Adaptive robust tracking of nonlinear systems and with an application ...

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Procaedingsof the 291h Conference on Declslon and Control Honolulu, Hawaii December 1990

FP-4-2 5:lO Adaptive Robust Tracking of Nonlinear Systems and with an Application to a Robotic Manipulator Teh-Lu Liao, Li-Chen Fu, and Chen-Fa Hsu Department of Electrical Engineering National Taiwan University Taipei, Taiwan, Republic of China ABSTRACT Based on the input-output linearization technique and the variable structure control strategy, an adaptive control law is developed so that no prior knowledge of the bounds on the plant uncertainties is required. It is shown that the outputs of the closed loop system asymptotically track the given output trajectories despite the uncertainties, and the tracking errors can be made arbitrarily small. The scheme is then applied to (DOF) robotic the control of a two degree-f-freedom manipulator with unknown payload.

I. INTRODUCTION Consider a class of nonlinear systems in the presence of uncertainties of the following form: i=O

=f(x)+Af(x)+(G(x)+AG(x)) U

,j

yj=hj(x) where x( .):R"

-t

= L m

R" is the system state,

(1.1) U(

e ) ,

y(

a )

: R,

+

Af(x)=G(x)D(x) ,AG(x)=G(x)E(x).

+

NOW let y =(y ,..., y )T be the desired output trajectories

R",

1

i=l, ..., m, are sufficiently smooth vector fields, h,( .): R" -+ R, 1 i = l , ...,m, are sufficiently smooth scalar functions, and Af(x), Agi(x), i=l, ...,m represent uncertainties continuously differentiable with respect to x. The nominal system is then defined as follows: (i.e. with Af(x)=O, AG(x)=O in (1.1)) U

yi=hi(x) , i = l , ...,m (1.2) There has been a great deal of excitement in recent years over the development of a rather complete theory for explicitly linearizing the input-output response of the nominal system (1.2) using state feedback. This has been explicitly worked out in several papers, like those of [1,5]. Variable structure control [2- 41 is a powerful method which can deal with nonlinear systems. It first defines the sliding surface in the error state space and forms a switching feedback control. Also using the bounds of the uncertainties, the fast switching control forces the error state to slide alon the surface until it converges and then the tracking is attainef. Based on the input-output linearization and variable structure control strate y, the previous works [2 - 41 explicitly rely on the knowledge of the bounds on the plant uncertainties, and then the tracking is achieved. Contribution of the present paper can be stated as follows: given desired output trajectories y,(t) E Rm design an adaptive robust controller, which neither requires the bounds on the uncertainties of the system nor uses discontinuous control, so that the outputs y=(y l,..., ym)T (T denotes transpose below) of the closed loop system will attain arbitrarily small tracking errors, i.e. e =y - y are arbitrarily d small, disregard the uncertainties Af a i d AG.

11. ROBUST CONTROL The uncertain nonlinear systems under consideration are of the form (1.1) but subject to the following conditions: (C1) The strong relative degree v i of the i-th output of the system (1.2) is a finite number, i=l,..,m, and B(x) (to be

CH2917-3/90/0OO0-3130$1.OO @ 1990 IEEE

(2.2)

Rm, are

system inputs and outputs respectively, f(.), g.(.),: R"

x=f(x)+G(x)

defined short1 in the sequel) is nonsingular for all x E R". (C2) The system 6 . 2 ) is exponentially minimum-phase, i.e. the equilibrium point 0 of the zero dynamics is exponentially stable. Moreover the zero dynamics satisfy a Lipschitz condition. Condition (Cl) implies that the nominal system ca.n be linearized by a nonlinear state feedback law of the following form: u=B( x)-'[ -A( x)+ W], (2.1) where A(x)=(a (x),..., am(x))T , B(x)=(b,(x),...,bm(x))T with V vi-1 v.-1 a.(x)=Lfih. and bi(x)=[L L hi ,...,L L hj. gl gm (C3) There exist continuous functions D(.): R" + Rm, E(.): R" + Rmxm such that the uncertainties Af(x) and AG(x) in (1.1) satisfy

dl

dm

where y 'j3, j=O ,...,v., i=l, ...,m, are bounded signals.. If we di

denote e.=y.-y 1

1

di'

i = l , ...,ni, the sliding surface is then defined

as where P..,j=l, ...,v-i, i=l, ...,m are chosen such that the 1 i following )polynomials:

are Hurwitz. Now let the input U=

U

(2.4) in (2.1) be chosen as follows

B(x)-l[-A( x)+ (( x, t )--K( x,t )sgn( S)]

(2.5)

where K(x,t) is a switching state feedback gain to be determined and sgn(S)=(sgn(sl), ...,sgn(s ))T where sgn( * ) denotes the standard sign function, then

6

> 0 (sufficiently small) such that for all x

>0

II B(x)E(x)B(x)-lIlim 5 I c p 5 1 - to

II B ( ~ ) D ( ~ ) + B ( ~ ) E ( X ) B ( ~ ) - ~ ( - A ( ~ ) II+ 5S $Jx7t) (~,~)) 3130

Lemma 1 [2]: Consider the system (1.1) under conditions (Cl)-(C4). Given the desired output trajectories yd (t) E [Rm with the aforementioned properties, the variable structure control law in (2.5) with the switching feedback gain K(x,t) given as follows: +,t) L (1-!y-?9 + li2(x,t)), 9 > 0 achieves the asymptotic tracking, i.e. y(t) + y d (t) as t

+

m.

Proof: For the detailed proof, please see [3]

IV. APPLICATION T O A TWO DOF ROBOTIC MANIPULATOR A two DOF robotic manipulator studied by [6] is used here to illustrate the efficiency of the adaptive robust control law proposed. The endeffector of the manipulator can be extended or extracted from and rotate about a vertical axis. Let (x , x ) 1 2 denote the position of the center of mass, C, of the motional link in polar coordinate so that the equations of motion are given by (Mc+M) 2 -R(x , M) X = V

111. ADAPTIVE ROBUST CONTROL In practical application, however, there may be situations in which the bounds (condition C4) are unknown. Therefore, the switching feedback gain k(x,t) can not be constructed to meet the sliding condition. One possible way to overcome this difficulty is to estimate the gain and update it by some adaptation law so that the sliding condition can be met and the error state reaches the sliding surface and sta.ys on it. To realize this idea, we first state the following condition which takes the place of the condition (C4).

I(x 1,M) x 2+2R(x 1,M) k1k 2=v 2 (4.1) where R(xl,M)=Mc xl+M(xl+a), I(xl,M)=J1+J2+M, x '+M(xl+a)? Suppose the payload inass M is not known. Let M O be the nominal mass and x , x be the output trajectories, i.e. y =x

(C4') There exist known functions @l(.):R"

and y =x

e

RmxR+-I R,

e

and 1>

t

+

R, ', Icl,(.,*):

> 0 (sufficiently small) such that for all

x E R" and t > 0

(I B(x)E(x)B(X)-llim 5 P;qX) 5 1 - to

11 B(~)D(~)+B(~)E(~)B(~)-~(--A(x)+s(~,~)) II 5 P2T11)2(X?t)

(3.1)

e

e

e

where p E R,' and p E R> are unknown parameters. Here, R,'

e

( IR? ) denotes the e ( e )-dimensional space in which each 1 2 entry is nonnegative. An adaptive control law is described as follows: u=B(x)"(-A(x)+t(x,t)-k(x,t)sat(T)) S ,@ > 0

1

1

qx,t)=(l-P

1

(3.3) where Q, is the thickness of the boundary layer, P and P are

'+

where S =(s ,...,S ~ ~ ) ~ = Ssat(T) -$ S 0

and

1

to

r =diag(yl ,.., -ye) >

01

7 0

1

B

4

6

8

10

0

1

B

6

d

0, ~ , = d i a g ( r ~ + l , . . , y e + e )> 0 are the adapta.tion gain 1

1

References Byrnes, C.I. and A. Isidori . "Local stabilization of minimum phase nonlinear system", System Control Lett.lO, 9-17, 1987. Fu, L.C. and T.L. Liao . "Globally stable robust tracking of nonlinear systems using variable structure control and with an application to a robotic manipulator", to appear an IEEE Trans. Automatzc Control 1990. Iyer, A. and S.N. Singh . Variable structure control of decouplable systems and attitude control of spacecraft in the presence of uncertainty", Proc. American Control Conference, Atlanta, G.A., 2238-2243111988. Liao, T.L., L.C. Fu, and C.F. Hsu , Adaptive robust tracking of nonlinear systems and with an application to a robotic manipulator", Technical Report, E.E., University of Taiwan, 1990. "Adaptive control of Sastry, S.S. and A. Isidori linearizable systems", IEEE Trans. Automatic Control 34, 1123-1131, 1989. Suhada, J. and N.H. Cheng . "Tracking controllers for robot manipulator: a high gain perspective", ASME J. Dynamac System Measurement, and Control 110, 39-46,

Ti 1(x))'(s+P;qX,t))

io

1

Then, the state re resentation of the system (4.1) can be written of the form 6.1). Condition. (CI)-(C4') are seen to hold here, and there exist two unknown parameters p and p 1 2' Simulation Results: To obtain numerical results, we set Mc=lOO kg, Mo=50 g , J1=100 k g d , J2=100 kgm2, and a=l m. The desired trajectories y =(y , y ) T are C2 function pair given by d d l d2 y =1+cos(0.2t), y =0.2sin(0.2t) dl d2 In Fig. 4.1, the payload mass M is set equal to 100 kg and the simulation shows that the tracking errors are bounded. Therefore, the controller is shown to be robust with respect to the uncertainties.

1

S sat($))T estimates of P and P , and sat(T)=(sat(q),...., with s sign(s ) if I s i / > @ i=1,2 ,....,m. sat($)=( si i -&-ifIs;l_ 1-6, O< : 1 1 < P=

2

. We now define x 3 =; 1 , x4 =k 2 2 2 ' U =I(x , MO)'% , U =I(x , MO)% 1 2 1

(3.2) and

2 1 '

1

10

/---

2

matrices. Now wc are ready to state our main result. 0

'J'hcorern : Considcr the system (1.1) satisfying (Cl)-(C3) and ( 0 4 ' ) . Given the desired output trajectories y (t) E Rm with the d afol.eiiieiitioiied propcrlies, the adaptive control law described by (3.2)(3.:1)(3.4)will force thc tracking error e (t)=y(t)-y (t) Y d t,o be i1.s m a l ! as dcsircd. 3131

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