simplified adaptive control of an orbiting flexible spacecraft

Acta Astronautica 61 (2007) 575–589www.elsevier.com/locate/actaastro

Simplified adaptive control of an orbiting flexible spacecraftGanesh B. Maganti, Sahjendra N. Singh∗

Department of Electrical and Computer Engineering, University of Nevada Las Vegas, Las Vegas, NV 89154-4026, USA

Received 17 April 2005Available online 17 April 2007

Abstract

The paper presents the design of a new simple adaptive system for the rotational maneuver and vibration suppression of anorbiting spacecraft with flexible appendages. A moment generating device located on the central rigid body of the spacecraft isused for the attitude control. It is assumed that the system parameters are unknown and the truncated model of the spacecraft hasfinite but arbitrary dimension. In addition, only the pitch angle and its derivative are measured and elastic modes are not availablefor feedback. The control output variable is chosen as the linear combination of the pitch angle and the pitch rate. Exploiting thehyper minimum phase nature of the spacecraft, a simple adaptive control law is derived for the pitch angle control and elasticmode stabilization. The adaptation rule requires only four adjustable parameters and the structure of the control system does notdepend on the order of the truncated spacecraft model. For the synthesis of control system, the measured output error and thestates of a third-order command generator are used. Simulation results are presented which show that in the closed-loop systemadaptive output regulation is accomplished in spite of large parameter uncertainties and disturbance input.© 2007 Elsevier Ltd. All rights reserved.

Keywords: Flexible spacecraft; Adaptive control; Output regulation

1. Introduction

Orbiting spacecraft with flexible appendages play animportant role in communication, remote sensing, andvariety of space related research. Appendages such asantennas and solar arrays attached to the central rigidbody of the spacecraft are flexible. The dynamic mod-els of elastic spacecraft include large number of weaklydamped elastic modes and the rigid modes. The pri-mary challenges in control system design for flexiblespacecraft are: (i) large model order; (ii) weakly dampedoscillating behavior of the elastic appendages, (iii) in-herent modeling errors due to finite order approximation

∗ Corresponding author. Tel.: +1 702 895 3417;fax: +1 702 895 4075.

E-mail address: [email protected] (S.N. Singh).

0094-5765/$ - see front matter © 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.actaastro.2007.02.004

of the partial differential equations describing the mo-tion of the flexible parts; (iv) parameter uncertainties,and (v) nonlinearities.

Advances in control theory has provided several de-sign techniques which have been applied to control flex-ible spacecraft. An excellent survey of research in thisarea has been published by Hyland et al. [1]. Optimaland suboptimal control systems for the control of flex-ible spacecraft have been developed [2–8]. Effort hasbeen also made to design robust and nonlinear controlsystems [9–18]. Vibration reduction for flexible space-craft by input shaping has been considered in [19–21].In [21], a new approach integrating component synthesisvibration suppression (CSVS) based command shapingtechnique and positive position feedback (PPF) controlis proposed for attitude maneuver. Using Lyapunovstability and dissipativity theory, control systems forslewing and vibration suppression have been developed

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576 G.B. Maganti, S.N. Singh / Acta Astronautica 61 (2007) 575–589

Nomenclature

A, B, H system matricesA1, F reference model matricesCpp, Kpp, Mpp, m�p modal integralse tracking errorh, �h nonlinear functionskx, kv, U feedback gainskx , U estimates of feedback gainsLb length of each appendagep elastic modes

P, Q positive definite symmetric matricesu, v control and reference model stateV0, V Lyapunov functionsx, x, x∗ state vectorsy, ym output, reference modal output� mass per unit length�c, �c reference model parameters�, � pitch angle and pitch rate� orbital rate

[22–24]. An approach to vibration suppression of flexi-ble spacecraft using pulse width frequency modulationduring attitude control has been presented [25]. A neuraloptimal controller for nonlinear slew maneuver of flex-ible spacecraft has been designed using state feedback[26]. Variety of adaptive control techniques have alsobeen used for designing controllers for flexible space-craft [27–29]. The adaptive design of [29] is based onthe command generator tracker design approach.

For nonlinear orbiting flexible spacecraft model of[14], feedback linearization, model reference variablestructure control [12], and modeling error compensationdesign [16] techniques have been applied and usefulresults have been obtained. However, the feedback lin-earization technique of [14] requires complete knowl-edge of system parameters and the variable structureadaptive control presented in [12] requires the knowl-edge of the bounds on uncertain parameters as well aslarge order filters for control. The modeling error com-pensation approach of [16] uses a high-gain feedbackobserver for the estimation of unmodeled functions forfeedback. Certainly, it will be interesting to design anadaptive control law which is simple, yet independentof the number of elastic modes retained in the space-craft model.

The contribution of this paper lies in the design of asimple new adaptive control law for rotational maneu-vers and elastic mode stabilization of a flexible space-craft. The derivation is based on the command generatortracker design method [29]. However, in the derivationof the control law, certain arguments from the outputregulation theory are used for clarity in this presenta-tion. The spacecraft model considered in this study issimilar to that of [14]. Although, the design approachis applicable to spacecraft of other configurations, fordefiniteness, in this study an orbiting spacecraft con-sisting of a rigid hub with two flexible appendages isconsidered. The spacecraft parameters are completelyunknown and its model is assumed to be of finite but

arbitrary order. This is important because flexible struc-tures have infinite dimension, but for simplicity finite di-mensional approximate models are often used for study.It is assumed that only the pitch angle and its deriva-tive are accessible for feedback and flexible modes arenot measured. A reference trajectory generator of third-order is chosen and a linear combination of the pitchangle and its derivative is selected for the output regula-tor design. Based on the output regulation theory [30], asimple structure of the control system is chosen whichinvolves only four gains. By the choice of the outputvariable, the system becomes hyper minimum phase(to be defined later) and this feature is exploited to de-sign an adaptive controller, which requires feedback ofonly the output (tracking) error and the three states of thereference model. Interestingly, the structure of the con-troller and the order of the dynamic feedback (adaptive)law is independent of the dimension of the spacecraftmodel; and moreover, the instability caused by controland observation spillover cannot occur in the closed-loop system. In the closed-loop system, the pitch an-gle asymptotically converges to the terminal value andthe flexible modes attain the equilibrium point. Simula-tion results are obtained which show precise pitch angletrajectory control and vibration suppression in spite oflarge uncertainties in the model.

The organization of the paper is as follows. Section 2presents the spacecraft model. The existence and designof an adaptive control system are treated in Section 3and Section 4 presents numerical results.

2. Elastic spacecraft model and control problem

Fig. 1 shows the model of an orbiting flexible space-craft. This model is representative of a relatively largeclass of spacecraft employed for communication, re-mote sensing, and numerous other applications. Themodel consists of a rigid central hub, which repre-sents the satellite body, and two flexible appendages.

G.B. Maganti, S.N. Singh / Acta Astronautica 61 (2007) 575–589 577

Fig. 1. Spacecraft with elastic appendages.

A right-handed coordinate frame with axes Xc, Yc, Zcis fixed to spacecraft with its origin at the mass centerof the spacecraft. X0, Y0, Z0 are the axes of an inertialframe, and the orbital frame is represented by the axesXs, Ys, Zs .

Let � be the true anomaly and � be the pitch an-gle which is to be controlled using a torque generatingdevise located at the center of the structure. When thespacecraft is slewed about the axis Zs which is normal tothe orbital plane, the flexible appendages are deformed.It is assumed that these appendages undergo elastictransverse bending only in the orbital plane Xs − Ys .Similar to the previous studies [14], it is assumed thatthe pitch maneuver excites the two flexible appendagesanti-symmetrically. For the configuration of spacecraft,this assumption of antisymmetric deformation is rea-sonable and has been validated in [14].

The equations of motion of the spacecraft are de-rived using the Lagrangian approach. Using assumedmode method, the transverse elastic deflection of theappendage along Yc in the orbital plane is expressed as

�(l, t) =N∑

i=1

�i (l − r)pi(t), (1)

where pi(t) are the generalized coordinates and r is theradius of the hub, l is the distance of a point chosen on

the appendage from the center of the hub, and �i are thechosen admissible functions which satisfy the geometricand physical boundary conditions. It is assumed herethat N mode are sufficient for the computation of elasticdeformation.

The nonlinear differential equations describing therotational and elastic dynamics in a circular orbit aregiven by (see [14] for the details)

(J + 2J1 + pTMppp)� + mT�p

p + 3�2

sin(2�)

× (J1 − 12pTMppp) + 2(� + �)pTMppp

+ 3�2

cos(2�)mT�p

p = u,

Mppp + m�p� + 32 �

2sin(2�)m�p + Cppp

+ [Kpp − (�2 + 2�� + 3�

2sin2 �)Mpp]p = 0, (2)

where p = (p1, . . . , pN)T is the vector of generalizedcoordinates, J and J1 are the moment of inertia of thecentral hub and each appendage, respectively, u is thecontrol torque, � is the orbital rate, and Mpp, m�p, Cpp

and Kpp are the following modal integrals:

[Mpp]ij = 2∫ r+Lb

r

�i (l − r)�j (l − r) dm,

[m�p]i = 2r

∫ r+Lb

r

l�i (l − r) dm,


[Cpp]ij = 2∫ r+Lb

r

CI�′′i (l − r)�′′

j (l − r) dl,

[Kpp]ij = 2∫ r+Lb

r

EI�′′i (l − r)�′′

j (l − r) dl, (3)

where r is the hub radius, �′′i = (�2�i/�l2), C and E

are the damping coefficient and modulus of elasticity,respectively, for the appendages and I is the sectionalarea moment of inertia with respect to the appendagebending axis.

Thus, the system (2) can be written in a compact formas

M

[�p

]+ C

[�p

]+ K

[�p

]=

[h1h2

]+

[10

]u, (4)

where (�, pT)T ∈ RN+1, 0 denotes a null vector ofappropriate dimension, and

M(p) =[J + 2J1 + pTMppp mT

�p

m�p Mpp

],

C = diag(0, Cpp),

K = diag(0, Kpp),

h1 = − 3�2[J1 − 1

2pTMppp] sin(2�)

− 3�2 cos(2�)mT�p

p − 2(� + �)pTMppp,

h2 = (�2+2��+3�

2sin2�)Mppp − 3

2 �2

sin(2�)m�p.

We shall be interested in the solution of (4) in a compactsubset of the state space Rn in which M−1 exist,where n = 2N + 2.

Defining the state vector x = (�, pT, �, pT) ∈ Rn,one obtains a state variable representation of (4) givenby

x =[

0 I

−M−1(p)K −M−1(p)C

]x

+[

0M−1(p)h(x)

]+

[0

M−1(p)b0

]u

� Ap(p)x + �h(x) + Bp(p)u, (5)

where h = (h1, hT2 )T, b0 = [1, 0T]T, I is an iden-

tity matrix of appropriate dimension, and �h(x) =[0, (M−1(p)h)T]T is the h-dependent nonlinear vectorfunction.

Suppose a reference trajectory generator is given ofthe form

v = A1v, (6)

where v = (v1, v2, v3)T ∈ R3

A1 =[ 0 0 0

0 0 1�2

c −�2c −2�c�c

](7)

and �c > 0, �c > 0. The matrix A1 has two stable eigen-values and one eigenvalue is at zero. Let v1(0) = �∗,the target pitch angle. Since A1 is marginally stable,v1(t) = �∗ and (v2(t), v3(t)) converges to (�∗, 0) ast → ∞. It is assumed that the system parameters arenot known, and moreover only the pitch angle and itsderivative are measured.

We select the controlled output variable as a linearcombination of the pitch angle and pitch rate given by

y = � + ��Hx, (8)

where � > 0 and H = [� 0 1 0] ∈ R1×n. Consider areference output trajectory of the form

ym = v3 + �v2 = v2 + �v2� − Fv, (9)

where F = −[0 � 1] ∈ R1×3.We are interested in designing an adaptive control law

u(y, v) for the output regulation of the uncertain sys-tem such that the closed-loop system has the followingproperties:

(P1) The closed-loop system is asymptotically stable ifv = 0; and

(P2) For all x(0) and v(0), the tracking error

e = y(t) − ym(t)

= Hx + Fv (10)

tends to zero as t → ∞.Moreover, the synthesis of the controller must be

done using only the measured signals � and �, sincethe elastic modes p and p are not available. In addition,it is desired that the structure of the control law must beindependent of the dimension of the spacecraft model.This is important since the flexible structures have in-finite dimension, but often finite dimensional truncatedmodels are considered for analysis and design.

Defining the pitch angle error � = � − v2, one hasfrom (8)–(10) that

˙� + �� = e (11)

which implies that if e(t) converges to zero, �(t)=�−v2tends to zero and the pitch angle is controlled to thetarget angle � = v2(∞) = �∗. Thus it is sufficient todesign a controller which regulates the error e(t) to zero.


3. Control law

In this section, the design of an adaptive controlsystem is considered. The nonlinear system can bewritten as

x = Ax + Bu + Ap(p)x + Bp(p)u + �h(x), (12)

where A=Ap(0), B =Bp(0), Ap(p)=Ap(p)−Ap(0)

and Bp(p)=Bp(p)−Bp(0). Since p-dependent term ofM(p) is of second-order, for small p, one has M(p) ≈M(0) and as such Ap(p) and Bp(p) are small and areignored for the purpose of control system design. Also�h(x) involves �-dependent small terms and higherorder nonlinear functions. For simplicity in design, weconsider a linear approximate model obtained from(12) by ignoring the nonlinear function �h(x) as wellgiven by

x = Ax + Bu,

y = Hx. (13)

However, later simulation is done using the completenonlinear model (5).

3.1. Hyper minimum phase linear model

The adaptive design considered here is based on thelinear model (13) using the output regulation theory[30]. Computing the transfer function of the system (13)gives

y(s)

u(s)= (s + �)np(s)

dp(s)= L(s), (14)

where ˆdenotes the Laplace transform, s is the Laplacevariable, and np(s) is a polynomial of degree 2N (twicethe number of elastic modes). For the spacecraft modelnp(s) is found to be Hurwitz matrix (i.e., np(s) has itsroots in the open left half complex plane (LHP)). Since� > 0, the polynomial (s + �)np(s) is Hurwitz, whichimplies that the system (13) is minimum phase.

Of course, the minimum phase property of the systemcan also be established in view of the fact that if the pitchangle is forced to be identically zero (i.e., �(t) ≡ 0) bythe choice of the control input, then the zero dynamicsobtained from (4) are governed by

Mppp + Cppp + Kppp = 0. (15)

The matrices Mpp, Cpp and Kpp are positive definiteand symmetric. Consider a positive definite Lyapunovfunction

V0 = pTMppp + pTKppp. (16)

Then its derivative along the solution of (15) is given by

V0 = −2pTCppp�0. (17)

Since V0 is positive definite and V0 is negative semidef-inite V0 is bounded. Note that V0=0 if p=0. The largestinvariant set contained in the set E = {[p, p] : V0 = 0}is (p, p) = 0. Now using LaSalle theorem [31] it fol-lows that (p, p) → 0 as t → ∞, which establishes thatthe origin of the zero dynamics is asymptotically stable;and therefore the system is minimum phase.

By the choice of the output y in (8), the transferfunction L(s) has relative degree one. For the spacecraftmodel

lims→∞ sL(s) = lim

s→∞ s[H(sI − A)−1B], (18)

lims→∞ s

[H

s

(I + A

s+ A2

s2+ · · ·

)B

]= HB > 0. (19)

Definition 1 (Hyper minimum phase system Fradkovet al. [32]). A multivariable system of the form givenin (13) (with m inputs and m outputs) is said to be hy-per minimum phase, if (i) the system has strictly stablezero dynamics (zeros of transfer matrix lie in the openLHP), and (ii) lims→∞[sH(sI − A)−1B] is a positivedefinite symmetric matrix.

According to this definition the spacecraft linearapproximate model is hyper minimum phase. Inter-estingly, hyper minimum phase systems have specialoutput feedback features as stated in Lemma 1 [32].

Lemma 1. For the hyper minimum phase spacecraftmodel (13), there exist a positive definite symmetricmatrix P = P T (denoted as P > 0) and gain ky ∈ R

satisfying

P(A + BkyH) + (A + BkyH)TP� − Q < 0, (20)

PB = HT. (21)

The implication of Lemma 1 is that by the outputfeedback (u = kyy), the system (13) can be stabilizedand, in addition, the input and output matrices B and Hare related as shown in (21). Indeed Lemma 1 is oftencalled Feedback Kalman–Yakubovich lemma [32].

3.2. Existence of regulator

We first consider the existence of a controller for theoutput regulation of the system (13) satisfying prop-erties (P1) and (P2), when the parameters are known.


In view of the hyper minimum phase nature of thespacecraft model, consider a control law

u = kyHx + kvv

� kxx + kvv (22)

where kx�kyH , ky is a real number, and kv =(kv1, kv2, kv3). Then the closed-loop system is

x = (A + Bkx)x + Bkvv

� Acx + Bcv, (23)

where Ac = A + Bkx and Bc = Bkv . For the hyperminimum phase system, there exists gain ky such thatthe closed-loop system matrix Ac is Hurwitz. This canbe easily established using simple arguments from theroot-locus technique since L(s) has stable zeros. Thissatisfies condition (P1) for v = 0.

According to the output regulation theory [30], theclosed-loop system has, in addition, property (P2) if andonly if there exist two matrices X ∈ Rn×3 and U ∈R1×3 such that the following matrix equations:

XA1 = AX + BU , (24)

0 = HX + F (25)

are satisfied. The feedforward matrix kv in (22) is givenby

kv = U − kxX. (26)

For the existence of solution for X and U of (24) and(25), the following assumption is made.

Assumption 1. For all ∈ �(A1), the spectrum of A1,

rank

[A − I B

H 0

]= n + 1.

Equivalently, the rank condition implies that ∈ �(A1)

is not a zero of the transfer function L(s) [30].

Since the designer has a choice of the matrix A1, andL(s) does not have a zero at the origin, one can alwaysselect �c and �c to satisfy Assumption 1. For the space-craft model, the assumption is satisfied by setting thenonzero eigenvalues of A1 sufficiently negative basedon the estimate of the zero locations of L(s).

Defining x∗=Xv, u∗=Uv, x=x−x∗, and u=u−u∗,the derivative of ˙x in view of (6), (13) and (24) is givenby

˙x = Ax + Bu − XA1v

= A(x + Xv) + B(u + Uv) − XA1v

= Ax + Bu, (27)

Let us choose the control u as

u = kyH x (28)

which yields

˙x = (A + BkyH)x = Acx. (29)

Since Ac is Hurwitz by the choice of ky (see (20)), x →0, as t → ∞. Using (25), the tracking error given in(10) can be written as

e = Hx + Fv = H(x + Xv) + Fv = Hx. (30)

Since x → 0 as t → ∞, e(t) = y − ym(t) converges tozero as well. Using (28) and (30), the control input u is

u = u∗ + u = Uv + kyH x = Uv + kye. (31)

The matrices X and U satisfying the regulator equa-tions (24) and (25) cannot be computed, because A andB are unknown; and therefore it is not possible to de-termine the control law (31). Of course, ky is also notknown. In the next subsection, we consider an adaptiveversion of the control law (31) for synthesis.

3.3. Adaptive control

Let us choose the control law

u = kye + Uv, (32)

where ky and U are the estimates of the parameters ky

and U. Noting that e = Hx and using the control law(32) in (27) gives

˙x = Ax + B(u − u∗)= Acx + B[−kye + u − Uv]= Acx + B[−kye − Uv], (33)

where ky =ky − ky and U =U −U are parameter errors.Now consider a Lyapunov function

V (x, ky, U ) = xTP x + �k2y + U UT, (34)

where � > 0 and ∈ R3×3 a positive definite symmetricmatrix. Differentiating (34) along the trajectory of (33),one has

V = xT[PAc + ATc P ]x + 2xTPB[−kye

− Uv] + 2�ky˙ky + 2U ˙

UT. (35)

Using Lemma 1 and noting that xTPB = xTHT = e,(35) gives

V = −xTQx+2ky(�˙ky − e2)+2U ( UT−ev). (36)


0 50 100 150 200-20

0

20

40

60

Pitch

an

gle

ψ (d

eg

)

Time (sec)

0 50 100 150 200-2

0

2x 104

x 105

Tra

ckin

g e

rro

r (d

eg

)

Time (sec)

0 50 100 150 200

Time (sec)

0 50 100 150 200

Time (sec)

0 50 100 150 200

Time (sec)

0 50 100 150 200

Time (sec)

-2

-1

0

1

2

δ (m

)

-20

0

20

40

Co

ntr

ol in

pu

t u

(N

.m)

-1

-1

-1

-1

-1

Estim

ate

of

ky

-0.05

0

0.05

Estim

ate

s o

f U

U3

U2

U 1

Fig. 2. Adaptive pitch angle controls: (a) pitch angle; (b) tracking error; (c) tip deflection (�); (d) control input; (e) estimate of ky ; (f) estimatesof U; (g) elastic mode p1; (h) elastic mode p2; (i) elastic mode p3; (j) elastic mode p4; (k) elastic mode p5; (l) phase plane plot (p, p).

In order to eliminate unknown parameter dependentterms in (36), one selects the adaptation law as

˙ky = −˙

ky = �−1e2,

˙UT = − ˙

UT = −1ev. (37)

Substituting (37) in (36) gives

V = −xTQx�0. (38)

Since V is positive definite, V �0 and x∗ and u∗ arebounded; x, ky and U ∈ L∞[0, ∞) (the set of boundedfunctions) and the limit of V (∞) exists. Integrating (38)gives

min(Q)

∫ ∞

0‖x‖2 dt

�∫ ∞

0xTQx dt = V (0) − V (∞) < ∞, (39)


0 50 100 150 200-0.4

-0.2

0

0.2

0.4

p1(m

)p

3(m

)p

5(m

)

Time (sec)

0 50 100 150 200-0.02

-0.01

0

0.01

p2(m

)

Time (sec)

0 50 100 150 200

Time (sec)

0 50 100 150 200

Time (sec)

0 50 100 150 200

Time (sec)

-4

-2

0

2

x 103 x 104

x 104

-10

-5

0

5

p4(m

)

-4

-2

0

2

-0.3 -0.2 -0.1 0 0.1 0.2

-0.1

-0.05

0

0.05

0.1

De

riva

tive

of

p1(m

/s)

p1(m)

Fig. 2. (continued).

where ‖.‖ denotes the norm of a vector or matrix andmin(Q) is the smallest eigenvalue of Q. In view of (39),x ∈ L2[0, ∞) (the set of square integrable functions).Boundedness of ky and U implies that u ∈ L∞[0, ∞)

and ˙x ∈ L∞[0, ∞). Now, invoking Barbalat’s lemma[33], it follows that x(t) → 0 as t → ∞, which impliesthat e(t) = y − ym tends to zero. This in turn impliesthat �(t) → �∗, as t → ∞.

The above derivation is based on the linear model(13). The closed-loop system, using the nonlinear model

x = Ax + Bu + �h(x) (40)

may cause parameter divergence. It is well known thatparameter divergence can be avoided using a modified

adaptation rule. For example, �-modification is often in-troduced for avoiding parameter divergence in the pres-ence of disturbance [34]. Such a modification gives amodified adaptation law given by

˙ky = −�−1(e2 + �ky),

˙UT

y = − −1(ev + �U )T, (41)

where � > 0.Consider a region ⊂ Rn surrounding the origin x=

0 and let ‖�h(x)‖�� for x ∈ , where � is a constant. Itcan be shown that for x ∈ , the trajectory in the closed-loop system remains bounded if one uses the modifiedadaptation law (41). For this, let us consider the state


0 50 100 150 2000

20

40

60

Pitch a

ngle

ψ (d

eg)

Time (sec) Time (sec)

0 50 100 150 200

0 50 100 150 200


0 50 100 150 200

0 50 100 150 200


0 50 100 150 200

-0.5

0

0.5

Tra

ckin

g e

rror

(deg)

-2

-1

0

1

2

δ (m

)

-50

0

50

100

Contr

ol in

put u (

N.m

)

-150

-100

-50

0

Estim

ate

of k

y

-20

-10

0

10

20

Estim

ate

s o

f U

U1

U2

U3

Fig. 3. Adaptive pitch angle control: effect of initial estimate of ky : (a) pitch angle; (b) tracking error; (c) tip deflection; (d) control input;(e) estimate of ky ; (f) estimates of U.

error equation in the closed-loop system obtained using(33) and including �h(x) which is

˙x = Acx + B[−kye − Uv] + �h(x). (42)

Then the derivative of the Lyapunov function V (x)

along the trajectory of (42) takes the form (see (38))

V = −xTQx + 2xP�h(x) + 2�ky ky + 2�U UT, (43)

where the additional �-dependent terms are due to useof the modified adaptation rule (41). Noting that for x ∈, ‖�h‖��, and using ky = ky − ky and U = U − U ,

(43) gives

V � − [min(Q)‖x‖ − 2‖P ‖�]‖x‖− 2�k2

y + 2�kyky − 2�U UT + 2�UUT

� − [min(Q)‖x‖ − 2‖P ‖�]‖x‖ − 2�(|ky |− |ky |)|ky | − 2�‖U‖(‖U‖ − ‖U‖). (44)

In view of (44) if

‖x‖�2‖P ‖�(min(Q))−1,

|ky |� |ky |,

‖U‖�‖U‖ (45)


-20

0

20

40

60

Pitch

an

gle

ψ (d

eg

)

-2

0

2x 104

Tra

ckin

g e

rro

r (d

eg

)

-2

-1

0

1

2

δ (m

)

-20

0

20

40

Co

ntr

ol in

pu

t u

(N

.m)

0 50 100 150 200

-1

-1

-1

-1

x 105

Estim

ate

of

ky

Time (sec)

0 50 100 150 200

-0.1

-0.05

0

0.05

0.1

Estim

ate

s o

f U

Time (sec)

0 50 100 150 200

Time (sec)

0 50 100 150 200

Time (sec)

0 50 100 150 200

Time (sec)

0 50 100 150 200

Time (sec)

U1

U3

U2

Fig. 4. Adaptive pitch angle control: effect of damping parameter (CI): (a) pitch angle; (b) tracking error; (c) tip deflection; (d) control input;(e) estimate of ky ; (f) estimates of U; (g) elastic mode p1; (h) elastic mode p2; (i) elastic mode p3; (j) elastic mode p4; (k) elastic modep5; (l) phase plane plot (p, p).

the derivative V �0. As such it follows that x, ky, U

are uniformly bounded and eventually converge to anellipsoid E = {V (x, ky, U ) = �∗}, where �∗ is suchthat (45) is satisfied. Indeed simulation results of thenext section show that in the closed-loop system us-ing the complete model (including �h(x) and also theneglected nonlinearity pTMppp of Mp(p)), bounded-ness of x, ky, U and pitch angle control are accom-plished. Furthermore, for simulation, disturbance d(t)

is assumed to be acting on the spacecraft. Of course, ifd �= 0, one can also show boundedness in the closed-

loop system following similar steps used here by simplylumping d and �h.

4. Simulation results

In this section, simulation results for the closed-loopsystem are presented. The spacecraft is assumed to be ina circular orbit at an altitude of 400 km. The parametersof the spacecraft given in [14] are used for simulation.Selected nominal parameters are J=3972 (kg m2), J1=500 (kg m2), �= 6/30 (kg m−1), CI = 545 (kg m3/s),


0 50 100 150 200-0.4

-0.2

0

0.2

0.4

p1(m

)

p2(m

)p

4(m

)

p3(m

)p

5(m

)

Time (sec)

0 50 100 150 200-0.02

-0.01

0

0.01

Time (sec)

0 50 100 150 200

Time (sec)

0 50 100 150 200

Time (sec)

0 50 100 150 200

Time (sec)

-4

-2

0

2

-1

-0.5

0

0.5

1

-4

-2

0

2

4x 104

x 103 x 103

-0.4 -0.2 0 0.2 0.4-0.1

-0.05

0

0.05

0.1

Derivative o

f p

1(m

/s)

p1(m)


EI = 1500 (kg m3/s2), Lb = 30 (m), r = 1 (m). Themode shapes given in [5] of the form (i = 1, 2, . . .)

�i (l − r) = 1 − cos

(i�(l − r)

Lb

)

+ 1

2(−1)i+1

(i�(l − r)

Lb

)2

(46)

are used to compute the modal integral in (3). The pa-rameters of the reference model given in (6) are �c =1 and �c = 0.07, yielding its nonzero eigenvalues at−0.07. The initial state of the reference model is v(0)=[�∗, 0, 0]T with �∗ = 1 (rad). The weighting parameterand the matrix in the adaptation law are � = 0.01 and = diag(0.02, 0.03, 0.04) ∈ R3×3.

We point out that for simplicity the adaptive con-troller has been designed ignoring the nonlinear func-tion �h(x) and pTMppp in the matrix M(p). Butthe simulation results presented here use the completespacecraft model (5) including �h(x) and pTMppp. Ithas been found that in spite of the inclusion of non-linear terms �h(x) and pTMppp in the model, theadaptation law without �-modification is adequate forthe pitch angle control. As such we have set � = 0 inall the simulations.

A1. Adaptive pitch angle control.

Simulation is done using the complete spacecraftmodel (5) including the adaptive control law (32) and


0 50 100 150 200-20

0

20

40

60

Pitch a

ngle

ψ (d

eg)

Time (sec)

-1

-0.5

0

0.5

1

Dis

turb

ance d

(N

.m)

0 50 100 150 200-2

-1

0

1

2

δ (m

)

Time (sec)

0 50 100 150 200

Time (sec)

0 50 100 150 200

Time (sec)

0 50 100 150 200

Time (sec)

0 50 100 150 200

Time (sec)

-20

0

20

40

Contr

ol in

put u (

N.m

)

-1

-1

-1

-1

-1

x 105

Estim

ate

of k

y

-0.05

0

0.05

Estim

ate

s o

f U

U3

U2

U1

Fig. 5. Adaptive pitch angle control: nonzero disturbance input: (a) pitch angle; (b) tracking error; (c) tip deflection; (d) disturbance input;(e) estimate of ky ; (f) estimates of U; (g) elastic mode p1; (h) elastic mode p2; (i) elastic mode p3; (j) elastic mode p4; (k) elastic modep5; (l) control input.

(37). It is desired to slew the spacecraft to a target pitchangle of 1 (rad). The initial conditions are assumedto be �(0) = 0, �(0) = 0, p(0), and p(0) = 0. Theinitial conditions of the estimates of ky and U are as-sumed to be ky(0) = 105 and U (0) = [0, 0, 0]. Selectedresponses are shown in Fig. 2. The control systemaccomplishes tracking of the reference pitch angle tra-jectory with a maximum tracking error of 1.25 × 10−4

(deg). The tracking error is very small for this choiceof large initial value for the estimate of ky . A smooth

control of the pitch angle in about 100 s is observedand the vibration is suppressed in nearly 200 s. Onlypitch angle and pitch angle rate feedback have beenused for control. The maximum control magnitude is21.63 (N m). The tip deflection, �, of the appendagehas maximum value 1.54 m. It is seen that the esti-mated parameters converge to certain constant values.It may be noted that the estimated parameters do notconverge to the actual values. Of course, it is wellknown that these estimates can converge to their actual


0 50 100 150 200-0.4

-0.2

0

0.2

0.4

p1(m

)

p2(m

)p

4(m

)

p3(m

)p

5(m

)

Time (sec)

0 50 100 150 200

-0.02

-0.01

0

0.01

Time (sec)

0 50 100 150 200

Time (sec)

0 50 100 150 200

Time (sec)

0 50 100 150 200

Time (sec)

0 50 100 150 200

Time (sec)

-4

-2

0

2x 103 x 104

x 104 x 104

-10

-5

0

5

-4

-2

0

2

-2

0

2

Tra

ckin

g e

rror

(deg)


values only when there is persistent excitation in thesystem [34].

A2. Adaptive pitch angle control: effect of initial esti-mate of ky .

In order to examine the effect of initial value ky(0)

of the feedback gain, simulations for different ky(0)=0are shown. But all the other parameters and the initialestimates of U (0) = 0 are retained. Thus all the initialestimates of the adjustable parameters are zero. This israther a worst choice of initial values of the estimates.However, these estimates have been selected to show therobustness of the controller. The selected responses are

shown in Fig. 3. It is seen that the pitch angle control andvibration suppression are accomplished. However, themaximum value of tracking error (0.4◦) in the transientperiod and the control magnitude (69.3 (N m)) are largercompared to case A1. The absolute value of maximumelastic tip deflection is 1.66 m.

A3. Adaptive pitch angle control: effect of dampingparameter (CI).

In order to examine the effect of damping coefficientCI, simulation is done with reduced damping coefficientof value CI = 272.5 (kg m3/s) (i.e., 50% lower thanthe nominal value), but the remaining parameters and


the initial conditions of case A1 were retained. Selectedresponses are shown in Fig. 4. Smooth pitch angle tra-jectory tracking is observed. As expected, compared tocase A1, lower damping caused slower convergence ofthe elastic modes to their equilibrium values due to weakstability of the zero dynamics and the response is moreoscillatory. The absolute value of maximum elastic tipdeflection � is 1.6 m.

A4. Adaptive pitch angle control: nonzero disturbanceinput.

The controller has been designed for the model with-out disturbance input. However, in order to examine therobustness of the designed control system, simulation isdone in the presence of disturbance torque d(t) actingon the spacecraft. Thus in (5) u is replaced by u + d

for simulation. A disturbance torque d(t) of mean valuezero is generated by passing a random signal through afirst-order filter of transfer function H(s)=1/(0.5s+1).The remaining parameters of case A1 are retained. Se-lected responses are shown in Fig. 5. It is observed thatin spite of the presence of the disturbance input, pitchangle control and vibration damping are accomplished.The absolute value of maximum elastic tip deflectionis 1.54 m.

5. Conclusion

In this paper, a new and simple adaptive control sys-tem for the attitude control of an orbiting spacecraftwith flexible appendages was derived. The system pa-rameters were completely unknown and only the pitchangle and pitch rate were measured for feedback andthe computation of elastic modes was not required. Forthe purpose of derivation a choice of a linear combina-tion of the pitch angle and its derivative was made. Bythis choice, the spacecraft model was found to be hy-per minimum phase. A reference command generatorof third-order was chosen for output tracking. Exploit-ing the hyper minimum phase property of the space-craft model and using output regulation theory, a struc-ture of controller requiring the feedback of only theoutput error and the states of the reference model wasderived. Then an adaptive controller was designed. In-terestingly, the controller has a simple form and its fourparameters were adaptively tuned for the output regu-lation. In the closed-loop system, asymptotic control ofthe pitch angle to the target value and stabilization ofthe elastic modes were accomplished. A novel featureof the controller is that it does not depend on the dimen-sion of the state space of the spacecraft model and the

problems arising from control and observation spilloveris nonexistent in this design.

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simplified adaptive control of an orbiting flexible spacecraft

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