Abstract—A nonlinear adaptive-robust scheme is presented to control a class of nonlinear systems. It effectively combines the design techniques of adaptive and robust control to improve performance by retaining the advantages of both techniques. It enhances transient performance and eliminates robustness problems compared to a parameter adaptive control. It also improves tracking performance by reducing modeling uncertainty and optimizes control effort by estimating bound uncertainty compared to a deterministic robust control. The controller is applied to a spacecraft making the body frame track a reference orientation, when there are parametric uncertainty, modeling errors, and external disturbances.

I. INTRODUCTIONrobust sliding mode controller attenuates the effect of model uncertainty coming from both parametric

uncertainty and unknown nonlinear functions. In general, it can guarantee transient performance and certain final tracking accuracy. However it does not discriminate between parametric uncertainty and unknown nonlinear functions and the control law uses fixed parameters. Model uncertainty coming from parametric errors cannot be reduced. In order to reduce tracking errors, the feedback gains must be increased, resulting in a high-gain feedback and requiring the increased bandwidths of closed-loop system [7], [9].

Main advantage of an adaptive control lies in fact that through online parameter adaptation, parametric uncertainty can be eliminated and thus, asymptotic stability or zero final tracking error can be achieved in the presence of parametric uncertainty without using a high-gain feedback. Despite this advantage, adaptive nonlinear controllers deal with the ideal case of parametric uncertainty only. System nonlinearity is assumed known and unknown parameters are assumed to appear linearly with respect to these known nonlinear functions. Adaptation law may be modified to achieve stability for bounded errors in the dynamic model, but tracking accuracy is no longer guaranteed. In this case, it can only be shown that the steady state tracking error stays within an unknown ball, whose size depends on the errors of system [3]. Moreover, transient performance is unknown and cannot be guaranteed [8].

Reza Banirazi was a graduate student in Electrical Engineering Department, Iran University of Science & Technology. He is now with the Department of Karun 3, Iran Water & Power Co, No 66 Bakht Yar St, 7th Tir Sqt, Tehran 15746, Iran (corresponding author to provide phone: 9821-883-3772; fax: 9821-883-36943; e-mail: banirazi@karun3.com).

Mohammad Jahed is an assistant professor of controls in Computer Engineering Department, Iran University of Science & Technology, Narmak, Tehran 16844, Iran (e-mail: jahedmr@iust.ac.ir).

Each type of control, an adaptive or a robust, when used alone has inherent limitations and problems to overcome, but when combined into a single controller the two techniques are complementary and the limitations and problems are mostly eliminated. This paper provides maximum quality of control while parameter adaptation is used to reduce model uncertainty and improve tracking performance and robust technique is used to guarantee the transient performance and certain final tracking accuracy.

II. PROBLEM FORMULATION Consider a general MIMO nonlinear system described by

a dynamic equation as ),,,(),(),,(),,( tttt uxxxx DuGfx

),( txhywhere my and mu are the output and input vectors respectively, nx is the state vector, p is the vector of unknown parameters, m),( txh ,

n),,( txf , mn),,( txG , and ln),( txD are known functions and l),,,( tux is a vector of unknown nonlinear terms such as modeling errors and unknown disturbances. It is assumed that and are bounded by some known parameters or functions as

maxmin and ),(),,,( tt xux

where pp1 ),,( minminmin and maxp

p1 ),,( maxmax are known vectors and ),( tx is a known scalar function. Assume that all functions in this paper are bounded with respect to time t and have finite values when all their variables except t are finite.

To derive adaptive control law, it is required that the state vector f and the input matrix G can be linearly parameterized by , i.e.

),(),(),,(p

1

0 ttt kk

k xxx fff

),(),(),,(p

1

0 ttt kk

k xxx GGG

To derive a deterministic robust control law, it is required that the perturbation matrix D satisfy the so-called matching condition. This simply means that the perturbation terms lie in a space spanned by the control inputs u.

To design control law in nonlinear systems, various state transformations are used to put the differential equations of system in one of several possible canonical forms [2]. Interest of this study is about systems that can be transformed into the input-output decoupled forms. In a MIMO nonlinear system and for a given output iy , define

Adaptive Robust Attitude Tracking Control of Spacecraft Reza Banirazi Motlagh and M. R. Jahed Motlagh

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Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

MC4.4

0-7803-9354-6/05/$20.00 ©2005 IEEE 498

relative degree ir to be the smallest integer such that at least one of the inputs appears explicitly in a differentiation. Since there are m outputs in the system, there are m such integers ir . The relative degree is denoted by set )r,,(r m1and total relative degree is defined by m1 rrr . If total relative degree r equals the system order n, then there are conditions such that the primary MIMO system can be partitioned into m subsystems, each of which be represented in the controllable canonical form [2].

Suppose that the system (1), under nominal conditions, is transformable into the input-output decoupled form. In the nominal conditions, is set into a clearly known value and

is null. Applying the same transformation to the real system (1) influenced by the uncertainty, yields a set of m decoupled higher-order differential equations perturbed by some nonlinear functions due to different forms of uncertainty as follows

u uxxx~ˆˆˆˆAˆ ˆˆˆ ),,,(),,(),,( ttt iiiiii

),,,(),,(),,( ttt ii uxxx D ˆˆ ˆˆˆ

where ˆ~ , and

iiii

rr1-r00

0 IA ,

m

1nr

ii

T1r ],,,[ ),(),(),(),,(),,( ttt iiii hhh i

ttLL xxx

xx ff ˆˆ ˆˆˆ

itt iirT

0 ],0,[0, ),,(),,( xx ˆˆ ˆˆmrT

0 ],0,[0, ),,(),,( itt ii xx ˆˆ ˆˆprT

r1 )],,,(,),,,,([),,,( ii

ttt iii uxuxux ˆˆˆˆˆˆprT

r1 ],,[ ),,(),,(),,( ii

ttt iii xxx ˆˆˆ ˆˆˆlrT

0 ],0,[0, ),,(),,( itt ii xx dD ˆˆ ˆˆ

The ^ sign denotes an evaluation of the quantity in ˆobtained by an appropriate adaptive technique. )(kL f is a kth order Lie derivative [2].

Finally, the overall system dynamics, after transformation, are given by

u uxxx~ˆˆˆˆAˆ ˆˆˆ ),,,(),,(),,( ttt

),,,(),,(),,( ttt uxxx D ˆˆ ˆˆˆwhere

nTm1 ],,[ ˆˆˆ

nnm21 ),,,( AAAA diag

nTm1 ],,[ ),,(),,(),,( ttt xxx ˆˆˆ ˆˆˆ

mnTm1 ],,[ ),,(),,(),,( ttt xxx ˆˆˆ ˆˆˆ

pnTm1 ],,[ ),,,(),,,(),,,( ttt uxuxux ˆˆˆ ˆˆˆ

pnTm1 ],,[ ),,(),,(),,( ttt xxx ˆˆˆ ˆˆˆ

lnTm1 ],,[ ),,(),,(),,( ttt xxx DDD ˆˆˆ ˆˆˆ

To derive independent control law for each subsystem, the following mm decoupling matrix ),,( txE ˆˆ needs being nonsingular.

Tm010 ],,[ ),,(),,(),,( ttt xxxE ˆˆˆ ˆˆˆ

Let md )(ty be the desired outputs at t, which is

assumed to be bounded with bounded derivatives up to a sufficient order. The control problem would be designing a control law for the inputs u such a way that the primary system (1), or equivalently the transferred system (2), is stable and the outputs y track desired outputs )(d ty .

III. ADAPTIVE-ROBUST CONTROLIf we can compensate the modeling errors coming from

the parametric uncertainty by an adaptive scheme, we may improve the performance of tracking accuracy. Moreover, if we can online estimate a bound for unknown nonlinear uncertainty in an adaptive manner, we may improve the gain value in the sliding mode that in turn will optimize the control effort. Also, since the bound of uncertainty is learned online, a prior knowledge about it will not be required, the thing that becomes complicated in many cases. Reference [5] realized that parameter adaptation can reduce the control effort but did not consider its destabilizing effect and the main advantages of adaptive and robust control methods. Reference [1] used adaptation to adjust some of the feedback gains to achieve stability when the bound of modeling uncertainty is unknown. The main goal of them was to relax the conditions under which stabilization is possible. In general, they do not provide better performance than their deterministic robust control partners when the bounds of modeling uncertainty are known.

The scheme here is formulated for general MIMO nonlinear systems in terms of the concept of Lyapunov functions. The algorithm uses true parameter adaptation to improve performance instead of relaxing the stabilizing conditions. The adaptive scheme proposed by [6] is used to compensate the uncertainty due to unknown constant parameters that appear linearly in the dynamics of system. To compensate other uncertainty in the system, an adaptive version of a sliding mode controller is used while it is derived from an asymptotic observer to eliminate chattering.

The adaptive-robust control law has a structure as

raf),,(1 uuuxEu ˆˆˆˆˆ t

where ˆ is a time-varying, online estimate of unknown constant parameters . The fu is a feedback control law to cancel nonlinear terms as

mTm010f ],,[ ),,(),,( tt xxu ˆˆ ˆˆˆ

where the values of its entries are continuously updated by an adaptive mechanism. The au is an adaptive control law to obtain zero steady state tracking error, in the form of

m

dmmTm

)(rdm

1d1T1

)(rd1

a)()(

)()(

m

1

y

yu

ˆc

ˆcˆ

ty

ty

with iiiiii yyy rT

dddd ],,,[ 1)(ry as a vector of ithdesired output and its first 1ri derivatives. i

irc is a

constant vector denoting coefficients of a Hurwitz polynomial and can be appropriately assigned to have a

(2)(3)

499

dy Adaptive Control

InverseDynamics

Sliding ModeGain

Observer

Parameter Estimation

Bound Estimation

u

x

yPlant

High FrequencyLoop

desired characteristic of dynamic behavior. Similar to fu ,the values of au are continuously updated too. ru is a robust control law to present later.

To design robust control law and parameters update law, a state observer form system is used to obtain a regressed form equation as

),,(),,( tt xx ˆˆ ˆˆˆAˆ

E uux ˆˆˆ ˆˆˆaf),,(1 t

where (0)(0) ˆ . nnm1 ),,(diag is a

design parameter with iii

rr as a Hurwitz matrix. Define an estimation error nn)(t as

)()()( ttt ˆ

where the dynamics of ˆ are given by (2). This error satisfies the following differential equation:

r1 ),,(),,,( uE xux ˆˆˆ~ˆ ˆˆ tt

),,,(),,( tt uxxD ˆˆ

To present the robust control law ru , define a reduced version of the ),,( txD ˆˆ as

lmTm010 ],,[ ),,(),,(),,(R ttt xxx ddD ˆˆˆ ˆˆˆ

where l),,(N txd ˆˆ is defined as a vector that its entries are the norms of column vectors of matrix RD . It is easy to see the validity of

dD xuxx ),,(),,,(),,( TR N ttt ˆˆˆ ˆ.ˆ

where l denotes an upper bound for the unknown vector ),,,( tux ˆ . The proposed robust adaptive control law for the ru is

Tm

Tm1

T1

T

r ],,[)(),,(Nd

ux

ccC

ˆˆˆ

ˆ tt

nmTm

T1 ),,( ccC diag

where )(tˆ is an adaptive estimate of .

Fig. 1. Architecture of adaptive-robust controller.

Remark: The robust control law (5) derives from the error signal generated by an asymptotic observer. The asymptotic observer bypasses high frequency components and prevents the excitation of un-modeled dynamics. A high frequency loop is occurred through the observer, instead of the plant. Therefore, an ideal sliding mode with the invariance property can be held without worrying about the high-frequency excitation of un-modeled dynamics.

Theorem: Consider the MIMO plant (1) transformable to the decoupled form (2) with a nonsingular system matrix E.Under linear parameterization for parametric uncertainty and matching condition for nonlinear uncertainty, the control law (3), along with the parameter updates

ux Pˆˆ ˆ ),,,(T t

d x Cˆˆ ˆ ),,(N t

where pp and ll are constant gain matrices and TPP is the solution to the Lyapunov equation IPPT , results in the stability of system and convergence of tracking errors.

Proof: To shorten the paper, we quietly summarize the proof and eliminate many mathematical manipulations. Consider a Lyapunov candidate as

~~~~P 1T1TTV

where ˆ~ . The derivation of V along the solution trajectories of (2) is

r1TTT 2)( uE ˆˆˆPPPV

D ~~~~ˆP~ˆP 1T1TTT2Substituting the robust adaptive control law (5) and the adaptation laws (6), yields

12 2- VV

dDdE C~ˆˆPC

CˆˆˆˆP TT

T1T

1 NN ))((

V

Now, we prove that 01V . Suppose that TCPB ,where 1EˆB is a constant matrix. It is easy to see that

RDD ˆBˆ . According to dD TNR

ˆˆ and after some matrix manipulations, we have

ddd C~ˆCˆC

CˆˆC TT

TTT

1 NNN )(

))((V

0)()( T

2T

NN d

dCˆˆ

CCˆˆ

Therefore 0-V 2 that implies 2 and ˆ ,ˆ . If it is proven that x , then Nd and the

robust adaptive control law (5) remains bounded, because , ˆ , Nd . Also, au since iyd and its

derivatives are bounded and the transformation between xand ˆ is uniformly continuous. So ˆ , ˆ , 1E , D ,and according to (4), . Therefore, the differentiable function V(t) has a finite limit as t , and its second derivative V exists and is bounded. This, according to the Barbalat’s lemma, implies that 0V , or equivalently

0 , as t .To prove x , let us define a tracking error vector as

nTdmm1d1 ],,[ yye ˆˆ

where the first entry of the vector ie is the ith input tracking error, i.e. iiiii yyye dd11

ˆ . Also define augmented error as e . Using the equations (2) and (4) and the adaptation laws (6), we can conclude a linear time-varying filter with dynamics , bounded input , and output e as

(4)

(5)

(6)

500

e])[( T PˆˆAA

where nnA is a stable constant matrix in the standard canonical form. Selecting a Lyapunov candidate function as

PTV with IAPPAT , differentiating Valong the solution trajectories of (7), and after some mathematical manipulations, one can prove that the above filter is exponentially stable, i.e. , e, . Since idy is bounded, ˆ and following it x . According to the first part of the proof, if x , then 0 as t .Therefore in the exponentially stable filter (7), the input converges to zero and hence its output e, i.e. the vector of input tracking errors, also converges to zero.

IV. SPACECRAFT CONTROL Orientation control of spacecraft is a key object in modern

spacecraft systems. Some difficulties in this control problem are the high nonlinearity of dynamic equations, incomplete state knowledge due to limitations of sensors, imprecise information about the parameters of spacecraft, influence of several disturbance torques and higher-order effects due to rigid body approximation.

A. Spacecraft ModelThree reaction wheels along orthogonal axes are assumed

to apply the control torques. Based on these axes, let us define the spacecraft frame as an orthogonal reference frame linked to the spacecraft body. For simplicity, assume that the origin of this frame is the spacecraft center of mass, and its axes are the principal axes of spacecraft. Note that these assumptions do not necessarily need and without them, the approach is still well applicable. Also define an internal frame with the same origin as the spacecraft frame, with respect to an inertial reference.

Let denote the angular velocity vector of spacecraft, expressed in the spacecraft frame. The equation describing the evolution of in time is

JJ )(where ),,( 321 JJJdiagJ is a diagonal matrix denoting the total central inertia of the spacecraft, J is the vector of total angular momentum of spacecraft and is the torque vector applied to the spacecraft by the reaction wheels, all expressed in the spacecraft coordinates. The notation ][vwill refer to the skew symmetric matrix defining the vector product.

The angular position of body may be described in various ways that are derived from different parameterization methods [4]. Here, attitude of body is described by the classical Euler angle representation, consisting of consecutive angular clockwise rotations of angles pitch, yaw, and roll that would bring the inertia frame in alignment with the spacecraft frame. Then

M )(where is the vector of attitude angles expressed in the inertial frame, and

3232

33

3232

sintancostan1cossin0

sinseccossec0)(M

Defining ];[x , the state space representation for the attitude of spacecraft may be written as

Ju

JJJM

x p133

133

1 )()( 00

where u is the torque vector as the control input and p is an unknown disturbance torque vector.

The aim is to establish a closed-loop control system in which the outputs )(t track the desired attitude trajectories

Td3d2d1dd ],,[ )()()()()( ttttty as soon as possible,

according to an appropriate regime, planed by a trajectory planner, in spite of imprecision mass specification and unknown disturbance torques. The desired regime for the attitude tracking is assumed to consist of five parts, namely, a 2-second constant acceleration phase to (2)dT],45,22.5[90 , a 2-second constant deceleration phase to

Td ],90,45[180(4) , a 3-second constant attitude

phase, and the reverse trajectory back to the zero attitude. Initial conditions of the spacecraft are 00)( (deg) and

00)( (deg/sec). Assume that random disturbance torques ip , 1,2,3i are acting on the spacecraft, where the mean

values and the standard deviations of them are 30,60)(70,)( pmean (N.m) and (50,50,20))( p

(N.m). It is considered relatively weak information about the mass properties of the whole spacecraft. Assume that the total inertia matrix of spacecraft has the nominal value

)(Kg.m95)(583,592,1 2n diagJ and its real value rJ

changes up to 50% about its nominal value. The effect of un-modeled dynamics is considered as the physical limitation of mechanical equipment. Assume that the torques generated by the reaction wheels and their variation rates are limited. This is possible in the simulation model by introducing a saturation filter in the controller output as

vp kddk

tu

u ii ,

where 2500kp and 15000kv in the simulation.

B. Simulation Results To obtain a comparison between the performance of

adaptive-robust controller and that of popular methods such as sliding mode and parameter adaptive controllers, the tracking problem is solved via three different methods as follows.

1) Sliding Mode Control: In this case, we need knowing a specified and predetermined bound for uncertainty. So assume 111 v JJ , 222 v JJ and 333 v JJ , where 1v , 2v , and 3v are positive scalars smaller than one. It is considered that 50vvv 321 . because of

%50nr JJ . The maximum value of p is also considered as )(3)( ppmean . The dynamics of the sliding mode are chosen as 1,2,394 iees iii where

is , ie and ie are sliding mode switching function, output error, and rate of output error in the ith loop, respectively.

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501

effects of

0 2 4 6 8 10 12 14-5

-4

-3

-2

-1

0

1

2

3

4

5

deg

Attitude Tracking Errors

Yaw

RollPitch

0 2 4 6 8 10 12 14-5

-4

-3

-2

-1

0

1

2

3

4

5

deg/

sec

Angular Velocity Errors

Pitch

Roll

Yaw

0 2 4 6 8 10 12 14-2000

-1500

-1000

-500

0

500

1000

1500

2000

N.m

Control Torques

Pitch

Roll

Yaw

0 2 4 6 8 10 12 14-200

0

200

400

600

800

1000

1200

1400

1600

sec

Kg.

m2

Inertia Moment Estimates

Yaw

Pitch

Roll

0 2 4 6 8 10 12 14-50

0

50

100

150

200

deg

Attitude Angles

Yaw

Roll

Pitch

0 2 4 6 8 10 12 14-5

-4

-3

-2

-1

0

1

2

3

4

5

sec

deg

Attitude Tracking Errors

Pitch

Roll Yaw

Fig. 2. Performance of sliding mode controller.

Performance of the sliding mode controller is shown by Fig. 2. Zero tracking error is obtained in the first 2-second leg. In the second leg, the zero tracking error is lost but the system is still stable. In the 3-second constant attitude leg, the system has a limit cycle. Beginning the next 2-second leg, the system becomes unstable. In fact, to maintain the zero tracking error, the controller needs applying control efforts with high rate variations and large amplitudes. If actual parts can not serve such a condition, the stability of system may fail. This shows the sensitivity of controller to the un-modeled dynamics and physical limitations in a real system. Results showed that the continuation method to remove chattering cannot solve the problem.

2) Parameter Adaptive Control: To derive the adaptive attitude tracking law, ),,( 321 JJJ is considered as an unknown parameter vector to estimate. Consider the initial values of parameters equal to their nominal values, while their real values are different as much as 50%. The constant coefficients specifying the tracking dynamics of the adaptive controller are chosen exactly equal to the sliding mode dynamics of the previous controller.

Performance of the adaptive controller is shown by Fig. 3. The variation of parameter estimations during large tracking error shows the attempt of controller to obtain zero tracking error via the new adjusted parameters. But the random nature of disturbances prevents it. Such behavior is a drawback of adaptive controllers that may cause the instability of control system in presence of big disturbances.

3) Adaptive-Robust Control: Performance of the new adaptive-robust controller is shown by Fig. 4. The tracking dynamics are exactly equal to those of two previous controllers. The initial conditions are also similar to those of the adaptive controller. The results show the ability of the controller to provide the desired transient performance, tracking quality and chattering elimination in the presence of parametric uncertainty, un-modeled dynamics and external

disturbances. This excellent performance is achieved thanks to the estimation of imprecision parameters to reduce system uncertainty, calculation of the upper bound of other uncertainty to optimize the control magnitudes, and having an observer-based structure to overcome the chattering of system.

The adaptive estimate of the bound of nonparametric uncertainties removes necessity of prior knowledge about these misgivings, which is complicated due to their unknown natures, and also optimizes the control torques required to compensate them.

C. Comparison and Discussion The simulation results explained that the new adaptive-

robust controller held the best specification from the aspects of the transient, steady state and stability performance. The adaptive controller efficiently compensated the

Fig. 3. Performance of adaptive controller.

502

0 2 4 6 8 10 12 14-5

-4

-3

-2

-1

0

1

2

3

4

5

deg

Attitude Tracking Errors

Yaw

Roll

Pitch

0 2 4 6 8 10 12 14-5

-4

-3

-2

-1

0

1

2

3

4

5

deg/

sec

Angular Velocity Errors

Yaw

Pitch

Roll

0 2 4 6 8 10 12 14-2000

-1500

-1000

-500

0

500

1000

1500

2000

N.m

Control Torques

Pitch

Roll

Yaw

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200

300

400

500

600

700

800

900

1000

Kg.

m 2

Inertia Moment Estimates

Yaw

Pitch

Roll

the parametric uncertainty but it was unable to overcome the effects of the unmodeled dynamics and unknown disturbances. The sliding mode controller suffered from large magnitudes in the chattering amplitude and it required that the closed-loop system had an infinitive bandwidth, thing that it may be not possible in a real system. In contrast, the new adaptive-robust controller was able to have a desired transient performance, tracking quality and chattering elimination in the presence of the real limitations and the unknown characteristic of the system.

Fig. 4. Performance of adaptive-robust controller.

TABLE I STATISTICAL COMPARISON FOR THREE CONTROLLERS

ROLLPITCHYAWmeanmeanmean

0.190.250.170.280.230.44AdaptiveRobust

0.850.900.690.781.181.26Adaptive

1.913.013.396.770.551.23SlidingMode

TRA

CK

ING

ERR

OR

S(deg)

5635470721389558AdaptiveRobust

5941470723400569Adaptive

4017621244146613201556SlidingMode

CO

NTR

OL

TOR

QU

ES(N

. m)

Table I compares performance of the three controllers based on the statistical results. While the adaptive-robust controller acts like as the adaptive one in making of control efforts, its performance in the reduction of output tracking errors is noticeably better.

V. CONCLUSIONIn the controller of this paper, the errors due to parametric

uncertainty are compensated by an adaptive mechanism and the errors due to other sources are eliminated by an adaptive version of a deterministic robust mechanism. The adaptive estimation of upper bound for nonparametric uncertainty removes the necessity of prior knowledge about it, which is complicated due to its unknown nature, and also optimizes the magnitude of control effort to compensate it. Moreover, by different classification of uncertainty, more improvement of control performance may be possible in the way of reducing online computation and system complexity.

The parameter adaptation is performed based on the error signal generated by an asymptotic observer. This signal is used to derive the sliding mode term and to estimate the upper bound of nonparametric uncertainty as well. This solves the problem of over parameterization in adaptive design and eliminates the chattering of sliding mode.

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