A novel approach to switching adaptive control of a spacecraft with time-varying uncertainties

Copyright © IFAC Automatic Systems for Building the Infrastructure in Developing Countries, Istanbul, Republic of Turkey, 2003

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A NOVEL APPROACH TO SWITCHING ADAPTIVE CONTROL OF A SPACECRAFT WITH TIMEVARYING UNCERTAINTIES

Mohammad J. Ameri

l

,

Mansour Kabganian

2

J Amirkabir University of Technology (Tehran Polytechnic), Ph.D Candidate ofMechanical Engineering, Tehran, Islamic Republic ofIran E-mail: [email protected]

2 Amirkabir University of Technology (Tehran Polytechnic), Faculty of Mechanical Engineering, Tehran. Islamic Republic ofIran E-mail: [email protected]

Abstract: The adaptive controller is very efficient in dealing with system containing uncertain parameters. One restriction for the adaptive scheme is that the unknown parameters should be constant, which is not always satisfied in practice. If bounds of uncertainties are available, the sliding controller might be designed. But for a nonlinear system with general uncertainties (i.e., time varying with unknown bounds), both traditional sliding control and adaptive control do not work properly. This paper proposes a new adaptive control scheme for non-linear systems containing time-varying uncertainties with unknown bounded in special case in which the uncertainties vary exponentially with time. The control law and parameter update law are obtained from the Lyapunov approach to guarantee the tracking errors converge asymptotically to zero. Computer simulations are performed to show efficacy of the proposed schemes. Copyright © 2003 lFAC Keywords: time-varying, spacecraft attitude control; adaptive control.

I. INTRODUCTION

mechanical manipulator using the computed torque adaptive control so that asymptotically tracking of desired path is achieved (Craig, J. 1. 1986). But there is a drawback, which is the existence of an acceleration term in the control law. This problem is remedied by Slotine et al by replacing the tracking error vector i and its rate i by combined tracking

In some control task of the spacecraft, the systems to be controlled contain parameter uncertainties such as the uncertainty in the moment of inertia, and the uncertainty in the position of center of mass. The traditional adaptive controller is very efficient in dealing with above mentioned system (Wen, 1. T-T., and K. Kreute-Delgado 1991; Lammerts, I. M. M, F. E. Veldpause, M. 1. G. Molengraf, and 1. J. Kok 1995; Singh, S. N. and M. Steinberg 1996; Yamada, K. and s. Yoshikava 1998; Abmed, J., V. T. Coppla, and D. S. Bemstein 1998; Slotine, 1. E. and W. Li 1991; Craig, 1. 1. 1986; Narendra, K. S. and A. M. Annaswamy 1989; Middleton, R. H. and G. C. Goodwin 1988; Astrom, J. K. and B. Wittenmark 1989). But they have, by and large, treated the control problem in the presence of parametric uncertainties with the assumption that the unknown parameters are constant. Craig controlled a

error s = ~ + AX, filter error, in the Iyapunov function (Craig, J. J. 1986; Slotine, 1. 1. E. and M. D. Dibenedetto 1990). But in these works, there is not any analytically discussion when the uncertain parameters vary with time. In some research, above mentioned limitation is eliminated, but other constraints are entered into the control problem (Narendra, K. S. and A. M. Annaswamy 1989; Middleton, R. H. and G. C. Goodwin 1988; Astrom, J. K. and B. Wittenmark 1989). Middleton et al presented an adaptive control of time-varying parameters but it was assumed that the plant was linear. Narandra et al studied the case in which the

45

uncertainties vary exponentially with time. But they did not discuss about stability or in the other words about the amount of the tracking error while the uncertain parameters have not converged to its steady state amount (Narendra, K. S. and A. M. Annaswamy 1989). Therefore in that interval the tracking error may be becomes more than allowable bound so that the structure of the spacecraft can not stand against it. So they guaranteed the stability only when the uncertain parameter becomes constant. Furthermore they assumed the exponential rate of variation. the power of exponential function. is known and only constant steady state value of uncertain parameter is unknown, which reduces the generality of exponential variation form.

This paper is organized as follows: The spacecraft model is reviewed in section 2 which derived motion equation by combining the dynamic and kinematic equations. Our main results, a new SAC for a spacecraft is presented in section 3. As mentioned previously, proposed control law employs the variable gain matrix to achieve the asymptotic stability of spacecraft with exponential time-varying moment of inertia. Some numerical simulations are proposed in section 4 to demonstrate the efficiency of the proposed new SAC scheme. Finally, conclusions are drawn in section 5. 2. THE SPACECRAFT MODEL AUTOMATION SYSTEM

For remedying these problems, a large amount of research into the area of sliding controller was reported in various publications [Slotine, 1. E. and W. Li 1991; Slotine, 1-1. E. and S. S. Sastry 1983; Fernanderz, B. and 1. K. Hedrik 1987; Won. M. and 1. K. Hedeik 1996; Chen, X. and T. Fukuda 1997). However the spacecraft with uncertain time-varying parameter can be controlled by this controller, it is necessary that the bounds of uncertainty must be known. Hang et al by combining adaptive control and sliding control resolved both disadvantages of two controllers (Slotine, 1-1. E. and S. S. Sastry 1983). In this work, the unknown bounds timevarying uncertainties were represented in finite-term Fourier series so that they were estimated by updating the Fourier constant coefficients. But because of simplifier assumptions, the generality reduced and if the procedure investigates carefully, it can be courageously say that the stability proof is invalid.

In this section. the attitude control of a spacecraft driven with a couple of thruster is studied. The spacecraft is treated as a rigid body whose attitude can be described by two sets of equations of kinematic and dynamic. The kinematic equations relate the time derivatives of the angular position coordinates to the angular velocity vector, in body coordinate. The dynamic equations describe the evolution of the angular velocity vector. The results are also directly applicable to a spacecraft having itself inadequately known mass properties, such as moment of inertia, due to, e.g., reconfiguration, fuel variation in the gas-jet systems, thermal deformation. and so on. 2.1 DYNAMIC EQUATIONS At first the reference frames in which the attitude control problem shall be described, is defined. The control torques are applied through a set of three couples of thrusters, gas jet, along orthogonal axes. Based on these axes, an arbitrary orthonormal reference frame linked to the spacecraft is defined. It shall be referred to as the spacecraft or body coordinates. The origin of the coordinates is not necessarily the center of mass of the system nor are the axes necessarily the principle axes of the spacecraft. It is also assumed that an arbitrary inertial frame has been defined, with respect to fixed stars. The angular velocity of spacecraft, expressed in the body coordinates is denoted by co. The Euler's equations describing the evolution of CO in time may be written as

The paper under study, proposed a novel switching adaptive control (SAC) for a spacecraft containing time-varying uncertainties with unknown bounds. It is assumed that the uncertainties, components of moment of inertia matrix, vary exponentially. In contrast last work «(Narendra, K. S. and A. M. Annaswamy 1989), the general form of exponential variation is considered. The complete form of exponential function contains three constant unknown parameters 11, m and H, which 11 and m are the rate and the coefficient of exponential function respectively and H is steady state or final value of uncertain parameter which is completely explained later. In this paper using the Lyapunov function and some lemmas such as Barbalat's lemma, the asymptotic stability is guaranteed. The used control law contains the regressor matrix, estimated parameter, combined tracking error and gain matrix. The main difference of the procedure used in this paper with other works in Literature, lies in definition of the gain matrix and uncertain parameters. Unlike traditional adaptive controllers, the gain matrix employed in this paper varies with time so that the tracking of desired path is obtained.

T =H10"Ho>xH\0>

(1)

where HI and Tare the moment of inertia and the control torque vector respectively and the X operator denotes the vector product operation. 2.2 KINEMATIC EQUATIONS The angular position of the body may be described in various ways. The main difference between each method refers to kind of their singularity. In this

46

paper, both Euler's vector representation and Gibbs vector representation with singularities in angles ±rt/2 and ±rt respectively can be used. In both cases, the kinematic equations may be written as follow = J(x)ro (2) Where depending on the representation used for the kinematic equations, the vector x is either Euler's vector, or the Gibbs vector, and J(x) represents the Jacobian matrix (Slotine, J. E. and W. Li 1991; Wie, B. 1998).

In this paper, the change of Euler's equation due to this fact is ignored. For designing of current adaptive control, it is necessary to define two sets of laws control law and adaptation law. The control law assigns various values for the torque control and the parameter update law adjusts the controller parameters, so that perfect tracking is asymptotically achieved.

x

The adaptation law has a structure similar to other adaptation laws reported in the Literature. It contains the terms of gain matrix, regressor matrix, and the combined error vector. But it estimate final value of uncertain parameter instead of whole one. The control law is also similar to some previous work (Slotine, J. E. and W. Li 1991). It includes the terms of regressor matrix, combined error vector and gain matrix. But as mentioned previously, the gain matrix varies with time.

2.3 MOTION EQUATIONS The equations (1) and (2) may be rewritten as

T} = H l &+ rox Him

(3)

x= Jro where vectors x and x are chosen as the state space coordinates. This choice is well-defined, since the matrix J remains invertible in the domain of validity of kinematic representation. By differentiating the expression in equation (2) and substituting in equation (I) instead of ro and the equations of motion can be written as follow (Slotine, J. E. and W. Li 1991)

x

In the beginning it is necessary to convert uncertain time-varying parameters to constant ones. Dividing two sides of the motion equation by the variable term, var, yields new coefficients for the motion equation as

ro,

H*J (x)i +C*I (x, i)i

=F I

H* (x)i +C* (x,

(4)

F T=JTF1

H* (x) = rTHlr

=-.!L var

H*(x) = rTHr l

(5)

I

(8)

where

With l

x)x = F

(9)

c*(x,x) = -rTHr1jr} -rT[HffiX]r

c; (x,x) = -rTH}rljr l - rT[H1ffiX)r l

1

As considered, the left side of new .equation of motion is similar to the case in which the uncertain parameters were constant. But right side has changed. Now the following Lyapunov function candidate is proposed:

where superscripts T and -1 denote the transpose and inverse operation. 3 DESIGN OF THE CONTROL LAW

IT. 2

Based on the dynamic formulation (4), it is very straightforward to derive the model reference adaptive control law similar to those obtained in other works (Slotine, J. E. and W. Li 1991; Craig, J. J. 1986; Slotine, J. E. and M. D. Dibenedetto 1990). But when HI varies with time, all the adaptive controllers used in the literature can not work properly. In this paper, a new SAC is presented to resolve mentioned problems when H I varies as follows (6) HI = H ·var

I -T 2

1-

V(t)=-s H (x)s+-a r-a

(10)

where the combined vector error s is

r and Aare constant positive definite symmetric matrices, i and xr are the tracking error vector and reference velocity respectively as follow

-x =

X-X d

x =x d -Ai

with

r

var = 1- me -TIt

m<1

(7)

x d is the desired path and the parameter estimated error

where m and 11 are constant positive unknown parameters. Without any reduction of generality of exponential variation, m has known upper bound. Final value of the moment of inertia matrix which is denoted by H is an unknown positive definite matrix.

a is defined as

a =a-a

a

where nxl is estimated value of a nxl' a nxl n x I matrix which its components are components of uncertain matrix H.

47

a the

IS

The Lyapunov function and its time derivative are investigated for studying of the stability of the system and boundedness of the parameters. Last quadratic tenn in the Lyapunov function is positive, because r is a positive definite symmetric matrix. The first tenn is composed of a quadratic tenn J -1 S and H as follows

Since 1 - me-'11 > 0 , it can be written as (l-var)sTYa~sTkos

If the upper bound of left side is less than right side, above mentioned inequality is satisfied. Therefore the sign of sTYa must be studied

Since H is a positive definite-symmetric matrix, then the first tenn is also positive. Thus the Lyapunov function candidate is positive for all non-zero state variables x V(x) > 0

Using adaptation law (IS) (21)

Since r is a positive definite matrix

(12)

sign(sTYa) =-sign(iTa)

Now by differentiating V(t), and using the skewsymmetry of the matrix H· - 2C' , lead to V(t)=sT(F+H'xr-c*xr)+~Tr-la

(20)

(22)

where sign functoin is defined as (13)

x~O

sign(x) ={I -I

Defining the regressor matrix Y as

a)sTya>O

(23)

x
iTi
Y is only a function of x, X, x r and x r and does not depend on X.

In this case, it is clear that

Now adding and subtracting Ya in the first tenn

As a result, k o must be a positive definite matrix.

V(t) = sT(F- Ya+ Ya- Ya)+ ~Tr-la = sT(F _ Ya+ Ya)+ ~Tr-la

0~e-'11 ~I

~ O~(l-var)sTYa~msTYa

(24)

The sufficient condition for V ~ 0 is as (14) (25)

Since the upper bound of m is equal to I, it can be concluded that

Choosing the adaptation law as

~ = i =-rvTs

(15)

yields

v(t) =s T (F - Ya)

s:;tO

=sT(~-Ya)

(16)

var

As considered there is no need for the lower bound of m. Also, the limited upper bound will not impose a disadvantage, either. Since m= I means the moment of inertia matrix can be zero which is impossible.

Taking the control law to be F1 = Ya-kos

(26)

s=O

(17)

yields

or . T Ya • kos V(t)=s ( - - Y a - - ) var var ST

The upper bound of left side can be derived by following computation

(18)

•

o~ e-'1

= -((l-var)Ya-kos) var

t

~ I ~ -s TYa ~ -e-'1 I S TYa ~ 0

(27)

So k o can not be a negative definite matrix because the minimum value ofleft side is zero. It means that k o can be any positive definite matrix such as P .

where k o is not necessarily a constant positive definite matrix and plays the most roll in this procedure.

ko~O ~ ko=P Now, the stability sufficient conditions may be written as

m(t) = 2t Sufficient condition for V ~ 0 is

48

where all parameters are defined as previously. As shown in figure 2, there is a region in which the stability and unstability do not determine by the mentioned Lyapunov function.

s TkDS ~ STYa

I I

or

sTkDs=sTYa+sTps s TkDS ~ 0

(28)

or Unstable Region

sTkDS=STpS

Where P is the positive definite matrix. Above relation can be rewritten as: (29) For removing the chattering phenomena due to nonsmooth sign function, this function is substituted by saturation one as follows: T T ~Ya • s kDs=s Ps+--(l-sat(sT ya » 2

3.1 THE ASYMPTOTIC STABILITY Semi-definiteness of V implies that the stability or in the other words boundedness of tracking error vector is obtained. But for asymptotic stability studying it is

(30)

If l( and cr are the maximum and minimum eigen values of k Dand P respectively, the following one can be replaced instead of equation (29)

necessary to apply another lemma such as the Barbalat's lemma. To apply this lemma to analysis of dynamic systems, one typically uses the following corollary, which looks very much like an invariant set theorem.

(31)

Lemma1: If a scalar function V(x, t) satisfies the following condition (Slotine, J. E. and W. Li 1991):

Then V can be rewritten as following

1- V(x, t) is lower bounded.

-(var).s TYa -s Tps V=

var { (l-var)sTYa-sTps

2- V(x, t) is negative semi-definite.

(32)

3- V(x, t) is uniformly continuous in time.

var

Then

Figure 1 illustrates V versus sTya . As shown, V is

V(x,t)~O as t~O

a continuous function but its derivative in sTYa = 0 does not exist. Unstable Region

(34)

As mentioned previously, using equations (10-26), the conditions 1 and 2 were satisfied. As a result of following lemma, the condition 3 will be also satisfied.

v

Lemma 2: A function which is continuous on a closed interval, is also uniformly continuous on that interval (Parzynski, W. R. and P. W. Zipse 1990; Malik, S.c. 1996; Gordon, R. A. 1997). As shown in figure 1, V(x, t) is a continuous

Stable Region

function even when s TYa = 0 . Indeed, according to two first conditions of the lemma 1, all terms of V(x, t) are bounded and belong to a closed interval.

Figure 1. V versus sTYa Using some algebraic operations, the unstability condition

It implied that V(x, t) is continuous on a closed

interval. Therefore V(x, t) is uniformly continuous on that interval. Now, using the Barbalat's lemma, it can be concluded that V(x, t) converges to zero as

(33)

49

5- There is not any acceleration term in the control law and adaptation law.

time tends to infinity. Equations (26) means that

V(x, t)

is zero only if only s equals zero (s -)

0 ).

6- The exponential form of uncertain parameter used in this paper, has a good generality in comparison to last research reported in the Literature (Narendra, K. S. and A. M. Annaswamy 1989).

Then the asymptotic stability is guaranteed.

4 SIMULATION

7- Since there is not any limitation on the uncertain parameters m, A. and H, it seems that other nonexponential forms of variation of the moment of inertia can be approximated by changing these parameters m, A. and H.

In this section, simulations of control of the spacecraft containing exponential time-varying uncertainty with unknown bounds are performed using proposed new SAC. All terms of moment of inertia matrix for the simulations is selected to be

l~ ~],

H =[1:

°

r The

=

°

[1~2 1~2l system,

xoT=[OI

H =[1;

10

°

1°1 °

~]

-.-."

11 ;. •

A. = 205,

11 =0.2,

f

m=.08

u

,

'"aT.

"

.'!

!

·'e

r

it

initially

0]andi/=[3

at °

the

state

of

> .

O],isrequiredto

",~,-':"-....,..........,-~~,.:--.,.,....~-~-=--!

track the desired trajectory x d T = pt 1 t 2 ]. Simulation results are shown in the figures (3-10) with the proposed control gain K d in equation (30). Figures (3-8) are shown the tracking errors and figures (9-10) are represented the parameter estimation errors. As expected, the tracking and estimated parameters errors converge to zero and bounded values respectively. Using the persistent excitation, the estimated parameter errors converge to zero too.

..... <-3

Figure 3. Roll Tracking Error

..:If It'" "

~ {\\

;. ~1U

i.;:

Q

. \... - -

u

..

'~,\-,-~-:---:,....--:-~ :---:-:---::---,:::--;:;~

5. CONCLUSION

r'"

t;',-';

Figure 4. Pitch Tracking Error The major assumption in other adaptive control is based on the fact that unknown parameters must be constant. In this paper, in the beginning the time varying terms of uncertain parameter were separated into a constant and variable term. In continuing the asymptotic stability was guaranteed by applying a novel switching adaptive control law containing the constant mentioned term and a time-varying gain. Simulation results approved the efficiency of the proposed new switching adaptive control scheme. There are many advantageous in this paper as follow:

.,..'

", I

... ,

.Pe

,r; !

"

"..,

n

., \

"

'-.

"u figure 5. Yaw Tracking Error

1- The main constrain of the traditional adaptive controllers when the uncertain parameter exponentially varies with time, is remedied.

'~r-'''~'- - - - - - - - - - - - - - - , o

2- There is not any knowledge about the bounds of uncertain parameter. This property remedies the drawback of the sliding mode controller.

:

.,~

~

I~

E

~1

I(

3- There is no need to estimate all the uncertain parameters, in this paper m and A..

·'n~-o-~~...---:--="'.--.......,-:;..---,"'",...--:'''"n~''' _.rt"'OJ'

4- All results were derived in an analytical approach.

Figure 6. Roll Rate Tracking Error

50

'/i_'b'll

~

J

m

iS .

J '[ i

: ," r-----------I..•• /

,>~,~'~'--:,I---r,,---.--.It<>,.--''''..,--,;--±e;-''' .... ,, ~X\ tt_~

.._:

Figure 7, Pitch Rate Tracking Error

.a.,..

f.3

•f

I

;

~

Il~

---,---t"',.--""'.,,--*,,--±..-..,'.----,,!XI

.c,.:i;~,-t--7----,.""",

...- (r.-:

Figure 8, Yaw Rate Tracking Error

. .t .. . 'J

...

.~

~

'"

i

"

-.

'.

Figure 9. Adaptation Error of Parameter #1

/

/

.'

/

/

~.//

. ~ ..."~i-/::.~ .. •...__... ~... _ .l

,;.

llr

.._.... tNtu

10·· !:.'

1.·· . :;. ;;,..

('lie<.:'

Figure 10, Adaptation Error of Parameter #2 6. REFERENCES

Wen. 1. T-T. and K. Kreute-Delgado (1991), The Attitude Control Problem, IEEE Transaction Automatic Control, Vo!. 36, No. 10, pp. 1148-1162. Lammerts, I. M. M, F. E. Veldpause, M. 1. G. Molengraf, and 1. 1. Kok (1995), Adaptive Computed Reference Computed Torque Control of Flexible Robots. ASME Journal Dynamic, System, Measurment and Control, Vo!. 117, pp. 31-36. Singh, S. N. and M. Steinberg (1996), Adaptire Control of Feadback Linearizable Nonlinear Systems with

Application to Flight Control, Journal Guidance Control & Dynamics, Vol. 19, No. 4, pp. 871-877. Yamada, K. and s. Yoshikava (1998), Adaptive Attitude Control ofan Artificial Satellite with Mobile Bodies, Journal Guidance, Control and Dynamics, Vol. 19, No. 4, pp. 948-953. Ahmed, 1.., V T Coppla, and D. S. Bernstein (1998), Adaptive Asymptotic Tracking of Spacecraft Attitude Motion with Inertial Matrix Identification, Journal Guidance Control and Dynamics, Vol. 21, No. 5, pp. 684-691. Slotine, 1. E. and W Li (1991), Applied Nonlinear Control, Prentice-Hall International. Craig, 1. 1. (1986), Adaptive Control of Mechanical Manipulator, Ph.D. Dissertation, University of Stanford. Narendra, K. S. and A. M. Annaswamy (1989), Stable Adaptive Systems, Prentice Hall International Editions. Middleton, R. H. and G. C. Goodwin (1988), Adaptive Control of Time-Varying Linear Systems, IEEE Transaction Automatic Control, Vo/. 23, No. 2, pp. 150-155. Astrom, J. K. and B. Wittenmark (1989), Adaptive Control, Addison-Wesley Publishing Company. Slotine, J. J. E. and M. D. Dibenedetto (1990), Hamiltonian Adaptive Control of spacecraft, IEEE transaction Automatic control, Vo!. 35, No. 7, pp. 848-852. Huang, A-c. and Y. S. Kuo( 2003), Sliding Control of Non-Linear Systems Containing Time-Varying Uncertainty with Unknown Bounds, International Journal ofControl, Vol. 74, No. J: pp. 253-264. Slotine, 1.-1. E. and S. S. Sastry(1983), Tracking Control of Nonlinear Systems using Sliding Surfaces with Application to Robot Manipulators, International Journal ofControl, Vol. 38, pp, 465-492. Fernanderz, B. and J. K. Hedrik(1987), Control of Multivariable Nonlinear Systems by the Sliding Mode Method, International Journal of Control, Vo!. 46, pp. 10/9-1040. Won, M. and 1. K. Hedeik (1996), Multiple Surface Sliding Control of a Class of Uncertain Nonlinear System, International Journal Control, pp. 693-706. Chen, X. and T Fukuda (1997), Variable Structure System Theory Based Disturbance Identifications, International Journal Control, Vol. 68, pp. 373-384. Oh, S. and H. K. KhaIil (1995), Output Feedback Stabilization using Variable Structure Control, International Journal Control, Vol. 62, pp. 831-848. Wie, B. (1998), Space Vehicle Dynamics and Control, AIAA Education Series. Parzynski, W R. and P. W Zipse (1990), Introduction to Mathematical Analysis, Translated by Talebian M. to Persian, Razavi Inc. Malik, S.c. (1996), Principles of Real Analysis, New Age International Limited Publishers, pp. 127-128. Gordon, R. A, (1997), Real Analysis a first Course. Addison Wesley Longman Inc, pp. 82-85.

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