Adaptive Tracking for a Class of Nonlinear Systems of Relative Degree Two

2b-06 5

Copyright c&'l 1996 IF AC I3lh Triennial World Congress, San francisco, USA

ADAPTIVE TRACKING FOR A CLASS OF NONLINEAR SYSTEMS OF RELATIVE DEGREE TWO E.P. Ryan School of Mathematical Sciences, University uf Bath . Bath BA2 'lA y, United Kin.Qdom Abstract. Th(, prohh~m of controlling a scalar ontput. v(l,l'iable to track any reference ~ignal of (Sobolev) das~ R = n r3 '=(lR), whil~t maintaining state houndedness, is adtlresspcl for a cla....,s .'\//j of single-input, siIlgle-output~ nonlinear, minimllmphase H,YStf'llb of rdative degree two (the prototype being a single-degree-of-freedom uncertain I!H'(~hanical systt~m with extraneous disturbances, wit.h damping, quantified hy t,lw par<-lmf'ter J. and with position, hut not vf'loc:ity, "lvailahk for feedback), An adaptive discontmuous output-feedback control is eonst-ruet.ed whkh is (R, N tJ ) uniV(~rsal in the sellS.> that, for any given reference sigTlal of chtss Rand e\'ery system of class .\",j. t h(> st.rategy ensnres that the tracking error is a,symptotir to ;~ero, Keywords .. \daptiH' systems: discontinuous control. I10nliIlCar systems, tracking, l111iv(,l"sal S(~I"\()JtH'("hiinisms;

1. INTRODUCTION

Approaches to adaptivE' cont.rol may be classified into methods that. - either implicit.ly or explicitly - exhibit some aspe<.:t of ident.itication of the process to be controlled, aIlcl methods that s(·;!t'k oHI\' to control. Here, the latter approach is adopt.ed. In common with related work in t.he area, (sc(" c.g., Ikhlllatlll (UHJ3) and bibliography t.her('in), thc piipcr is ('OlH:enu'd with demonstrating the existence UlHipr weak assllmptions - of a single controller that achi~~yes some prescril )~ld objedive for every system of a gi yen class. T1H' papl'r is focussed 011 the servonwchallism prohlem in the nHltf'Xt. of a class of Ilonlinear minimum-phase SYS\PIIIS Olt RV of relative degree two, the prototype lH'lug a Ct)Htrolled lUl(".ertain mechauir:nl ~ystell1 on ~~ of the form :ij = dfJ + j(t, u) + bu, \\o'ith qllantifiahl(~ damping (ill r-h" S(~llS(' that d < -6 < 0, 6 a known constant) and with position:tJ t), but not velocity ,j(t), availabh' for fe(slbadc

two. Every system L: E L ha.,<, (proper rational) transfer fUIlt:tiOIl G of the form

1,1 Lhtea',. system das8 L;' and nn11 t:1'sal stabilization

By \.,-ay of rnotivar-ioll, first cousider the class L of singlcinput (u(t) E JR).

sin~le-olltpllt (y(t) E

dimensionaL minimlllll-phas('

s~ ' st(>nls

JR). linear. finiteof relative degree

+ {3(8)

I

b ." 0,

where /1 and 11, arc' monk polynomials with degCB) < deg(n) (with t.hE:' interpretation that /3 == 0 if 11. =: 1). Let (AI, ([2, a;l) be a minimal state space (of dimension N - 22: 0) realLmtion of tlH' transfer funct.ion G I (,) = ,tJ(-)jn(·): by the minimUIIlRvtlase assumption, the sp(~e­ trllm aC-t I) lies ill the open left-half complex plane C_. It follows th:;t (""rh 1: E [. has a state space (JRN) representation of tlw form

+ a"J(t). x(t) E JR N-' "Iif(t) + ""/(t) + a3x(t) + bu(f)

i(t) = .411:(t)

iiU)

=

with o-(A,) c C_ and /1." o. d('IlOle the subclass £:, = {1: E £:1 f.>1 < -6 < O} _ This suhcla..'iS has the property that eaeh ~ E CfJ is individually stabilizable b.v some static output. feedback (but. no siugle static output feedbaek will stabilize' every member of C,j), The pl'Oblem uf L,s-universal stabilizatioTl may be posed loosf'l;y as follows: determine an

Let £:,

1

b'lI(s) (8 2 - ",8 - "2)"(8)

') __ T,~ G(

2084

adaptive output-feedback strategy that renders the zero ::;tate a gluual attractor for all E E [ 6 . Let " be a.ny comillUOUS funetion IR ---) 1R with the N u s~ ­ haum prop ert.h ~s (s('(', e.g .. B yrl1(,~ &. vVillems, 1084 ; N II Ss hallJlI ~ 1 98~1 : \Yillcms &: Byrnf':-<, 1984) (a) Hm sup -J 1/-+ 'X,

1]

1"

11" =

v = +x. (h ) iim inf -

0

1)---f"X::

v

71

- 00

(2)

0

For example, 'TJ I-t IJ "}, cos 1] suffici'S. It is kllown (Corless & Rra n, 1903) that the following simple ada ptive Ollt.Pllt. - f( ~(~ dback strat('~:v (£,;-llni\'ersal stabili",er) u(t)

= v(~(t))y(l).

1/(1)

= iil:(/) + !'1I"(t),

k(t)

= y'(t)

ellsures g io baJ attradivity of the zero state of r. vr.ry system 1 in C/i aud b()undedlle~~ of tlu ' function k(· ) (and of V(I/( ' )) )' III (Carless &. Ryall, 1993) it is also shown t.hat weakly nOlllilwar perturiJations can be tolerated. In (Ryan . 1995), t ill.' res lIlt i:"o gelwrali:leU to eucompass a much wider <:lass of lloniilH·arit,il's. Here: the latter stabilizatiou re mits an' ext.(,l1d, ~d [0 enc.ompass a problem of asymptotic Olltput lmekinq of reference signals o f a prescribed. da..'i~ 'R a.nd gcncrali zes, t.o N-dilllellSiuDti. rela t ed results (Ryall. 1996) fo r plrm(l1' systems.

which guarantees that, for e very (r (· ) , E) E 'R x N, and fOf each (to , xO , yO, -u°,kO) ER x RN + J , evefY solution (x(-),y(-).Y( ·), k(-) ) of t h.. initial-val uc problem (3-4) has ma..'X imal int.efval of existence [to ,oo), is bountled , and. is such that the output y(.) asympt.otically t.racks thc signal ,.(.) ill tI,e se,,,e Iha t. t.he error c( t ) = y(i) r (t) a.nd its derivative i'(t ) t fmd, as t -4 00, to zero? Remarks. Universal servonH'chanisms for d a..'5ses of linear syst.pms haw' been consitlered previollsly in the case of reference signals t.hat corn ~spolld to solutions of known lineaT differential equations (e.g., Helmke et al., 1990; Mareels. 1984~ Town1cy & Owens, 1991). In essence, these earlier studies invoke an inter'nul m odel principle. Adaptive tra.cking, wit.h,mt usi ng an iutcrnal moor! and thereby allowing a iar g~ ~r d ass of reference ~ignals, is considf'red in (Ilchrnann & Ryan, 1994 j Ryan, 1992, 1994): tllf' latter result.s difft'r from those of the present. paper insofar as here sys tems of relative degree t wo are <:() n s idef(~ I. 2. 1 Class 1? of t-e/crence. si,q-(wl!:> Th~

cIa.·,;s 'R of reference signals r.ompri:";es those func .. tions r : IR; -----t IFr. of class C'l nL'X) with a.bsolutely continlIOUS second dnivative and ;"' , f, ~i: E L''lO. Thus, may be identified as the (Sobolev I space W ",~ (IJ!.), equipped with thf' lWI'm 11 Ila.= given by

n

2. TH E :-iONLINEAIl TRACKING PROBLEM The abo\'e tineal' class Ll! will now be subsumed by a of single-iIlput (u(t) E II!.) , single-output. large r cl .." (y(t) E R), systems ~ = (~[, L]), encompassing nonlinearities ,md BOIl-allto nomy in thp subsystem 1: 2 :

N,

1:1 : x (/ ) = A , x(t)

E,: Wl

+ "oy (l) .

.rU) E

II!.N - 2

= a !i(t ) + j(/ , y(/ j) + " ".e(/.) + 1",(/) ,

)

k(/ol = k O Ell!.}

usual Ilorm 011 L'" (II!.).

2.2 System cia8s }\,/o

Tht' dass ;\flJ of systems E := (El l L2) is implicit1y defined through Assumptions l-4 below.

The output y(t) E IR is the only state information avail· able for f(·f~l back. TIH' ('OlH'em (.f I.h(, paper is to address a scrvo prohl em (\vit It iutern a.l hOtllldedll p.ss) of CR, ,J\ffj) _ universal tracking; of referenc(' signals 1'(.) of class 1<. (to be spedfil'Ci) , pos{~ d n." follows. Does t here exist 811 adap .. tin~ Oll f,J)1lt fp p.rlhack stra.t.q!,y

k(t) = [{(!(I), yil), 1'(1)),

1111 = dellot.es the

(3)

(.r(/n). y(/,,). )i(/o )) = (.r" , 11" . ,,0) E !RN

u(l) - [ ' (k(t),y(t).r(t))

wheT<'

(4)

1 Univer:;al ~tabiliz('rs ha\'(' Iwell developed (see, e.g. ) IlclunaIlll & Townip.v, 1!)9:3~ LogelllCtIlIl & Zwart , 1991 ; MartellssolL. 1985, 1986; !'diJl cr & Dav isoll) 1991 ; Towllley, 1995) for das ses of lillf'a!" s~:stems of greater generality than ( /j: hOwf·' vpr~ it. is 110' f'vidclll that these fC!suits

can IJe exl.elld~ to i'lH:ompa....s o utpllt, tracking of refer· enee Signals ;:\'(ld systE:'nI llun IiIH'ari t ie.s of thf' gelleralit.y allowed lU'n ~.

a(Ad C C _. . A,.",,,pf.ion 2 (Effective control action): b ¥ O. A,~.'iHmph:(m :1 ( Quantifiab le damping): n < - 6 for SOIne A.'lSlLt7I.l,ti nu

1 ( Minimum phase):

known sc.llaI' J

> O.

Ass""'ption 4 (Regulality of j): j: II!. x IJ!. -t IJ!. is a Cl function and t.here exists a knowIl, eont.inuotls, n011decreasillg function 1> : [0, 01.') -t [0, 00) with t.he property that, for ever.Y R 2:' 0, th ere exist:; (unknown) scalar 'YR ~ () slIch t.hat. for all (t, cL

Ij(t, c +1')1+ ID,j(t, e +')1+ ID,](t, e + r)1 S ')'R ~(le l ) V r E [- R, R)

where

DJ,

i = L 2, denotes the partia1 derivative of with resped. t.o it.s it-It arglUHent.

j

Rema rk.<;. Ry Vola.\" of iIIustr;.ttion , Stlppos(' t.hat j has polynomial dependence (of degree Hot exc~edil1 g p , say)

2085

on the output variable, with t-depeJldent coefficients of

and so, by Assllmption 4 and (8) ,

Cl () L::IO class ha,\'ing bounded deri va.tive , then Assump-

tion 4 holds with

If(t,e )1+ ID,/ (t ,e)1

J, : 1"1>-t 1 + lel" : if pis "nknow n, then

+ 1) + "to )R < "i1 4>( lel) It (t, el

::; (1 + llhR';'( lcl) i · [2(1<>1

'" : lel >-t oxp(lel) suffices. :3. THE

SERVOMECH .~ N 1 SM

Define th~' (;OntillllOllS map (;): [0 . .x)

--t

where / 1 := (i + R hR + [2(1,.. 1+ 1) +'}o]R . Observe (for later use) that , for all (t, X", e, w) E IR X RN ,

(0.00) by

q,(~) :~ 1 + ~ + ~U;I

wh(t, :1,'"

( G)

~ >-t

xl.

::; -f1ll'

l'


(6)

Clai m: tlw follm"ing Htratl~gy i~ a (R., A'J }-universal servome<:hanism ,

= v(l,(t))d>(ly(11 -

u(l)

r(t)l)sgll(y(l) - 1'(1»

7,(1 ) = 6k(t ) + (ly(l) - r(tll)

= 1' (ly(1) -

k(t)

r(lllI ly(t ) -

'0')

+ "12'" + '1, 1I ,",(t1 11' + wf(l, e) (10) and '" := 110311/(2<).

a nd let 4f> denot.(' it.s indefinite intf'gral : [0, ccl --> [0,

C,

1

with" := 2f6' Noting the discontinuous nat ure of the proposeu f~d­ back, the first f'Xpression ill (7) is interpreted in the following set-valued sense: u(1) E <\I (e (I) , k(I)I ,

<\1(1', kl := v(6k

(7)

((Ill

wherE' l ' E 'R. anti v is a ll,Y cUIItiIlllOU:; function !R -+ lR wit.h t.he \'lI~Shl\lIm properties (2),

+ 1>(lel)l4>(l el h~ (e)

with e >-t
,'I. J A nal]j~i.';

r.~t r E .VIi' l' E R and to E IR. be arbitrary. Define constants f., R and function .r /' (.) as follows F:= - ~("

.c, : t>-t

j

where

Z: (t,z)

R:= 11"lh,,~

+ J) > 0,

.,

x Z,(t, z ) x {1>(leI1Iel}

exp( .-l , (t - 81 )a,r(s)ds.

Z3(t,zl:= {h(t , x .. ,', w)

Since u(A , I

c (1:- , la,x, (Il l + la,,:i:, (t)1 <:

'}oR

It I ~ 10

(8)

for some t onst·ant. 10 > O. DeIlote I he output tracking y(t) - 1'(1). Writill!\ mU) t U) + & (1) error by ('(11

=

= r(l ) -

H

{A,x,. + a·,e} < {- 6e+w}

I"

and ,r .. (tj form

= (L.r"e,'W,k)

=

,J: ,- (t}, (:J) may

:;',.(1. 1 = .1, .r,.(I.)

lu~

expressed iu t.lIe

+ ",,<"(t l,

= - 01' (1) + w(l), ,;,(t) = h(t,.I:,(t),c(II,w(l)j +bu(t),

I' (t)

(9)

(:r,.(tol , "' (0 ) , w(lolI = (;rul'll, wo) whr.rr.

It is readily verified that Z3 (and hence Z) is upper semiIR x '\RN + I and takes Ilon· empty, convex and eomJla.ct values in Ill'''' + 1 . T herefore, the initial-value prohlem ( ll) has a solution and every solut iOIl can be extemlei.1 juto a maximal so l1lt.i on z : [to,w ) -4- 1R.N+ 1 (a. function. absolutely cont,illUOWi on compact subintervals of it.s maximal (forward s time) int.erval of existence [to, w), :o;atisfying t.he differeudal iudusion in (11) almost everywh(~rc and wit.h z(to) = ,iJ).

Theorem: Let ee('1 = (x,(-), e(·),w(·),k(·»: [Io,w)--> IltN +1 ue a maximal Hoilltion of (11). Then (i) w = (0,0) as t --> 00. De-fill(> t.he IOl:aHy

Lip ~chit.7.

map

+ 7'(1)) + oj-(I ) - let) + "31',.(1 ) (I, ,,) >-t F (I , t"}: = 1'(lel) - 1'

\'0" , for all (I , d .

0, f(l ,,') = D,;(I. " + 7'(11:1

D,](I ,e + ,·(t))i-(l) + "i'(I) - ,,'(t) + a,±,(t) +

<\I(e,k)}

nmtiullOIlS 011

Proof.

f(t · - I := j et , e

+ Iwl" E

where P= i~

'}l

+ii., with

jj.

= ("

(12)

+'}2 + <+()/6 and (> 0

a. parameter which will pla y its role la ter in t he proof.

Note thl:' following properties of F:-

2086

f(I, Od~

1. F (t ,e) ::> 1~C2 for all (t,")'

for almos t all t E [Io ,w). Int"I)rating ( 16),

2. For each e E IR; F (-, r.) is diff('rc~lltiahlp. with derivative

D, F(t , e) = - 1"D,f(t, Od€

011

~ F(T, efT), w(.)) + "13/.'lI x' (') [["d.<

R\{O} with

+ /' ['I(t :'

= 11
e

# O.

f

+ ()'M Ic(t) I)Htll -- /(t, e(t))w(t) + /'cJ (l e(t)llsg,,(e(l))w(l)

Moreover. since d,) '" 0

0"

11,,· SI'I

1/0. ",,)\E,

= () = wIt)

F(Lc(t)) = F(f ,c (l))

V I E [lo, w)\E

F1lrthermore, for almost all tEE.

.,

r'

/

~ 74[.",1.)11'

for all I, T E [Io,w), with t 2:

o~

'1(t)

1

+ II,[.,U) - '/(T)] + b

(a)

for

a lm o~1 AliI E It(l ~ ;k').

(13)

Abo llot-f' that. for almost all

t E [to ,w). 11",(1) = U(1,(t)ri(t)

V

It

E 'P(!:(t.),c(l))

Define the map , ': (f..e,lI') -+ F(f.,) by propert.y 1 of F,

\ ' (t,e.,,'):O ~11C' + ~It,"

+

(b)

1'1" v(ll)dB iil

>-t - ex::

1 -c-

~",2 for

(14 )

1

(18)

a.o;; 11 '4 ~)O , Witho ut loss of generality, wc ma}' assume tj, ,-rh 2': 1, l'iow, -r/ (' ) is boun, led from below (in particUlar, 1](t) :> oko for alii 2: 10 ) alld so, by the s"pposition , rt(·) is ullbollml~1 from a.bov'~. Therefore, there exist increasing sequenc(,s (t n ) , (s,,) C [t.Q.w) such that.

which,

V (l, e,1O )

(15)

Case I: I, > O.

o ~ \' h

+ lIIaxjwltlblll u E \I>(e(t) , !:(t) ) I (e + O
By (17),

, ,.(" d. 1II(st!) + 1l1'411.>:, (8,) [[ 2

+ l:n.,[k(s, . ) - k(s,)]

hl"lt)) - ,,,,"(t) + "12.'(1) + , ,,lIl ,-ltW + 1/ ,(t)!(I , f.(t))

~ -

-+ + 00

.lilt

Clearly, t >-t \-,(1, 1'( 1), "'(I)) is ails
\:'(I,e(t ).'II'(I)) ~

~ r"" v(i/)dO "h, 11n

+ /lr) (t)

(17)

for all t, T E [t o,w), wit.h I 2: T. It wiJl 110\1,' be shown that the function fI(') (and hence k(·)) is hounded. Seeking a contradiction, suppose that TIC) is Illlbolm
have liO :; 0 on Itu ~w)\E. It may now hp COlldlld(~d that - ",(1)/(1. P(I))

u(8)d8

T}( T)

Wt'

+ ()o(j.>(t)ll ldl)1

Theretore,

+ 1'3 ~.d l x,(T) 1I 2 + 1',1'5 [k(t) - k(T)]

=
FU , e(t) ~ - (1'2 +,

T.

+ ,:,s[k(t) - kiT)]

V (t , e(I),1JI(t)) ~ V(T, e(T) , w(T))

1j(t) = ok(t) + ([e(t) [)sgll(dt)),'(I) and , sincr d,) '" 0 on [to ,,,i l\ £,

v(8)d8

., 11:1,·(·')11'(/8 ~ :41Ix,.IT)[[" + 7, j , e'(.<) d.<

= D , F(t , e(t))+D"F(U(t)) [ - be(l) + w(I) ]

-«., +

l

, {t]

for all I. T E [to,w), with t 2: T. Now, sillce x,(t) = A 1 x,(t) + (I,dl) with o-(A,) C
I:

~

+b

- 'I( T)]

t}{T)

Sincp p,(.) is nhsolnteiy cOlltinuolls OIl compact subin· tervals of (to,:.I.:) and F is locall~' Lipschit~1 the function t H F(t :c(t)) is absolutel,\' cominuous on compact subintervals of [tu. w) and so is diH·(~r~llt.iable almost everywhere with F(T,c(T)) - F (n-, ri,,)) = F(t , e(t))dt for all T," E [to, "-'). DefilH' t :.= slIpp(I<)n[to , w), where su pp(e) denotes the support of thp solution component d ') ' tbat is, the dosure of th e set (t E [to ,w)1eft ) # 01 . Clearly, c(t) # 0 for almost
V(l,e(t) . wit ))

~ 1', (le llle l

3. For each t E liL F(t,, ) is differelltiahle derivative D .,F(t,te)

o~

+ 1'["" - ii,] + b

~"" v(O)d8

Jtil

(19)

for an n. Di viding by Inin 2': b111 ;::: b > 0 , Hoting that. k(s,,) ~ 1111: then passing t.o the limi t 11 -+ 00 and invoking ( 18 b) yiel
Gase 2: IJ < O.

2087

Agaill . hy (17).

0::5 1I (1" e(t ,),1I.'(t, »

+ ,,,TdII ..(t I W

+ 1:0, [k(I .. ) -

+ I'[.i .. - li, l - 1"1 Ibllj"

for a ll n. Dividing by

By boundedness of z(·), it is rea.dily vprified that. there exist s p > 0 s uch t hat, for a ll I 2:

t".

k(t, )]

Iblll,

i,1t

Ihl >

0 , noting that k (t n ) :5 11n. t hf'1l pa,,~ing t.o the limi t n --+ 00 and invoking (18a) yidds t.he requisite clltltradict ion. Therefore , ~(-) (and hence k(·) is bonmhl. Boundedne" of ry( .) and k( · ), wgctlH'r wirh (17). illlply boundeclno,s of V(· ,e(-),1i{». By (15), wc llIay condude that c (·) a nd 111( -) are hound(~d. This establishes a."isertions (i) and (ii). A s~~rtioll (Hi) follO\\'s by lIlonotoIlicily of If(·). It remains to prove assertion (iv). )0

Z( t , z (l» ) C

r"" v(6)diI (~U)

)0

Boundednt:'Hs of thl:' solutioll .:::;(.) l ' n s ur e~ that it ha~ non-e mpty ..,;-Hmit se t n c a N + 1 . Since the solution approach('s it s ....;- lilllit set.. assertioll (iv) i:; proved by showing tha r n is ("omailwd in tlw set

-.f

;1: z - (x, .. e, W, k) ,..., (

(and so,/(t,,) -t

for all n suffici,!ntly large. L'!L n · b('

Set.t.ing (= , - ' (2(

,,(1)1.1 11' as t .... oc

=

=

Suppose!1 0 such (hat A(z) > 2;:. Dy contiuuity of :\ , thf'rf~ exis(.s (~ > {) s uch that.

11 2 - i ll < J

=

.\( zl >, .

:~

(24)

t.hat

+ (J/(3p»)]

whi ch holds for all n > n*. TherefoTt!, by IIlolloroni<:ity and using (21 ) and (22 ), i( follow s th at, for ail n suffici ently large, W(t",.1:,(t,,).e(t,,) ,w(I,,),I,(t,,)

~.

-IT'

.f",

A{ z(t»dl

>

,3 3p

TWO-DI MENSION AL EX AMPLE

Suppose t hat N :::: 2 (in which CagC\ Assumption 1 holds vacuously) _ Let Assumption 3 hold wit.h {, = 1- and Assumpt.ion <1 hold wit.h ~( I ell = 1 + kl". The following ill ustra tive systelll (with llllknow ll parameters b, Co. Cl, C2, Il, and disturhallces fj() , f}1)

(21)

Thus, t .... 11-' (1.,.10,(1),1'(1),11'(1),1,(/) is monotone a nd is bounded (by bOlludcducss of z(·) and F (·,e(·),w(·»)). TherefoI'{~, t here pxists 11- SHell that

=

SUdl

fo r a ll I E [t" , t"

)0

+ 'f:J)' IIPII'lIa,II'.

J :•. (I ). ,·(I),w(/) .

W<

which contradierB inequa li ty (24). Therefore, 0 C nand so (x •.(t ),e(t),w(t» -t (0,0 ,0) a8 1 -t 00 Q.E.D.

1;' (/ , :1:•. (1). " (1).,,,(1).1,(11) S-A( z(I»::5 0

1l'(I.,

whence

Dy (11 ) and (23 ), it follows t.hat.

[11" , 11' + e' + w' ]

+ 'f:d' IIPII' lla, II') ,,'(I )

' (2,

00,

t ,,+(6/3iJ)

IV(t, x.(/), e(I) , ",(I).I, (t )) ::5 - E [.'(1) +",' (1)] + ,,, II:r,{tJlI' - (e 2 (t) + 2(2f + " d ( P:I:. (t ). /1, ;10, (t ) + "2~(t)) ::5 - £ [IIJA/)II" + (" {I) + (""(11] _. fll.c, (I ) 11' - (.'( /) + 2(2< + '):dllPlllIll,lI l1 x.(!lll le(t)1

- A( ,(I)) - [e -,

ry:= Jk + (]ell) os n ....

W(I",l;,(t"),e(t,, ),u:(/,,j,-I,(t,,» -

[I' + bv(8)ldO

llecalling (16), t'Jr a lmost all I )0 10 •

::5

= z( I ,,) -4 Z

(x, (t,,),e(t,,),w(t,, ), k(t,,»

II z( t,,) - i ll < S

W : (t, .10" re, ,/'. '/1 <-+ 1'(1,,,, w) + (2, + ''t3 ) (x" F:e,l

(23)

where Bp is t.he closed ball of radius f1 centred a t 0 E lJtN + l . Since Z = (xe1e, ill, k) is an "1..1 ~ limit point. then~ exists a sequence (t n ) with t '! -+ 00 and

n := It = (O.O, O. k) 1 k E tR) . Since 0(04,) cC . , there exists p~. pT > 0 such t ha t FA., + Ar p + J = O. Denn,'

Bp .

fi(l)

= dil(l) + (eo + 1/0(1 )) ' in (y(t)) + ",y(1)

t r:2Y"( t)

+ Ifl(t) + Im(t),

is admisHihle provided that b:j:. 0, d < - t, qo. qI E (Cl n L~)(ll!.) and qo.q, E U"(tR). Let. the referen,,, signal to be t.r ackf~d hp l' := cos E R. Two partic:lllar syst.em l"ealizatiolls (each unknown 10 t he controiler) are:

4,

= =

(a) d = Co = 2, (" I 0 (;2 : Qo .; 1/ a mi ql == 0 : yielcls a eOIlt,rolJt~d non linear in vert.ed penduluIIl with dampillg a nd disturbed support.:-

(22)

ijU)

2088

=-

tli{t) - (2 -- q(l)) ' in (y( I»)

+ bu(t)

=

=

(b) d = -~, Co = 0, Cl = 1 = -c" qu 0 and q, q, yields a controlled equation of Duffing type with extraIleous disturbancc:y(t) = -hi(t)

y(t) - y"(t)

T

+ q(t) + bu(t)

In each case, it is supposed that. the disturbance q(.) coincides with the first component. :r:] (.) of the solution of the init.ial-problclTl for the Lo}'(~nl' equations: .il (t) ~. 10[x2(1) i,,(t)~.

.i,(t)

(t)],

,CI

"I

(0)

00

1

28xtll) - :1:2(1) - lOXI(I)1',,(I), X2(0) = 0

= lOx,(I)"'2(1)

- 8x,,(I)/3, ",,,(0)

=3

This system exhibits chaotic: lwhavionr: the solution component Xl (.) = q(.) is depicted in Figure 1.

40

time

Fig. 1: Disturbance q(.) In each of cases (a) alld (b), the I'Qlltrolled evolution of outpllt" for initial data. (!f(O)' y(O), k(O)) = (0,0,0), is depicted in Figure 2 \vhereiu the convergence to the reference signalr : t H cas(t) is evideut.

:~\ i0tr~"'~J\I

-'['V

w

~

Iv

\)

:~Nhl\t\ /~7\~»)/\ r\JV \J

2

o

-

V

j

'v

---'-

10

--'--------'----

15

20

25

30

35

40

time

50

Fig. 2: Controlled Olltput evolution y(.)

ftEFEREI\CES Byrnes, C.1. and .l.C. Willellls (1984). Adaptive stabilizatioIl of ruultiva.riable lillPar systems. In: Proc. 23rd IEEE Con!tTt:no; on Dcci.':lioH f!i Control, pp.1574-1577, IEEE, York. Corless, \1. and E.P. Ryan (199.3). Adaptive control of a class of Il()nliJl(~arly Iwrt.urhC'd linea.r systems of

""W

I,

I

-2'------"---~-- ~- - ----'------------'---- -----'----- ----'----------~ o 5 10 15 20 2~ 3(J 35 40 tintf' !j(l

-) V U

relative degree two: SY8r.(';ms f!<} Control Letters, 21, 59-64. Helmke, U., D. Priitzel-Woltcrs and S. Schmidt (1990). Adaptive tracking for ~calar minimum phase systems. In: Control of Uw'erto,in SYBtemB (D. Hinrichsen & B. !vl
2089