Ilchmann, Achim :
Non-identifier-based adaptive control of dynamical systems: a survey
Zuerst erschienen in: IMA Journal of Mathematical Control and Information 8 (1991), S. 321-366
IMA Journal of Mathemalical Control & Infc.,rmaiion(1991) 8, 321-366
Adaptive Control Non-Identifier-Based of Dvnamical Svstems: A Survev Acsrrr,rL,cuuaruN Uniaersttyof Ereler, Cenlrefor Syslemsand ControlEngineering, Norlh Park Road, Schoolof Engineering, Ereter, DeuonEX4 4QF, U.K. ilchnaan.
[email protected] . uk
fReceived28 August 1991 and in revisedform 12 November 1991]
In adaptive control, the objective is to provide a single controller (consisting of a feedback law and a parameter adaptation law) which can control each system belonging to a certain class of systems. The systems are not known precisely: only structural properties (e.g. minimality, minimum phase, known relative degree) are assumed to hold. The control objectives are stabilization, tracking or servomechanism action. The paper surveys those aspects of the field of adaptive control which started in the 1970s wherein no parameter estimators are used. In addition to universal adaptive controllers for finite dimensional minimum phase systems of relative degree 1, controllers for higher relative degree, non minimum phase, infinite dimensional, and nonlinear systems are also presented. 1. Introduction A wrno range of control theory deals with the problem that, for a knownplant, a controller has to be designed in order that the feedback system achieves a prespecified control objective. The fundamental difference between this approach and that of ailapliae control is that the plqnt is nol known exactly, only structural information is available. The aim is therefore to design a single controller which can be applied to a variety of systems belonging to a certain class. The control law has to be designed so that the controller learns from the behaviour of the system, and based on this information, it adjusts its parameters. This area has been intensively studied over the last 40 years. See Aström (1987) for a survey article. Up to the end of the 1970s, most adaptive control mechanisms would attempt to identify or to estimate certain parameters of the plant, and then design a feedback In this survey, an overview is given controller on the basis of this information. on adaptive controllers which are not based on any parameter identification or estimation algorithm or injections of probing signals. The objective is not to obtain information about the plant, but simply to control the unknown plant or process. For a conceptual framework, containing the controllers described in the present paper and, in addition, adaptive control systems formulated in terms of enor models
321
@ Oxford University Press 1991
Y
322
A. ILCHMANN
based on identification mechanisms,see Morse (lgg0), (1ggOa). Most of the adaptive controllerssurveyedin the present paper fit into the following general description. SupposeX is a certain classof linear finite dimensional time-invariant systems of the form it(t)=Ax(t)+Bu(t)+Esa, u(t)=Cr(t)+E2ut, r(0) e R" L1 ( A , B , C , E t , E z )€ R ' " t x l R n x m x l R p x m x I R n x xz R l Px'', t
( r 'ir\ J
rr1,Trr,pare usually fixed, but n is an arbitrary and unknown number, g"u1belonging to a (known) class of reference signals Ure1, and tr belongs to a (known) .lus" oi disturbancesignals2. It is desiredto desiqn a feedbacklaw
u ( t ) = f ( k ( t ) .y ( t ) .y , " 1 ( t. t )
( 12 )
depending on the referencesignal, the system output, and a'tuning'parameter fr generatedby
k ( t ) = s ( k ( t ) , a ( t ) , y , q ( t ) ) , f r ( 0 )€ R '
(1.3)
so that there exists a (unique) solution of the closed loop system (1.1)-(1.3) on [0, *), the internal variablesare bounded, and most importantly g(r) "ry*piotically tracks y""y (t). D p p r r u t r l o u 1 . 1 . L e t f : l R r x I R px I R P - R - , g : l R . r xl R px l R p - R r b e c o n t i n u o u s in g and areJ, a.\d piecewiseright continuous in #. The controller consistingof the feedbacklaw (1.2) and the adaptation taw (I.3) is called a uniuersaladaptiue regulatorsolving the seruomechanismproblem for the classof systemsX, the class o f d i s t u r b a n c e s2 a n d o f r e f e r e n c es i g n a l sU , " y , i f f o r e v e r y w e D , U r e J e lrey, a n d e v e r y s y s t e m ( 1 . 1 ) b e l o n g i n gt o E t h e c l o s e ds y s t e m( r . 1 ) - ( 1 . 3 )s a t i s f i e s (i) there exists a (unique) solution on 1R.. (ii) z, A,u are bounded if y,"y and ur are bounded (iii) liml*-[y(t) (iv) liml*-
- a,"J(t)] = 0
k(t) = &- € lRrexists.
/ is called the order of the controller. A (universal) adaptive regulator is called a (uniaersal) adaptiuetraching controller i f D - { 0 } , a n d a ( u n i u e r s a l )a d a p t i u es t a b i l r z e r i D f = {0} and.}""1 : {0}. If the closed-loopsystem does not have the property of uniquenessof solutions, then (ii) and (iii) must be valid for ezer3rsolution. We also introduce adaptive controllerswhich do not fit into the desoiptions given in Definition 1.1, i.e. ß( ) in (1.3) not being generatedby a differentialequation, / and g depend on time, or X is a classof infinite-dimensionalor nonlineai systems. However, an extension of the above definitions to these cases is straightforward. The model referenceproblemis also coveredby the tracking problem sincethe class of referencesignalscan be identified with a classof referencemodels and its inouts.
A D A P T I V E C O N T R O L- A S U R V E Y
323
The knowledgeof the system,disturbanceand referenceclassesis crucial for the design of simple adaptive controllers. If the system class consistsof single input single output systemsof the form t(l) = Ar(t) + bu(t), y(t) = cr(t), ( A , b , c )€ I R nx n x R ' x I R l " '
r(0) e R'
\ J
,' - ^' '\
and the problem of adaptive stabilization is studied then the following assumptions are known as slandard assumplions (Ai)
The sign of the high frequencygain is known.
( A 2 ) A n u p p e r b o u n d o n t h e o r d e r n o f t h e p r o c e s si s k n o w n . (A3) The relative degreeof the plant is known. ( A 4 ) T h e s y s t e m ( 4 , 0 , c ) i s m i n i m u m p h a s e ( s e eD e f i n i t i o n 3 . 1 ) . Over the last l5 years, various authors have investigatedthe necessityof these conclitions,how they can be generalizedfor larger classes,if they can be relaxed and |ow to design simple universaladaptive controllers. A chronologicallist of the most important contribut,ionsis as follows. The first adaptive stabilizer, not based on identification of the system parameter and being universalfor the classof single irrput single output systemssatisfying only the assurnptions(A1)-(A4), was given by Feuer and Morse (1978). This approach was irnproved in the following years; however, lhe controllersuse full state 'lhe first very simple controllerexplorobserversand are thus cornplicatedin nature. ing the high gain propertiesof minimum phasesystemswas introduced by Willems ' l h e y s h o w e t lt h a t t h e c o n t r o l l e rk = ! J 2 ,u = - s g n ( c b ) k g t s a anrl Byrnes (1984). u n i v e r s a la d a p t i v es t , a b i l i z efro r a l l s y s t e m so f t h e f o r m ( 1 . 4 ) s a t i s f y i n g( A 1 ) , ( A a ) , 'Ihe o p e n q u e s t i o n ,s e eM o r s e ( 1 9 8 3 ) ,a s t o w h e t h e r a n d h a v i n g r e l a t i v ed e g r e e1 . thr: knowlcdgeof the sign of the high frequencygain (A1) is a necessarycondition for a 0 'fhat
or
s@,u) < 0
(2'I4)
this condition is almost sufficientis shown in the following theorem'
2.2. supposef ,g are analgTzc functions, g salisfieslhe necessarycondz' THE,ORo\,{ z n a d d t l t o n ' , ( 2 . 1 3 ) , ( 2 . 1 1 ) a n d , lions !J(k,ü > ^
for alt (k,y) e R'\ rc
f o r s o m ec o m p a c l s e tK C R " 2 a n d s o m e m )
f
'.= A 4-
0 , S u p p o s ef u r l h e r m o r e ,l h a l
, f(k'u) 1t
into i1r,ü1sft,ü is suchthati lg can be ilecomposeil l h ( k , y ) lt M
(2.i5)
= ifu) + h(k,v) with
f o ra t t ( k , v ) e R ' \ r c , f o r s o m e M > 0
328
A. IL(]HMANN
and tht follou,ing l\-ussbttu,ntlupe r:ondzl,ions art
satisfit:tl
supr>o I [,i itrl clr = !:xt inf6;s * f,i itrldr 't'ht:n
artrt supr>o i f,i jtrl d,r - 1.cxt : -,cx, anrt, inf6;6 i f,i jtrlr/r - -oo.
( 2 .I 6 )
(2.12) zs a unire,srtl adaplit,t: st.abil,i:er fo, thc class (l.l). Exanrplcs for these c.nr,rollers are the Nussbaunr cont,roller (2.7), the wiilt_.rns B y r n e s < : o n t r o l l e r( 2 . 9 ) , a n < 1t h c f o r l o w i r g l l e y r n a . n Le.wis Mcye,rcontrolrer f r ^l? / t \ u ( t ) - l l ' ft t + - , t u \ 2| , . o ,{ f r ( / ) F I u u ) : ) t 1 ( t ) . i . ( t J= , ( t ) . . , t ( u € ) _ 2' L ) t''-'','' /
IIeyma'ntrt al' (1985)provedct-'rtain terminalbehaviour of t,hiscont,roller.l,,flill
I n g o n t h c i n i t i a l c o n d i t i o n so f t h e s y s t e n r .I t i s s h o w n t i r a t ,t h e t r a n s r e n tb e h a v i o u r o f ( 2 . 1 7 )i s b e t t e r t h a n t h a t o f t h c w i l l e r n s R y r n t : s c o n t r o l r e r( 2 . 9 ) o r o f t h e M o r s e c o r t t r o l l e r( 2 ' 1 1 ) w h o s e t e r m i n a i b e h a v i o r r r is unprcclict,aba l en d d e p e n < l e . t i,n a r r r : r r a t , i cw a y , o n t , h ci n i t i a l d a t a . A n o t h r ] ri t n p r o v c m c n to, f t h e l o c a l b e h a v i o u ro f t h e c o n l , r o l l c(r2 . 9 ) ,i n t h r : c a s eo f k n o w . h i g h f r e q u e n . l g 1 i n , . i sa c h i e v e t br y c . b . r c . r a a n d F . r r r u t ai i v s o ; w h o m o d i f y t h c i r d ; r p t , a t i o nl a w i n ( 2 . 9 ) t o i = _ o l k + y 2 f n r r o r n , ,o ' r ' 0 . ljndcr certain assunrptionson r,h. system 0 such that (3.7) has a unique solution on [0,]/). Suppose k O e L * ( 0 , 1 / ) . T h e n L e m m a 3 . 4 i m p l i e st h a t r ( . ) s a t i s f i e s
1 r ( r ) l l< M e - ^ 'l l " ( O ) l l f o r a l l r € [ 0 ,r ' ) . and the adaptation mle implies fr(.) e ,-(0,l/). Thereforethe solution of (3.7) does not have a finite escapetime, i.e. t' = r::n.Since g(.) € Lp(0,oo), it follows from t h e s e c < r nedq u a t i o ni n ( 3 . 3 ) t h a t z ( . ) € L p ( 0 , o o ) , t h u s ( 3 . 3 ) y i e l d s ; ( . ) e I o ( 0 , o o ) . : 0 . T h i s c o m p l e t e st h e p r o o f . N o w r ( . ) , ä ( . ) e l o ( 0 , o o ) g i v e sl i m l - - z ( l ) D Using the basic ideas of the previous proof, it can be shown that the classof adaptation rules can be extended as follows. T s n o R o t r ,3t . 7 . L e t p Z l
and consider
u ( r ): - f r ( r ) s ( r ) ,
k ( t ) = s ( t , k ( t ) , y ( t ) ) , ß ( 0 )= & o€ R
(38)
where g:lR 1x lR.. x lR.-*iR is a Carath4.odorgfunction, locally Ltpschitz in lhe second and thzrd argumenl, and locally integrablein I f R*. Supposethat the solulion k(.) of the clr.,sed-loopsyslem (3 1), (3.8) salisfies, on ils marimal inlerual of eristence l0.t'), the followrng condilions
A ( l ) > 0 a n d n o n d e c r e a s z ntgn t y(.) e Li(O,t') for all i€ [p,"o] + ß(.)e I-(0,t')
(3e) ß(.)€ 1,-(0,1')
+ aO € Lp(O,t')
(3 10) (3.11)
Then (3.8) is a uniaersal adaptiueconlrolk:r for lhe class (3.5). E x a N r p r , s3 . 8 . p'
*(,) = I
i l r ( r ) l l ' r ' ( v ( rr)1) ,p 1 p ' K c x r*,( 0 )> 0
z=p
where -F:lR-lR. is a polynomial such that l'())
) F0 ) 0 for all ) € lR.
T'hat the feedback law z(t) : -k(t)y(t) together with an appropriate adaptation law leads to a universal adaptive stabilizer for the class (3.5) has been shown for
aat
A. ILCHMANN
rn = L and ,t(t) = llg(t)ll, by Willenrsand Byrnes(19g4) m ) 1 and ß(t) = lly(r)l12 by ByrnesanrlWillems(1984) -
r n ) I a n d ß ( l ) : a l l y ( t ) l l 2+ t | l l " ( t ) l l z , r) r 0 , 0 > 0 byMärrensso ( 1n9 8 6 ) r n ) 1 , k ( t ) s a r i s f y i n(g: t . 9 ) _ ( 3 . 1f o _ 1 r) p 2 b y O w e n se t a t .( 1 9 8 7 ) . owens (1991)has provedthat,the wilrems-Byrnescontroller is arsoapplicableto
a certain class of singular systems.
3.3. o(C R) c C1
or
o(C R\ c C._
If the sign of cä is unknown or the spectrum of cB is known to lie either in the open right or left half planc, then Nussbaum'sidea of implementing a switching function (seeSt-'ction2) carries over to the n -th order ca^se.Consider the following classof allrninimum phase systems of the form (3.1) with unknown state climensionand o(C B) C C1 or o(C B) C rC_ i.e. , r(t) = Ar(t) + Bu(t), s(t) = Cx(t), r(0) e tR" ) ( 4 , B , C ) € l R ' x ' x R T l x ' rX R - " , oGb) C Ca or o(CB) c C_ l ( A , B , C ) i s r n i n i m u m p h a s e ,r ua r b i t r a r y
fr.rrt
I)uc to the multivariablesituation, we needto introduce sr:alinginuarianlNussbaum f u n c t i o n s ,i . e . p i e c e w i s er i g h t - c o n t i n u o u sf u n c t i o n s 1/(.):lR_ R.o that for every o, ß ) 0, the function
,rr1r;: { ;;[;]
if if
N(t) > 0 /t/(t) < 0
satisfies(2 10) as well. (The concept of scaling invariant switching functions was o r i g i n a l l yi n t r o d u c e db y L o g e m a n na n d O w e n r l f g S S ) . ) 3.9. Letp) TI inTeLn" tl h ( c o n t r o l l e r
7 a n d l y ' : l R - R b e a s c a l t n gi n u a r i a n t 1 r { u s s b a u m functzon.
u ( t )- - N ( k ( t ) ) k ( t ) s ( t ) , is a uniuersal adapliue controller
L 1 t 1 =l l y ( r ) l l p , # ( 0 )€ R
(3.13)
for lhe class(5.1g)
Pror''f. The proof is simirar to that of rheorem 3.6, only the step that ß(.) ( tr- (0, t/) needsa modification. without lossof generality, assumethat o(cB) c c1 (otherwise consider -CB) Let p € Rmxm-be posiiive definite and such that PC B + (c B)T P - 1-. Inserting the feedbackraw into the inequality (3.4) yields, f o r s o m eM > 0 a n d ß ( t s ) 1 0 , t
t f
f
t
;llu{t)l(r0 if N(t)< 0 s.,iu(P) denotesthc snrallestsingular value of P. Now the right hand side becomes negativcby the property of the Nussbaumfunction which contradicts thc positivity of the left hand sicle.This con'rpletesthe proof. 'l'hat
t h e f e e d b a c kl a w z ( t ) = - 1 / ( ß ( t ) ) y ( l ) , w h e r e 1 i ( . ) i s a N u s s b a u mg a i n , t o gether with an appropriate adaptation law leads to a universal adaptive stabilizer for the class (3.12) has been shown for m:
I a n d f r ( t ) = A ( t ) 2 b y W i l l e m s a n d B y r n e s( 1 9 3 4 )
r n ) _ | a n d i n t r o d u c i n ga g e n e r a lf u n c t i o n N ( k ) b y M ä r t e n s s o n( 1 9 8 6 ) m ) I and more general adaptation laws and switching functions by Owens el a/. (1989) Ioannou (1986) considcredsystems belonging to (3.12) which are coupled with a 'parasitic slow' linear system. He showed that under certain assumptionsthe Willems Byrnes controller (2.9) is a universal adaptive stabilizer if the initial state of the unknown system lies in a certain bounded region. An alternative approach to Nussbaum's switching strategy makes use of the following switching decision function which determines the switching ti;res 0 = l o ( t r ( . . . o f t h e s w i t c h i n g f u n c t i o n N : i R 1 * { - 1 , * 1 } i n t h e f o l l o w i n gw a y . C o n s i d e rt h e s w r l c h r n gd e c t s t o nf u n c l r o n d ( . ) t!(l):=l
[f' a'G)d',= o n , r t t . ( r ) u 2 ( , \s02, t r ) a r ) fia2(,)d,I o
[ * t tl ' t-
t J
ft-
l-l
(3'15)
LI
w i t h ß - A 2 . f i { ) , } ; e n i s a s t r i c t l y i n c r e a s i n gu, n b o u n d e ds e q u e n c e o f r e a l ,p o s i t i v e numbers or 'thresholds',then l/(l) is defined by the following algorithm:
(*)
i = 0 N(l;) := I l;+r := min {l > ti I N(ti){(t) < ),+rß(t0)} lü(r) := IV(ti), t € [l;,11*t) l{(t,;+r ) := *I/(1, ) i :=j *l go to (*)
The algorit,hrnis well-definedbecause (i) l(l)
is monotonic on any interval I > 0 where N(t) is constant
( i i ) l ( 1 0 ) : , t ( 1 0 )e n s u r e sc o r r e c ti n i t i a l i z a t i o no f t h e a l g o r i t h m .
(3.16)
A. ILCHMANN
u s i n g t h e i d e : r sp r e s c n t e di n I l c h m a n na n d o w e n s ( l g g l ) , t h r : I b l l o w i n gt h e o r e m can be shown. Notc that the switching pararneter 1/(A) is adjusted in finite time and that it is in the hand of the designer to choosean appropriate sequenceof thresholds. T'ntonnu 3.10. Täe controlle.r.
u ( t ): - n ( ß ( t ) ) a ( t ) y ( t ) ,
k ( t )- l l v ( r ) l l n , ß ( 0 )€ r R
uthere'l{(') is produced hy (3.15), is a uniuersal adaptiue stabilrzerfor the class (3 12). Moreouer,d(t) has a finite timit {- as |-lut, and the swrtchtngfunctton ' Y ( t ) s w t t c h e so n l y a f i n i t e n u m b e r o f t i m e st t , t 2 , . . . , t y , s o t h a t l / ( r ) i s c o n s l a n l for t ) ty. 3 . / . d e t ( C B )l 0 If it is only known that the systenr has an invert,iblehigh frequency gain but the s p e c t r r r mi s m i x e d , i . e . d e t ( C B ) f 0 , t h e n t h e c o n s t r u c t i o no f a u n i v e r s a la d a p t i v e stabilizer is basedon the following result from linear algebraproved by Märtensson [ 5 ] , S e c t i o n8 . l , p u u a i J . l l . T h e r e e r i s l sa f i n z t es e l { 1 { , , . . . , 1 { r u } C G r _ ( R ) w i t h t h e p r o p e r l g that,for any M € Gr-(R), thereeüsts i e N suchthat o(M K;) C (r_ Now the feedbacklaw is given by u(t) - k(t)Ii s(k(tDy(t)
(3.17)
S:lR-{1,...,N}=lrl
(3 18)
where il ke (-oo,11) [t (3 1e) t i f kefnN+t,rrN+i+1) forsome /gNo, ig1/ I is a switching function driven by ft(l) so that lis(,t(r)) cycles thro'gh the spect r u m u n r n i x i n gs e t { Ä r , . . . , 1 { r y } a n d { q } ; 6 1 i s a n r o n o t o n ei n c r e a s i n gs e q u e n c e of switching points which satisfy 'q' \r' ä ' t)
,
Ti-t
i-m
Ti
=
U
(3.20)
The switching sequencenecessarilyfulfils liml* a ri = oo . The classof switching sequencessatisfying (3.19) is more restrictivethan in the single-input single-output ca^se, sincefor the sequencer; ..= i2 (3.1g) doesnot hold true. Iloweve,r,r;a1 ..= f a n d 4 1 1 ' . = r i * e x p { f 2 } s a t i s f y( 3 . 1 9 ) . Under the above assumptions,the following result is available. T H E , o R o N!41 . I 2 . T h e f e e d b a c ka n d a d a p t a t i o nl a u
u ( t ) : ß ( l ) K 5 1 5 1 , ) ; e ( t ) , ß ( r )= l l y ( r ) l l o ,
ß ( 0 )e R
is a uniuersal adaptzue stabilizer for the class of mullivarzable minimum lems of the form (3.1) which satisfy det(CB) 10.
(3.21) phase sgs-
A D A P T I V E C O N T R O L_ A S U R V E Y
99tr
The intuition behind this control strategy is similar to Nussbaum's idea. If the 'correct' /i; is hit, the gain is large enough, and the time interval until the next possibleswitch is long enough (which is ensured by condition (3.19)), then the system settles down and no more switchingsoccur. This result was claimed by Byrnes and Willems (198a) and by Märtensson (1986). However,both proofs are incomplete, a correct proof is given in Ilchmann andLogemann (1991). 3 . 5 . E r y o n e n l i a ls l a b z l i z a l t o n For first-order systems it has been shown, in Section 2, that the trajectory u(.) of the closed-loopadaptive control system (2.5) decaysexponentially to zero. It also follows that the t,errninalsystem defined by z(t) = [o - ]-öc]z(t), fr- := limr-.o ,t(l) is exponentially stable. This was not shown for higher-ordersystems, where only asymptotic decay to zero was proved. Note that we did not show that the lerminal sqslem
i = IA - k- BC]r(r),
ß- =
,lim
ß(l)
(3.'22)
is exponentially stable, but only that each trajectory of the closed-loopsystem tends to zero asymptotically. Counterexampleswhere (3.21) is unstable can easily be constmcted. However, computer simulations have shown that the controller (3.13) produces in most casesan exponentially stable terminal system. But, to the author's knowledge, it is still an open problem if generically,with respect to t h e i n i t i a l c o n d i t i o n sr ( 0 ) € R ' , f r ( 0 ) € R , t h e t e r m i n a l s y s t e m p r o d u c e db y t h e universaa l d a p t i v ec o n t r o l l e r( 3 . 1 3 ) i s e x p o n e n t i a l l ys t a b l e . To overcontet,helack of exponentialdecay,it is possibleeither to strengthenthe minimum phaseassumptionon the system classor to introduce additional dynamics int,othe adapt,ationlaw. I f ( / , 0 , c ) i s i n t h e c l a s s( 2 . 3 ) , t h e n f o r c u ) 0 s u f f i c i e n t l ys m a l l ( , 4 * u l n , b , c ) belongsalso to (2.3). If the adaptation rnechanismis chosento ensurethat r.(.) is an asymptotically stable (and hence bounded) solution of the closed-loopsystem
j,-(r) = l(A + ut") - fr(r)örl z.(r)
(3.23)
then the solution of
t(r) = IA-k(t)bc)r(t),
(324)
g i v e nb y r ( t ) = e - ' ' x , ( t ) , m u s t b e o f e x p o n e n t i a ld e c a y . E x a m p l e so f s u c h a d a p tation mechanismsare the so-called'exponentially weighted' controllers
k(t) = /t0+
Ji"tä
t'" lls(")|l,
k(l) : ""llv(t)ll''
seeowensel nl' ( 1987)
s e eL o g e m a n(n1 9 9 0 )
which consequently yield the desired stabilization result. However, it does require knowledge of a suitable value of r..,.
A. ILCHMANN
.1.1())
In order to apply the strategy explainedabove,one possibility is (seeLogemann (1990)) to strengthen the minimumphase condition defining the system classx to satisfy
0", "t"t , [
äl
+ 0 forall s€{.\€Cl Re)>-r}
(3.25)
for some known cu > 0 Another possibility is to considerschemesthat adaptively find a suitable value for c..'on-line. This idea was introduced for a special control iaw in Ilchmann and Owens (1990), where it has been shown that exponential stabilization can be achieved by choosing cu adaptively using the control scheme defined by f 1r;:
e 2 a ( t ) t l l y ( r ) l l ' ,ß ( 0 ) > - 1 ,
I , ( r ./ : l f
r++T
f o r ' € [ 0 ,ä )
for t>h
(3.26) where h ) 0 is arbitrary. The idea behind this is the knowledgettrat, for some 0 r * > 0 , t h e a d a p t i v ec o n t r o ll a w & ( l ) : e x p ( 2 a r * r ) l l y ( t ) l l ' w i l l e x p o n e n t i a l l y s t a b i l i z e the system. Thus, as long as c..,(l)is too large, r(l) will increaseand the gain grows whencer..,(l)becomessmaller. Eventually c,r(t)is small enough to guarantee convergenceof ,t(t). Now it follows from (3.25) that u,,(t)convergesitself. In fact, the example (3.25) can be extendedsince we only use that qr :lR1+R* is a continuously differentiablefunction which satisfiesthe conditions "(k) "(k) lim6-*"u(k) 'I'his
=
is non-increasingin ß € R+
l
0
)
puts us into a position to prove a more generalresult.
T s t o R r ; n t 3 . 7 3 . S u p p o s e1 / ( . ) i s a N u s s b a u r ng a i n , .T h e n t h e f e e d b a c kl a w u(l) - l/(A(r))y(l)
( r e s p e c t z u e l y ,u ( l ) = - s s n ( C B ) y ( t ) )
a n d l h e a d a p t a t i o nl a w ß ( t ) = " z ' ( t ) t 1 1 y Q ) 1 1 2k,( 0 ) > - 1 ,
u s a t t s f i e(s5 . 2 6 )
is a uniuersal adaptiuestabilizerfor the class (2.1) (respectiuelg, (3.5)), uthichproducesan erponenlially decagingsolulion of the closed-loopsgslem. A proof is given in Ilchmann and Owens (1990). A version for the non differential gain adaptation using the switching decisionfunction (3.1a)is presentedin Ilchmann a n d O w e n s ( 1 9 9 1 , 1 9 9 1 a ) . I f ß ( l ) i n T h e o r e m 3 . 1 2 i s s u b s t i t u t e db y ( 3 . 2 5 ) ,t h e n exponential decay of the solution of the closed-loopsystem holds true. This has b e c n p r o v e d b y I l c h m a n n a n d L o g e m a n n( 1 9 9 1 ) . lJnfortunately,all contributions describedin this subsectionhave the disadvant,agethat the gain adaptation y - k is achievedby an unbounded function. Por a rnore satisfying approach see Sectioir5.1.
ADAPTIVE CONTROL - A SURVEY
.t.t r
3.6. Tracktng In this section, we consider the tracking problem for the following classof multiinput, multi output, linear, minimum phasesystems
i(t) : Ax(t)+ Bu(t), r(0) € R" l u(t): Cr(t), ( A , B , C ) € J R " x "x R n x mX R - " ' , d e t ( CB ) I 0 | (A, B,C) is minimumphase,n arbitrary )
1:.ZS;
and the class of referencesignals
) l(f r l r , " r ( r:)0 } ! , " y : : { a , " y€ C - ( R , R - o
(3.29)
where a(s) e R[s] is a monic polonomial with zerosin C1 only. Note that 0 € !,"J, thereforeit is not relevant to considerthe casethat a(s) has zerosin C- since the correspondingmodes are decayingexponentially. One possibility of handling this problem is to make use of the internal model principle, that is, a reduplicated model of the dynamic referencesignals is incorporated as a precornpensatorin the feedback loop, see Wonham (1979), Section 8.8. For a different approach, see Section 4.3 and 5.2. Here, the precornpensator is chosen as follows. Let B(s) e R[s] be a monic Hurwitz polynomial of dggreep = deg(rr), and choose a minimal realization of p(s)lo(s), denoted by ( A , 8 , C , 1 ) € R l X p x R p x 1 R . 1 xxp R , a n d t h e p r e c o m p e n s a t oirs g i v e n b y € ( t ) = , 4 . € ( l )* B - u ( 1 ) ,
u(t) - C.((l) * u(t),
€(0)€ lR''
(3.30)
wnere
A. = diag{A,...,A}€ lR-px-p, B* = diag{B,...,8} € R-P"', ] Rm'mP. C = diaglC.....C€ Then the input output behaviour a e ( 3 . 2 7 )a n d ( 3 . 2 9 ) i s d e s c r i b e db y r1t; = Ä"rU)+ Ilu(t),
y of the seriesinterconnectionformed by aU) = Cr(t).
r(t))€ Rn+'np
(3.31)
where
^ : l t " f .l ., B = l ; . ] c - r c , o r , ' = l ; ] In order to rewrite this as a stabilization problem, the following two lemmata are needed. L n n t N , t . q3.. 7 4 . ( A , B , C ) andCB:
b e l o n g st o ( 3 . 9 7 ) z f a n d o n l y ü @ , 8 , ( ' 7
betongt to (3.27)
CB.
L p n t n r , q3 . .15. For eaery !/reJ €U,.J a,"1(t) = -i1L).
lhere erisls a io € IRz*mp such thal .l(/)= /r(1).
.i(0)= i,11.
(3.32)
,t.tö
{. IL(-'ilMANN"
\ o r v , . r , . ( / ) : - r l ( 1 )- t ( l ) s a t i s f i c s i,1t1 -
At:,(t)t Bu(t).
! , t ,J, ( t ) - y ( t )
- ( 1 . r ,( t ) ,
r : " ( l ) -) t ( 0 ) - t ( 0 ) . ( 3 . 3 3 )
'l'his
- y i c l r l st l r c l o l l o w i r r gt l r e o r c n r . 'l'IJI,;otu,,l\,t :1.16 If
? r ( 1-) / ( Ä . ( t )y.( t ) ) ,
f r ( t )= c ( # ( r )y, ( r ) ) . t l ( Q e) t R
t . t e u n , t r ( r ' s e la d a p l i t , t s t a b i l i : e r f o r t h e c l a s s( . 1 . J 7 ) , t l t c n
t.(t.)-y,,Jft)_u(t) ' , ( l ) / ( { : ( t ) ,(, 1 ) ) . t . ( 1=) a ( I . ( r ) . r ( t ) ) f, ( 0 ) € t R : u ( r )- a ' . € ( 1! ) I ) ' t ' ( t ) , € ( / )- . 1 . € ( t ) r R . r ( r ) , € ( 0 )e r P . - p t,, o ttTt,tr:(rsaltdasttittc Lratki,nq corttroller for t.hc ttass (,1.17) aud Lht: class of ref, ( r ' ( n ( r s t g r t a l s! , , 1 r l i r r : n h u ( , 1. t 7 ) . 'l
h r ' 1 > r c v i o r r sp r e s e n t a l , i o n i s g i v c n b y M i l l e r a n d l ) a v i s o n ( l 9 g 1 h ) , a n d i n c l e p e n t l e r r l , l .tv. , y ' l b w n l t : y a n d o w e n s ( l g g 1 ) . T h e r c s u l t s i r r M i l l c r a n d D a v i s o n (1gg1b) covcr it rnor(' goneral forln including nt ) p t,o sorne r:xtent,, ccrtain disturbanccs 1, s ; r l i s f y i n g o ( f ; ) u , = 0 a r e a l l o w e d i n t , h e s t , a t e a n < 1o u t y r r r t e q u a t , i o n ,a n d t,heyshow t,h;rt int t i v ter a c k i r r g c o n t r o l l e r . l V l : r r e e l s( - l 9 8 4 ) w a s t h e f i r s t w h o u s c d t h e i r r t e r n a l r n o t l e l p r i n c i p l e t o c o n s t r u c t a u n i v r : r s a l a d a p t i v c t r a c k i r r g c o n t r o l l e r f o r s i n g l e - i n p r r t s i n g l e - o u t p u t ,s y s t e m s of r e l ; r l , i v t :d e g r c e p ) l, st:cSection 4.3. 'l'ao a n d l o a n t i o t r ( 1 9 9 1 ) h a v t - ' i n t r o d u r : e c lt h e f o l l o w i n g < l i f f e r c n t t r a c k i l g c o n t r o l l t : r f o r s i n g l e i n p u 1 ,s i r r g l e o u t , p u t ,s y s t e n r s . P t t o p < r s r r r < l x l l . i 7 . T h c : a d a p t a t i o n ,l a u
0(t) = ,(t)lv,"t\) - aftl, , ( 0 ) € r R 2 ( r + rI) k ( t ) = ( B ( t ) , u , ( t ) ) [ y , . _r (at )0 ) , k ( 0 )€ R . /
(3 34)
u h e r ru: ( t ) = l t / , , 1 f t-1 a ( ) , 1 , s i n o 1 l ,. . . , s i nu i , . . . J c o s L r r l , . . c. ,o suft)'t logether , u;ilh Lht:feedhncklau
u ( t ) = f r ( t ) zs i nf r ( r ) ( p ( r ) , . r ( t ) )
(3.3b)
zs a un.iuersal adaptiue tracking conlroller for the class (2.3) antl the class of reference signals conststtng of
9,"1ft) = oo+ fj=r nr sincrit* ö,cosr,;;l 1 r 1 , . . . , o t€ l R 'I'he
a r e l ; n o w n , e o , . . . , a t , b t , . . . , ö r€ R
\
(3.36)
orcunknorr. J
proof by Tao and loannou (1991) is not based on a conversion to an ad aptive stabilization problem.
A D A P T I V E C O N T R O L- A S U R V E Y
r)J v
3.7. Robustness Robustnessfor the adaptive controllerssurveyedin the previous sections has been consicleredfor state and input nonlinearitiesentering a system b e l o n g i n g t o ( 3 . 1 ) in the following form
; ( l ) = A r ( t ) + e Q ,x ( t ) ) + d ( r )+ a l u ( t )+ \ r ( t ,r ( t ) ) + r t 2 f tr,( t ) ) + a 3 ( 1r (, l ) ,u ( l ) ) l
u(t) = Cr(t)
and also for sector-boundedinput and output nonlinearities( and ( so that the real input z can enter the system via u(l) = €(r, t(t)) and the reai output measurement is given bV y(t) : ((l, g(l)). All nonlinear functions are appropriately defined in order to ensure uniquenessand no finite escapetime, we omit details for brevity. T h e t e r m p ( t , r ) r e p r e s e n t st i m e v a r y i n g s t a t e d e p e n d i n gp e r t u r b a t i o n sw h i c h are assumedto be of sulficinetly smallfinite gain, thus proving well posedness.d(l) representsan arbitrary tro(0,oo) function. The time-varying input perturbation are of bounded growth or can be unboundedif they are of 'correct'sign. More precisely, the following results have bcen achieved. flelmke and Prätzel Wolters (1988) showedthat the Willems-tsyrnescontroller (2.9) for ff(fr) : -sgn(cb) is a universal adaptive stabilizer if for all (C,r) € lR2 we nave (3.37) W ( t , r ) 1 5 e l " l f o r s o m e( u n k n o w n ) , p > 0 and d(l) is an .Ln(0,oo) function. An improvement of the local behaviour of the controller (2.9) is caseof known high frequency gain is achieved by Cabrera and Furuta (1989) who modify the adaptation law in (2.9) to ,t = -o k -t y2 for sorneo ) 0. lInder certain assumptions on the system classthe closedloop system is robust against bounded disturbances. Theorem 3.7 holds true for multivariablesystemsif for all (t, r, u) € IRx IR'x Ra n d s o m e( u n k n o w n ), i r , i t , i z , ? s > 0 w e h a v e
l l p ( t ," ) l l < P l l " l l ,
Q s u f f i c i e n tsl ym a l l
l l ' r ' ( t , r ; 14 lrSl l " l l y ( t ) r C B \ 2 ( 1 ,r ) ( 0
y ( t ) rCB q s(t, r, u ) < l l yl4l ' fll " l l+ llulll
(3.38) ( 3 . 3)e (3.40) (3.41)
s e eO w e n s e Co / . ( 1 9 8 7 ) . In the single input, single output caseTheorem 3.9 is valid, if for all (t, r) e R . x R ' a n d s o m e ( u r r k n o w n )i r , i z > . 0 , w e h a v e
4tll"ll, ilr1(r,r)llS
v ( ü r C n q 2 ( l , rS) 0
This has been proved in Prätzel Wolters el a/. (1989). Theorem 3.9 holds also true if the Nussbaumfunction is scalinginvariant and if the classof single-input single-output systemsis subjected to actuator and sensor nonlinearities((1, ü) and ((1, y) which are seclor-bounded,i.e. for all I € R. and some (unknown) B ) n ) 0 we have aü < ((t,ü) < pit
for all
u € R1,
p u S € ( t . u )( o u
forall u€R-
:J40
A. II,(IIIMANN
'this and fbr ( :rlalogously. is a const:r1ut:rrcc of a gent:r:r,lrcsrrlt for r 0
ü(r) < l-cAbk(t) + r{)y(t)z and integration yields rt
a s= y ( o ) - , o b v ( t )s r ( 0 ) + t l d t l + ^ ' l f r ( t ) -f t ( o ) ] J"l-raut{')+nly1s12 f orr'r',,' ( 4 . 11 ) wherewe made useof the substitution ft(s) = 1t. If k(t) is unbounded,the right hand side of (4.11) becomesnegative, hence producing a contradiction. The remainder of the proof usessimilar arguments as in Theorem 3.6. rr Byrnes and Isidori (1986) gave a different (and incomplete) proof of Theorem 4.2. For higher relative degree minimum phase systems, the intuition arises from the non-'adaptive case as well. It is shown in Märtensson (1986) that the linear time invarianlcomDensator ü(s) :
-k2r-1
( s+ t ; r - t ii(") ( s * f r 2 ) (+ s f r ' ) .. . ( " * k 2 o - ' )
(4.12)
stabilizesa minimum phasesystemswith positive high frequencygain and relative degreeless or equal to p for k sufficiently large. The same result has been shown by Khalil and Saberi (1987) for the different compensator
=_'tt2*'##o't'l u(s)
(4.13)
Then the problem is to determine a suitable adaptive controllerin the time domain. Note that the transformation from (4.7) to (a.8) is meaninglessif ß(.) is depending on f . However, the Lyapunov function candidate sometimes gives a hint for the correct t,ime-domainrealization. A very early contribution to solvea certain adaptive tracking problem was made by Mareels (1984). He considersthe following class of single input single output systemswith arbitrary but known relative degree,known sign of the high frequency gain, and known upper bound for its magnitude. y(t) : m(t), t(l) : Äz(t) + öu(t), r(0) € lR" ) =cAp-2b=0, ch=cAb= 0 0 s u c h t h a t t h e c l o s e d , loop syst,emis erponcntially stablefor all k ) fr0. The adaptivc versiorrof the previous lenirna is as follows. 'I'sr 0 , o ) 0 . T h i s r e s u l t h a s b e e n e x t e n d e db y C o r l e s s( 1 9 8 8 ,1 9 9 1 )a s f o l l o w s . T I r p o R o u 4 . 5 . T h e c o n t r o l l e ru ( t ) = k ( t ) y ( t ) , f r ( / ) = y ( t ) 2 i s a u n i a e r s a la d , a p t i a e s l a b r l t z e rf o r t h e c l a s s o f c o n t r o l l a b l ea n d o b s e r u a b l es y s t e m s( A , b , c ) 6 l p n x n y lR.x iRrx" which are uniformly stabilzzable ura highgazn feedback, t,.e. there exisle, f r * > 0 ( d e p e n d i n go n ( A , b , c ) ) s o t h a t m a x Ä e \ t ( A - k b c )< - e w h e r e) 6 ( A -
for all
k b c ) d e n o l e st h e e r g e n u a l u eosf A -
k > k*
kbc.
Morse (1988) introduces a stabilizer for the following class of relative 1 or 2 systentswhere the exact relative degreeis unknown. 'l i(t) = Ar(t) + bu(t), y(t) = ü(t), r(0) e R" ( A , b ,c ) € I R " ' x I R ' x R t " ' , , , i s u n k n o w n l cb>0 orifcö:0then cAb>0 ) He has proved the following theorem.
t+.tSl
ADAPTIVE CONTR,OL_ A STIRVEY
345
T s E , o R s t r 4 . 6 . T h e f e e d b a c kl a w u(t) : -k(t)0(t) - k(t)2y(t)
(4.1e)
l o g e t h e rw i t h t h e a d a p l a l i o nl a w 0 ( t ) = - ( k ( , ) + ) ) d ( l ) - k ( t ) 2 y ( t ) , d ( 0 )€ l R , ) > 0 k(t)=s(t)2, k(0)€R
\ J
(' -4' r- o" '\
i s a u n i u e r s a la d a p t i u es t a b z l z z ef or r t h e c l a s s( / . 1 8 ) . By increasingthe dimension of the compensatorby one, Morse (1987a)generalized Theorern 4.6 to the classof relative degree3 systems. T o r e l a x t h e k n o w n s i g n c o n d i t i o ni n t h e c l a s s( 4 . 1 8 ) ,M o r s e ( 1 9 8 5 )i n t r o d u c e d a two-parameter adapt,ationlaw of the following form .
I
n f ' . , . k r ( t )= O ( t ) v ( r+) z l ( t ) + ,y(t)' Jn u(")'d". 20(t): \0(t)y(t)- y(r)r'/(|l*(t)ll)10(t)k0e) + ae)kve]nn, = [)s(r)- 1/(||ß(,)||)s(t)ks(ü]a(t) - s(r)'1/( ||e(r)II)ßy(,)
k o ? l=
(4.2t) 9.22)
w h e r e l l ß l l , : r / k u + k o , ) > 0 , a n d 1 ü ( ' ) i s a N u s s b a u mf u n c t i o n ,s e e( 2 . 1 0 ) .T h e n the following result is obtained. T s o o R p n a4 . 7 .
'I'he a d a p t a t i o nl a w ( 1 . 2 1 ) , ( / . 2 2 ) l o g e l h e rw i t h
u ( r )= N ( l l ß ( T ) l l ) 1 0 ( t ) k+0 y( t(), ) k y ( , ) 1 , 0 ( t )= - i ^ e ( t ) + u ( , )
(4.23)
i s a u n i u e r s a la d a p t t u es l a b z l z z ef or r l h e c l a s so f m t n t m u m p h a s es y s l e m s( , 4 ,ö ,c ) € i R n x n x R " x l R l x n o f r e l a l z u ed e g r e e1 o r 2 . To get an intuition for this controller, considerthe fact that the output feedback compensator
/ v ( l l k l l ) k q+( s) ) y(s)=;-iffi,f')
U-24)
yields closedloop stability for a suitable constant fry,&e € R. Mudgett and Morse (1989) introduced an alternative stabilizer to that given by Mareels (1984) for the class (4.14) and p - 2. 1.3. Tracking wilhtn a ball Miller and Davison (199i) considereda modification of the usual adaptive tracking problem. Instead of forcing the error between the plant output and the reference signal asymptotically to zero, it is desired to force the error to be less than an arbitrarily small prespecifiedconstant after an arbitrarily short prespecifiedperiod of time with an arbitrarily short upper bound on the overshoot. More precisely they have studied the following problem. Let PC* PCl*::
:: the set of piecewiseconstant bounded functions / : IR-Rq the set of continuousf e PC* which have derivativesin PC*
:t46
A. IL'IIIMANN
P n o g l t ' r t u r4 . 8
S u p l r o s t 'e , 6 . ' t ' > 0 a r e p r . s p c c i f i e d . l ' i n r l a n a < l a p t a t i o ' l a w
( u ( ' )u, , , , 1 ( 'l)t )o , *, t * ( t ) s u < : i trh a t t h e f c e d b a c kl a w u ( l ) = f r ( t ) a p p l i r : ctlo a n y s y s t e n rb e f t r n g i l gt o a p r t , specifiedsyst,enrclass ancl any '!!r.:J ( ) € pall. yieltls for thc solut,ionof the closed loop systerriand the error e(l) :- y,"1ft)- y(t) (i) -,
< s s ? ) ( e ( 0 ) ) € ( r 1 and sufficiently large i € N
ra l l t ) t ; , t ; ) 0 l l t ( r ) l l a g e sl i r T ] J _ l J - r 7 g l g, f i g ) . \ ' l t t , t . r ; t r ,I ) . 1 , 1 .& : l ) a v l s o w , I i . J . 1 9 g 1 b A n a r l ; r p l i v t , t r a c k i r r gp r o b l c r r r . [)rcprirrt. \ l t t . t , t , ; n .l ) . 1 r . , ( : l ) Ä v r S o N . t , l. . 1. l g g l c A r r a c l : r p t , i vltr,a r : k i r r gp r o b [ , r L rr v i l h a
