Asymptotic observers for a class of evolution equations. A nonlinear approach

C. R. Acad. Automatique

Sci. Paris, t. 324, thkorique/Automation

Asymptotic equations. Philippe

and

dr Rourn,

1997

for a class of evolution approach

Jean-Franqois

LMI,

76131 Mont-Saint-,&pan [email protected]

J.-F. C. : Univrrsitb Ilr du Sad-y, 5404.5 E-mail: coucht,urQloria.fr

I, p. 355-360, (theoretical)

observers A nonlinear

LIGARIUS

Ph. I,. : INSA BP 8, E-mail:

SCrie

URA CEDEX,

dr Mrtz. INRIA Metz, Franw.

COUCHOURON

CNRS

137X,

Franc-r: Lorrainr

Projet

CONGE,

We consider in a Hilbert space H the system (E,,) = {.i: = 74~43.+B(x) : !/ = (:I:. c)[, }, where the control ‘(1, E L”([O. +cu[, Rf) multiplies a possibly unbounded 711dissipative linear operator A. The operator I3 is nonlinear dissipative, and y stands for the output of the system. We prove, in this nonlinear framework, the existence of a suitable Luenberger-like observer. For this purpose, we show that the usual notions of regularly persistent inputs proposed in [7] or [4] are the appropriate concepts that allow one to generalize the main results of [9] and [S] or [7] for bilinear systems to our nonlinear general system: For each regularly persistent input, the estimation error of the observer converges weakly to zero. If in addition A generates a compact semigroup, the estimation error converges strongly to zero. A prototype of such a system is the heat exchanger system described in [9] or 181.

Observateurs d’&olution. Une approche RCsumC.

asymptotiques non

pour

une classe

d’&quations

lin&aire

On consid&e duns un espace de Hilhert H. le syt&ne (E,, ) = { :f = ,uA:I. + B( :I:) ; g = (.I:. f,),, }, oi le contrcile 11.E L” ([0, +i70[, W) agit sur un ope’ruteur, Pventuellement non hot& A linkire et m-dissiputij’: L’oppPrateur B est non 1inPaire dissiputiji et ~1 dkigne 1u sortie du .syst&v~e. Nous prouvons, duns ce cadre non linkaire, l’rxistence d’un ohsrrvateur de &pe Luenher:yer. Pour celu, nou.s montrons que les notions usuelles d’entrkes r&uliPrement persistantes, proposkes dans [4] ou 171, sent les concepts approprils qui permettent de g&Pruliser les principuux rtCsultats de [9] et [8] ou [7] pour les syt&nes hiline’uires Li nos syst2mes non linkuires : Pour chuque entr& rPguliPrernmt persistunte, I’erreur d’estimution de l’observateur converge jiriblement vers :Pro. Si de plus A engendre un semi-groupe compact, 1 ‘erreur d’estimation cornverge ,fiwterncnt vers :kro. L’rxemple type de ces syst2mes est l’e’changeur thermique dkrit duns 191 014 [Xl.

Note pr&entCe 0764.4442/97/03240355

par Roland

GLOWINSKI.

0 AcadCmie

des Sciences/Elsevier.

Paris

355

P. Ligarius

and

J.-F.

Version

franqaise

Couchouron

abr&gke

1. PrCsentation Dam cette Note, nous considerons le systeme suivant, dam un espace de Hilbert H:

E,,(.I.(‘) = E,,=

i(t)

=

‘?J.(f)h(f)

y/(t) = (x(t). c:)ff.

+

n(:I:(t))

+

h,,(t),

.r(O) = .I.‘) E H t E [o. +x[

oh le controle ‘11E L” ([O, ,x)[; W+), I’operateur lineaire A est m-dissipatif, l’operateur I3 est non lintaire dissipatif et IV,qui est une fonction de II dans I? (11est le nombre de capteurs du systeme, avec, p = 1 par souci de simplicite), represente la sortie du systeme; c E H est le vecteur d’observation. Rappelons qu’en dimension finie, pour les systttmes non lineaires et deja pour les systemes bilineaires (A et B lineaires dam E,,), I’une des difficultes majeures de la synthese d’observateurs est I’existence de (( mauvaises s entrees (ou entrees non universelles au sens de Sussman [IO]) qui rendent le systeme inobservable. L’un des outils usuels utilise pour les systcmes bilintaires, est le Grammien d’observabilite qui donne, grossierement, une c( mesure d’observabilite )) du systeme. Lorsque A et B sont des operateurs lineaires et lorsque la sortie j/ = C’(X) a valeurs dam un espace de Hilbert est lineaire non compacte, le Grammien d’observabilite peut-2tre coercif et de nombreuses solutions sont alors possibles (v.g. l’observateur de Kalman [3]). Mais notre contexte presente une double difficult& a savoir d’une part la non-lintarite de B qui rend caduc le concept de Grammien, et d’autre part, la compacite des sorties qui correspond a la realite physique des applications (en fait, le nombre de capteurs doit &tre fini). Nous montrons dans cette Note que les notions usuelles d’entrees universelles [lo], d’entrees regulierement persistantes 171, et les inegalitts integrales de Benilan adaptees a notre contexte, sont suffisantes pour resoudre le probleme non lineaire d’observation pour la classe des systemes E,, ci-dessus. Nos resultats etendent ceux concernant le probleme de I’observabilite connus dans le cas des systkmes bilineaires (wir- ]9] ou IS). wi/- encore ]7] ou ] 11)). Des applications aux phenomenes de transfert thermique ayant motive ce travail sont presentees dans 191, [6] ou [S].

2. Hypothkses Nous supposerons que : (i) L’operateur A a domaine dense est nr-dissipatif. Le controle ‘U appartient a Li”( [O?x[; W+). (ii) L’operateur non lineaire dissipatif B defini sur H est : LI) Condensant continu, i.e. B est continu et 3C > 0, constante telle que ‘d&?borne de H, la relation x(B(ft)) _< Cx(12) est realisee. oti ?( est la mesure de non compacite de Hausdorff; h) Faiblement continu de II vers II. (iii) L’application f ++ b,(t) = [I(U)(~) appartient a L”([O, c0[; H) et pour tout 1’ > 0, pour tout 9 E H. si la suite (u,,) converge vers 76, dans la topologie faible” de L”( [0, T]; Iw), alors (!I,,,, .:q) converge vers (6,, x . y/) dam la topologie faible* de L”( [O, T]; !R). (iv) La solution .I’ de &(:r:“) consideree ici est bornee sur [0, x[. Nous proposons un observateur simple vtrihant, avec 7’ > 0, l’equation (2). On appelle erreur d’estimation (ou erreur d’observation) -c = .k - .I:, avec .i~solution de I’equation de I’observateur (2) et :I: solution du systeme E,,, et E verifiant I’equation (3). Precisons par ailleurs la notation suivante : la fonction u?;] (.) = ,f/.(r + .) represente la r-translatee de TJ sur [O, T].

356

Asymptotic

DEFINITION

I.

-

observers

Une en&e ‘(L, E L”([tl,T]:

(2 # ,?I)

=+

for

a class

of evolution

equations.

A nonlinear

approach

If%+) est dite universelle sur [O. Z’] si,

((qz()).c)

# (sy+)).

oti ST(.r:“) represente la restriction sur [O. T] de la solution de E,,_ (.r:“). DEFINITION 2. - Une entree 11,E L”([O. x[; R+) est dite r6gulii3rement persistante s’il existe T > 0 et une suite croissante (T,,),, de rCels positifs. Gritiant : (i) lini TV, = +w avec (7,,+l - 7,,),, bornCe, R+), (ii) UT -2 Il., E L”([O.T]; ir-1 r,--toc (iii) TL, est universelle sur [O. ?‘I.

3. Les rthltats Nous avons alors obtenu le r&hat

suivant :

THI~ORBME 1. - Pour une entree r6guMrement persistante ‘~1E L” ([O. x[; W+), I ‘erreur d’estimation c(t) converge faihlement vers ztro quund 1 tend vers +w. Ce thCor2me a pour consequences : COROLLAIRE 1. - Si pour UIZ certain T > 0. toure entrke u E L”( [0, xs[: R+) esr universelle. ulors, pour route entrPe 71.E L”( [O. ‘x[; R+), 1 ‘erreur d’observution &(t) converge faiblement vcrs :Pro quand t tend vers 1‘iyfini. THI%RI?ME 2. - Soit une entrke rc;KuliPremetlt persistante IL E L”( [0, xc[: R+). Si en outre .4 engendre un semi-groupe compact et s ‘il existe ‘lo > 0. telle que 1‘on ait : ,u,(t ) > ‘r/o,pour presque tolct 1: E [01 +co[, alors I’erreur d’observatiorz E(L) converge fortement vers :e’ro quarzd t tend vers SK.

Introduction In tinite dimension, for nonlinear systems and already for bilinear systems, one of the major difficulties for the synthesis of observers, is the existence of “bad” inputs (or non universal inputs, in the sense of Sussman [IO]) which make the system unobservable. For finite dimensional linear systems, the much simpler solution of the observer problem is given by the Luenberger observer. From the point of view of observability, bilinear systems are nothing but linear time dependent systems. For linear time dependent systems, Kalman’s observer provides a solution. One can easily show, for linear time dependent systems that are dissipative for all values of t > 0, that the Luenberger observer also works. It is this last result that we will generalize in infinite dimension. even into the care of some nonlinear cases. In infinite dimension, the observer problem has already been examined from this point of view for a few years, in the case of skew adjoint or more general dissipative bilinear systems (see [7]? [I 11, 191 or IS]), for which the authors exhibit the observer used in this Note. The engineering problem of the heat exchanger process (see [8] or [9]) motivates our study of the following nonlinear system E,, in a Hilbert space H (with its inner product written (., .),) :

E&T”)

= Eu =

i:(t) = u(t)A:r:(t) + L?(:r:(t)) + b,,(i). ;//l(k) = (;r:(t). (gH.

r(O) = :I:” E H t E [o: +cc[

357

P. Ligarius

and

J.-F.

Couchouron

where the control ,/i(t) E Lm([O, s[; R8+) multiplies a possibly unbounded linear m-dissipative operator .4. The operator 13 is nonlinear dissipative and y, which is a function from H to FP’ (with 1~the number of sensors of the system, which we set to I here for the sake of simplicity), stands for the output of the system. c E 1f being the “observation vector”. The assumptions on A, B3.b,, are stated precisely in the next section. Let us point out some fundamental aspects of our observation problem. Recall first, that the Gramobservability operator is usually used for bilinear systems (i.e. A, B linear in E,,) to give, roughly speaking. a “measure of observability” of the system. Second, observe that the output y of E,, is R-valued. which physically makes sense. In the bilinear case for instance, when the observation ,y = C’(J) has values in an infinite dimensional Hilbert space, and when the Gram-observability operator is not compact. it may be coercive. Therefore, as in finite dimension. many solutions are possible and the observation problem can be solved by an infinite dimensional Kalman’s observer for instance (XC 131). In our context, there are two main difficulties : the nonlinearity of L? which excludes the use of the Gram-observability operator, and the compactness of the outputs (in fact, only a finite number of observations physically makes sense). We prove in this Note that the classical notions of “universal inputs” [IO] and “regularly persistent inputs” (171, [41). and adapted Benilan’s inequalities are the appropriate concepts that allow one to give a IVJ:\’ .si~l/Jr solution, of Luenberger type, of the observer problem for the above class E,, of nonlinear evolution systems. The results presented here, extend to our nonlinear framework those given in 181. 191 or similar ones in [ 1 I] or 171 for bilinear systems.

Definitions,

Notations

and Assumptions

Our assumptions on 21, 13. h,, are: (i) The operator .1 is a densely defined In-dissipative operator. The control IL(~) belongs to L’([O. n[; R+). (ii) The nonlinear dissipative operator U has domain H and is : (I) continuous condensing, i.e. B is continuous and there exists a constant 6’ > 0, s.t. for all bounded subset It of Il. the relation x(U(0)) 5 C,‘X(~) holds, with y the Hausdorff non compactness measure, and h) weakly-weakly continuous from H to H. (iii) We assume that the map f H /j,,(t) = b(~)(t) belongs to L”( [0, m[; H) and that for all T > 0: and all !/ E H. if the sequence (u,~ ) converges towards I/,, in the weak* topology of L”([O, T]; W), then (C,,, , !I) converges towards (6,, ~ , y) in the weak* topology of L”( [O,T]; R). (iv) The solution .I’ of E,,(.r”) is bounded on [O, a[. In this paper, all the solutions of the evolution equations are mild solutions (see [ 11). More precisely, if we denote by 4):( I, .s) = ( r’(‘,‘)-r. with r~(t. s) = I1’1 the evolution operator associated with h MU&, //(/),i. the solution 01‘ E,, (:l.“,) for instance is given by the classical Duhamel formula,

(1)

.r(t) = (P;l’(t,s).r” +

.t I .,

‘D;;(s,r)[B(.r:(r)) + b,,(T)]fh

Keimrk I I. If 13 is a bounded linear operator, the assumptions in (ii) are automatically satisfied. 2. If dim H < +oc. the assumptions in (ii) are obvious for a continuous operator B. .?. Assumption (iii) is satisfied if ‘II H b,, is sequentially continuous from ,C-( [0, T]; R) to L”( [O, 7’1: H) en d owed with their respective weak* topologies.

358

Asymptotic

observers

for

a class

of evolution

equations.

A nonlinear

approach

4. Clearly, if there exists 2” E H such that the solution of E,(z”) is bounded, then for all J:” E H the solutions of E,,(n:O) are bounded. The equation of our “candidate” Luenberger-like observer is, with r > 0 : F

(2)

1,

=

.&(t) = v(t)At(t) i(0)

and the observer error I

+ B(?(t))

+ b,(t) - Y( (i(t), C) - ;y(t))~

= :F:OE H = Y(t) - s,:(t) satisfies the following

equation :

Er = i(f) = 4t)A~(f) + [W(f)) - B(d(t) - c(t))] - 7.(E(t),(.)I:

(3)

E(O) = E’ E H

We set qIl,,3(t,.s) the evolution operator associated with < H u(t)A< + l?@(t)) - U(z(t) - 0, where z is a continuous function from [O: +m[ to H. Basic properties of a;: and Q!,,, are precisely given in [6]. For II E La([O, oo[; W+) and 7 > 0, we set I$,(.) = ~(7 + .), the r-trunslated input function on [0, T]. Let us recall the classical notions of suitable inputs, namely universal and regularly persistent inputs: DEFINITION 1. - An input 11, E L”(

(20 # 2)

[O. T]; W+) IS said to be universal on [(I. T] if, *

((ST(z”),

c) # (S’(zl).

c)).

where, ST(zo) denotes the restriction on [O. T] of the mild solution of E,,_ (:I:“). Another equivalent definition of universal inputs can be given as follows. 1. - An input u, E L”( [O,T]; R+) is universal on [0, T] if and only $ .for all mild solution t E [O: T]) of E,,_ (z”)> we have the following implication, .for all E’ E H:

LEMMA

(z(t),

(Vt E [O:T], (P,,_,&~~)E’),c)

= 0)

=+

(E” = 0)

DEFINITION 2. - An input 11E L”( [O. rx;[; Rt) is said to be regularly persistent if there exist T > 0 and a strictly increasing sequence (TV)n of real positive numbers, satisfying (i) lim 7,L = SOC; with (r,+r - 7,L)ll bounded, (ii) u;,,] nyX ‘u, E L”([O, T]; B;p+), (iii) U, is universal on [I). T]. Regularly persistent inputs are sufficiently rich to guarantee an “asymptotic estimation” of the state of the system, as we shall see. For instance, a periodic input with period T, which is universal on [0, T], is regularly persistent.

Main

Results

We can state now the main results of this Note: 1. - For a regularly persistent input ?I. E L”( [O, nc[; W+), the estimation error e(t) converges weakly to zero in H as t goes to +x. As a consequence of theorem 1. we can get further results with universal inputs or with additional assumptions on the system: THEOREM

COROLLARY 1. - If evev positive u E L”( [0, KI[: R+) is a universal input for some jxed T > 0, then the observer error c(f) converges weakly to zero in H (HIhen t goes to +x8) for any positive 11,E L”([O, ca[; I%+).

359

P. Ligarius

and

J.-F.

Couchouron

THEOREM 2. - ij’irz uddition. we msum that A ,generate.s u compact semigroup and if there exists ‘rjrlo > 0. .YUC.II that the ,fi)IIoMiq estimation holds: rl[j (a.e.), t E [O, +w[, where ‘u. is a regular!\. persistent input OH [(I. + x[. then the ohserver error ~(1) conveyqcs strongly to zero in 1-I (1.S t g0c.r to +‘x’.

Sketch of’pro$ - The proof of theorem I is based up on the three following main ingredients : (u) the weak uniform continuity of the estimation error C; (h) a theorem of continuity with respect to the controls II endowed with the weak* topology : (c) Benilan’s inequalities (st’e 121, [6]). Given a regularly persistent input II E L”([O. XN[: iw’.). we begin with a sequence (T,!) T +cx;, associated with II (,\,ee definition 2). Then ( 1). u[;,,] is weakly” convcrgcnt towards a universal II,with II K t L’“‘([O, 7’1: W). and for all ,z E II. CD,,,(t, 0)~ converges uniformly in t E [0, T] towards CD,,I (1.0); : (2) thanks to a weak Ascoli-ArzelB theorem, we show that there exists a subsequence on [O! T] with 2, ( T,,~ ) verifying : (i) ,?;I,, , is weakly pointwise convergent towards some &

solution of f:‘,,, (:PI(0))

1 (ii) EF“i I is weakly pointwise convergent towards some E,, on [0,7’] ,

(iii) Vi E [O. T], we have (Q(, \,, . (~.O)E,,,(()), C) = 0 . Then, since ‘II, is universal, WC obtain, t ‘i z 0 on [0,7’]. Thus the unique cluster value of $iVzl

is the null function on [0, T]. And it remains to prove that we have E(t,,) 2 t,, -

4-x.

This

last step is achieved

by using

0 for any sequence

II + 7u of the sequence (T,,+~ - T,,)~). We

the boundedness

refer to [ 61 for complete proofs. (‘) Ici 2

signifie, convergence danh la lopolugie laible’ It is a pleasure to acknowledge preparation of this work

Acknowlegment.

discussions during Note

ret&e

le

sur LY ([II.T];

Pr. J.-P. Gauthier

Wj

for his valuable

suggestions

and helpful

semi-lineaire.

C. R. Acud.

the

15 juin

1996.

acceptCe

le 22

novembre

IY96

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