1. Introduction - CORE

ESAIM: Control, Optimisation and Calculus of Variations URL: http://www.emath.fr/cocv/

May 1998, Vol. 3, 133{162

ADAPTIVE PARAMETER ESTIMATION OF HYPERBOLIC DISTRIBUTED PARAMETER SYSTEMS: NON-SYMMETRIC DAMPING AND SLOWLY TIME VARYING SYSTEMS H.T. BANKS AND M.A. DEMETRIOU

Abstract. In this paper a model reference-based adaptive parameter

estimator for a wide class of hyperbolic distributed parameter systems is considered. The proposed state and parameter estimator can handle hyperbolic systems in which the damping sesquilinear form may not be symmetric (or even present) and a modication to the standard adaptive law is introduced to account for this lack of symmetry (or absence) in the damping form. In addition, the proposed scheme is modied for systems in which the input operator, bounded or unbounded, is also unknown. Parameters that are slowly time varying are also considered in this scheme via an extension of nite dimensional results. Using a Lyapunov type argument, state convergence is established and with the additional assumption of persistence of excitation, parameter convergence is shown. An approximation theory necessary for numerical implementation is established and numerical results are presented to demonstrate the applicability of the above parameter estimators.

Introduction The adaptive parameter estimation of second order distributed parameter systems was initially studied by Demetriou and Rosen, 15], written in a second order setting and by Scondo 25], Demetriou 14] and Baumeister et al. 12] written as a rst order system having strong damping. The scheme presented there did not account for systems with non-symmetric damping bilinear forms. Such a system, which motivated the current work, was observed in the adaptive estimation of structural acoustic models 4] where lack of symmetry in the damping form was observed and thus the scheme presented in 15] was no longer applicable. A modi cation to the adaptive scheme was therefore required in order to account for the lack of such a symmetry. In this note we propose various modi cations to account for this (lack of symmetry) and therefore rendering the scheme implementable. In addition, 1.

H.T. Banks: Center for Research in Scientic Computation, North Carolina State University, Raleigh, NC 27695-8205, USA. E-mail: [email protected]. M.A. Demetriou: Department of Mechanical Engineering, Worcester Polytechnic Institute, Worcester, MA 01609, USA. E-mail: [email protected]. Research supported in part by the Air Force Oce of Scientic Research under grant AFOSR F49620-95-1-0236 and in part by NASA under grant NAG-1-1600. Received by the journal December 7, 1996. Revised November 6, 1997. Accepted for publication February 19, 1998. c Societe de Mathematiques Appliquees et Industrielles. Typeset by LATEX.

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H.T. BANKS AND M.A. DEMETRIOU

if it is known that the parameters are (slowly) time varying, then a state and parameter estimator is proposed to estimate these time varying parameters. Speci cally, if it is known that the (unknown) parameters converge to a steady state value, then with the additional assumption of persistence of excitation, the scheme can guarantee that the parameter estimates converge to the steady state value of the system's parameters. Another issue that arises in many hyperbolic distributed parameter systems is when the system has no damping at all. While the \no-damping" case does not occur physically, it is often the model used for systems with very light and/or poorly understood damping. This case is also treated here, and the persistence of excitation condition, required for parameter convergence, is modi ed in order to account for the absence of the damping term. It is worth mentioning that even though the system under consideration does not have a damping term, the proposed estimator requires one in order to guarantee convergence. Furthermore, this estimator appears to be more general than the one presented in 14] (systems with strong damping) or in 25] (systems with no damping), because the estimator operators here are not simply chosen as the plant's operators evaluated at some optimal parameter, but rather some other general and not necessarily parameterized operators. In all cases above, unlike 15] and 25], the parameters in the input operator are assumed to be unknown and therefore the adaptive schemes can identify these parameters in the input operator. In addition, these schemes can handle the case where the input operator is unbounded. A similar result was established in 14] for the case with symmetric damping form. The issue of unknown parameters in the source term is of great importance in the control of exible structures and especially when control is implemented via smart actuators. In this case, parameters related to the geometric and physical properties of these smart actuators are usually known to within a percentage of the actual value and often vary (slowly) with time. These uctuations exhibit destabilizing eects in controlling these structures with the end result of poor, if not inadequate, control performance. The following section introduces the underlying spaces that are needed to analyze the system and its estimator and gives conditions imposed on the system that are required for parameter convergence. This essentially de nes the class of systems for which the adaptive parameter estimation schemes are applicable. Section 3 provides a summary of the stability and convergence arguments for such an estimator. The adaptive parameter estimation scheme for systems with (slowly) time varying parameters is presented in Section 4. Since the proposed state and parameter estimators are in nite dimensional, an approximation theory, which will be used for implementation purposes, is developed in Section 5. Section 6, which includes examples and numerical simulations of the structural acoustic system 4], is added to demonstrate the applicability of the above theoretical results while Section 7 summarizes the results and concludes with further open problems in the area of adaptive parameter identi cation of distributed parameter systems.

ESAIM: Cocv, May 1998, Vol. 3, 133{162

ADAPTIVE PARAMETER ESTIMATION OF HYPERBOLIC SYSTEMS

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The Plant and the Estimator Let H be a real Hilbert space with inner product h i and corresponding induced norm j j, and let V1 and V2 be real reexive Banach spaces with norms denoted by k kV1 and k kV2 , respectively. We assume that V1 is embedded densely and continuously in V2 and that V2 is embedded densely and continuously in H . It then follows that (see, for example, 6, 10]) V1 ,! V2 ,! H (2.1) = H ,! V2 ,! V1 where H , V2 , and V1 denote the conjugate duals of H , V2, and V1, respectively. Since the embeddings in (2.1) are dense and continuous, 21, 26, 27, 28], we assume that there exist positive constants, KV2 , KV1 , and KV2V1 , such that j'j KV2 k'kV2 , ' 2 V2, that j'j KV1 k'kV1 , ' 2 V1, and that k'kV2 KV2V1 k'kV1 , ' 2 V1. We denote the usual operator norms on V2 and V1 by k kV2 and k kV1 , respectively. The duality pairing, denoted by h iV1 V1 , is the extension by continuity of the inner product h i from H V1 to V1 V1 hence, elements ' 2 V1 have the representation ' (') = h' 'iV1 V1 , see 7, 10]. As it was pointed out in 4, 6, 7, 15], V2 can either be V1, H or some intermediate space, depending on the damping form chosen. Let Q be a real Hilbert space (henceforth the parameter space) with inner product h iQ and corresponding induced norm jjQ, and for each q 2 Q let 1(q ) : V1 V1 ! R 1 be a bilinear form on V1 satisfying (A1) (Symmetry) 1 (q ' ) = 1 (q '), ' 2 V1 , and q 2 Q, for at least one tuning parameter q 2 Q, we have (A2) (Boundedness) j1(q ' )j 0(q )k'kV1 kkV1 ' 2 V1 with 0(q ) > 0 (A3) (Coercivity) 1(q ' ') 0(q )k'k2V1 ' 2 V1 with 0(q ) > 0 (A4) (Linearity) the map q ! 1 (q ' ) from Q into R 1 is linear for each ' 2 V1. Similarly, for each q 2 Q, let 2 (q ) : V2 V2 ! R 1 be a bilinear form on V2 satisfying (B 1) (Symmetry) 2 (q ' ) = 2 (q '), ' 2 V2, (B 2) (Boundedness) j2(q ' )j 0(q )k'kV2kkV2 ' 2 V2 (B 3) (Coercivity) 2 (q ' ') 0 (q )k'k2V2 , ' 2 V2, (B 4) (Linearity) The map q ! 2 (q ' ) from Q into R 1 is linear for each ' 2 V2, where 0 (q ), 0 (q ) > 0, and q is the same xed element of Q appearing in Assumptions (A2) and (A3) above. For each q 2 Q, let b(q ) : H V1 ! R 1 be a bilinear form satisfying (C 1) (Boundedness) jb(q # )j jqjQj#jkkV1 # 2 H 2 V1 2.


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(C 2) (Linearity) The map q ! b(q # ) from Q into R 1 is linear for # 2 H and 2 V1. For q 2 Q de ne the linear operators A1(q ) and A2 (q ) from V1 into V1 and V2 into V2 by hA1(q)' i = 1(q ' ) ' 2 V1

hA2(q)' i = 2(q ' )

' 2 V2: In addition, we de ne the input operator B (q ) : H ! V1 by hB(q)# iV1 V = b(q # ) # 2 H 2 V1: For q 2 Q we consider the second order linear initial value problem given

in variational form by hztt(t) 'iV1 V1 + 2(q zt(t) ') + 1(q z(t) ') = b(q f (t) ') (2.2) ' 2 V1 t > 0

z(0) = z0

zt(0) = z1 (2.3) where zt denotes time dierentiation, z0 2 V1, z1 2 H , and f 2 L2(0 1 H ). The well posedness of the system (2.2), (2.3) with symmetric 2 was established in 6] using linear semigroup theory. The case where 2 is not

symmetric was treated in 5, 8] for structural acoustic interaction models. The identi cation objective is to design a state and parameter estimator using the plant state (z (t) zt(t)) and the plant input f (t) in order to identify the unknown plant parameter q adaptively. We now make the assumption of an admissible plant as it was de ned in 15], namely a boundedness condition on the state of the system. Definition 2.1. An admissible plant is a triple (q z f ) with q 2 Q and (z zt) a solution to the initial value problem (2.2), (2.3) corresponding to q and f , for which there exists constants > 0 such that j2(p zt(t) ') + 1(p z(t) ') ; b(p f (t) ')j jpjQk'kV2 for almost every t > 0, p 2 Q and ' 2 V1. Following the results in 14], it is possible to specify sucient conditions for a solution (z zt) to the initial value problem (2.2), (2.3) to have the necessary regularity for (q z f ) to be an admissible plant.

2.1. Non-symmetric Damping Given an admissible plant (q z f ) and a tuning parameter q 2 Q, we de ne in the same manner the estimator for q and z , denoted by q^ and z^ respectively, in the form of the initial value problem hz^tt(t) 'i + 2(q z^t(t) ') + 1(q z^(t) ') + 2(^q(t) zt(t) ') + 1 (^q(t) z (t) ') (2.4) = b(^q(t) f (t) ') + 2 (q zt(t) ') + 1 (q z (t) ') ' 2 V1 t > 0



137

hq^t(t) piQ + 2(p zt(t) z(t) ; z^(t)) + 1(p z(t) z(t) ; z^(t)) ; b(p f (t) z(t) ; z^(t)) + 2(p zt(t) zt(t) ; z^t(t)) + 1(p z (t) zt(t) ; z^t(t)) ; b(p f (t) zt(t) ; z^t (t)) = 0 (2.5) p 2 Q t > 0 z^(0) 2 V1 z^t(0) 2 H q^(0) 2 Q: (2.6) Remark 2.2. In the case that condition (B 1) is not satis ed, i.e. 2 (q ) is

not symmetric, the above adaptive estimator does not satisfy the conditions presented in 15] and therefore it cannot be implemented. We propose two ways to alleviate such a problem. (S1) We can impose the condition that the bilinear form satis es the symmetry condition (B 1) only when it is evaluated at the tuning parameter q , namely 2 (q ' ) = 2(q ') 8' 2 V1 and some q 2 Q: (2.7) (S2) Replace the non-symmetric term 2 (q ' ) in (2.4) with the symmetric term 1 (q ' ) + (q ')] ' 2 V1: (2.8) 2 2 2 The rst option seems more attractive for implementational purposes, but it might happen that equation (2.7) can never be satis ed for any tuning parameter q 2 Q that simultaneously satis es (S1), (A2), (A3), (B 2) and (B 3). Of course, if there exists such a q , then (S1) is the preferred modi cation. If, on the other hand, there does not exist a q 2 Q such that (S1 ) is satis ed, then (S2) must be implemented. With the second option taken, equation (2.4) then becomes hz^tt(t) 'i + 12 2(q z^t(t) ') + 2(q ' z^t(t))] + 1 (q z^(t) ') + 2(^q (t) zt(t) ') + 1 (^q(t) z (t) ') = b(^q(t) f (t) ') + 12 2 (q zt(t) ') + 2 (q ' zt(t))] + 1 (q z (t) ') ' 2 V1 t > 0 We note that the parameters q 2 Q and > 0 can be thought of as gains, or tuning parameters, which are used to \tune" the estimator, see 15]. Of course q must be such that Assumptions (A2), (A3), (B 2), and (B 3) (and (S1) when applicable) are satis ed. We note as well, that weighting the Q inner product can also serve to tune the scheme, see 15, 16]. A third option seems to be the most general and will be the one treated throughout these pages. Instead of searching for an optimal parameter q 2 Q such that Assumptions (A2)-(A3), (B2)-(B3) and (S1) or (S2) are satis ed, one can introduce bilinear forms i ( ) : Vi Vi ! R 1 that do not depend on any parameter q 2 Q or necessarily bear any resemblance to the bilinear forms i (q ), and that satisfy Assumptions (A1)-(A3) and (B 1)-(B 3) with known bounds. That is, let 1 ( ) : V1 V1 ! R 1 be a bilinear form on V1 satisfying ' 2 V1, (A1) (Symmetry) 1 (' ) = 1 ( '), 0 (A2) (Boundedness) j1(' )j k'kV1 k kV1 , 0 > 0, ' 2 V1 , ESAIM: Cocv, May 1998, Vol. 3, 133{162

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(A3) (Coercivity) 1 (' ') 0 k'k2V1 , 0 > 0, ' 2 V1, and let 2 ( ) : V2 V2 ! R 1 be a bilinear form on V2 satisfying (B 1) (Symmetry) 2 (' ) = 2 ( '), ' 2 V2, 0 (B 2) (Boundedness) j2(' )j k'kV2 k kV2 , 0 > 0, ' 2 V2, (B 3) (Coercivity) 2 (' ') 0 k'k2V2 , 0 > 0, ' 2 V2. This can also be viewed as being a more general case of the adaptive scheme presented in 15, 25] where i (q ) is a speci c form of i ( ), i = 1 2. Then, the state and parameter estimator equivalents of (2.4) - (2.6) are now given by hz^tt(t) 'i + 2(^zt(t) ') + 1(^z(t) ') + 2(^q(t) zt(t) ') + 1(^q(t) z(t) ') (2.9) = b(^q(t) f (t) ') + 2 (zt(t) ') + 1 (z (t) ') ' 2 V1 t > 0 hq^t(t) piQ + 2(p zt(t) z(t) ; z^(t)) + 1(p z(t) z(t) ; z^(t)) ; b(p f (t) z(t) ; z^(t)) + 2(p zt(t) zt(t) ; z^t(t)) + 1(p z (t) zt(t) ; z^t (t)) ; b(p f (t) zt(t) ; z^t (t)) = 0 (2.10) p 2 Q t > 0 z^(0) 2 V1 z^t(0) 2 H q^(0) 2 Q: (2.11) Assumptions (A4) and (B 4) imply that the system (2.9), (2.10) is linear in (^z q^). One way that the initial value problem (2.9), (2.10) and (2.11) can be shown to be well posed is in the sense of the existence of a unique mild or generalized solution. This can be argued via the theory of in nite dimensional evolution equations as presented in, for example, Pazy 23], Tanabe 27] or the more relevant treatment by Demetriou and Rosen 15] and Scondo 25] for the symmetric or absent 2 case. We duplicate, for sake of completeness, the procedure used in 15] and present the required modi cations. Let fX h iX g be the Hilbert space de ned by X = V1 H and introducing an additional tuning parameter , we de ne the X -inner product h' iX = f1('1 1) + h'2 2ig + h'1 2i + h1 '2i + 2('1 1) for ' = ('1 '2), = (1 2) 2 X , 2 R. It can easily be shown 15] that if > 0 is suciently large, then h iX is in fact an inner product on X and moreover, that the corresponding induced norm, j jX , is equivalent to the norm on X induced by the more standard inner product on X given by (' )X = 1('1 1) + h'2 2i for ' = ('1 '2), = (1 2) 2 X . In fact, these arguments will be given in the proofs of Lemma 3.1 and convergence results in the next section (see also 15]). Now let Y = V1 V1 be endowed with the norm 1 k'kY = k'1k2V1 + k'2k2V1 2 for ' = ('1 '2) 2 Y . Then Y is a reexive Banach space and Y ,! X ,! Y with the embeddings dense and continuous. We rewrite (2.9), (2.10) in rst order form as d x(t) = A B(t) x(t) + F (t) t > 0 (2.12) q^(t) ;B(t) 0 dt q^(t) ESAIM: Cocv, May 1998, Vol. 3, 133{162


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where A 2 L(Y Y ) and B(t) 2 L(Q Y ), are given by

I

0

A' = ;A1 ;A2 and

'1 ' = (' ' ) 2 Y 1 2 '2

B(t)q = ;A2(q)zt(t) ; A10(q)z(t) + B(q)f (t) q 2 Q

(2.13) (2.14)

respectively (the operators A1 (q ) and A2 (q ) are as they have been de ned above), A1 2 L(V1 V1 ) and A2 2 L(V2 V2 ) are the operators that correspond to the bilinear forms 1 2 and are de ned similar to the operators A1 (q) A2(q), B(t) 2 L(Y Q) is the Banach space adjoint of B(t) (that is, recalling that above embeddings, that hB(t) ' q iQ = h' B(t)q iX , ' 2 Y , q 2 Q), and the forcing term is 3

2

0 F (t) = 4 A2zt(t) + A1z(t) 5 t > 0: B(t) (z(t) zt(t))

(2.15)

Since (q z f ) is assumed to be an admissible plant, then F 2 L2 (0 T (Y Q) ) for all T > 0. We assume that z is such that A2 (q)zt(t)+A1 (q)z (t) 2 H , t 0, q 2 Q, and that A2 zt(t) + A1 z(t) 2 H , t 0. In addition we assume that the map t ! A2 (q )zt(t) + A1(q )z (t), for each q 2 Q, is strongly continuously dierentiable in H . Let X = X Q, let the domain D be

D = f(' q) 2 X : ' = ('1 '2) 2 Y and A1'1 + A2'2 2 H g n

and de ne the family of operators A~(t)

o

, A~(t) : D X ! X by

t0

A~(t) = ;BA(t) B0(t) t 0: Using the results in 15] it can be shown that fA~(t)gt0 is a stable family of in nitesimal generators of C0 semigroups on X and that the map t ! A~(t)(' q) for (' q) 2 D is strongly continuously dierentiable in X . It then follows, 23], that the system (2.9), (2.10), (2.11) admits a unique mild solution. When the initial data, (x(0) q^(0)), is suciently regular (i.e. (x(0) q^(0)) 2 D) and F is suciently regular (which essentially depends upon the regularity of the plant state, z ) we have a strong solution to the initial value problem (2.9) - (2.11). In the rest of this note, we assume that the initial data and the plant are suciently regular to ensure that the initial value problem (2.9) - (2.11) has a strong solution. Let e(t) := z^(t) ; z (t) denote the state error and r(t) := q^(t) ; q the parameter error, with (q z f ) a plant and (^q z^) a solution to the initial value problem (2.9) - (2.11). In the next section we will show that under no additional assumptions we have state error convergence and that under the additional assumption of persistence of excitation can guarantee parameter convergence. ESAIM: Cocv, May 1998, Vol. 3, 133{162

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2.2. Systems with no Damping In this subsection we consider systems with no damping term, given in weak form by hztt(t) 'iV1 V1 + 1(q z(t) ') = b(q f (t) ') ' 2 V1 t > 0 (2.16)

z(0) = z0 zt(0) = z1 (2.17) where z0 2 V1 , z1 2 H , and f 2 L2(0 1 H ). In this case we only utilize a Gelfand triple with V1 ,! H ' H ,! V1 , 28].

The well posedness of the system (2.16), (2.17) with bounded input operator can easily be established using linear semigroup theory, see 10, 20, 21, 23, 26, 28]. The case of unbounded input operator is treated in 7]. Once again, it is possible to specify sucient conditions for a solution (z zt) to (2.16), (2.17) to have the necessary regularity for (q z f ) to be an admissible plant. The corresponding de nition for an admissible plant, in this case, is the triple (q z f ) with (z zt) a solution to (2.16), for which there exist > 0 such that j1(p z(t) ') ; b(p f (t) ')j jpjQk'kV2 (2.18) for almost every t > 0, p 2 Q and ' 2 V1. In the proposed state and parameter estimator below, we utilize a damping sesquilinear form 2 even though 2 does not occur naturally in the system (2.16), (2.17). The motivation behind this is to enhance the convergence properties of the state error (both ke(t)kV1 and jet(t)j) to zero this will become clearer in the ensuing stability analysis. In this case the proposed state and parameter estimators take the form hz^tt(t) 'i + 2(q z^t(t) ') + 1(q z^(t) ') + 1(^q(t) z(t) ') = b(^q(t) f (t) ') + 2 (q zt(t) ') + 1 (q z (t) ') ' 2 V t > 0 (2.19)

hqt(t) piQ + 1(p z(t) z(t) ; z^(t)) ; b(p f (t) z(t) ; z^(t)) + 1(p z (t) zt(t) ; z^t (t)) ; b(p f (t) zt(t) ; z^t (t))] = 0 (2.20) p 2 Q t > 0 z^(0) 2 V1 z^t(0) 2 H q^(0) 2 Q (2.21)

where 2 (a design term) is chosen to satisfy Assumptions (A1) - (A4). We note, however, that by employing a 5-space setting as in (2.1), 2 could alternatively be chosen to satisfy Assumptions (B 1) - (B 4) which permits one to obtain the desired results under, in general, much weaker assumptions on 2. The reader is also directed to 25] for a similar treatment of second order systems with no damping in a 3-space setting where that author chooses 2(q ) of the estimator to be the same as 1 (q ) of the system. As in the case of subsection 2.1, we de ne the space X = V1 H with the same inner product h' iX = f1(q '1 1) + h'2 2ig+ h'1 2i + h1 '2i + 2(q '1 1) for ' = ('1 '2), = (1 2) 2 X , where q 2 Q is as it was de ned in Assumptions (A2), (A3) (or equivalently in Assumptions (B 2), (B 3)). ESAIM: Cocv, May 1998, Vol. 3, 133{162


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Remark 2.3. In the no damping case the bilinear form 2 , proposed in

(2.19), (2.20), is chosen to satisfy all four conditions (A1) - (A4), and this of course results in a Gelfand triple setting rather than a Gelfand quintuple setting. The symmetry condition is also imposed even when 2 (q ) is not evaluated at the tuning parameter q . This can be done since 2 is a purely design term. One can further modify the above estimator by using the results of the third option (general option) to replace 1 (q ) by 1 ( ) and 2 (q ) by 2( ) if more design exibility is desired, at the expense of increased complexity via a Gelfand quintuple. This then becomes a more general case that includes the treatment of symmetric damping in 12, 15] and of no damping in 25]. Similarly, we de ne the reexive Banach space Y = V1 V1 , endowed with the norm 1 k'kY = k'1k2V1 + k'2k2V1 2 for ('1 '2) 2 Y with Y ,! H ,! Y and the embeddings dense and continuous. We rewrite (2.19)-(2.21) in rst order form as d x(t) = A B(t) x(t) + F (t) t > 0 ;B(t) 0 q^(t) dt q^(t) where in this case A 2 L(Y Y ) and B(t) 2 L(Q Y ), are now given by 0 I ' 1 A' = ;A1(q ) ;A2(q ) '2 ' = ('1 '2) 2 Y and 0 B(t)q = ;A (q)z(t) + B(q)f (t) q 2 Q 1

respectively (once again the operators A1 (q ) and A2 (q ) are as they have been de ned in subsection 2.1 with A2 (q ) now in L(V1 V )), B(t) 2 L(Y Q) is again the Banach space adjoint of B(t), and 2 3 0 F (t) = 4 A2(q )zt(t) + A1(q )z(t) 5 t > 0: B(t) (z(t) zt(t)) From the equivalent de nition of the plant, equation (2.18), we have F 2 L2 (0 T Y ) for all T > 0. Using similar arguments as in Section 2, we can show the well posedness of the state and parameter estimator, (2.19)-(2.21). Note that if the general case is used (Remark 2.3) then the A operator is now given by (2.13) and F (t) by (2.15)

Stability and Convergence We now establish the convergence of the state estimator lim ke(t)kV1 = 0 and tlim jet(t)j = 0 t!1 !1 and, with the additional assumption of persistence of excitation, parameter convergence. That is, lim t!1 jr(t)jQ = limt!1 jq^(t) ; q jQ = 0. We assume throughout this section that (q z f ) is an admissible plant (see De nition 2.1). 3.


142


3.1. Non-symmetric Damping Using the plant equation (2.2) with non-symmetric 2 (i.e. (B 1) not satis ed), the state estimator (2.9), and the parameter estimator (2.10), we have that e and r satisfy the linear, homogeneous, non-autonomous, initial value problem given in weak or variational form by hett(t) 'i + 2(et(t) ') + 1(e(t) ') + 2(r(t) zt(t) ') + 1(r(t) z(t) ') (3.1) = b(r(t) f (t) ') ' 2 V1 t > 0

hrt(t) piQ ; 2(p zt(t) e(t)) ; 1(p z(t) e(t)) + b(p f (t) e(t)) ; 2(p zt(t) et(t)) ; 1(p z(t) et(t)) + b(p f (t) et(t)) = 0 (3.2) p 2 Q t > 0 e(0) 2 V1 et(0) 2 H r(0) 2 Q: (3.3)

We next establish a Lyapunov-like estimate for the system (3.1) - (3.3). In essence, it is shown that the derivative of an energy function is non-positive along the trajectories of the error equations (3.1), (3.2). Lemma 3.1. (15]) If the tuning parameter satises (

)

K2

> max KV1 KV1 V2 0 0 then there exist constants , k > 0 such that for all t > 0

(3.4)

Zt 2 2 2 ke(t)kV1 + jet(t)j + jr(t)jQ + ke(s)k2V1 + kes (s)k2V2 ds 0 n o where = k ke(0)k2V1 + jet (0)j2 + jr(0)j2Q .

Proof. Following the treatment in 15], we de ne the energy functional, E : 0 1) ! R 1, for the system (3.1), (3.2) by E (t) = 1(e(t) e(t)) + jet (t)j2 + 2he(t) et(t)i + 2 (e(t) e(t)) + jr(t)j2Q

2 e ( t ) = e (t) + jr(t)j2Q: (3.5) t X Then, using (3.1), (3.2), and assumptions (A3) and (B 3), we obtain Et(t) = f21(e(t) et(t)) + 2hett(t) et(t)ig + 2het(t) et(t)i + 2he(t) ett(t)i + 22(et (t) e(t)) + 2hrt(t) r(t)iQ = ; 2 2(et (t) et(t)) ; 21(e(t) e(t)) + 2jet(t)j2 (3.6) 2 2 2 ; 20ke(t)kV1 ; 2 0 ; 2KV2 ket(t)kV2 ; 0 ke(t)k2V1 + ket(t)k2V2 for some 0 = minf20 2 0 ; 2KV22 g > 0, since, by assumption, > KV22 = 0. Consequently,

E (t) + 0

Zt 0

ke(s)k2V1 + kes(s)k2V2 ds E (0):


(3.7)


143

From assumptions (A2) and (B 2), see also 15], we nd that

E (t) R fke(t)k2V1 + jet(t)j2 + jr(t)j2Qg for some R > 0 and from (3.5) that

(3.8)

L ke(t)k2V1 + jet(t)j2 + jr(t)j2Q E (t)

(3.9)

for some L > 0. The desired result then follows from (3.7), (3.8), (3.9), and the assumed lower bound on . The convergence of the state estimate is established below without imposing any additional assumptions. Let (q z f ) be an admissible plant. If

> maxfKV1 KV1 =0 KV22= 0 g, then the energy functional E given by (3.5) is non-increasing, lim ke(t)kV1 = 0 and tlim jet(t)j = 0: (3.10) !1 The proof of the state convergence is similar to the one given in 15] for the symmetric case and is given in greater detail in Appendix A of the companion report 3]. Remark 3.2. (15]) In the case that V1 = V2, inspection of the de nition of E given in (3.5) reveals that Lemma 3.1 and the error convergence (3.10) can be proved with the somewhat weaker assumption on that t!1

)

(

K2

> max KV1 KV1; 0 V2 : 0 0 In order to establish parameter convergence we require the notion of persistence of excitation. Definition 3.3. A plant (q z f ) is said to be persistently excited via the input f , if there exists T0 0 0 > 0 such that for each p 2 Q with jpjQ = 1 and each t1 > 0 suciently large, there exists a t~ 2 t1 t1 + T0] such that Z ~ t+0 h A ( p ) z ( ) i + h A ( p ) z ( ) i; h B ( p )( ) i d 2 1 ~t V1

0:

With the above condition assumed, we have that if the plant (q z f ) is persistently excited then lim jr(t)jQ = 0:

t!1

The proof of parameter convergence follows from two technical lemmas which are stated and proven in Appendix A of 3]. It is almost identical to the symmetric case presented in 15]. Remark 3.4. Continuing with possible extensions for the non-symmetric case as was presented in Remark 2.2, we also present a fourth possible way which is less complicated but changes the bound on the constant . If the original state estimator (with no modi cations) given by (2.4) is used, then ESAIM: Cocv, May 1998, Vol. 3, 133{162

144


in the proof of Lemma 3.1 we have that Et(t) ; 20 (q )ke(t)k2V1 ; 2 0(q ) ; 2KV22 ke(t)k2V1 + 2 (q et (t) e(t)) ; 2 (q e(t) et(t)) ; 20(q ) ; 0(q ) ke(t)k2V1 n o ; 2 0(q ) ; 2KV22 ; 0(q )KV22V1 ke(t)k2V1

which imposes the additional condition 20(q ) ; 0 (q ) > 0 and decreases 2K 22 ; 0 (q )K 22 1 . These changes will aect the lower bound of to 20(q ) the choice of the constant c2 and 0 in the proof of parameter convergence, (see Appendix A of 3]) which will eventually aect the rate of convergence

0 of E (t) to zero, see 16]. Concluding about this option, either the level of excitation, 0 must increase or the length of the time window 0 for De nition 3.3 must decrease. V

V V

3.2. Systems with no Damping Assuming that the purely designed term 2 is used (Remark 2.3) then the resulting state and parameter error equations become hett(t) 'i + 2(q et(t) ') + 1(q e(t) ') + 1(r(t) z(t) ') = b(r(t) f (t) ') ' 2 V1 t > 0 (3.11)

hrt(t) piQ ; 1(p z(t) e(t)) + b(p f (t) e(t)) ; f1(p z(t) et(t)) ; b(p f (t) et(t))g = 0 p 2 Q t > 0 (3.12) e(0) 2 V1 et(0) 2 H r(0) 2 Q: (3.13)

The persistence of excitation condition reduces to requiring that for T0 , 0 0 > 0, for each p 2 Q with jpjQ = 1 and each t1 > suciently large, there exists a t~ 2 t1 t1 + T ] such that Z ~ t+0 hA (p)z ( ) i; hB (p)f ( ) i d 1 t~ V1

0:

(3.14)

The corresponding result in state error convergence is then summarized. Let (q z f ) be an admissible plant. If > maxfKV1 KV1 =0 (q ) KV21 =0 (q )g, then the energy functional E given by (3.5) is non-increasing, lim ke(t)kV1 = 0 and tlim jet(t)j = 0: t!1 !1 The arguments leading to the proof of the state error convergence are similar to the case with damping presented in Appendix A of 3] and are therefore omitted. The dierences are in the de nition of the (admissible) plant given by (2.18). Once again, in order to achieve parameter convergence we impose the persistence of excitation condition given by (3.14). If the condition of persistence of excitation, given by (3.14), is satis ed, then lim jr(t)jQ = 0: t!1 ESAIM: Cocv, May 1998, Vol. 3, 133{162


145

The proof of parameter convergence once again resembles the arguments used in Appendix A of 3] for the case with damping. The dierences are in the choice of the constant used in the proof of Lemma A.1 and in the choice of c1 , c2, 0 and g (t) used in the proof of state error convergence with damping. Remark 3.5. Following Remark 2.3, if the general option with a 5-space setting is used, then the parameter required to guarantee convergence is given by (3.4). Slowly Time-Varying Systems We now turn our attention to systems with time varying parameters. Speci cally, we treat systems whose parameters are assumed to asymptotically converge to an (unknown) steady state value. Consider the system (with a symmetric 2 (q ) assumed here for simplicity, i.e. satisfying Assumption (B 1)) hztt(t) 'iV1 V1 + 2(q(t) zt(t) ') + 1(q(t) z(t) ') = b(q(t) f (t) ') (4.1) ' 2 V1 t > 0 4.

z (0) = z0 2 V1 zt(0) = z1 2 H q(0) = q0 2 Q (4.2) where z0 2 V1, z1 2 H , f 2 L2(0 1 H ), and q (0) 2 Q is some unknown initial condition of the parameter q (t). We assume that the parameter q (t) satis es

hqt(t) piQ = hAq q(t) piQ + hcss piQ

(4.3) where q (0) 2 Q, Aq is the in nitesimal generator of an exponentially stable C0 semigroup on Q satisfying hAq p piQ ;q jpj2Q p 2 Q some q > 0 (4.4) and css is related to the steady state value of q (t). Following the results in 1, 2] for nite dimensional adaptive systems and in 23] for Q being an in nite dimensional parameter space with q (t) now denoting the mild solution, we have that (4.3) satis es lim q (t) = qss = ;A;q 1 css t!1

where qss is the steady state value of q (t). It is assumed in this case that both q (0) and qss (equivalently css ) are unknown. The only parameter assumed known here is the dissipation bound q . Remark 4.1. We can also assume that css is a function of time and that there exists a steady state value for css (t) (assumed bounded and measurable ;1 1 on 0 1)), namely, c1 ss = limt!1 css (t), so that qss = ;Aq css , see 18, 23]. The de nition of an admissible plant will be taken to be the same presented in Section 2, namely De nition 2.1. We can follow the procedure given in 20], 23], or 28] to establish the well posedness of the system (4.1) - (4.3). ESAIM: Cocv, May 1998, Vol. 3, 133{162

146


We now proceed to establish a state and parameter estimator that will guarantee the convergence of the state error to zero, and by imposing the additional condition of persistence of excitation, parameter convergence. Following the treatment of nite dimensional results for slowly time varying parameters, see for example 22], we re-introduce the parameter error given by q~(t) := q^(t) ; qss , instead of the more conventional term, q^(t) ; q (t), and de ne q (t) := q (t) ; qss . The equations for the state and parameter estimator are the same as the ones given for hyperbolic systems with symmetric damping and time invariant parameters by (2.4) - (2.6), namely hz^tt(t) 'i + 2(q z^t(t) ') + 1(q z^(t) ') + 2(^q(t) zt(t) ') + 1 (^q(t) z (t) ') = b(^q (t) f (t) ') + 2(q zt(t) ') + 1(q z (t) ') ' 2 V t > 0 (4.5)

hq^t(t) piQ + 2(p zt(t) z(t) ; z^(t)) + 1(p z(t) z(t) ; z^(t)) ; b(p f (t) z(t) ; z^(t)) + 2(p zt(t) zt(t) ; z^t(t)) + 1(p z (t) zt(t) ; z^t(t)) ; b(p f (t) zt(t) ; z^t (t)) = 0 (4.6) p 2 Q t > 0 z^(0) 2 V1

z^t(0) 2 H

q^(0) 2 Q:

(4.7) Following the procedure given in Section 2, we duplicate the steps taken for establishing the well posedness of the state and parameter estimator, (4.5) - (4.7). We rst assume that Assumptions (A1) - (A4), (B 1) - (B 4) are satis ed for q q 2 Q. Let fX h iXg be the Hilbert space de ned by X = V1 H and now de ne h' iX = f1(q '1 1) + h'2 2ig + h'1 2i + h1 '2i + 2(q '1 1) for ' = ('1 '2), = (1 2) 2 X , where q 2 Q is as it was de ned in Assumptions (A2), (A3), (B 2), and (B 3). We rewrite (4.5), (4.6) in rst order form as d x(t) = A B(t) x(t) + F (t) t > 0 ;B(t) 0 q^(t) dt q^(t) where A 2 L(Y Y ) and B(t) 2 L(Q Y ), are now given by A' = ;A10(q ) ;A2I(q ) ''12 ' = ('1 '2) 2 Y and 0 B(t)q = ;A2(q)zt(t) ; A1(q)z(t) + B(q)f (t) q 2 Q respectively , B(t) 2 L(Y Q), and F (t) is now given by 3 2 0 F (t) = 4 A2(q )zt(t) + A1(q )z(t) 5 t > 0: B(t) (z(t) zt(t)) ESAIM: Cocv, May 1998, Vol. 3, 133{162


147

The rest of the arguments are similar to the ones used in Section 2 and are therefore omitted. The resulting error equations are now given by hett(t) 'i + 2(q et(t) ') + 1(q e(t) ') + 2(~q(t) zt(t) ') + 1 (~q (t) z (t) ') ; 2(q (t) zt(t) ') ; 1 (q(t) z (t) ') (4.8) = b(~q(t) f (t) ') ; b(q(t) f (t) ') ' 2 V t > 0

hq~t(t) piQ ; 2(p zt(t) e(t)) ; 1(p z(t) e(t)) + b(p f (t) e(t)) ; 2(p zt(t) et(t)) ; 1(p z(t) et(t)) + b(p f (t) et(t)) = 0 (4.9) p 2 Q t > 0 hqt(t) pi = h dtd q(t) ; dtd qss pi = hAq q(t) pi + hcss pi = hAq q (t) ; qss + qss ] pi + hcss pi = hAq q (t) pi + hAq qss + css pi = hAq q(t) pi

(4.10)

(4.11) e(0) 2 V1 et (0) 2 H q~(0) q(0) 2 Q where we used the fact dtd q~(t) = dtd q^(t) ; dtd qss = q^t (t) and that qss = ;A;q 1 css.

The dierence in the (slowly) time varying case arises in the de nition of the energy functional which is now given by E (t) = 1(q e(t) e(t)) + jet(t)j2 + 2he(t) et(t)i (4.12) + 2 (q e(t) e(t)) + jq~(t)j2Q + jq (t)j2Q for some > 0 to be de ned later. The derivative Et(t) is given by Et(t) = f21 (q e(t) et(t)) + 2hett(t) et(t)ig + 2het(t) et(t)i + 2he(t) ett(t)i + 22(q et(t) e(t)) + 2hq~t(t) q~(t)iQ + 2 hq t (t) q(t)iQ = ; 2 2(q et(t) et(t)) ; 21 (q e(t) e(t)) + 2jet (t)j2 + 22 (q(t) zt(t) e(t)) + 21 (q(t) z (t) e(t)) + 2 2(q(t) zt(t) et(t)) + 2 1(q(t) z (t) et(t)) ; 2 b(q(t) f (t) et(t)) ; 2b(q(t) f (t) e(t)) + 2 hAq q(t) q(t)iQ ; 20(q )ke(t)k2V1 + ;2 0(q ) + 2KV22 ket(t)k2V2 + 2 jq(t)jQke(t)kV2 + 2 jq(t)jQket (t)kV2 ; q jq (t)j2Q ; 20(q ) ; KV22V1 1 ke(t)k2V1 ; 2 0(q ) ; 2KV22 ; 2 ket(t)k2V2

1 ; q ; 4 + 4 jq(t)j2Q 1 2 2 ;0 ke(t)kV1 + ket(t)k2V2 + jq(t)j2Q (4.13) where we used the identity a b a2 =41 + b2 1 . The equivalent of Lemma 3.1 is ESAIM: Cocv, May 1998, Vol. 3, 133{162

148


Lemma 4.2. If , 1 , 2 are chosen so that the following are satised

0 < 1 < 20(q2 ) 0 < 2 < 2 0(q )

KV2

and

"

where now satises

#

2 K2 > 8 (Vq2 ) + ( q ) q 0 0

(

)

2K 2

> max KV1 K(Vq1 ) 2 (q ) V;2 0 0 2 then there exist constants , > 0 such that for all t > 0 ke(t)k2V1 + jet(t)j2 + jq~(t)j2Q + jq(t)j2Q n

+

Zt 0

(4.14)

ke(s)k2V1 + kes (s)k2V2 + jq(s)j2Q ds o

where = ke(0)k2V1 + jet(0)j2 + jq~(0)j2Q + jq(0)j2Q . The proof of Lemma 4.2 readily follows from equation (4.13) and using arguments similar to the ones in the proof of Lemma 3.1. We now establish the convergence of the state error. Theorem 4.3. Let (q z f ) be an admissible plant. If satises the conditions in Lemma 4.2, then the energy functional E given by (4.12) is nonincreasing, and lim ke(t)kV1 = 0 and tlim jet(t)j = 0: t!1 !1

Proof. The proof of Theorem 4.3 is similar to the one given for the time invariant case with the exception of minor modi cations and is therefore omitted. Parameter convergence is again achieved by imposing the additional condition of persistence of excitation. If the plant (q z f ) is persistently excited (see De nition 3.3) then lim jq~(t)jQ = 0: t!1

The proof of parameter convergence is similar to the previous case of time invariant parameters and is summarized in Appendix B of 3]. Remark 4.4. The above theorem shows convergence of the parameter estimate q^(t) to the steady state value qss of the unknown parameter q . Since jr(t)j2Q = j(^q(t) ; qss ) ; (q(t) ; qss )j2Q = jq~(t) ; q(t)j2Q 2jq~(t)j2Q + 2jq(t)j2Q and since lim jq (t) ; qss jQ = 0 by (4.10) and the assumption on Aq t!1 then we have that lim jq^(t) ; q (t)jQ = 0: t!1 ESAIM: Cocv, May 1998, Vol. 3, 133{162


149

Further study is needed to see if the parameter estimate, q^(t), can converge to the parameter q (t) (in nite time) instead of its steady state value qss . Remark 4.5. Using equations (4.4), (4.10) and (4.13) it can be seen that it is not Aq that is required to be known, but rather its coercivity bound q that is used in (4.13). Remark 4.6. A similar result can be established for the case of no damping and time varying parameters. Combination of results from the previous sections can give the conditions needed for the estimator of time varying systems with no damping. Finite Dimensional Approximation All the adaptive identi cation schemes presented in the previous sections are in nite dimensional and their application necessitates a nite dimensional approximation. We will outline a Galerkin approach in order to implement nite dimensional adaptive estimators and establish abstract convergence results. Even though the adaptive estimators require information on the full state, a rather unlikely scenario, the approximate estimators will be shown to adequately function with a nite dimensional information of the state of the system. This, in a way, can be viewed as an adaptive ( nite dimensional) state and parameter estimator that uses a nite dimensional output of the system (this being a nite dimensional approximation of the system) to yield estimates of the state and the parameters of the actual in nite dimensional system. For each n = 1 2 : : : let H n be a nite dimensional subspace of H with n H V1, n = 1 2 : : : and let Qn be a nite dimensional subspace of Q. We consider only the case of the non-symmetric bilinear form 2 , the other two cases, the slowly time-varying systems and systems with no damping, being similar and therefore omitted. The Galerkin equations for z^n and q^n in H n and Qn corresponding to (2.9) and (2.10) are given by hz^ttn (t) 'ni + 2(^ztn(t) 'n) + 1(^zn(t) 'n) + 2(^qn(t) zt(t) 'n) + 1 (^q n (t) z (t) 'n) = b(^q n(t) f (t) 'n) + 2(zt (t) 'n) + 1 (z (t) 'n) 'n 2 H n t > 0 (5.1) 5.

hq^tn(t) pniQ + 2(pn zt(t) z(t) ; z^n (t)) + 1(pn z(t) z(t) ; z^n (t)) ; b(pn f (t) z(t) ; z^n (t)) + 2(pn zt(t) zt(t) ; z^tn (t)) + 1(pn z (t) zt(t) ; z^tn (t)) ; b(pn f (t) zt(t) ; z^tn (t)) = 0 (5.2) pn 2 Qn t > 0 z^n (0) 2 H n z^tn (0) 2 H n q^n(0) 2 Qn: (5.3)

We make the following standard Galerkin approximation assumptions. First de ne the orthogonal projections P n : H ! H n of H onto H n and PQn : Q ! Qn of Q onto Qn . (H 1) The nite dimensional subspaces H n satisfy H n V1 (H 2) The functions ^n := P n z^ and ^n := PQn q^ with ^n 2 L2(0 T H n) and ^n := PQn q^ 2 L2(0 T Qn) are such that ESAIM: Cocv, May 1998, Vol. 3, 133{162

150


(i) ^n ! z^ in C (0 T V1), (ii) dtd ^n ! dtd z^ in C (0 T H ) and L2 (0 T V2), (iii) ^n ! q^ in C (0 T Q). Using the above assumptions we prove the following convergence result. Theorem 5.1. Assume that assumptions (H 1) - (H 2) hold, and (q z f ) satises De nition 2.1. Let (^q z^) be the solution to the initial value problem (2.9) - (2.11), and for each n = 1 2 : : : let (^q n z^n ) be the solution to the initial value problem (5.1) - (5.3) with z^n (0) = ^n (0) = P n z^(0) dtd z^n(0) = dtd ^n(0) = P n dtd z^(0) (5.4) q^n (0) = ^n (0) = PQn q^(0): Then (i) z^n ! z^ in C (0 T V1) (ii) d z^n ! d z^ in C (0 T H ) and L2 (0 T V2) and dtn dt (iii) q^ ! q^ in C (0 T Q): Proof. Since the proof is rather lengthy and technical, it is summarized in Appendix C of 3]. Even though the scheme here is similar to the 2 one presented in 15], Theorem 5.1 does not impose any assumptions on dtd 2 ^n , and dtd ^n . The proof can be given under relaxed conditions by following ideas in 11] (see also Chapter 5 of 10]).

The above nite dimensional estimator requires the state of the system, z, which lies in the in nite dimensional space V1. Therefore, it will be more convenient to replace it in (5.1) - (5.3) by a nite dimensional approximation, zn . In order to present an approximation result that would use a nite dimensional approximation of the plant, we need to impose some conditions on the plant, as was similarly done for the symmetric case in 15]. (H 3) There exists a constant s1 > 0 such that j1(p ' )j s1jpjQk'kV1 kkV1 p 2 Q ' 2 V1: (H 4) There exists a constant s2 > 0 such that j2(p ' )j s2jpjQk'kV2 kkV2 p 2 Q ' 2 V2: (H 5) There exists a constant b1 > 0 such that jb(p ' )j b1jpjQj'jkkV1 p 2 Q ' 2 H 2 V1: (H 6) For the plant (q z f ) there exists z n 2 C 1 (0 T H n) such that (i) z n ! z in C (0 T V1), (ii) z_ n ! z_ in C (0 T V1). Theorem 5.2. Assume that (q z f ) is an admissible plant and that (H 1) (H 6) are satised. Let (^q z^) be the solution to (2.9) - (2.11), and for each n = 1 2 : : : let (^qn z^n ) be the solution to (5.1) - (5.3) with z replaced by ESAIM: Cocv, May 1998, Vol. 3, 133{162


151

z n and z_ by z_n . Then (i) z^n ! z^ in C (0 T V1) (ii) d z^n ! d z^ in C (0 T H ) and L2 (0 T V2) and dt dt (iii) q^n ! q^ in C (0 T Q): Proof. The proof is also summarized in Appendix C of 3]. Remark 5.3. As it was observed in 15], if we choose PVn2 : V2 ! H n be the

orthogonal projection of V2 onto H n with respect to the standard V2 inner product and set z n := PVn2 z then we see that 1 (p PVn2 z ') 1(p z ') for ' 2 H n 2 (p PVn2 z_ ') 2 (p z_ ') for ' 2 H n for ' 2 H n and 1 (PVn2 z ') 1 (z ') 2 (PVn2 z_ ') 2 (z_ ') for ' 2 H n : Indeed, the above make sense since z 2 L2 (0 T V1) (and z 2 L2(0 T V2)), and z_ 2 L2(0 T V2) were assumed as part of the existence of a weak solution to the plant (2.2), (2.3). Remark 5.4. From the conclusion of Theorem 5.2 and Remark 5.3 above, we can see that if only a nite dimensional approximation (z n z_ n ) of the plant (z z_ ) is available and used in (5.1) - (5.3), then this may be viewed as an adaptive state and parameter estimator that uses a part of the state (z z_ ) with the output y (t) 2 H n H n of the system having the speci c form

n

n P z ( t ) z ( t ) V 2 y (t) = C z_(t) = P n z_ (t) = zz_n ((tt)) 2 H n H n V2 where the output (or observation) operator C : V2 V2 ! H n H n . Examples and Numerical Simulations As an example of a system with non-symmetric damping, we study the 2-D structural acoustic model with piezoceramic actuators presented in 4] which indeed motivated the eorts of this paper. This 2-D structural acoustic model investigated here represents a \linearized slice" of a full 3-D acoustic cavity/plate structure currently being used in experiments in the Acoustics Division at NASA Langley Research Center, see 10] for further details. 6.

The equations of motion for the 2-D linearized structural acoustic model are given by tt = c2 # + d#t (x y) 2 $ t > 0 r n^ = 0 (x y) 2 ; t > 0 r(t x 0) n^ = wt(t x) 0 < x < a t > 0


152


y

Γ

l

+ -

Ω

+

Γ0

a

0

x

(a)

(b)

Figure 1. (a) Acoustic cavity with piezoceramic patches,

(b) Piezoceramic patch excitation.

2 @ 2w 3w @ @ b wtt + @x2 EI @x2 + cDI @x2@t @ 2 ;KB = ;f t (t x 0) + f (t x) + @x x1 x2 ] u(t) 0 < x < a t > 0 2

w(t 0) = @w (t 0) = w(t a) = @w (t a) = 0 t > 0 @x @x

(0 x y ) = 0(x y ) w(0 x) = w0(x) t (0 x y) = 1(x y ) wt(0 x) = w1(x) where (t x y ) denotes the cavity potential, f t (t x y ) the cavity pressure with f being the equilibrium uid density. The uid constants c and d

denote the speed of sound in the uid and damping coecient, respectively. The patch parameters u(t), x1 x2 ] (x) and KB denote the voltage applied to the patch, the characteristic function for the location of the patch and patch parameter, see 5, 9]. The beam transverse displacement is denoted by w(t x) and the beam parameters b, EI and cD I represent the linear mass density, stiness and damping, respectively. In this note the piezoceramic material parameter KB , as well as the beam parameters EI and cD I are considered to be unknown and must be estimated adaptively (on-line). The domain is given by $ = 0 a] 0 l] and the boundary is denoted by ; (hard walls) and ;0 (perturbable boundary{beam), see Figure 1. In order to pose the problem in a variational form which is required for our approach to adaptive parameter estimation and approximation, (see 10]), the state is taken to be z = ( w) in the state space H = L2 ($) L2 (;0 ) ESAIM: Cocv, May 1998, Vol. 3, 133{162


with the energy inner product

w

H

Z

=

f d! +

Z ;0

153

bwd :

Here, L2 ($) is the quotient space of L2 over the constant functions. The quotient space is used because the potentials are determined only up to a constant, again see 10]. We also de ne the Hilbert space V1 = H 1 ($) H02(;0 ) where H 1($) is the quotient space of H 1 ($) over the constant functions and H02(;) = f 2 H 2(;) : (x) = 0(x) = 0 at x = 0 ag. The V1 inner product is taken as

w

V1

=

Z

r rd! +

Z

;0

D2wD2 d :

In this case the parameter q is taken to be q = fEI () cDI () KBg. For q 2 Q, we de ne bilinear forms i (q ) : V1 V1 ! R1, i = 1 2, on V by

1 (q ' () = 2 (q ' () =

Z

Z

f c2r rd! +

f dr rd! +

Z

Z

;0

EID2wD2 d

;0

cD ID2wD2 + f ( ; w ) d

where ' = ( w) and ( = ( ) are in V1 . In addition, for each q 2 Q, we de ne the bilinear form b(q ) : H V1 ! R1 by

b(q ) () =

Z

;0

KB D2d

) = ( ) 2 H ( 2 V1:

In the work reported here, we followed the approach in 5], 8], 9], using a Galerkin approximation scheme to approximate the beam/cavity system. This was achieved by discretizing the potential and beam displacement in terms of spectral and spline expansions, respectively. Our scheme for the second order formulation approximating (2.2) was of dimension N where N = m + n ; 1 with n ; 1 modi ed cubic splines used for the beam approximation and m tensor products of Legendre polynomials used for the cavity approximation (see 5] for complete details). Using a standard Galerkin scheme, we choose a sequence of nite dimensional subspaces H N V1 with projections P N : H N ! H . ;1 To illustrate the approximation technique mentioned above let fBin gni=1 denote the 1-D basis functions which are used to discretize the beam and let fBim gmi=1 , m = (mx + 1) (my + 1) ; 1, denote the 2-D basis functions which are used in the cavity. The n ; 1 and m dimensional approximating ;1 subspaces are then taken to be Hbn = span fBin gni=1 and Hcm = fBim gmi=1 , respectively. Using the de nition of N above, the approximating state space is H N = Hcm Hbn . The restriction of the in nite dimensional system to the space H N yields for ( = ( ) the following

hzttN (t) (iX + 2(q ztN (t) () + 1(q zN (t) () =

Z

KB x1x2](x)u(t)D2 d

;0


154


When ( is chosen in H N and the approximate beam and cavity solutions are taken to be nX ;1 N w (t x) = wiN (t)Bin (x) i=1

m X N (t x y) = Ni (t)Bim (x y) i=1

and

respectively, this yields the system M2N zttN (t) + AN2 (q )ztN (t) + AN1 (q )zN (t) = BN (q)u(t) + F N (t) Here

M1N zN (0) = g1N

M2N ztN (0) = g2N :

z N (t) = N1 (t) N2 (t) : : : Nm (t) w1N (t) w2N (t) : : : wnN;1 (t) T with ztN (t) de ned analogously, denote the N 1 = (m + n ; 1) 1 approx-

imate state vector coecients. The structure of the approximate system's matrices are given in detail in 3] (see also 5]). For our computational tests, we chose physical parameters based on our experimental eorts. The parameter choices a = 0:6 m, l = 1 m, f = 1:21 kg=m3, c2 = 117649 m2=sec2, b = 1:35 kg=m, x1 = 0:20 and x2 = 0:40 are reasonable for a :6 m by 1 m cavity in which the bounding end beam has a centered piezoceramic patch covering 1=3 of the beam. The beam is assumed to have width and thickness :1 m and 0:005 m, respectively. For a simple set of runs, we used 4 cavity basis functions (mx = my = 4) and 7 beam basis functions. Therefore m = 24 and n = 8 thus giving N = 31. The dimension of the overall system (both plant and state estimator) is 4N . It is worth mentioning that due to the structure of the bilinear form 2, the adaptive estimator utilized in the numerical studies below, was constructed via the method described by (S1 ) in Remark 2.2 with f 0 and a nonzero cD I . 6.1. Constant parameters As a rst test of our scheme, we attempt to identify all three parameters, fEI () cDI () KB()g, which are assumed to be constant in time and space. In this case the actual parameters to be identi ed adaptively are EI = 73:96 Nm2 cDI = 0:001 kgm3=sec KB = 0:0067 N=m2V and the tuning parameters EI (x) = 80 Nm2 cD I (x) = 0:1 kgm3=sec 0 x 0:6 m = 500: Due to the structure of the damping form 2 , we used the scheme (S1) in Remark 2.2 with f = 0 in the selection of the tuning parameters. The Q-inner product is taken to be the weighted Euclidean product on R 3 given by hq piQ = 1 q1 p1 + 1 q2 p2 + 1 q3 p3 1

2

3

q = (q1 q2 q3) p = (p1 p2 p3) 2 Q, where the weights i , i = 1 2 3 are

given by

1 = 2000 2 = 0:02 3 = 5:



155

The initial conditions for the plant state and the state estimator are taken to be zero and the initial guesses for the three parameters are c (0) = 55 Nm2 cd EI D I (0) = 0:0005 kgm3=sec KcB (0) = 0:0090 N=m2V:

In Figure 2(a)-(c), we see both the actual (dashed) and the estimated (solid) values of the three parameters. All three parameters converge after three seconds to within 2% of the actual value. The overshoot appearing in the estimate of cd D I (t) can be reduced by tuning the parameters EI (x) cDI (x) and the gains and i , i = 1 2 3. The high frequency oscillations observed in the rst two seconds of the time history of KcB (t) was also observed in a similar study by Rosen and Demetriou in 24] for the adaptive identi cation of a exible cantilevered Euler-Bernoulli beam. 6.2. Functional parameters Continuing with our test, we now assume that EI () and cD I () are spatially dependent and that KB is a constant. The actual values are taken to be 8 < 73:96 Nm2 if x 2 0:0 0:2) EI (x) = : 125:4 Nm2 if x 2 0:2 0:4) 73:96 Nm2 if x 2 (0:4 0:6] 8 < 0:00100 kgm3=sec if x 2 0:0 0:2) cD I (x) = : 0:00125 kgm3=sec if x 2 0:2 0:4) 0:00100 kgm3=sec if x 2 (0:4 0:6]

and KB = 0:0067 N=m2V . The tuning parameters are

EI

8 < 80:00 Nm2 if x 2 0:0 0:2) (x) = : 135:0 Nm2 if x 2 0:2 0:4) 80:00 Nm2 if x 2 (0:4 0:6]

8 < 0:01 kgm3=sec if x 2 0:0 0:2) cD I (x) = : 0:01 kgm3=sec if x 2 0:2 0:4) 0:01 kgm3=sec if x 2 (0:4 0:6]

and = 500, while the initial estimates are

8 < 66:00 Nm2 if x 2 0:0 0:2) c EI(x 0) = : 110:0 Nm2 if x 2 0:2 0:4) 66:00 Nm2 if x 2 (0:4 0:6]

8 < 0:0009 kgm3=sec if x 2 0:0 0:2) cd D I (x 0) = : 0:0011 kgm3=sec if x 2 0:2 0:4) 0:0009 kgm3=sec if x 2 (0:4 0:6]


156


and KcB (0) = 0:0075 N=m2V . For this case, the Q- inner product is taken to be

hq(x) p(x)iQ =

Z 0:6 q11p11

q12p12 ( x ) + 0 0 : 2] 11 12 0:20:4](x) 0 + q13 p13 0:40:6](x) + q21 p21 00:2](x) 13 21

q p q p 23 23 22 22 + 0:20:4](x) + 0:40:6](x) dx + q3 p3 22

23

3

with the weights given by 11 = 12 = 13 = 3500 21 = 22 = 23 = 0:2 3 = 0:5: Figure 3(a)-(c) shows the actual (solid) value for EI (), cD I (), and KB , the time history for KcB (t) and the time estimates at 0 (dashed) and at 2.5 sec c (x t) and cd (dotted) for EI D I (x t). The functional parameters converge to the true values after about 2.5 seconds within a 2% of the actual values. It c (x t) (Figure 3(a)) should be noted that the stiness functional parameter EI converges to the actual parameter EI (x) to within a 0:1% in less that 2.5 seconds which explains why the two graphs are indistinguishable. Finally, the piezoceramic parameter KcB (t) converges within a 5% of the actual value in 3 sec. Once again, the high frequency oscillations observed above, are also present in the time estimate of KB . 6.3. Slowly time varying parameters In this example, we attempt to identify the two time varying parameters d 1 d 1 dt EI (t) = ; 4 EI (t) + 20 dt cD I (t) = ; 4 cD I (t) + 0:000275 c (0) = 68:00 Nm2, and where EIss = 80 Nm2, EI (0) = 73:96 Nm2, EI ; 4 3 cD Iss = 11:0 10 kgm =sec, cD I (0) = 10:0 10;4 kgm3=sec, cd D I (0) = ; 4 3 B 2 11:5 10 kgm =sec. The parameter K = 0:0067 N=m V is assumed to be known. In this case the tuning parameters and adaptive gains are EI = 80 Nm2 cD I = 0:01 kgm3=sec = 1 = 1000 2 = 0:0003 Figure 4 shows the time estimates of EI (t) and cD I (t). The time estimate c EI (t) of the time varying stiness parameter converges to the actual value within 1 sec whereas the time estimate cd D I (t) of the damping parameter converges much later at around 5 sec to within a 2% of the actual time varying cD I (t). 6.4. Systems with no damping The absence of damping allowed the use of the adaptive scheme (2.19), (2.20) with 2 = 1 . For this case the parameter cD I (x) 0 for 0 x L, and KB = 0:0067 N=m2V is assumed to be known. The parameter to be c (0) = 68 Nm2. The estimated is EI = 73:96 Nm2 and the initial guess is EI tuning parameters and gains (using the scheme given by (2.19), (2.20)) are EI = 80 Nm2 cD I = 0:01 kgm3=sec = 1 = 1000: ESAIM: Cocv, May 1998, Vol. 3, 133{162


157

80

75

70

stiffness parameter

65

60

55

50 0

1

2

3

4

5

6

7

8

5

6

7

8

5

6

7

8

(a) 0.03 0.025 0.02 0.015 damping parameter

0.01 0.005 0 -0.005 -0.01 0

1

2

3

4

(b) 0.05 0.04 0.03 input parameter

0.02 0.01 0 -0.01 -0.02 -0.03 0

1

2

3

4

(c) Figure 2. Actual and estimated parameters constant (a)-

(c) (example 6.1).


158

H.T. BANKS AND M.A. DEMETRIOU 130

120

110 solid: actual

100

dashed: t=0 sec 90

dotted: t=2.5 sec

80

70

60 0

0.1

0.2

0.3 x

0.4

0.5

0.6

(a) -3

1.4

x 10

solid: actual 1.3

dashed: t=0 sec

1.2

dotted: t=2.5sec 1.1

1

0.9

0.8 0

0.1

0.2

0.3 x

0.4

0.5

0.6

(b) 0.06 0.05 0.04 0.03 0.02 0.01 0 -0.01 -0.02 -0.03 -0.04 0

1

2

3

4 Time (sec)

5

6

7

8

(c) Figure 3. Actual and estimated parameters functional (a)-

(c) (example 6.2).



159

80

78

76

74

(a)

72

70

68 0

1

2

3

4 Time (sec)

5

6

7

8

-3

2.4

x 10

2.2 2 1.8

(b)

1.6 1.4 1.2 1 0.8 0

1

2

3

4 Time (sec)

5

6

7

8

Figure 4. Actual and estimated values of time varying pac (t) (solid), (b) rameters (a) stiness EI (t) (dashed) and EI d damping cD I (t) (dashed) and cD I (t) (solid).

The time estimate of EI converges to a 0:5% of the actual value in 1/2 sec as seen in Figure 5. Even when the system has no damping at all, the estimator can still estimate the parameter. ESAIM: Cocv, May 1998, Vol. 3, 133{162

160

H.T. BANKS AND M.A. DEMETRIOU 80

78

76

74

72

70

68 0

0.5

1 Time (sec)

1.5

2

Figure 5. Actual and estimated values of EI - no damping.

Conclusions In this paper, we have extended and modi ed previous adaptive estimation schemes for hyperbolic systems that have non-symmetric, or even absent, damping bilinear form. In addition, a modi cation was added for systems that had slowly time varying parameters which converge to their steady state values asymptotically (or exponentially) in time. Of course, for all modi cations, the adaptive estimator can identify both constant (in space) and functional parameters. The estimation of parameters in the input operator, bounded or unbounded, was also incorporated to the above modi cations. The proposed estimator, like the one given in 15], is in nite dimensional and therefore cannot be implemented. An approximation scheme was presented, which was similar to the one in 15]. Since the estimator required full plant data, a modi cation to the nite dimensional scheme was added to include a nite dimensional approximation to the plant data. This in a way can be viewed as an adaptive state and parameter estimator that uses input and output (through the nite dimensional approximation assumptions on the plant) information only. A possible extension and further study to the above is the incorporation of an adaptive observer which adaptively estimates both the parameters and the state using only input and output data through a more general output operator. A possible point of reference would be the work of Lilly in 19], wherein the system is divided in two subsystems, a nite dimensional dominant system and a stable in nite dimensional residual system. Further yet, another direction towards adaptive observers would be the one used 7.



161

in 13] for a special class of in nite dimensional systems. In addition, the eect of noisy data can be taken into account by introducing robust adaptive schemes, as this was studied for the parabolic case in 17]. Further extension is the improvement of the numerical algorithms so that for fast systems the adaptive estimator can be applicable in real time. One such extension might be the incorporation of a hybrid adaptive parameter estimator in which the parameters are updated not in a single time unit but every several time units. References

1] A.M. Annaswamy, K.S. Narendra: Adaptive control of a rst order plant with timevarying parameter, In Proceedings of the 1989 American Control Conference, 1989, 975{980. 2] A.M. Annaswamy, K.S. Narendra: Adaptive control of simple time-varying systems, In Proceedings of the 28th IEEE Conference on Decision and Control, Tampa, Florida, 1989, 1014{1018. 3] H.T. Banks, M.A. Demetriou: Adaptive parameter estimation of hyperbolic distributed parameter systems with non-symmetric damping and slowly time varying parameters: Convergence proofs and numerical results, Center for Research in Scientic Computation Report 96-31, North Carolina State University, Raleigh, 1996. 4] H.T. Banks, M.A. Demetriou, R.C. Smith: Adaptive parameter estimation in a 2d structural acoustic model with piezoceramic actuator, In Proceedings of the 1994 American Control Conference, Baltimore, Maryland, 1994. 5] H.T. Banks, W. Fang, R.J. Silcox, R.C. Smith: Approximation methods for control of structural acoustics models with piezoceramic actuators, Journal of Intelligent Material Systems and Structures, 4, 1993, 98{116. 6] H.T. Banks, K. Ito: A unied framework for approximation and inverse problems for distributed parameter systems, Control-Theory and Advanced Technology, 4, 1988, 73{90. 7] H.T. Banks, K. Ito, Y. Wang: Well posedness for damped second order systems with unbounded input operators, Dierential and Integral Equations, 8(3), 1995, 587{606. 8] H.T. Banks, R.J. Silcox, R.C. Smith: The modeling and control of acoustic/structure interaction problems via piezoceramic actuators: 2-d numerical examples, ASME Journal of Vibration and Acoustics, 116, 1994, 386{396. 9] H.T. Banks, R.C. Smith: Feedback control of noise in a 2-d nonlinear structural acoustics model, Continuous and Discrete Dynamical Systems, 1, 1995, 119{149. 10] H.T. Banks, R.C. Smith, Y. Wang: Smart Material Structures: Modeling, Estimation and Control, Wiley-Masson, New York, 1996. 11] H.T. Banks, Y. Wang, D.J. Inman, J.C. Slater: Approximation and parameter identication for damped second order systems with unbounded input operators, Control: Theory and Advanced Technology, 10, 1994, 873{892. 12] J. Baumeister, W. Scondo, M.A. Demetriou, I.G. Rosen: On-line parameter estimation for innite dimensional dynamical systems, SIAM J. Control and Optimization, 35(2), 1997, 678{713. 13] R.F. Curtain, M.A. Demetriou, K. Ito: Adaptive observers for structurally perturbed innite dimensional systems, In Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, California, 1997. 14] M.A. Demetriou: Adaptive Parameter Estimation of Abstract Parabolic and Hyperbolic Distributed Parameter Systems, PhD thesis, Department of Electrical Engineering - Systems, University of Southern California, Los Angeles, 1993. 15] M.A. Demetriou, I.G. Rosen: Adaptive identication of second order distributed parameter systems, Inverse Problems, 10(2), 1994, 261{294. 16] M.A. Demetriou, I.G. Rosen: On the persistence of excitation in the adaptive identication of distributed parameter systems, IEEE Trans. on Automatic Control, 39(5), 1994, 1117{1123. ESAIM: Cocv, May 1998, Vol. 3, 133{162

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17] M.A. Demetriou, I.G. Rosen: Robust adaptive estimation schemes for parabolic distributed parameter systems, In Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, California, 1997. 18] J.K. Hale: Ordinary Dierential Equations, Wiley-Interscience, John Wiley and Sons, New York, 1969. 19] J.H. Lilly: Finite dimensional adaptive observers applied to distributed parameter systems, IEEE Trans. on Automatic Control, 38(3), 1993, 469{474. 20] J.L. Lions: Optimal Control of Systems Governed by Partial Dierential Equations, Springer-Verlag, New York, 1971. 21] J.L. Lions, E. Magenes: Non-Homogeneous Boundary Value Problems, I, SpringerVerlag, New York, 1972. 22] K.S. Narendra, A.M. Annaswamy: Stable Adaptive Systems, Prentice Hall, Englewood Clis, 1989. 23] A. Pazy: Semigroups of Linear Operators and Applications to Partial Dierential Equations, Springer-Verlag, New York, 1983. 24] I.G. Rosen, M.A. Demetriou: Adaptive estimation of a exible beam, In Proceedings of the 1993 IEEE Regional Conference on Aerospace Control Systems, Westlake Village, California, 1993. 25] W. Scondo: Ein Modellabgleichsverfahren zur adaptiven Parameteridentikation in Evolutionsgleichungen, PhD thesis, Johann Wolfgang Goethe-Universitat zu Frankfurt am Main, Frankfurt am Main, Germany, 1987. 26] R.E. Showalter: Hilbert Space Methods for Partial Dierential Equations, Pitman, London, 1977. 27] H. Tanabe: Equations of Evolution, Pitman, London, 1979. 28] J. Wloka: Partial dierential equations, Cambridge University Press, Cambridge, 1987.


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