Adaptive λ-tracking for polynomial minimum phase

Ilchmann, Achim :

Adaptive λ-[lambda]-tracking for polynomial minimum phase systems

Zuerst erschienen in: Dynamics and Stability of Systems 13 (1998), Nr.4, S.341-371

D)'rtanticsand Stability of Systcnts,Vol. 13, No. 4, 1998

341

Adaptive 2-tracking for polynomial minimum phasesystems*

Achirn Ilchrnann School of Mathenmtical Sciences,Uniaersity of Exet.er,North Park Road, Exeter EX4 4QE, UK (RcceivedNoaember1997; Jinal uersion.fune 1998)

Abstract.

It is proaed that the adaptit:e contoller u(t) :

'c(t), - k(t) llc(t) jl'

it t)', tro: [t tt"ctl

lo,

c(t) :: y(t) - y..,(0 - n(t)

lle(0ll> ?. l e ( 0 l l< I

uhen applied to certain classesof muhiaariable nonlinear systems,achieves)"-tracking, i.e. the trajectoriesof the closed-loop systemare bowtded and lle(t) l] - [0,].1 as t+ co. )'">0 and r)s are desigrtparameters,and s) I * some(known) upper bound of the po[ntontial degreeo.f tlze right-hand side of the plant. The crucial assuntptionson the system classesare ntinittum phase and stong relative degree-one.Classesencontpass systerns with sectorboundedinput nonlinearities,systcmsin input affineform, and systems not in input ffine lornt but zpitlzbotntdedtrajectories.The referencesignalsy,"1() and noisesignalsn(') are only assumedto be absolutelycontinuouson boundedinteraalsand bounded with essentially bounded deriaatizrc. lYe also intoduce ntodifications of the feedback strategy which preserztethe sintplicity of the controller but improae the tansient behaaiour.

Nornenclature [R, ([R ) thc set of non-negative (non-positive) real numbers flR1 the set of positive real numbers C, (C ) open right- (left-) half complex plane o(A) thc spcctrum of the matrixAeC"'n I xll n JQ,Px) for xe[R",P: Pt-e[t""" positive-definite

l l x l L : Ix l l ' . ,

E-rnail: ilchmann(rlmaths. ex.ac.uk. *Supported in part by thc Univcrsity of Ilxctcr Ilcscarch Fund. 0268-l I 101981040341-31

O 1998 Carfax Publishine Ltd

342

A. ILCHMANN

9 4 ' ( 0 ) : { e e [ R " 'tllll > ) l f o r m e N , r . > 0 tr(1,[R) the vector spaceof measurablcfunctionsf; I- [t',, 1c [R an interval, suchthat lf(.) l,,,rn< co, wherc

i/(.) irr.,o I[ i

't')ll'o'-l

l.",,.rrp t,.rG) | 7//"'

for pell, n) for p: cr:

the Sobolev spacc of functions /:[Rr -[t,,, which are absolutcly conrinuous on compact intcrvals and /(.), jO u f , ([R' R-)

For ), p > 0 and positive-definite P: P'e R. ' ',,we introducc. for e e llt,,,.thc distancc functions

d , ( e ) : { J , " * ' , l l c ) l I> l l e l ) ll and r is large, then the gain ä(r) increasesfast and hence tracking is achieved faster. This is illustrated in Section 7. \Wc would like to finalize this introduction by illustrating the concept of highgain 2-tracking for the simple cxample of scalar nonlincar systems of rhe form: x(t) : f(x(t)) + g(x(t))u(t),

y(t) : cx(t),

x(0) : 1,,u pX

(4)

whcre the only structural assumptions bcing made arc: ce R and the continuous functions /(.) and g(.) satisfy, for all r e R there cxists some (unknown) o > 0 such tbat o < cg(x) there exists some (unknown) polynomial p(.) such rhat lf(x)l < p(xl)

(5) (6)

only thc upper bound s ) dcgp(') needs to bc known. we want to show that thc time-varying and nonlinear feedback law u(t) : * k(r) lle(t)ll' 'e(r),

e(t): y(r) -y.,(r)

(7)

with the simple gain adaptation (3) for r)s achieves/.-tracking (i.e. le(Dl__+[0,,i] as t+oe) for arbitrary initial conditions Ao,xoeR and refcrencc signals -!."r(.),as long as y."r(.) and y."r(.) are essentiallybounded. I-ct [0, ro) denote the maximal interval of existence of thc solution of the closedloop system (4), (7), (3) for some ür€ (0, m l. \üe first prove k(') e L" ( [0, ar), [R). Seeking a conrradiction, suppose lim,-,,, A(r) : co. (Note rhat tr-+k(t) is monotonically non-decreasing by construction-) Differentiation of the Lyapunov like candidate v;(e)::Ld,(e), (seeNomenclature) along the solurion of the closed-loop system (4), (7), (3) yields, for almosr all re [0, ro)

!,-v,1r1r11 ,e: d,(r)3t f@) , kcg(x)lel, !,*l or lel { - kod;(e)lel' + d, (e)lcf(x)* j,,*l

(8)

ADAPTIVE2-TRACKING

345

By (6) and boundedness of 1,"1 änd y,"1, there exist some M > O such that lcf(x) -j)*,1-A l l e ( t ) 0 there cxists some l'> 0 so that, for almost all re [0,ro) l l y ( r ) l l< L

+

lljt(t)ll i rr,iG),E(t)) d -..

iual

:. : fr1t, 21t11 + h(t,t(r),z(t))

where j(t, i, Z) i : f(t, t I !", 2 * z ") * f(t, y", z") * [G(r, g * y., E * e") - G(t, y", z ")lu. GG,i, E) :: G(t,j * y", E * z") ü(u);: Y- u.

frQ,fl::h(t,y",z*2") h!,i, 2) :: h(t,j * y",Z+ e")* h(t,!",E* z") This follows easily by rearranging (17) and using the fact that (y",e",2") is an equilibrium point. (Al), (A2) and (A4) yield that /1r,0,0; :9,

ö1r,0,0):G(t,y",2"),

fre,01:h(t,O,E) :0,

and

ü ( u " ): g

whence (0,0,0) is an equilibrium point. Since (a-tb)q 0 and for all (r, (, 4) e Rr x R", x Rt

-l njl,(,Lr) lr+ llGG,(,,r) ll 0 is arbitrary. Before we proceed in severalsteps, we record, also for later use, the following facts: Ifp,::

llP

'll ''',p.i:llPll''',

t h e n p , l l e l l< l l e l l u 4 p r l l e l lf o r a l l e e R " ' ( 2 6 )

lf D,,(e) > o, thent .

r

ll,11,,