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 adap- tive 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 con- vergence 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.
1. 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where lack of symmetry in the damping form was observed and thus the scheme presented in 15] was no longer applicable.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,
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 Insti- tute, 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.
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 ex- citation, 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 ac- count 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. Further- more, 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 gen- eral and not necessarily parameterized operators. In all cases above, unlike 15] and 25], the parameters in the input operator are assumed to be un- known 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
2. The Plant and the Estimator
Let
H
be a real Hilbert space with inner product hiand corresponding induced norm jj, and letV
1 andV
2 be real reexive Banach spaces with norms denoted by kkV1 and kkV2, respectively. We assume thatV
1 is embedded densely and continuously inV
2 and thatV
2 is embedded densely and continuously inH
. It then follows that (see, for example, 6, 10])V
1,
!V
2,
!H
=H ,
!V
2,
!V
1 (2.1) whereH
,V
2, andV
1 denote the conjugate duals ofH
,V
2, andV
1, respec- tively. Since the embeddings in (2.1) are dense and continuous, 21, 26, 27, 28], we assume that there exist positive constants,K
V2,K
V1, andK
V2V1, such that j'
jK
V2k'
kV2,'
2V
2, that j'
jK
V1k'
kV1,'
2V
1, and thatk
'
kV2K
V2V1k'
kV1,'
2V
1. We denote the usual operator norms onV
2 andV
1 by kkV2 and kkV1, respectively. The duality pairing, de- noted by hiV1V1, is the extension by continuity of the inner product hi fromH
V
1 toV
1V
1 hence, elements'
2V
1 have the representation'
('
) = h' '
iV1V1, see 7, 10]. As it was pointed out in 4, 6, 7, 15],V
2 can either beV
1,H
or some intermediate space, depending on the damping form chosen.Let
Q
be a real Hilbert space (henceforth the parameter space) with inner producthiQand corresponding induced normjjQ, and for eachq
2Q
let 1(q
) :V
1V
1 !R1be a bilinear form onV
1 satisfying(
A
1) (Symmetry)1(q '
) =1(q '
),'
2V
1, andq
2Q
, for at least one tuning parameterq
2Q
, we have(
A
2) (Boundedness)j
1(q '
)j0(q
)k'
kV1kkV1'
2V
1 with 0(q
)>
0 (A
3) (Coercivity) 1(q ''
)0(q
)k'
k2V1'
2V
1 with 0(q
)>
0(
A
4) (Linearity) the mapq
!1(q '
) fromQ
into R1 is linear for each'
2V
1.Similarly, for each
q
2Q
, let 2(q
) :V
2V
2 !R1be a bilinear form onV
2 satisfying(
B
1) (Symmetry)2(q '
) =2(q '
),'
2V
2, (B
2) (Boundedness)j
2(q '
)j 0(q
)k'
kV2kkV2'
2V
2 (B
3) (Coercivity)2(q ''
) 0(q
)k'
k2V2,'
2V
2,(
B
4) (Linearity) The mapq
!2(q '
) fromQ
intoR1 is linear for each'
2V
2,where 0(
q
), 0(q
)>
0, andq
is the same xed element ofQ
appearing in Assumptions (A
2) and (A
3) above.For each
q
2Q
, letb
(q
) :H
V
1!R1be a bilinear form satisfying (C
1) (Boundedness)j
b
(q #
)jjq
jQj#
jkkV1#
2H
2V
1(
C
2) (Linearity) The mapq
!b
(q #
) fromQ
intoR is linear for#
2H
and2V
1.For
q
2Q
de ne the linear operatorsA
1(q
) andA
2(q
) fromV
1 intoV
1 andV
2 intoV
2 byh
A
1(q
)'
i = 1(q '
)'
2V
1h
A
2(q
)'
i = 2(q '
)'
2V
2:
In addition, we de ne the input operatorB
(q
) :H
!V
1 byh
B
(q
)#
iV1V =b
(q #
)#
2H
2V
1:
For
q
2Q
we consider the second order linear initial value problem given in variational form byh
z
tt(t
)'
iV1V1+2(q z
t(t
)'
) +1(q z
(t
)'
) =b
(q f
(t
)'
) (2.2)'
2V
1t >
0z
(0) =z
0z
t(0) =z
1 (2.3) wherez
t denotes time dierentiation,z
0 2V
1,z
1 2H
, andf
2L
2(01H
).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
)z
t(t
)) and the plant inputf
(t
) in order to identify the unknown plant parameterq
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 (
qzf
) withq
2Q
and (zz
t) a solution to the initial value problem (2.2), (2.3) corresponding toq
andf
, for which there exists constants>
0 such thatj
2(p z
t(t
)'
)+1(p z
(t
)'
);b
(p f
(t
)'
)jjp
jQk'
kV2 for almost everyt >
0,p
2Q
and'
2V
1.Following the results in 14], it is possible to specify sucient conditions for a solution (
zz
t) to the initial value problem (2.2), (2.3) to have the necessary regularity for (qzf
) to be an admissible plant.2.1. Non-symmetric Damping
Given an admissible plant (
qzf
) and a tuning parameterq
2Q
, we de ne in the same manner the estimator forq
andz
, denoted by ^q
and ^z
respectively, in the form of the initial value problemh
z
^tt(t
)'
i+2(q
^z
t(t
)'
)+1(q
^z
(t
)'
) +2(^q
(t
)z
t(t
)'
) +1(^q
(t
)z
(t
)'
)=
b
(^q
(t
)f
(t
)'
)+2(q z
t(t
)'
) +1(q z
(t
)'
) (2.4)'
2V
1t >
0ESAIM: Cocv, May 1998, Vol. 3, 133{162
h
q
^t(t
)p
iQ+2(p z
t(t
)z
(t
);^z
(t
)) +1(p z
(t
)z
(t
);z
^(t
));
b
(p f
(t
)z
(t
);z
^(t
)) +2(p z
t(t
)z
t(t
);z
^t(t
))+
1(p z
(t
)z
t(t
);z
^t(t
));b
(p f
(t
)z
t(t
);z
^t(t
)) = 0 (2.5)p
2Q t >
0z
^(0)2V
1z
^t(0)2H q
^(0)2Q:
(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.(
S
1) We can impose the condition that the bilinear form satis es the sym- metry condition (B
1) only when it is evaluated at the tuning parameterq
, namely 2(q '
) =2(q '
) 8'
2V
1 and someq
2Q:
(2.7) (S
2) Replace the non-symmetric term 2(q '
) in (2.4) with the sym-metric term
12
2(q '
)+2(q '
)]'
2V
1:
(2.8) 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 parameterq
2Q
that simultaneously satis es (S
1), (A
2), (A
3), (B
2) and (B
3). Of course, if there exists such aq
, then (S
1) is the preferred modi - cation. If, on the other hand, there does not exist aq
2Q
such that (S
1) is satis ed, then (S
2) must be implemented. With the second option taken, equation (2.4) then becomesh
z
^tt(t
)'
i+ 12 2(q
^z
t(t
)'
)+2(q ' z
^t(t
))]+
1(q
^z
(t
)'
)+2(^q
(t
)z
t(t
)'
)+1(^q
(t
)z
(t
)'
)=
b
(^q
(t
)f
(t
)'
)+ 12 2(q z
t(t
)'
) +2(q 'z
t(t
))]+
1(q z
(t
)'
)'
2V
1t >
0We note that the parameters
q
2Q
and>
0 can be thought of as gains, or tuning parameters, which are used to \tune" the estimator, see 15]. Of courseq
must be such that Assumptions (A
2), (A
3), (B
2), and (B
3) (and (S
1) when applicable) are satis ed. We note as well, that weighting theQ
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
2Q
such that Assumptions (A
2)-(A
3), (B
2)-(B
3) and (S
1) or (S
2) are satis ed, one can introduce bilinear forms i() :V
iV
i ! R1 that do not depend on any parameterq
2Q
or necessarily bear any resemblance to the bilinear formsi(q
), and that satisfy Assumptions (A
1)-(A
3) and (B
1)-(B
3) with known bounds. That is, let 1() :V
1V
1 ! R1 be a bilinear form onV
1 satisfying(
A
1) (Symmetry) 1('
) = 1('
),'
2V
1,(
A
2) (Boundedness)j1('
)j0k'
kV1kkV1, 0>
0,'
2V
1,(
A
3) (Coercivity) 1(''
)0k'
kV1, 0>
0,'
2V
1, and let 2() :V
2V
2!R1 be a bilinear form onV
2 satisfying (B
1) (Symmetry) 2('
) = 2('
),'
2V
2,(
B
2) (Boundedness)j2('
)j 0k'
kV2kkV2, 0>
0,'
2V
2, (B
3) (Coercivity) 2(''
) 0k'
k2V2, 0>
0,'
2V
2.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
= 12.Then, the state and parameter estimator equivalents of (2.4) - (2.6) are now given by
h
z
^tt(t
)'
i+ 2(^z
t(t
)'
)+ 1(^z
(t
)'
) +2(^q
(t
)z
t(t
)'
) +1(^q
(t
)z
(t
)'
)=
b
(^q
(t
)f
(t
)'
)+ 2(z
t(t
)'
) + 1(z
(t
)'
)'
2V
1t >
0 (2.9)h
q
^t(t
)p
iQ+2(p z
t(t
)z
(t
);z
^(t
)) +1(p z
(t
)z
(t
);z
^(t
));
b
(p f
(t
)z
(t
);z
^(t
)) +2(p z
t(t
)z
t(t
);z
^t(t
))+
1(p z
(t
)z
t(t
);z
^t(t
));b
(p f
(t
)z
t(t
);z
^t(t
)) = 0 (2.10)p
2Q t >
0z
^(0)2V
1z
^t(0)2H q
^(0)2Q:
(2.11) Assumptions (A
4) 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 f
X
hiXg be the Hilbert space de ned byX
=V
1H
and intro- ducing an additional tuning parameter, we de ne theX
-inner producth
'
iX =f1('
11) +h'
22ig+h'
12i+h1'
2i+ 2('
11) for'
= ('
1'
2),= (12) 2X
,2R. It can easily be shown 15] that if>
0 is suciently large, then hiX is in fact an inner product onX
and moreover, that the corresponding induced norm, jjX, is equivalent to the norm onX
induced by the more standard inner product onX
given by(
'
)X = 1('
11) +h'
22ifor
'
= ('
1'
2), = (12) 2X
. 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 letY
=V
1V
1 be endowed with the normk
'
kY = k'
1k2V1 +k'
2k2V112for
'
= ('
1'
2)2Y
. ThenY
is a reexive Banach space andY ,
!X ,
!Y
with the embeddings dense and continuous.We rewrite (2.9), (2.10) in rst order form as
dt d
x
(t
)q
^(t
)
=
A B(
t
);B(
t
) 0
x
(t
)q
^(t
)
+F(
t
)t >
0 (2.12)ESAIM: Cocv, May 1998, Vol. 3, 133{162
where A2L(
YY
) andB(t
)2L(QY
), are given byA
'
=0
I
;A
1
;A
2
'
1'
2
'
= ('
1'
2)2Y
(2.13) andB(
t
)q
=0
;
A
2(q
)z
t(t
);A
1(q
)z
(t
) +B
(q
)f
(t
)
q
2Q
(2.14)respectively (the operators
A
1(q
) andA
2(q
) are as they have been de ned above), A1 2 L(V
1V
1) and A2 2 L(V
2V
2) are the operators that corre- spond to the bilinear forms 12 and are de ned similar to the operatorsA
1(q
)A
2(q
), B(t
) 2 L(YQ
) is the Banach space adjoint of B(t
) (that is, recalling that above embeddings, that hB(t
)'q
iQ = h'
B(t
)q
iX,'
2Y
,q
2Q
), and the forcing term isF(
t
) =2
4
0
A
2
z
t(t
) +A1z
(t
)B(
t
) (z
(t
)z
t(t
))3
5
t >
0:
(2.15) Since (qzf
) is assumed to be an admissible plant, then F 2L
2(0T
(Y
Q
) ) for allT >
0. We assume thatz
is such thatA
2(q
)z
t(t
)+A
1(q
)z
(t
)2H
,t
0,q
2Q
, and that A2z
t(t
) +A1z
(t
) 2H
,t
0. In addition we assume that the mapt
!A
2(q
)z
t(t
)+A
1(q
)z
(t
), for eachq
2Q
, is strongly continuously dierentiable inH
. Let X =X
Q
, let the domainD beD=f(
'q
)2X :'
= ('
1'
2)2Y
and A1'
1+A2'
2 2H
g and de ne the family of operators nA~(t
)ot0
, ~A(
t
) :DX !X by~
A(
t
) =
A B(
t
);B(
t
) 0
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 mapt
!~
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 andr
(t
) := ^q
(t
);q
the parameter error, with (qzf
) 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.2.2. Systems with no Damping
In this subsection we consider systems with no damping term, given in weak form by
h
z
tt(t
)'
iV1V1 +1(q z
(t
)'
) =b
(q f
(t
)'
)'
2V
1t >
0 (2.16)z
(0) =z
0z
t(0) =z
1 (2.17) wherez
0 2V
1,z
1 2H
, andf
2L
2(01H
). In this case we only utilize a Gelfand triple withV
1,
!H
'H ,
!V
1, 28].The well posedness of the system (2.16), (2.17) with bounded input oper- ator 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 (
zz
t) to (2.16), (2.17) to have the necessary regularity for (qzf
) to be an admissible plant.The corresponding de nition for an admissible plant, in this case, is the triple (
qzf
) with (zz
t) a solution to (2.16), for which there exist>
0 such thatj
1(p z
(t
)'
);b
(p f
(t
)'
)jjp
jQk'
kV2 (2.18) for almost everyt >
0,p
2Q
and'
2V
1. In the proposed state and param- eter estimator below, we utilize a damping sesquilinear form2 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 (bothk
e
(t
)kV1 and je
t(t
)j) to zero this will become clearer in the ensuing stability analysis. In this case the proposed state and parameter estimators take the formh
z
^tt(t
)'
i+2(q
^z
t(t
)'
)+1(q
^z
(t
)'
) +1(^q
(t
)z
(t
)'
)=
b
(^q
(t
)f
(t
)'
)+2(q z
t(t
)'
) +1(q z
(t
)'
)'
2V t >
0(2.19)h
q
t(t
)p
iQ+1(p z
(t
)z
(t
);z
^(t
));b
(p f
(t
)z
(t
);z
^(t
))+
1(
p z
(t
)z
t(t
);z
^t(t
));b
(p f
(t
)z
t(t
);z
^t(t
))] = 0 (2.20)p
2Q t >
0z
^(0)2V
1z
^t(0)2H q
^(0)2Q
(2.21) where 2 (a design term) is chosen to satisfy Assumptions (A
1) - (A
4). We note, however, that by employing a 5-space setting as in (2.1),2 could al- ternatively 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 or- der systems with no damping in a 3-space setting where that author chooses 2(q
) of the estimator to be the same as1(q
) of the system.As in the case of subsection 2.1, we de ne the space
X
=V
1H
with the same inner producth
'
iX=f1(q '
11) +h'
22ig+h'
12i+h1'
2i+2(q '
11) for'
= ('
1'
2), = (12) 2X
, whereq
2Q
is as it was de ned in Assumptions (A
2), (A
3) (or equivalently in Assumptions (B
2), (B
3)).ESAIM: Cocv, May 1998, Vol. 3, 133{162
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 (A
1) - (A
4), 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 parameterq
. This can be done since2 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
=V
1V
1, endowed with the normk
'
kY = k'
1k2V1 +k'
2k2V112for (
'
1'
2)2Y
withY ,
!H ,
!Y
and the embeddings dense and contin- uous. We rewrite (2.19)-(2.21) in rst order form asdt d
x
(t
)q
^(t
)
=
A B(
t
);B(
t
) 0
x
(t
)q
^(t
)
+F(
t
)t >
0 where in this case A2L(YY
) and B(t
)2L(QY
), are now given byA
'
=0
I
;
A
1(q
) ;A
2(q
)
'
1'
2
'
= ('
1'
2)2Y
andB(
t
)q
=0
;
A
1(q
)z
(t
) +B
(q
)f
(t
)
q
2Q
respectively (once again the operators
A
1(q
) andA
2(q
) are as they have been de ned in subsection 2.1 withA
2(q
) now in L(V
1V
)), B(t
) 2 L(YQ
) is again the Banach space adjoint of B(t
), andF(
t
) =2
4
A
2(q
)z
t(t
) +0A
1(q
)z
(t
)B(
t
) (z
(t
)z
t(t
))3
5
t >
0:
From the equivalent de nition of the plant, equation (2.18), we have F 2
L
2(0T Y
) for allT >
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 Aoperator is now given by (2.13) and F(
t
) by (2.15)3. Stability and Convergence We now establish the convergence of the state estimator
tlim!1k
e
(t
)kV1 = 0 and limt!1
j
e
t(t
)j= 0and, with the additional assumption of persistence of excitation, parame- ter convergence. That is, limt!1j
r
(t
)jQ = limt!1jq
^(t
);q
jQ = 0. We assume throughout this section that (qzf
) is an admissible plant (see De nition 2.1).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 thate
andr
satisfy the linear, homogeneous, non-autonomous, initial value problem given in weak or variational form byh
e
tt(t
)'
i+ 2(e
t(t
)'
)+ 1(e
(t
)'
)+2(r
(t
)z
t(t
)'
)+1(r
(t
)z
(t
)'
)=
b
(r
(t
)f
(t
)'
)'
2V
1t >
0 (3.1)h
r
t(t
)p
iQ;2(p z
t(t
)e
(t
));1(p z
(t
)e
(t
))+b
(p f
(t
)e
(t
));
2(p z
t(t
)e
t(t
));1(p z
(t
)e
t(t
)) +b
(p f
(t
)e
t(t
)) = 0 (3.2)p
2Qt >
0e
(0)2V
1e
t(0)2H r
(0)2Q:
(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>
max(
K
V1K
V01K
V220 )
(3.4)then there exist constants
,k >
0 such that for allt >
0k
e
(t
)k2V1+je
t(t
)j2+jr
(t
)j2Q+Z t0
k
e
(s
)k2V1 +ke
s(s
)k2V2ds
where =k
nke
(0)k2V1+je
t(0)j2+jr
(0)j2Qo.Proof. Following the treatment in 15], we de ne the energy functional,
E
: 01)!R1, for the system (3.1), (3.2) byE
(t
) = 1(e
(t
)e
(t
)) +je
t(t
)j2+ 2he
(t
)e
t(t
)i + 2(e
(t
)e
(t
))+jr
(t
)j2Q=
e
(t
)e
t(t
)
2
X +j
r
(t
)j2Q:
(3.5) Then, using (3.1), (3.2), and assumptions (A
3) and (B
3), we obtainE
t(t
) = f21(e
(t
)e
t(t
)) + 2he
tt(t
)e
t(t
)ig+ 2he
t(t
)e
t(t
)i + 2he
(t
)e
tt(t
)i+ 22(e
t(t
)e
(t
)) + 2hr
t(t
)r
(t
)iQ= ;2
2(e
t(t
)e
t(t
));21(e
(t
)e
(t
)) + 2je
t(t
)j2 (3.6);2
0ke
(t
)k2V1; 2 0;2K
V22ke
t(t
)k2V2;
0 ke
(t
)k2V1+ke
t(t
)k2V2for some
0 = minf202 0;2K
V22g>
0, since, by assumption,>
K
V22=
0. Consequently,E
(t
) +0Z t0
k
e
(s
)k2V1+ke
s(s
)k2V2ds
E
(0):
(3.7)ESAIM: Cocv, May 1998, Vol. 3, 133{162
From assumptions (
A
2) and (B
2), see also 15], we nd thatE
(t
)Rfke
(t
)k2V1+je
t(t
)j2+jr
(t
)j2Qg (3.8) for someR>
0 and from (3.5) that L ke
(t
)k2V1+je
t(t
)j2+jr
(t
)j2QE
(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 im- posing any additional assumptions. Let (
qzf
) be an admissible plant. If>
maxfK
V1K
V1=
0K
V22=
0g, then the energy functionalE
given by (3.5) is non-increasing,tlim!1k
e
(t
)kV1 = 0 and limt!1
j
e
t(t
)j= 0:
(3.10) 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
V
1 =V
2, inspection of the de nition ofE
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>
max(
K
V1K
V1;0 0K
V220 )
:
In order to establish parameter convergence we require the notion of per- sistence of excitation.
Definition 3.3. A plant (
qzf
) is said to be persistently excited via the inputf
, if there existsT
000>
0 such that for eachp
2Q
withjp
jQ= 1 and eacht
1>
0 suciently large, there exists a ~t
2t
1t
1+T
0] such that
Z
~t+0
~t h
A
2(p
)z
()i+hA
1(p
)z
()i;hB
(p
)()id
V1
0:
With the above condition assumed, we have that if the plant (qzf
) is persistently excited thentlim!1j
r
(t
)jQ= 0:
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