Adaptive Robust Control of a Class of Multivariable Nonlinear Systems

Adaptive Robust Control of a Class of Multivariable Nonlinear Systems

Copyright © 1996 IF AC 13th Triennial World Congress , San Frallcis';o, USA 2b-25 5 ADAPT IVE ROBUS T CONTR OL OF A CLASS OF MULTI VARIA BLE NON LIN...

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Copyright © 1996 IF AC 13th Triennial World Congress , San Frallcis';o, USA

2b-25 5

ADAPT IVE ROBUS T CONTR OL OF A CLASS OF MULTI VARIA BLE NON LINEAR SYSTE MS

Bin Yao and Masayoshi Tomizuka Mechanical Engineering Departm ent University of California at Berke/ey. CA 94720. USA byaoem echatro 2.me.b erkeley .edu tomizu kaGeu ler.ber keley.e du

Abstra ct. This paper considers the adaptiv e robust control of a class of multi· input multioutput (MIMO) nonlinear systems with arbitrary known relative degrees transfor mable to a semi-str ict feedback form. The form allows paramet ric uncertainties, unknow n nontinear functions such as modeling errors and external disturbances. Parame tric uncertainties can appear in the input channels. The last layer's state equatio ns do not have to be completely linearly paramet rized . The method achieves a guarant eed transien t performance and a prescribed final tracking accuracy in the presence of both paramet and unknown non/inear functions . In addition, the method achieves ric uncertainties ity in the pr~sence of paramet ric uncertainties without using hign-ga asympto tic stabilin The applications of the method include the motion and force trackingfeedback control. control of robot manipu lators in contact with unknown stiffness environments. Keywo rds. Adaptive Control. Nonlinear System s, Robust Control , Robotic manipulators, Uncertain Dynamic System s . l. INTRODUCTION The following two types of uncertainties are of major concern in control of uncertain nonlinear systems: parame tric uncertainties (e.g., gravitat ional load for robots) and general uncertainties coming from modeling errors ( e.g. , ignored nonlinear friction) . To account for tnese uncertainties. two nontinear control method s have been popular : adaptive control (Krstic et al., 1995: Krstic et al., 1992; Sastry and Isidor; . 1989: Marino and Tomei , 1993; Pomet and Praly. 1992) and deterministic robust control (Utkin. 1992; Corless and Leimann. 1981). The adaptive control achieves asympto tic tracking for reasonably large classes of nonlinear systems without using high gain (Krslic et al" 1995). However, adaptive controllers only deal with the ideal case of constan t parame tric uncertainties and the adaptat ion law may lose stability when even a small dis· turbance appears (Reed and loannou. 1989). In contras t , the deterministic robust control- e.g., sliding mode control (Utkin, 1992)-a chieves guarant eed transien t performance and final tracking accuracy in the presence of both parametric uncertainties and unknown nonlinear functions . However, it usually involves switching, which introduc es chatt@ring. Chatter ing may be avoided at the @xpense of degraded tracking performance.

Recently . Yao and Tomizuka (19940) presented a systematic way to combine the adaptiv e control and the sliding mode control for tlle trajecto ry tracking control of robot manipulators to preserve the advanta ges of the two meth-

ads while eliminating tlleir drawbacks. Compar ative experimental results for the motion control of robot manipu lators (Yao and Tomizuka. 1994b) have demons trated the effectiveness of the suggest ed method s. In the motion control of rigid robots (Yao and Tomizuka. 1994b) . the design was for multivariable non linear differential equations with relative degree of one. In (Yao and Tomizuka. 1995a) . the method ology was extended to a class of 5150 nonlinear systems with arbitrary known relative degrees in a semi-strict feedback form (Polycarpou and loannou. 1993) by combining the backstepping adaptive control (Krstic et al.. 1992) with the determi nistic robust control method . In (Vao and Tomizuka . 1995b). the adaptiv e robust control (ARC) Lyapunov functions were introduced and a general framework was presented . Specific ARC Lyapunov functions were obtained for a class of MIMO "onlinear systems with arbitrary known relative degrees in a semi-str ict feedback form, in which paramet ric uncertainties were allowed in the input channel of each layer. However, the form does not apply to the control of mechanical systems such as robot manipulators since the state equatio ns are required to be linearly parametrized by the unknown paramet ers. In this paper, this requirement is relaxed to the extent that the form can be applied to the control of mechanical systems . It is shown that one importa nt application, the motion and force tracking control of robot manipu lators in contact with unknown stiffness environ ments where the system has relative degree two (Yao and Tomizuka . 1993). falls into the form . Thus. the applicability of the suggest ed method s is enlarged .

2732

2. ARC LYAPUNOV FUNCTION In this section. the general framework of the proposed adaptiv e robust control (ARC) (Yao and Tomizu ka , 1995b; Vaa, 19961 is ?riefly reviewed . The fra,mework is formula ted for the fol oWing general MIMO nonlmear system : ± ~ J(x ,6, 1)+ B(x,6. I)u+ /)(x . ')I'>(x ,6, u,l) (l)

accurac y. Specifically, the control law we seek consists of two parts given by u(x, 9~u ) t) = u~(x . 6~u), t) + t1 . (x. O~/u) , t) (3) where 111 is an index, !la function s as an adaptive control law and tl .1 a robust control law to be designed within an allowable set nu. The estimat ed parame ter is updated by the following adaptat ion law

where y E Rtn and u E RJfI are the output and input vectors respectively, x E J?!1 is the state vector, 9 E RP is the vector of unknown constan t parame ters, h , I, B. and D

§ = _r,..(x, B~ r J , u(x, O~") , t) , t) (4 ) where r is any symmet ric positive definite (s.p.d.) matrix and lr is an index.

y ~ h(x.l)

are known, and

d E

represents the vector of unknow n

RCd

non linear functions such as disturba nces and modeling error$ . All the functions used in the design are assumed to be bounded with respect to (w.r.t.) t g " there exists Cl function f,,(x ,O) such that Iflx, 0, t) I ::; fn(",O)) with partial derivatives bounded w.r.t . l up to Cl sufficient order . The functions are assumed to have finite values if all their variables except t are finite. The following reasonable and practical assump tions about the parame tric uncertai nties and the unknown non linear function s are made o E ne= { B: 8m ,,,, < 8 < Om o. x, }

le.

I'>

E n~={"

,

1II'>(x , 0, ,,, ,," ~ « x , I)}

(2)

where 8m , !} I (J",ar E RP and J'(x , t) are known . In general, represents the IAh compon ent of the vector _ and the operatio n < for two vectors is perform ed in terms of the corresponding element s of the vectors. let Yd(t) E R m be the desired output vector at t and let the output tracking errors be denoted as ey = Y - Yd(t) . The adaptive robust control problem can now be stated as that of designing a control law for the inputs u such that, under the assumptio n of (2), the system is globally stable and the output tracking has a prescribed transient performance and final track.ing accuracy. Furthermore , ;n the presence of parametric uncertainties only, asympto tic output tracking should be achieve d.

I

The control function s fla and U 6 and the adaptat ion function T are obtaine d through ARC lyapuno v functions . An ARC Lyapunov function for the system (1) is defined to be a positive semi-de finite (p.s d.) function V(x , 8 , 8~v) , t) with contirlUous partial derivatives (lv is any index) that satisfies the following three assump tions: Al. Bounded V means bounded x and the guarant eed transien t perform ance of V(t ) means the guarant eed tran sient perform ance of the output tracking error c . y A2. There exists a continu ous control law U(I (x , 8~1 .. ), l) such that V tl6 E nu:

_j

Let e denote the estimat e of _ (e.g .. iJ for e). For any unknown parame ter \I'~tor _ lying in a known bounded region U.={_ : - m in < - < - m u.:.: , } (e.g., ne), a simple smooth projection map 11' can be defined for. and satisfies the following properti es: (PI) . 'le En., "(0)

=

e; (P2) . '10 . ,,(e) E n.~(1' 0min - E. ::; I' S -nl ll .r + 6.} where E. is a known vector of arbitrarily small positive numbers; (P3). 11'"i(.d ;s a nondecreasing function

0(_. ; (P4) . The derivatives o( the projection are bounde d up to a sufficien tly high-order. i.e ., Vj :5 11., 1I'( j )(e);s bounded. See (Yao and Tomizu ka , 1995b) for details. For convenience, define -11' as e1l" = 11'(_) and the projected estImati on error as -11" :=: -.,.. - _ . For any index j, let denote ~ ["(el T , .. .. ,,(j )(e)Tj"' .

.!I)

.!I)

Through using smooth projections, for any 8, 8 and w "..(j) (9) belong to some known compac t sets. Instead of using 8 , we only use the bounded O!,il ~ [0;, ... , Jr(j)(9) Tf in designing control laws and adaptat ion laws. Such a chOice enables us to use the determi nistic robust control techniqu e to design a baseline control1 aw to guarant ee transien t performance and prescribed final tracking accu racy for both paramet ric uncertai nties and unknown nonlinear function s . In addition. we can use the adaptive control techniqu e to update the parame ter estimat es to improve the tracking

8 1/ (Jx

[J + B(U a + u~))

eN

+ -:;t"

~

or, equivalently.

-r

8V

- w + (J r. '" + ae r,.. av ,

• -T vl~ ~ o < -W+t'.T+~(o+rT)

-

80

(5 )

(6)

wnere T(x , 8~"), H,t) is a known function . V !.:l=o represents tne derivative of V under the conditio n tnat 8 O. and W(x, f} , B~~), t) is any continu ously differentiable p.s.d . function which satisfies the conditio n that asympto tic convergence of W means asympto tic output tracking . A3. Here exists a u,(""ii~") . t) E nu such that '10 E and '16. E

=

n,

av 8x

n"

8V

U + B(u" + u . ) + D.aJ + at

or, equivalently~

V

< -~vV + cv(l) +

-

av

- 0; r"':$ av :. a6

- '\v V + r. v (7 )

-,,(6 + rT)

(8)

where .Av > 0 and cv is a bounded positive scalar, Le .. 0::; cv(t) ::; cv"'" •. Both AV and ev(t) should be freely adjusted by some controll er parame ters in a known form without afrectin g V (0) .• (0) represen ts the initial value of o.

Theorem 1. If there exists an ARC Lyapunov function V for (1) , then, by using the control law (3) with the adaptation law (4), the following results hold (Y.o and Tomizu ka , 1995b; Yao , 1996): A. In general, the control input and tne state J: are bounded . V is bounded above by V(rl

5 ex p(-..\v/) V{O) -f

>.:a1

V

[1- exp(->'v t)) (9) i.e., the system is exponen tially stable at large. The exponenti al converg ence rate AV and and the bound of tne final tracking error , ~ , can be freely adjusted (e.g., made arbitrarily small) by the control paramet ers in a known form. This in turn guarant ees that output tracking can have arbitrarily good transien t perform ance and final tracking accurac y.

2733

C

B. If, after a finite time, tt.ere are no unknow n nonlinea r function s, i.e., ~ == 0, 'Vt > to , for some finite to. in addition to tt.e results in A~ the system outputs track the desired outputs asympto tically.

=

Remark 1. In the absence of adaptat ion (i .e., r 0), the proposed ARC law reduces to a DRC law and Result A of Theorem 1 still holds. Thus, the adaptat ion loop can be switched off at any time without affectin g the stability. Remark 2. Althoug h jj is nol guarant eed to be bounded in the presence of unknown nonlinea r function s, the stability and the perform ance of the co ntroller is not affected since only the bounded projection is used in the design . Furthermore, fro m (4), 0 is bounded . T hus . for any finite time. 8(t) is bounded . In applicat ion , the execution time is al ~ ways finite , so the issue of bounded ness of 6 is not essentia l here . In addition , in view of Remark I, either reinitialization or switchin g off the adaptat ion can be used in case that unbeara ble wrong adaptat ion is observe d .

3. BACKSTEP PING DESIGN VIA ARC LYAPUNOV FUNCT IONS Theorem 1 reduces the ARC problem to the problem of finding an ARC Lyapunov Functio n . In this section , We assume t"'at an ARC lyapuno v function is known for an initial system and constru ct a new ARC Lyapunov fundion for the augmen ted system by adding a non linear system to the back of the initial system . Conside r the followin g initial MIMO system J~(x"q+Fd·l: I,t) B+B I( XJ , fJ,t)"UJ+J), 6., (10) Yt :::: hr(xl , t) U/ , YI E H'''I, J . J € Rn , 2.;, ::::

which satisfies the general setting in (\) and RI = BJ(XI, t) +BJ(x/ ,O,t) where each element of H} is linear w.r.t. 8. Since H} is linear w.r.t . to O. there exist known matrices G/(2:[, e , t) and Gj( XI •• , t ), whose element s are linear

and the output vector Ye = 'C e E R m r c = M-I (XI ./3 , t) [J~(~ , t) + F. (x, t)9 + F, (x, tW +B. (x, 0, p, t)u. + D. (x, t)A.(x ,8,P,u" tll (13) ,j =

Bp X1.6, t)vr ::: Gf(x/,tJ r., c)8

"fur E Hmi "IvI

=

=

From (A5) and (A6), the system (\3) has relative degree one from U r. to Ye. 1} function s as the vector of its internal states and (AS) assures that the internal dynamic s of the system (13) is stable . Since f29 and nil can be chosen arbitrarily dose to Oe and ftp, from (A5), we assume that Be (= B~( x.91r.li1f' t)) is nonsingular also . For convenience. N e and U t< denote N e = [0 Ifn-n1rJT E jiffl x(m-m, ) and U. = [/m, DJ E Rm,xm in the following . Now. augment the system by connect ing the first mJ outputs of the system 13). to the inputs of the . initial system ~l0l : I.e ., UI == .:ternr Uex tl • The remainI ng outputs of 13, Xe(ml+ l ), . ' . ,Xem . are co mbined with the outputs 0 the initial system to form the new outputs of the augmen ted system and the inputs of (13) become the inputs of the augmen ted system . The augmen ted system thus has the dimension n and is describe d by

J =

= I~ + Fre + BJ( XI .8,(tXem l + DIAl X, = M-I [f~ + j<~ 9 + F.fi + B , (z, B, I1, t)u i, == 4>'1(x,B ,#,t) y= (yT , (N;[ :te }Tf :x't

(11)

ERn,

We call Cl and G~ the right and left substitu tion matrice s of R} w . r .~ .. O, res~ec.tively. In the following, (or a system matrix e . • IS obtaine d by substitu ting the projecte d esti~ated parame, ters for the unknow n parame ters in • (e.g., .B, ~ BI!J: / ,01f,i )) . • refers to the estimat ion error of e , I.e., • == • - •. We assume that there exists an ARC LyaM punov function Vdx [, 9~'r), t} for the above system with the associat ed control u! = al(X [ , O~kd, t) = ClI'! + ClI., and the adaptat ion function TI (x I • 9~k d ) 0 ·1 , t) . Therefo re. 1/, satisfies the assump tions 1-3, in which the assump tions 2 and 3 are rewritte n as B2 .

.

B 3.

. •

-7

,iV

~

r +r VIIll. , = o<;-WI+fJtrTI+-~(8 TI)

. r) v}"0 , VI ::; ->'v, \.·,+ cv,(t)+ - , (8+r-rd .•

.

OB

(12 )

=

where V/ denotes VI under the conditio n that Ut Cl} , and AV/ and cv, can be freely adjusted by some controll er paramet ers, say Cl . Conside r the following MIMO nonlinea r system with the slate vector Ye [X; , 1]'1'JT , the input vector "U~ E Rm .

=

nE R."'1

X,

w.r .t . • , such that

VI B; (x {,8,t):::: fJ Gj( XI . V/, t)

4>,,( x. 9, (J, /)

where x :::: (,:j , x;, 7)T ]T EH", n ::: nJ+m+ nQ.fJ ER" is a vector of some unknow n parame ters. and ~ e is the vector of unknown nonlinea r function s . The matrice s in (13) can be function s of as well as Xe. We make the following assump tions: (A4) 13 E n~={j3: j3m," < 13 < j3m •• } and 11<1.11 :0; d, (x , t) where flp and cS. ar. known. (AS). 8, is nonsingular for any 0 E fl, and 13 E fl p . In addition, B, B~(x,t)+R.,(x,O,t)+B.p(z,j3,t) wher. B" and B, p are linear w.r.t . 0 and 13 respectively. (A6) . M is a s.p.d. matrix and there exist positive scalars k m and kM such that km/m $ M(XI,j 3,t) $ kMlm. (A7) . M M U (XI,t )+Mp(XI , j3, t) where Mo is linear w.r.!. 13. (AS). The ~- subsyst em is bounde d-input bounded-state (BIBS) stable w.r.t. the input (:<1 , :
+ D.A. j ( 14)

We make the following compati bility assump tion about the connect ion : (A9). i.m,(O) <>1(0). In the case that f1 is known and can be omitted in the expressi ons (M = Ee where Ee is a known s .p.d. matrix) , (14) reduces to the augmen ted system studied in (Yao and Tomilu ka, 1995b) where th~ added system' s state equatio n can be linearly parame tnzed by 0 when ~e == O. Here, althoug h M is assumed to be linearly parame t rized in terms of fi. in general, M-I cannot be linearly paramet rized . Thus, the state equatio ns of (14) cannot be Ijnearly paramet rized when Ll., O. Introducing M greatly expands the applicability of the met."od since m?st mechan ical systems , including robot manipul ators. satisfy (14 ) but not the form studied before as will be shown later.

=

=

Similar to. (11), we let C:WJ",o,t) and C~(:< , o,t) denote the right .and left substitu tion matrice s of the matrix Mp(x,j3 ,t). C;,(x , o,l ) and G;,(x, o,t) denote the right and left substitu tion matrice s of the matrix lJes(J.;, 0, t) (in terms of D). and G;p (x , 0 , t) and G~p (x , 0 , t) for B,p (:<, {3 , t) (in terms of /1). For simplicity, in the following , we will omit variable s for substitu tion matrices except . (e.g., G"M(.)

2734

instead of a:;" {x , . , t)) . The key point in the backste pping design is to design a control law for the system (13) such that i~"2:J tra cks O J and other outputs track their desired values_ 10 this end . define the tracking error z E R m and V as

= T t - a (X I , e~k/) , t) = Vr( x [ , o~{) t) +

z{x , o!,.k£l, t)

V {x . eL1v},13, t )

I

av ,

~ z'r M(:q, P , t )z

'1'00. 1

'

From (14), each compon ent of

== ~~(x.li!.lv) • t) + G~9 ( u ) ~OT( .-(I v i ) , t ) -_ TJ (:r. ,U,(k.J ... ,0/, t) - o/(J X , ,, , t z (23) T!l (X, 9~l v) , u , t) == T2 (x , B~lv), t ) (u}z

av} rT av T - --;:-- r r8 :::: aB a9

~ (O'(l 'f'~ x , 11" vl ,u.t ) (iv)

.M is

T ",

8;:;1

i ll) , M ( X I .~, t ) can be linearly paramet rized in terms of p, B and .1, where {) = [/i, OT ,ti,OT , ... ,PI, OT f E RI,p . For simplici ty, let 0. denote 0, = [UT, tiT , '1T f .

= VJ

+ - B /U"z + z

-

where

= \:' i"

+ zT

+B .. "

+ O,tl." -

(av/ BJU.. )T+

[ao./ 0 -, - ( j + FIO + SIr e"" 8a I . :: } M -+-M

[h]

T

MU"

+V ,A, z) + -BC)(l. f}:] -

ae

Since Mp is linea r w.r.t . Dp.{x , /) such that M tJ (x/,{J , t) UJ' [F,6

{ f~ + Fee + Fp {3

p,

1


at

)

19

+ B }(r/ .9.t)x" m/ l =

Dp,, (x, I) t1 (2 0 )

av . + ( -::;- - -+)8 + ~ T { -
h were

o

(a .. 0U.-) -. B 8:£ 1

a,,}

• • l] T + /vlo e --;;;-- [10 1 Vr }

o

- (lvl

f",, (x , 9.,.

. t)

o

~



(

' Ta'l l M oV(' -. -fFr

-;:,(/XI

If

OJ; I

0 -

-

TOO'} fix!

(h )

at

.

· DI A , + 'u'AI

I

(27)

) )

=

u.. (r , 8~tQ) , .0 .... J 1f , t) = L - \ (- Qz + Cl!]

(28)

a

When Ll 0 , from (22) , ,= O. Since V does not depend on and d, from (2 6) and B2 of (12) , we have that VU$ E n", . or av -. .

/3

- w + ge1r + 06 "T(



(Be + f" T.J)

(29)

where W == W J + ;: TQ :: and f .. :: diag{ f , r~ , f t')} . Thus . Assumption 2 is satisfied by V in terms of the augmen ted unknown parame ter vector Be. In general , noting (23) , (25) , (28) , and tU) , from (26) and B3 of (12), , oV ~ T T V - - . (9" + r ,or!!) < -'xVI \ r[ c v! - Z IP" Bur +;: { a Be . (30) - Qz + Lu : + 6.} = - AVV cv[ zT [(L - Lu )u$ + a~J

+

-

-( 10 L .. ( x ,9,..

3

,

+

2.\ ''k'~( Q» ) and

)'

'( la)

0',,(x , 9,..

,. + G'(i: em , )1

0

' ~,/J..,,,

tu :: max{l v, k, + l} and the equality ~ r G~T z has been used . let fl. lu, zTL Il, S; OJ . If L is nonsingular. we can choose tt(\ as

t)

+ Bf xe m,] + -) + D M ~

(j'.l~ ::: DM .J(.l:f • .!,t) - Dpi/(x , tj

.6. == De 4 ~ - /\tIll", -

<:; ,



= Znu

where >'v = miTlPv I

(BVl)T a:;- 1+ P'.C

TiJ (J~I [ 0

On :::: f'P - GM U e

d M{ XI,

T


+

at

.

O- m/ J + H 1Xe -

=- U<,T G IT l (

+D M9 ( X [ , z, I) -

- I~

1

( 26)

in which

(2 1)

0+ Mo(la-

T

I

=- -

ro

7'

8V

- (l v / c)e (x , Btr ,t)

die = [tP~ , !/J~, ~~7'f

,T

V It\ =o=

there exists a known ma trix

(2,,)

T

'1

Substitu ting (18) and (20) into (1 9) and noting (16) and the definition of substitu tion matrice s , W~ have ~' ::::: i~;



, (1,.,t ) ::::

=

(

J!f

. (aV') - " + 0"° a8 Vr)T) - MU" . TZB L(r, li,.O' B e + B ".(f (a - . ao zT Bo, (r (: 0;:,) )1 ZB ( X, B!:o), t) ~ [ B,,( r ( a"';; '

2

;'1'

OT = - 1/1-8 z

0.. ,, :::: (6~' . if;;,

ll ,. , ·

T

fl )

(1 8 )

T }

~cq~ r r o0 -M- U t 00

1 et " ::::

(M;1:: ... - Mr_, + -M :: )

3xJ

T"

rJ'f.

. - . av , ' v,' - ~ ( 8 +r T, ,+ -aV .r(8+ ['rr) O(j T r - a"1 + Lu + - ) = 911" + T "r B z A

"Iv

Pr

aB

J.:r - fl' ~Z,

V

T

where ~M linearly depends on t:l.[ and can be bounded by 3 known function J M , i.e., lIaM (x, TI, 0" t)1I S; JM (x , v, t) . Viewing (2) and (A4) , ,1 Efl. =Iii : 11min < 11 < d mar } where n,~ is a known bounded set. So we can define = "d(li) , the projecti on of ii, and ,j, = Jr,/ (,i) , the projecti on of ,1, in t he same way as before . From (14), noting (15) , . . • av! T . 1· V

(24 )

Then , substitu ting (24) into (21) , we have

Thus. there exist known vectors or matrice s dM( Xl , . , t ), OM8(X " . , t), OM ' { X[,. , t ) and DM,/ (X"., t) such that v [d M (XI, v, t ) + D M j3 ( x T, v , I)~ + D ME/ ( :q , v, t 19 + D M ~ (·1: 1 ' v , t )iJ + ~M]

80

+ ~o r (BVr) - . T}

ae

,u,e ) =

= [ri . T;;,

Mo U.,T -aD/I . r To

.0 + •• (uT O"} r ) N =""j3 (' M e TO + 1'ej3 ( U- )

d>~ :::: (q)~ T, rP~T , 1P~7'IT

(17 )

at

(}x{

T {

• Tae"

TfJ ( x, O...

Since lvf(x / , tJ, f. ) can be linea rly paramet rized in terms of ,) , so do and a~~ ':1. . Therefore, when tl./ = 0, from

~

+GM (U e - . f ro )}3 Further , define 80

( 16)

Ih\l' , 8 + C"/ (f'c. I Af. q = a M.j -- i [/ "r + f[ .... / )e+v,a r+ -

' 2"I "T Mv =

G:r

~

(1 5)

"e - = M( ,·},p , tjU•. ae

a8 =

T

"'o(x , ~v) • u , t)

o( X ,U. tr(Iv ) TO

Then , from (16), it can be verified that

where (j ;::::[a:T
Define

' ,(J7I',t) Be,8+ B. fj -Mfj (!!! Z u ~ . _ - (Iv)

,.6.... 0,.. ,u . t) _ -IP~(:r,0 1l" If we can choose u ~ E nu such that (22)

, T [( L -

i . )., +

"'~)

-

.u .. , e) 9" ...

:> <.(t)

+ ~-

(31)

(32)

where Ce is a design parame ter. then , Assump tion 3 will be satisfied by V for ~v == cv! + e ~ . V is then a valid ARC l yapunov function for t he augmen ted system (14) with the control (unctions tt a given by (28) and u" determi ned from (32) and the adaptat ion fun ction T, by (25) .

2735

Remark 3 . . One solution to (32) can be found in the fOllowing w-ay. Let h(:z:,8\'''' ),.o,.., ,7 ,,-, t) be a known function satisfying h ? sUP8Efl, . AEfll't, fla~ 1I . For example , let h

O(lv)

] 5 ) ,U(I.. tlll + kMllUeT aU iJx l DIll I( X [ , t (33) +lI n . 1I5,(r,t) + 'M(X, ', t)

2:

OeMII~e( r.

11"

where (JeM ::::: !j8...mtl 2: - Bemin + £6~1I . Choose P.u such that p.( x , 8~' o l ,J .. t) ~ , up II Lu L-'1I (34) OEn e

which is not difficult to calculatE since Lo is linear w .r.t. the uncertai nties O<'! 1f ' We assume that Pu < 1. In the absence of input channel paramet ric uncertai nties (i .e., B~ == 0), we can set Pu = O. So as long as the input channel uncertainties are not big, pu < 1 can be satisfied . Choose -(I .. )

u ! ( x,6T<

,

-

, f.~trl tJ ... , t) .:::: -

I

<1

·1(1 - " u) ~"

h L

- \

z

(35)

Then, 01'«/- - t U)ll , + ai l S ,T Lu. + 1I'lIp.. IILu.1I +11'11"= - ( ~hll z ll _~ 2 + €" :$ e~ and (32) is satisfied .

Remark 4. Noticing that the last two terms of L in (27) are linear w.r.t. r and Be is nonsing ular, nonsing ularity of L can be guarant eed by using a small adaptat ion rate r . Also , since our controll er guarant ees transien t performance , the states can be restricte d with a known compac t region . Thus, an allowable range of r without making L singular may be calculat ed off-line. 4. ADAPTIVE ROBUS T CONTR OL OF A CLASS OF MIMO NONLlNEAR SYSTE MS

By recursively applying the backste pping design results in section 3, we obtain an ARC controll er for a class of MIMO nonlinear systems transfor mable to the following semi-str ict feedback form: r, == f?( x•• t) + Fdxi,t)O + B . (x;, 0, t) Xj+I ,m. + D,Ai tl. = ~?( x., t) + ~~h::.,t}8 I ~ i:$ r - l d;,- == M- 1 CX,._ I . ;'1, t )[f~ + f' 6 + F jj/i + Br u + Drar) (,36) Tj•. == 4" {:\:' 8./~, t) T y = (yTb >" " Y;bJ , Y,b = N,T x. where Xi ERn" TU E n(1l, . U E R n1 is the input vector, y E Rm is the output vector, 0 == nto < nq :5 m~ ~ < " l r ~ 111, Ai is the vector of unknown non linear T function s , Xi = [xT, 'IIJ T . \i Xn , X == .\1" i'i+ l, m, = [Ji+IXi+ l , Ui+l [lm, OJ , and Ni [0 Im,_m' _I]T . The assump tions about the form are: (AIO) . Vi , Bi is nonsingular. In addition, Vi ~ r - 1, Bi ::::: BP(X; ,t) + Bl LXi,O, t) wher~ Bl is linear w.r.t. 8, and B,. B~(X ,l) + 8,'(X , 0,/) + B, p( \ ,13,t), where B" and B,-p are linear w.r.t . and jJ respectively. (All). AI is a s.p.d. matrix and there exist positive scalars km r

= [xL " ' ,

=

=

=

e

and kM such that km /m ::S ,'\1(:\.r_l:i1,t) :S kM/m. In addition. M = MO(:\:, _I, t) + M !,(:i:,_I , (3, t) in which Mp is lin.ar w.r.t . 11. (A12) . The Iji- subsyst em is BIBS stable w.r. t. the input (Xi-I , x,J . (A13) Vi. ther. exists a known functions "'( Xi, t) such that 11 "';h , 0, l) 11 "i (Xi, l) .

s:

In Eq . (36), the output vector y is partition ed into r blocks, and the outputs of the i-th block, Yif> (empty if mj mi_l) . have a relative degree T - i + I . In this way, we can have relative degrees ranging from 1 to T. (36) is

=

called a semi-str ict feedbac k form in that only the bounding function s of the unknow n nonlinea r function s 6, are required to be the function s of Xi and t only, and 6.'(X , (J, t) can contain bounded function s of Xi , j > i and thus violate the strict-fe edback property. In the case that f3 is known and can thus be omitted , and ~r can be linearly paramet rized. (36) reduces to the form studied in (Vao and Tomizu ka, 1995b) . The detailed design procedu re and the exact form of the control law can b. found in (Yao, 1996) and are omitted here because of the space limit . In the following , we only show how the problem is solved .

= [yalb' ... ,

Instead of tracking the desired outputs Yd(t) is designe d to track the filtered outputs y,(t ) = [yit. , ... , in which the i-th block of outputs , Ytih, are created by a (r - i + 1)-th order stable system. Such a procedu re enables us to choose the initial conditio ns, YtiI,(O) " .. , Y~~b-i)(O), freely to guarantee the transien t perform ance. The design proceed s in th.e follow ing steps :

ya,.V directly , the controller

yi:.V,

Step 1 . The first system is defined to be the XI dynamics in (36) with output ill == YLb == X l and input 1:2,mj ' Compar ing it with the system (14) and noting Assump tions (AI0). (A12), and (Al3), we can .ee that the first system can be conside red as a special case of (14) with no XI dynamics (Tnt = 0), no /3, and Al RI where El is any s .p.d . constan t matrix. Xl , 1/1, x2,ml' fP, Fi t 8 , DJ, 1 and ~l in the first system corresp ond to Xe! 1], tt, Elf~, EIFel El Be. , E l De. , and 6 e in (14) , respectively. Therefore , we can apply the backste pping design procedu re in section 3 to find an ARC lyapuno v function VI for the first system . VI is given by (15), i.e"

=

V1

=

1 TE 1 %\

2" ZI

z,

=Xl -Ytlb

(37)

with the associat ed control adXl , Blr , t) and the adaptation function TI (XI , 8. , t) for B. By choosin g YIIb(O) =

z I (0), from (37) , ;,(0)

Step i . The

i~th

= O.

system is defined to be the Xi dynamic s in (36) with output iii [yr. ,· . . , Yf~V and input X;+I ,m, ' If we define the i-th subsyst em to be the Xi dynamic s in (36), then , the i-th system is the same as the system obtained by augmen ting the (i -- 1)-th system by the i-th subsyst em in the same way as in section 3. Furtherm ore, all the Assump tions in section 3 except (A9) art. satisfied . For any j and k.let h jbk( XU +k-l ), Xj +k ,rn(i+.t_I P 8 , l ) be the expression of YJ~) ' the k-th derivati ve of Y}b , calculat ed from (36) under the conditio n that "" 0, 'VI. It can be proved (Yao, 1996) that if we choose y~;~j)(O), 'Vj < i. as

=

=

y!;~' J) (0) = "']/>(i- j ) (X;-1 (n), x i,m'_l (0),8 11 (0) , 0) (38) (A9) is satisfied . Thus, we can apply the backste pping design results in section 3 to find an ARC lyapuno v function Vi for the 1'-th system. For 2 < i < r - 1, since the added i-th subsyst em does not have-ft. Vi is given by • lrl

1 T

= 1/, _ 1 + 2" Zi

.::, ==

3: , - 0. _ 1

-

1 T

Et z, = E;::l '2z] EJz, a i- l

(39)

= (aT_ 1 ,y:'"l T .

with the associat ed control law Oi(Xi , 9~' ) , t) and the adaptation function Ti (;~;' i , 8~i),t) for O. When i::::: T . since the

2736

added r-tn subsyst em contain s 8, Vr is given by .. _1 1 T

,

~ ., = E, =l

'iZj

1 T

.

+ 2" '::"

F..Jz)

M

- 1-

the system can be represen ted by

( Xr _I,j3 , t)~r

-

l) r _ l

X l:;::

(40)

= [a;'_l,y;"bl T, with the associat ed control (t,.( X, O!: ), iJ", ~l'I" ) t) and the adaptati on function Te (X , ()~r ) . /i ill!" t) for the augmen ted

z,. = $ "

X,

& ,-_1

1f ,

Y

_ [ Ke B 1-

=

_

- { - . {H,,,, IQ'I} " ",,, llId} - 'C"" v - tHlIl mm'Sr-1 \rn.u;(E ,)' kM ,r:v - L.,..,=l e,.

and V(O) = O. Q, is the feedback gain Q in (28) and <, is the design parame ter C'" in (32) at each step . Thus, V

or Zj can be made arbitrari ly small by adjustin g controll er paramet ers Qi and e j in a known form

5. ARC MOTION AND FORCE TRACKING CONTROL OF ROBOT MANIPULATORS The form (36) includes some importa nt practica l applications such as the trajecto ry tracking control of robot manipulato rs (Yao and Tomizu ka , 1994b) wnere the design is essentia lly a relative degree one synthes is techniqu e . Here, a relative degree two applicat ion is briefly mention ed . The

deta ils are given in (Yao, 1996). When the end-effector of

a robot comes in contact with a stiff environ ment . in the task space l' E Rm. the robot dynamic s can be represen ted

by (Yao and Tomizuka . 1993) ,""f( r. f, O)i: + ? (r. r , B}T + (;(t', t , /1) + Dt(r, ':', t, ,I3 ) (41)

t:

+/(r, r,l )+ f r

=

11.

where AI represen ts the inertia matrix , C is the Coriolis and centrifu gal force , G is the gravitio nal force, D represen ts t the effect of time-va rying transfor mation between the task coordin ates and the joint coordin ates. j represen ts external disturba nces and unmode led joint friction , Pr is the interaction force between the end-effe ctor and the environ ment, f3 is the vector of unknow n robot parame ters, and u E Rm. is the applied control torque . J\1, C, G . and D can be linearly paramet rized by /:J. The task space coordin ates can be decomp osed into two groups, i.e., r:::::: [1';, where 1'} E R n ! are aligned with the outer normal directio ns of the contact surfaces , in which motion is constrai ned . and l ' p can be conside red as the unconst rained subspac e. Along normal directio ns of the contact surfaces . the environ ment is assumed to be represen ted by an elastic model with an unknown constan t s .p.d . stiffness matrix K e ( either from the force sensor or from the contact surfaces ), i.e.,

rJJT

in = K'[ ' I -

'.1. (1))

(42)

where In E Rn! is the vector of normal contact force components and r/~ (t) represen ts the unknow n equilibr ium position . The interact ion force is

= L( rJ. t)f" + L,(r.r, /" ,t )

(4 3 )

where LIn represen ts the modelin g part of the interact ion force and LJ represen ts the modelin g error of the surface friction . The objectiv e is to d~sign a control Jaw for u so

that the robot followstl;e des.ired motion trajectory rpdttJ in the unconst rained subspac e J'p and applies the desired force JrHI (l) in the constra ined 5ubspac e

61 ::::

+ DILlI

(44)

Xl

o

Since 1·-th system is also the entire system, by usins: the real control law II :::: ('\'r and the adaptat ion law 8 - r .. rr~ ' we have tt,e results in Theorem 1 where V ::::e v,. ,

Fr

=

where

paramet er vector Of, '

A

B\ X 2

= M-'(r, t, fJ )[- Cl r,x" t,Il}. , - C[" t ,il ) -Dc( r , x2 . t,m - L(r, x:J , t)Xl , t + U + 6,,1 0 'm -ra J

-K .. ': , .. (L) ,

1 ,

~2

[), = [i., of (45 ) = .- j(r, x:J , t) - t,(r, x') . XI.I. t)

let 8 be the vector of unknow n parame ters in l(c . Then Bl (0) can be linearly paramet rized by O. If we treat r in the second equatio n of (44) as a known function of t in the design since rand rare meil5urable t , it can be checked

out that (44) is in the semi-str ict feedback form (36) and

satisfies all the Assump tions in section 4 . Thus , we can apply the general results in section 4 to obtain an ARC motion and force tracking controll er. 6. REFERENCES

Corless. M. J . and G. Leimann (1981). Continuo us state feedback guarante eing uniform ultimate bounded ness for uncertain dynamic systems. IEEE Trans. on Autom~tic Control 26, 11391144. KrstK;, M.. I. Kanellak opoulos and P. v . Kokotovi c (1992) . Adaptive nonlinear control without overpara metrizati on. Systems and Control Letters 19, 177-185 . Kntk. M .. I . Kanellak opoulos and P. V . Kokotolli c (1995). Nonlinear and adaptive control design. Wiley. New York . Marino. R. and P. Tomei (1993) . Global adaptille output-fe edba ck control of non linear systems, part i: Linear paramete rization: part ii: nonlinear paramete rization . IEEE Trans. on Automat ;c Control 38(1), 17-49. Po lycarpou. M. M. and P. A. /oannClu (1993) . A robust adaptive nonlinear control desisn . In : Proc. of American Control Conferen ce. pp . 1365-136 9 . Pomet. J . B . and l. Praly (1992). Adaptive nonlinear regulatio n : estimation from the Iyapunov equation . IEEE Trans. on Automat ic Contro/3 7,729-1 40. Reed, J . S. and P. A. loannou (1989) . Instabilit y analysis and robust adaptive control of robotic manipula tors. IEEE Tr~llS . on Robotics and Aut9mat ion. Sastry. S. and A. Isidorl {1989}. Adaptive control of linearizable systems. IEEE Trans . on Automat ic Control S4 1123- 1131 . r Utkin. V. 1. (1992) . Sliding modes in control optImizat ion. Springer Verlag. Yao . Bin and M. Tomizuka (1993) . Robust adaptive motion and force control of robot manipula tors in unknown stiffness environm ent . In: Proe. of IEEE Con'. on De-cision and Control. San Antonio, Texas. pp. 142-147. Yao . Bin and M . Tomizuka (1994a) . Smooth robust adaptive sliding mode control of robot manipula tors with guarante ed tran· sient pe rformanc e . In: Proc. of Am~rician Control Conferen ce. pp . 1176- 1180. submitte d to ASME J. DSMC. Yao, Bin and M . Tomizuka (1994b). Compara tive experime nts of robust and adaptille control witt. new robust adaptive controlle rs for robot manipula tors. In: Proc. IEEE Conf. on Decision iJnd Control. Florida . pp. 1290-129 !'). Yao, Bin and M. Tomizuk a (l995a) . Robust adaptive nonrinear con· trol with guarante ed transient performa nce . In: Proe. of Americian Control Conference. pp . 2500-25 05. Yao, Bin and M. Tomizuka (1995b). Adaptive robust control of mimo nonlincar systems with guarante ed transient performa nce. In : Proc. IEEE Conf. on Decision and Control. pp. 2346-235 1. Yao , Bin (1996). Adaptive robust control of nonlinear systems with application to control of mechanic al syst~ms. Ph.D Dissertat ion . Mechanic al Engineer ing Departm ent . University of California at Berkeley.

1",.

Define xtand x 2asxJ Tp

and

= [XT,l'

x, = r, respectively_

xT.2f , XI ,} =fn!

Xl , 2:::::

Noting (41) , (42) , and (43),

1 O t herwise, we have t o write r as a fUll c tion of 01:'1 . Th e relationshi p '1'!(~1,1 ) it; unknown because of thr: unknown sti ffnes.."I and t he unknown cquililorium . Then , te rms like M ( r ( l£l) , t, fJ) cannot be linearl y param e ttiz~ d .

2737