2. The first-order model showed superiorperformance to the second-order model in numericalsimulations (Hu and Zhang, 2014) and hardware-in-the-loop experiments. Most importantly, this guid-ance and control algorithm was finally uploaded tothe Chang’e 5T test vehicle, and the opening pointaccuracy was about 500 m from the expected point.

6 Spacecraft with high-level require-ments

In this section, four kinds of spacecraft with dif-ferent kinds of mission requirements are introduced.The classical PID controller can provide only a sys-tem with limited stability margin. Improved missionrequirements bring high demands on the robustnessand accuracy of the control system, which pushed usto the optimal control methods.

6.1 Reorientation of the International SpaceStation

There are a few ways for the ISS to performrotation maneuvers while in space. For the short-est path rotation, the cost is high because of thekinematically nonlinear dynamics. When the controlmoment gyroscopes (CMGs) are inadequate to pro-vide sufficient torques, thrusters are used. Becauseof CMGs’ lifetime issues, momentum desaturationusing thrusters is not a wise option. To maintain theCMGs within their operational limits while execut-ing large-angle attitude maneuvers, a practical way isthe zero-propellant maneuver (ZPM) method. ZPMgenerates a rotation trajectory, in which the rota-tion uses only the naturally occurring environmentaltorques without reaching the limits of CMGs.

The ZPM attitude control concept was devel-oped at the Draper Laboratory in the mid-1990s.The trajectory for ZPM was solved through an opti-mal control problem with the constraints of systemdynamics, initial and terminal states, and CMGs’capacity. Thanks to the advances in PS methods,which can solve this optimal control problem in anefficient and rapid way, the optimal trajectory canbe generated fast and is suitable for engineeringapplications.

The PS method uses the Lagrange interpolat-ing polynomials over Gaussian nodes to discretizethe optimal problem, and ensures a faster conver-gence rate than the previous fourth-order conver-gence (Betts and Kolmanovsky, 2002). Meanwhile,by using the covector mapping principle, the PSmethod allows checking the feasibility and optimal-ity of the optimal solution, which is quite valuableto ZPM. The control loop is a feed-forward openloop, as the carefully designed trajectory requires nothrusters or other external torques.

Two subsequential flight tests were scheduled totestify, for the first time in history, ZPM in Nov. 2006and Mar. 2007, which reoriented the ISS 90◦ and180◦, respectively. The ZPM trajectory was gener-ated a month before the flight date. First, an ini-tial trajectory was generated and tested in simula-tion. Parameter uncertainties were then consideredto improve the robustness of the trajectory. In bothflight tests, the trajectories of ZPM were completedsuccessfully and no propellant was used. Accord-ing to the data provided by the Mission EvaluationRoom, the actual attitude ideally fitted the com-manded one. According to the report, in the firstflight, ZPM saved 50 lbs of propellant, and the sec-ond flight saved 100 lbs. This was the first timethat the PS optimal control theory was ever used ina space mission (Bedrossian et al., 2007; Bedrossianand Bhatt, 2008).

6.2 Reorientation of the Transition Regionand Coronal Explorer

PS optimal control techniques have beensuccessfully applied to design and implement aminimum-time reorientation maneuver on NASA’sspace telescope, Transition Region and Coronal Ex-plorer (TRACE), in 2010 (Karpenko et al., 2012).TRACE was designed to document the magneticfeatures of the solar surface, transition region, and


corona. The mission requires its attitude control sys-tem be able to maintain the pitch and yaw pointingaccuracy of 20 arcseconds and to maneuver up to180◦ between targets. Practical optimal control thatis suitable for on-orbit implementation is needed toimprove the agility of the satellite.

The minimum-time reorientation optimal con-trol by the PS method was first addressed by Proulxand Ross (2001). It could overcome many numeri-cal difficulties associated with finding optimal solu-tions. Many ground experiments and flight imple-mentations (Ross and Karpenko, 2012) showed thereliability and suitability for generating practical so-lutions. Besides, most steps in the method were au-tomatic. It could provide the possibility of a fullyautomated design process for spacecraft operations.

The time-optimal reorientation maneuver of theTRACE spacecraft is obtained by solving the fol-lowing state-constrained optimal control problem(Karpenko et al., 2012):

BR :

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

minu

J(x,u, t) = tf

s.t. ẋ(t) =

[ 12Q(ω)q

I−1(ω × Iω − u)

],

x0 = [e0 sin(Φ0/2), cos(Φ0/2), ω0]T,

xf = [ef sin(Φf/2), cos(Φf/2), ωf]T,

‖q‖2 = 1,|ωi| ≤ ωmax, i = 1, 2, 3,|ui| ≤ ui,max, i = 1, 2, 3.

The problem aims to find the control u thatdrives the rigid body from the initial conditionsx0 at t = 0 to the final conditions xf att = tf, and to minimize the time needed. De-note q0 = [e0 sin(Φ0/2), cos(Φ0/2)]T and qf =[ef sin(Φf/2), cos(Φf/2)]

T as the initial and final atti-tudes of the rigid body, respectively, and ω0 = ωf = 0are the angular velocities. The state and control forthe optimal control problem satisfy the constraints‖q‖2 = 1, |ωi| ≤ ωmax, and |ui| ≤ ui,max (i =1, 2, 3). The angular velocity must be limited toa certain range to avoid the saturation of the rategyros, which would lead the satellite out of control.

The object of on-orbit time-optimal control is toperform a reorientation maneuver of 100◦. Through-out the maneuver, the body rate was limited within0.5◦/s to avoid gyro saturation. The PS-based time-optimal reorientation maneuver was solved for thespacecraft using the DIDO software (Ross, 2007) and

was directly implemented to the onboard attitudecontrol system of the TRACE. The maneuver in thePS-based method was completed in 181.4 s. More-over, a standard eigenaxis maneuver, which took205.5 s to complete, was implemented for compar-ison. Thus, the PS-based time-optimal reorientationwas about 12% faster than the conventional ma-neuver. More effective usage of the actuators wasobtained by building up angular rates around allthree body axes. This enabled the TRACE to rotatethrough a longer path more quickly than the con-ventional method. The flight results showed evidentimprovement in agility compared with the conven-tional method.

6.3 Fast maneuver of the SSTL microsatellite

The three-axis attitude control system for mi-crosatellites is based mainly on the PID controller,although the performance of the attitude controlhardware has been greatly improved during the lastdecades. For a given level of energy consumption, thePID controllers limit the attitude response rapidity.On the other hand, to obtain the global optimal feed-back control, it is necessary to solve an HJB partialdifferential equation. Generally, the online calcula-tion of the optimal control for operational attitudecontrol systems is infeasible. The inverse optimalcontrol method was originally developed for this kindof problem (Freeman and Kokotovic, 1996).

The inverse optimal control was originated byAnderson and Moore (1990) to establish certainstable margins for linear systems and was intro-duced into nonlinear control by Moylan and Ander-son (1973). Freeman and Kokotovic (1996) proposeda systematic robust inverse optimal control methodthat could circumvent the task of solving an HJBequation. The main idea of inverse optimality isbased on the fact that the steady-state solution tothe HJB equation is a control Lyapunov functionobtained from the stabilization problem of the non-linear system. Theoretically, the solution is globallystable and optimal, and the stability margins can becalculated for the input-to-state stable system.

The inverse optimal controller was validated bya software satellite simulator developed by SurreySatellite Technology Ltd. (SSTL) for microsatellites(Horri et al., 2011). The simulator incorporates fullattitude and orbital dynamics with precise externaldisturbance, gravity, atmospheric drag, etc. It was


designed to validate attitude determination and con-trol methods before uploading them to on-orbit satel-lites. In the experiments, a maneuver from the initialattitude to 30◦ off-pointing was first carried out, fol-lowed by a maneuver from 30◦ off-pointing to theinitial attitude, which can be based on either the PDcontroller with gyro-compensation or an inverse op-timal gain scheduled minimum norm controller. Theinverse optimal controller was simple to implementand achieved a faster attitude pointing than the PDcontroller. The settling time was significantly re-duced (Horri et al., 2011).

6.4 Accurate pointing of FASTSAT

The Fast Affordable Science and TechnologySatellite (FASTSAT) (DeKock et al., 2011) is a risk-tolerant, small-budget microsatellite program. Thefirst FASTSAT is the Huntsville-01 (HSV-01), whichwas developed collaboratively by the NASA Mar-shal Space Flight Center, Dynetics, the Universityof Alabama at Huntsville, and several other indus-try partners in Huntsville, Alabama. The satellite,whose sides are about 30 inches tall, was launched inNov. 2010 and operated in the 650 km, 72◦ inclina-tion orbit.

For a microsatellite like HSV-01, the actuatorsare magnetic torque rods, which can save a lot ofweight and power compared with a wheel-based sys-tem. Therefore, the satellite is a roughly periodicsystem due to the geomagnetic field the satellite fliesin.

The scientific experiments assigned to FAST-SAT require various attitudes be held. Therefore,the conventional spin-stabilizing method is inappro-priate. To achieve high pointing accuracy at differentattitudes, the periodic asymptotic LQR controllerwas chosen. The periodic LQR problem is formal-ized as⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

minu

J =1

2

∫ T0

[(x(τ))TQx(τ)

+(u(τ))TRu(τ)]dτ +

1

2(x(T ))TPx(T )

s.t. ẋ = Ax+B(t)u,B(t) = B(t+ T ),

x(0) is given.

The solution to the above problem is u(t) =−R−1(B(t))TP (t)x(t), where P (t) is periodic withperiod T . To facilitate practical implementation in

satellites, the periodic P (t) is approximated by aconstant matrix when a series of constraints on ma-trices R, [A, B], and [A, C], as well as the eigen-values of A are satisfied (Psiaki, 2001).

The periodic LQR implemented onboard FAST-SAT was nine-state periodic. According to the flightdata that were downlinked, FASTSAT reached ahigher pointing accuracy than with the previous bestdemonstrated local vertical local horizontal relativeall-magnetic attitude controller. Meanwhile, it al-lowed the satellite to point at specific ground tar-gets, instead of using large satellites with complexattitude control systems. It is hoped that this peri-odic LQR controller can be applied in a nanosatelliteplatform such as a CubeSat in the future.

7 Conclusions

Space activities of humans are becoming fre-quent with emerging new types of spacecrafts andstringent performance requirements like never be-fore. The traditional PID controller is graduallybecoming incapable of meeting the increasing de-mands. Robust control, adaptive control, and op-timal control are undoubtedly the most frequentlyconsidered modern control methods in aerospace en-gineering due to their robustness, adaptability, andoptimality. The biggest issue during controller de-sign would be the lack of an accurate mathemati-cal model, and the extremely high-order and non-linear character of those models. Furthermore, itis extremely difficult to simulate the space environ-ment, which is full of uncertainties and disturbance.How do we guarantee the on-orbit closed-loop per-formance of the controllers that are developed andtested on the ground? Engineers have been seekingfeasible ways to deal with this problem for quite along time. Meanwhile, rigorous theoretical results inrobust and adaptive control are always accompaniedby assumptions such as known parameter bounds orappropriate initial values, which have brought trou-bles to aerospace engineers when designing and tun-ing those controllers.

Fortunately, aerospace scientists and engineersall over the world have realized the necessity of in-troducing advanced control to aerospace engineer-ing. A few satellites and reentry vehicles have al-ready experimented with some advanced control lawsand have benefited from them. Higher pointing


accuracy and stronger robustness to disturbancesand unstructured dynamics have been validatedby some spacecrafts on-orbit with convincing flightdata. Motivated by this inspiring trend in aerospacecontrol, we provided a thorough review on the practi-cal applications of advanced control methods in satel-lites and reentry spacecrafts. Most of these controlmethods were validated by actual flight experiments,which are particularly valuable to both engineers andscientists. These spacecrafts and their advanced con-trol laws are summarized in Table 1. This table re-veals some valuable and interesting disciplines:

1. FlexibilityTo deal with the flexible modes, engineers from

Spacebus 4000, ETS-VI/VIII, and HST reached aconsensus on the H∞/H2 robust control that pro-vided satisfactory performance on-orbit and on theground. These successful applications suggestedthat, when dealing with flexible satellites, the robustcontrol method is usually the first choice.

2. UncertaintiesIn face of large parameter uncertainties, adap-

tive control methods, self-tuning and CM-GSACin particular, outperform other control methods asshown in Shenzhou spacecrafts and DRTS.

3. Changing parametersDuring the reentry of a spacecraft, a controller

with fixed parameters is incapable of dealing withthe large variations of parameters. Flight vehicle

F/A-18 has shown the adaptability of MRAC, whileShenzhou and Chang’e 5T spacecrafts have provedthat CM-GSAC can deal with the large parameterschange during reentry.

Although there have been several successful ap-plications of advanced control laws in aerospace en-gineering, there is still a long way to go beforethe new era of aerospace advanced control. Chal-lenges, both theoretical and practical, are broughtby the high-level stringent performance requirementsin the presence of large flexibility, large parame-ter changes, structural uncertainties, and high-ordermodes. Meanwhile, it is indispensable to develop asystematic design process based on modeling, syn-thesis, verification, and validation. This processshould be able to distinguish between controllersand help pick up the optimal ones. To be morespecific, such a systematic design process should becarried out through two manifolds, both of which de-pend highly on spacecraft characteristics and missionrequirements.

The first manifold is the modeling of the space-craft, which should be categorized by the main char-acters of the spacecraft, such as flexibility or differentkinds of uncertainties, and by the priority of missionrequirements, such as fast maneuverability or highpointing accuracy.

In the second manifold, controller design shouldbe distinguished among different kinds of plants.

Table 1 Spacecrafts with advanced control methods and the time when they were applied to the correspondingspacecrafts

Character SpacecraftOptimal control Adaptive control Robust control

LQRInverse

PS MRACSelf- CM-

H∞/H2 µoptimal tuning GSAC

Flexiblestructure

Spacebus 4000 2003+ 2003+ETS-VI 1995 1995ETS-VIII 2009; 2010 2009; 2010HST (simulation) 1995 1995

Unknownparameters

Shenzhou RVD 2011; 2012MACE 1995 1995DRTS 2002

Changingparameters

Space Launch System 2009+Chang’e 5T 2014Shenzhou reentry 1992+

High-levelrequirements

ISS 2007SSTL (simulation) 2011TRACE 2010FASTSAT 2010


Different control methods are competent with dif-ferent kinds of models and different control objec-tives. For instance, optimal control depends highlyon the plant models. According to Table 1, this kindof control method performs quite well in spacecraftswith high-level requirements. In particular, the PSmethod is quite handy in satellite reorientation mis-sions. Robust control allows a limited range of pa-rameter variations, as well as external disturbance.The changing rates of uncertainties could be large aslong as they do not exceed the designed bounds. Thisis the reason why in case of flexible structure, robustcontrol becomes quite useful. On the other hand,adaptive control can identify slowly time-varying pa-rameters that change over a large range. However,its robustness is not quantified. The most promi-nent field for adaptive control should be the reentryof spacecrafts, where the large parameter variationsare beyond the capability of robust control. Accord-ing to Table 1, it is worth noting that among allthe well-known adaptive control methods, the char-acteristic model-based golden-section adaptive con-trol method has received successful applications inas many as 10 reentry capsules in China. This kindof adaptive control method is different from the gen-eral adaptive control method in several ways. First,the characteristic model is built based on a completeanalysis of the plant and control objective. Second,its robustness is guaranteed by the golden-sectioncoefficients and the fixed parameter bounds that arecalculated beforehand. It is suitable particularly fora system that is slowly time-varying and can be mod-eled by a second-order or a first-order characteristicmodel. This is the reason why this simple but effec-tive adaptive control method has served so well inthese capsules in China.

We believe that the ultimate goal in aerospacecontrol engineering is a highly automatic systemwith great intelligence that can deal with all kindsof uncertainties, disturbance, failures, and missionchanges on its own. Building a comprehensivedatabase that contains the general models ofthe representative spacecraft and their candidateadvanced controllers is the foundation to this goal.Future artificial intelligence will learn from thisdatabase, reason from this knowledge base, andfinally evolve to the stage that provides humanswith a highly intelligent aerospace control system.

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Applicationsofadvancedcontrolmethodsinspacecrafts ......progress,challenges,andfutureprospects∗ Yong-chun XIE †‡ 1,2 , Huang HUANG † ,YongHU 1,2 , Guo-qi ZHANG ( 1 Science

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