A Class of Robust Adaptive Controllers for Innite
Dimensional Dynamical Systems
M. A. Demetriou
K. Ito
Center for Research in Scientic Computation
Department of Mathematics
North Carolina State University
Raleigh, NC 27695-8205
August 25, 1995
Abstract
An adaptive controller for a perturbed innite dimensional plant is developed to force the state of the plant track the state of a reference model. The reference model is based on the nom-inal plant that has a physical similarity with the plant. Using a Lyapunov stability argument, which is based on theH
1-Riccati equation of the nominal plant, an adaptive law is developed
for the adjustment of the feedback gain. It is proved that the closed-loop system is stable with the tracking error remaining bounded, and converging to zero provided that the norm of the structured perturbation is less than a specied attenuation bound. Results of numerical studies regarding a heat equation and a beam equation are presented to demonstrate the applicability of the proposed control algorithm.
1 Introduction
We consider an innite dimensional plant whose dynamics are governed by a linear evolution equation of the form
d
dt
xp(t) =Apxp(t) +Bu(t); xp(0) =x 0
2X : (1.1)
The linear operatorAp is an unknown innitesimal generator of aC 0
?semigroup on a Hilbert space
X with inner product and normh;iandjjX, respectively. The input operator B is known and it
is assumed thatB 2L(RIm;X) has nite rank. The objective is to nd a robust dynamic feedback
control law of the form
u(t) =?K(t)xp(t) +r(t) (1.2)
such that the state of the plant xp(t) tracks the state of a reference model xm(t). This reference
model is based on the nominal closed-loop dynamics and is given by
d
dt
xm(t) = (A 0
?BKm)xm(t) +Br(t) (1.3)
Researchsupp ortedinpartbytheAirForceOceofScienticResearchundergrantAFOSRF49620-93-1-0198,
wherer()2L 1
loc(0;1;RIm). The open loop operatorA
0 is the nominal plant generator in the sense
that the unknown plant generatorAp has the following perturbation form
Ap =A 0+
A; A2L(X);
and Km 2L(X ;RIm) is an appropriately chosen stabilizing feedback gain operator for the nominal
plant that will be dened below. The reference signal r(t) may be determined by designing an
optimal control for the reference dynamics (1.3) with certain performance index. For example, it might be chosen to minimize the performance index
Z T
0
jxm(t)?xj 2
+jr(t)j 2
dt+jxm(T)?xj 2
associated with the problem of regulating the statexm(t) to a desired state x2X. Following the
model reference adaptive feedback law [3, 9], the dynamic feedback gain operatorK(t) is given by
d
dt
K(t) =B
0(
xp(t)?xm(t))xp(t) (1.4)
where > 0 is the acceleration parameter. The derivation of the adaptation rule (1.4) is based
on the Lyapunov synthesis approach, [3, 9]. In our approach we choose the nonnegative denite self-adjoint operator 0 to be a solution to the
H
1 Riccati equation (1.8), discussed below, and
Km =B
0
: (1.5)
It is shown in Section 3 that, under certain assumptions on the norm of A and the reference
trajectoryxm(t), the closed-loop system (1.1) - (1.5) has a unique global solution (xp(t);K(t)) and
that the asymptotic tracking property
Z
1
0
jxp(t)?xm(t)j 2
Xdt<1; and e(t) =xp(t)?xm(t)!0 as t!1 (1.6)
is satised (see Theorem 3.2). The control law (1.2) provides a sub-optimal control law for the uncertain plant (1.1) in the sense of nding an optimal control law for the reference model (1.3) and an asymptotic regulator via the adaptive feedback law (1.4). Thus, the problem of constructing a feedback law for the uncertain plant (1.1) can be decomposed into (i) the optimal control problem of r(t) for the reference model (1.3) and (ii) the asymptotic tracking problem in the sense of (1.6).
The following matching condition is normally assumed in the adaptive control literature [9]. There exists a feedback operator, K : RIm !X such that
Ap+BK=Am=A 0
?BKm; (1.7)
which is equivalent to assume that
AR(B):
the adaptive law (1.4) is based on the fact that theH
1feedback control is developed to construct
a feedback law that stabilizes all perturbed plants with a specied uncertainty bound for kAk.
That is, if for given>0, there exists a nonnegative, self-adjoint operator 0 on
X satisfying the
game-theoretic Riccati equation
A
0 0+ 0
A
0 ?
0
BR ?1
B
?
1
2
D D
0+ C
C
x= 0; (1.8)
for all x2D(A
0), then for
K =R ?1
B
0, the closed-loop plant operator
Ap?BK=A 0+
D?C?BK
generates an exponentially stable semigroup on X provided that
k?k<
1
:
Here we assume the structured perturbation
A=D?C ; ?2L(X)
with known operatorsD2L(X) andC 2L(X). It is shown (e.g. [4, 12, 13, 14, 16]) that the least
> 0, such that (1.8) has a nonnegative solution, gives the best stability bound in the robust
linear feedback control, i.e.
= inf
8
>
>
<
>
>
:
>0
9 a linear feedback K2L(RIm;X) such that A
0
?BK+D?C generates an exponentially
stable semigroup, provided thatk?k<
1
9
>
>
=
>
>
; :
Moreover, it can be seen that this robust stabilization problem is equivalent to the standard H 1
-attenuation problem for the state feedback case (see [6, 7]). Throughout this paper we consider the case when D ; C are chosen to be the identity operators in order to account for unstructured
perturbation, and the control weight is taken, for simplicity, to be R =I. All discussions can be
simply extended to the structured perturbation case.
For the sake of completeness of our discussions, the asymptotic tracking property of the constant feedback synthesis
u(t) =?Kmxp(t) +r(t) with Km=B
0 (1.9)
is given in Section 2. Comparison of the performance between the constant feedback synthesis (1.9) and the adaptive control law (1.2), (1.4) is of our interest and will be analyzed in Section 3 in terms of the Lyapunov stability argument, and in Section 5 through numerical testings.
studies of a heat equation with spatially varying parameter and of an Euler-Bernoulli beam with piezoceramic actuators are shown in Section 5 and conclusions and discussion of possible extensions of this work are given in Section 6.
2 Robust Model Reference Control
When the control law (1.9) is applied to the plant equation (1.1), it yields the closed loop system
d
dt
xp(t) = (Ap?BKm)xp(t) +Br(t): (2.1)
The closed loop system (2.1) combined with the reference model (1.3) give, fore(t) =xp(t)?xm(t),
the following state error equation
d
dt
e(t) = (Ap?BB
0)
xp(t) +Br(t)?(A 0
?BB
0)
xm(t)?Br(t)
= (Ap?BB
0)
e(t) + (Ap?A 0)
xm(t) =Ape(t) + Axm(t)
= (A 0
?BB
0)
e(t) + (Ap?A 0)
xp(t) =A 0
e(t) + Axp(t); (2.2)
where Ap = Ap?BB
0, A
0 = A
0 ?BB
0 and
A = Ap ?A 0 =
Ap?A
0. Here, solutions
to (1.3), (2.1), (2.2) are understood in the sense of the mild solution, see [10]. In addition, if
xp(0);xm(0) 2 D(A 0) and
r(t) is continuously dierentiable, then equation (1.3) as well as (2.1)
have a classical solution x(t)2C 1(0
;;X)\C(0;;D(A
0)). Hence, we have that (2.2) holds in the
strong sense. The corresponding Lyapunov function is given by
V(t) =h 0
e(t);e(t)i: (2.3)
We assume here that 0
0 is the solution to the Riccati equation
A
0 0+ 0
A
0 ?
0
BB
?
1
2
I
0+ Q
x= 0 (2.4)
for all x2D(A 0) and
QI. Or, equivalently
A
0 0+ 0
A
0
x=?
0
BB + 1
2
I
0+ Q
x: (2.5)
We assume that
kAk<
1
(2.6) and thus we have that
kAk= 1?; 0<1: (2.7)
The following calculations are understood in the sense of Theorem 3.1 presented in x 3.
Taking the derivative of (2.3) along the solutions of (2.2) and using (2.5) - (2.7), we have that
d
dt
V(t) = ?h( 0
BB
0+ 1
2 00+
Q)e(t);e(t)i+h(Ap?A 0)
e(t); 0
+h 0
e(t);(Ap?A 0)
e(t)i+h(A 0
?Ap)xm(t); 0
e(t)i
+h 0
e(t);(A 0
?Ap)xm(t)i
? 1 2 j 0
e(t)j 2
X ?je(t)j 2
X+ 2j 0
e(t)jX(kAkje(t)jX+kAkjxm(t)jX)
?(1?)
je(t)jX ?
1
j
0 e(t)jX
2
?je(t)j 2 X? 2 j 0
e(t)j 2
X
+2j 0
e(t)jXjxm(t)jXkAk
?je(t)j 2 X? 2 2 j 0
e(t)j 2
X+ 2 2
kAk 2
jxm(t)j 2 X ?min 1; 1 2 2
je(t)j 2
X +j 0
e(t)j 2 X + 2 2
kAk 2
jxm(t)j 2
X
?2min 1; 1 2 2 1 2
je(t)j 2
X+j 0
e(t)j 2 X + 2 2
(1?) 2
2
jxm(t)j 2
X
?c
1
je(t)j 2
X +h 0
e(t);e(t)i
+ 2(1?) 2
jxm(t)j 2
X;
= ?c 1
je(t)j 2
X +V(t)
+c 2
jxm(t)j 2
X; (2.8)
for some positive constantsc 1
; c
2, where we used the completion of squares and the fact that
V(t) =h 0
e(t);e(t)ij 0
e(t)jXje(t)jX
1 2
j 0
e(t)j 2
X +je(t)j 2
X
: (2.9)
This implies that for all t0, the tracking errore(t) satises
V(t)e ?c
1t
V(0) +c 2 Z t 0 e ?c 1
(t?s)
jxm(s)j 2
Xds (2.10)
and Z t
0 je(t)j
2 Xdt 1 c 1
V(0) +c 2
Z t
0
jxm(s)j 2
Xds
: (2.11)
By imposing additional conditions on the state of the reference model, we can show that the function
je(t)j 2
X 2L 1(0
;1) with
dtdh 0
e(t);e(t)iX
<1. Ifje(t)j 2
X is uniformly continuous and integrable,
then it follows from Barbalat's lemma ([3, 9, 11]) that je(t)jX ! 0 as t !1. If the operator 0
is coercive, i.e. h 0
e(t);e(t)i 0
je(t)j 2
X, > 0, then we have that je(t)jX is indeed uniformly
continuous and integrable and thus we can apply Barbalat's lemma to conclude convergence ofe(t)
to zero. While this coercivity condition is true in nite dimensional systems, it is not necessarilly satised by innite dimensional systems. Since no such information of coercivity of 0 can be
extracted from the game-theoretic Riccati equation (2.4), we must nd an alternate route to show convergence of the state error to zero. Thus, assuming
Z
1
0
jxm(t)j 2
Xdt<1; (2.12)
i.e. xm() 2 L 2(0
;1;X), we can show that V(t) ! 0 as t ! 1. Moreover, we can prove that p
h 0
We present this in the form of the theorem below, but rst we prove that p h
0
x;xi indeed
denes a norm on X.
Lemma 2.1
The seminorm p h0
x;xiis a norm on X.
Proof.
We rst note thath 0 x;xi Z 1 0
jS(t)xj 2
Xdt (2.13)
forx2X, provided thatQI, whereS(t) is the semigroup generated byA 0 on
X. Suppose there
exists an element x2X such that h 0
x;xi= 0. Then, from (2.13) we have that Z
1
0
jS(t)xj 2
dt = 0
which implies thatS(t)x= 0, 0t1 since t!jS(t)xjis continuous in [0;1]. Hence, S(0)x= 0
and therefore x= 0. 2
Theorem 2.1
Given >0, assume that the Riccati equation (2.4) has a nonnegative solution 0.Then, if the control (1.9) is applied to the plant (2.1), the tracking error e(t) satises Z
1
0
je(t)j 2 Xdt 1 c 1
V(0) +c 2
Z
1
0
jxm(s)j 2
Xds
and limt!1
V(t) = 0, provided that xm()2L 2(0
;1;X). Moreover,
lim
t!1
e(t) = 0
in the sense that the norm p h
0
e(t);e(t)i!0 as t!1.
Proof.
We rst show that the normV(t)!0 ast!1 using the estimates (2.8) - (2.11). Then,an application of Lemma 2.1 would give the desired results. In addition to the estimate (2.11), using (2.8), we get that
Z t
0 j
0 e(t)j
2 Xdt 1 c 1
V(0) +c 2
Z t
0
jxm(s)j 2
Xds
: (2.14)
which implies that R 1
0
V(t)dt<1. Using the above (and implicitly (2.12)), we get that Z 1 0 d dt V(t)
dtM Z
1
0
je(t)j 2
X+j 0
e(t)j 2 X dt+ Z 1 0
jxm(t)j 2
Xdt
; (2.15)
for some positive constantM >0. It thus follows from (2.9), (2.11), (2.12), (2.14) and (2.15) that Z 1 0 d dt V(t)
dt<1; (2.16)
and Z
1
0
jV(t)jdt<1: (2.17)
Equation (2.17) implies that lim infV(t) = 0. But, (2.16) implies that limV(t) exists since V(t) = V(0) +
Rt 0
d
dtV(s)ds. Hence, limV(t) = lim infV(t) = 0.
Using Lemma 2.1 we have that
lim
t!1
e(t) = 0;
3 Model Reference Adaptive Control
We now proceed, as in the nite dimensional case, see [3, 9], to introduce the adaptive control law for the plant given by equation (1.1). Let us identify the space L(X ;RIm) with Xm. We consider
the adaptive control law of the form
u(t) =?K(t)xp(t) +r(t); (3.1)
where we aim to adjust K(t) 2Xm to the optimal feedback operator of the uncertain plant (1.1)
and not to the optimal feedback operator for the nominal plant, (i.e. Km =B
0). The rationale
behind such a choice is the desire to improve the performance of the nominal controller Km. The
update law for K(t) is given by
d
dt
K(t) =B
0
e(t)xp(t) (3.2)
where > 0; in this case can be viewed as the weighting on the Xm-inner product, or as the
reciprocal of the adaptive gain, see [3, 9]. In the adaptive law (3.1), (3.2), the parameter is a
design parameter that is not necessarily the optimal one as in the non-adaptive case, i.e. (2.4), because it is desired to study the combined eect of robust feedback and adaptation.
The Lyapunov stability argument used below requires regularity of solutions. This is guranteed under the following theorem.
Theorem 3.1 (Ito-Powers, [5])
Consider the abstract evolution equation in X ddt
x(t) = (A?BK(t))x(t) +f(t) (3.3)
with x 0
2 D(A), f 2 C 1(0
;T;X), where we assume that A is the innitesimal generator of C
0
?semigroup S(t) on X and t! BK(t)x; x 2X is continuously dierentiable for each x2 X.
Then, the mild solution dened by
x(t) =S(t)x+ Z t
0
S(t?s)(?BK(s)x(s) +f(s))ds (3.4)
is a strong (classical) solution [10], in the sense that x(t) 2 C(0;T;D(A)), x(t) 2 AC(0;T;X),
ddtx(t)2C(0;T;X).
From Theorem 3.1 all functions that appear in the following stability arguments are strongly dif-ferentiable, assumingx
0
2D(A),r2C 1(0
;T;RIm) and thus the calculations make sense. Then, we
use the continuity of solutionsx(t)2C(0;T;X)to (1.1) with respect to (x 0
;f()) inXL 2(0
;T;X)
and the Lyapunov functions to generalize the stability results to the general initial conditionx 0
2X
and the reference signal r()2L 2(0
;T;RIm). It follows from (1.1), (1.2) and (3.1) that d
dt
e(t) = (Ap?BK(t))xp(t) +Br(t)?(A 0
= (Ap?BK(t) +BKm?BKm)xp(t)?(A 0
?BKm)xm(t)
= (Ap?BKm)xp(t)?(A 0
?BKm)xm(t)?B(K(t)?Km)xp(t)
= Apxp(t)?Apxm(t)?B(K(t)?Km)xp(t) + Axm(t)
= Ape(t)?B(t)xp(t) + Axm(t)
= A 0
e(t)?B(t)xp(t) + Ae(t) + Axm(t); (3.5)
where (t) =K(t)?Km2Xm and Km=B
0. Let us dene the Lyapunov function
V(t) =h 0
e(t);e(t)i+j(t)j 2
Xm: (3.6)
Then, we have
d
dt
V(t) = ?h( 0
BB
0+ 1
2 00+
Q)e(t);e(t)i?2h 0
e(t);Ae(t)i
+2h 0
e(t);Axm(t)i?2h 0
e(t);B(t)xp(t)i+ 2h d
dt
(t);(t)iXm
?min
1;
1 2 2
je(t)j 2
X+j 0
e(t)j 2
X
+ 2(1?) 2
jxm(t)j 2
X
= ?
1
je(t)j 2
X +j 0
e(t)j 2
X
+c 2
jxm(t)j 2
X; (3.7)
for some positive constants 1,
c
2, where we used the fact that
h d
dt
(t);(t)iXm=h 0
e(t);B(t)xp(t)i: (3.8)
This implies that
V(t) + 1
Z t
0
je(s)j 2
X +j 0
e(s)j 2
X
dsV(0) +c 2
Z t
0
jxm(t)j 2
Xds (3.9)
for all t>0. If we assume
Z
1
0
jxm(t)j 2
Xds<1 (3.10)
thenj(t)jXmconst. fort0. Moreover, we can show state error convergence to zero using the
same arguments as in the proof of Theorem 2.1.
Theorem 3.2
Assume that for given>0 andQI, the Riccati equation(2.4) has a nonnegativeself-adjoint solution 0, that the uncertainty bound
kAk< 1
is satised, and that the state of the
reference model has nite energy (i.e. equation(3.10)), then the closed loop system for the adaptive law (3.1) - (3.2) satises
lim
t!1
e(t) = 0;
in the sense that the norm p h
0
e(t);e(t)i!0 as t!1, and u()2L 2(0
;1;RIm).
Proof.
DeneW(t) =h 0
Using (3.7), (3.9) and (3.10) we have that
Z
1
0
d
dt W(t)
dtM Z
1
0
je(t)j 2
X+j 0
e(t)j 2
X
dt+ Z
1
0
jxm(t)j 2
Xdt
;
for some positive constantM, where we used the fact that j(t)jXm const. It thus follows from
(3.9), (3.10) that Z
1
0
d
dt W(t)
dt<1; (3.11)
and that Z
1
0
jW(t)jdt<1: (3.12)
Using arguments similar to those for Theorem 2.1, we conclude that lim
t!1
W(t) = lim
t!1 h
0
e(t);e(t)i= 0: (3.13)
It thus follows from Lemma 2.1 that
lim
t!1
e(t) = 0: (3.14)
Finally, it follows from (3.9) and (3.10) that xp() 2 L 2(0
;1;X) and j(t)jXm < 1 and thus u()2L
2(0
;1;RIm). 2
It is not easy to analyze the performance of the adaptive feedback synthesis against the one of the constant feedback law Km = B
0 in the quantitative manner. However, we can argue
the qualitative improvement of the adaptive law (3.1) over the constant feedback (1.9) as follows. Assume that there exists a feedback lawKp 2Xm such that
kA?BKkkAk; 0 <1: (3.15)
where K =Kp?Km. Then, as above we have
d
dt
e(t) = (Ap?BKp)e(t)?B b
(t)xp(t) + (A?B(Kp?Km))xm(t)
= (A 0
?BKm)e(t)?B b
(t)xp(t) + (A?BK)(e(t) +xm(t)); (3.16)
where (b
t) =K(t)?Kp. Let b
V(t) =h 0
e(t);e(t)i+j b
(t)j 2
Xm (3.17)
Using the same calculation as above, we obtain
d
dt b
V(t) = ?h( 0
BB
0+ 1
2 00+
Q)e(t);e(t)i
?2h 0
e(t);(A?BK)e(t)i+ 2h 0
e(t);(A?BK)xm(t)i
?
1
2
j 0
e(t)j 2
X ?je(t)j 2
X + 2j 0
e(t)jX(kA?BKkje(t)jX +kA?BKkjxm(t)jX)
?
1
2
j 0
e(t)j 2
X ?je(t)j 2
X + 2j 0
? 1 2 j 0
e(t)j 2
X ?je(t)j 2
X + 2
(1?)
j 0
e(t)jXje(t)jX
+2j 0
e(t)jXkAkjxm(t)jX
?(1?)
je(t)jX ?
1
j
0 e(t)jX
2
?(1?(1?))je(t)j 2
X
?(1?(1?)) 1
2 j
0 e(t)j
2
X + 2j 0
e(t)jXkAkjxm(t)jX
?(1?(1?))je(t)j 2
X?(1?(1?)) 1
2 2
j 0
e(t)j 2
X
+ 2
2
(1?(1?))
2
kAk 2
jxm(t)j 2
X
?(1?(1?))min 1; 1 2 2
je(t)j 2
X+j 0
e(t)j 2
X
+ 2
2
(1?(1?))
(1?) 2
2
2
jxm(t)j 2
X
= ?(1?(1?))min 1; 1 2 2
je(t)j 2
X+j 0
e(t)j 2
X
+ 2(1?) 2
(1?(1?))
2
jxm(t)j 2
X
= ?~ 1
je(t)j 2
X +j 0
e(t)j 2
X
+ ~c 2
jxm(t)j 2
X; (3.18)
where we used (3.15) and (2.6), (2.7). The positive constants ~
1 and ~ c
2 are given by
~
1 = (1
?(1?))min 1; 1 2 2
; and ~c
2= 2(1 ?)
2
(1?(1?))
2
:
The equivalent constants from equation (2.8) are given (as they were explicitly found in equation (3.7), or simply using the above with 1) by
1 = min 1; 1 2 2
; and c
2= 2(1 ?)
2
:
Using the fact that 0 < 1 (from (2.7)) and that 0 < 1 (from (3.15)) along with the
inequality 1?(1?), we can show that ~ 1
1 and ~ c
2 <c
2. Hence, it would mean better
tracking and asymptotic behavior of the adaptive law (3.1) comparing to the constant feedback synthesis control law (1.9).
Remark 3.1
In regard to Theorem 3.2, the existence of the nonnegative solution for the Riccati equation (2.4) implies that the pair (A0
;B) is stabilizable.
Remark 3.2
Concerning possible limit of K(t), a more precise condition than (3.15) can beas-sumed. According to our energy estimate (3.18), the best Kp can be chosen such that
(A?BK)
0+ 0(
A?BK)I
is satised for the least 0. In the event that the matching condition (1.7) is satised, we then
Remark 3.3
For the nite dimensional case with the matching condition (1.7) satised and with persistence of excitation, [3, 9, 15], we have K(t)!K. But, Theorem 3.2 only shows that K(t) isbounded, which implies thatK(t) converges to someK
1subsequentially, i.e. there exists a sequence ftng such that K(tn)!K
1 as
n!1 and tn !1.
So, a natural question to ask is what could be a set of subsequential limit of K(t) as t ! 1?
According to the energy estimate (3.18), dtdV(t) is more negative if K(t) = Kp provides the least
upper bound for the following inequality (A?BK)
+ 0(
A?BK)I; (3.19)
for0. These heuristic observations make sense in the following argument. Suppose the matching
condition (1.7) is satised and (3.19) is satised with = 0 andK =K where Ap?BK =Am in
(1.7). This matches with the fact that the asymptotic limit of K(t) isK if it converges.
4 Well-Posedness
In this section we establish the well-posedness of the adaptive law (3.1), i.e., the existence of solutions to the closed loop (2.1), (3.1), (3.2) and (3.5).
Theorem 4.1
For arbitrary initial condition (x 0;K(0)) 2 X Xm, there exists a > 0 such
that the system of equations (2.1), (3.1) (3.2) and (3.5) has a solution (x(t);K(t)) in C(0;;X) AC(0;;Xm) with t!K(t)x being continuous for eachx2X. Moreover, if(2.4) is satised then
there exists a unique global solution.
Proof:
The proof is based on the Banach xed-point theorem. Let Y = C(0;;Xm) and denethe mapping on Y by c
K = (K) where
x(t) =S(t)x 0
? Z t
0
S(t?s)(BK(s)x(s)?Br(s))ds (4.1)
and
c
K(t) =K(0) + Z t
0
(B
0
e(s)x(s))ds; (4.2)
with e(t) =x(t)?xm(t). Let S be the closed subset inY dened by
S = (
max
t2[0;]
jK(t)jXm )
Then, we prove
(i) that forjK(0)jXmthere exists a >0 such that :S !S
(ii) and that is a contraction,
j (K 1)
? (K 2)
jY jK 1
?K
2
jY; K 1
;K
2
Proof of (i): Let kS(t)kMe!t where S(t) is the semigroup generated byAp. Then, from (4.1)
jx(t)jX Me!tjx 0
jX+ Z t
0 Me!
(t?s)
jBj(jx(s)jX+jr(s)jRm) ds
M~
jx
0
jX +jBj Z t
0
jr(s)jRmds
+MjBj~ Z t
0
jx(s)jXds
c 1+ c 2 Z t 0
jx(s)jXds:
where ~M =Me! and thusMe!tM~ fort2[0;],
c
1 = ~ M
jx
0
jX +jBj Z
0
jr(s)jRmds
; c
2 = ~ MjBj:
By Gronwall's inequality, we have
jx(t)jX c 1
ec 2
; 0t: (4.4)
From (4.2)
j c
K(t)jXm jK(0)jXm+c Z t
0
je(s)jXjx(s)jXds
jK(0)jXm+c(c 3+
c
1 ec
2) c
1 ec
2
F
():where c =jB
0
jXm and we assumed that jxm(t)jX c 3 on [0
;]. Since !
F
() is continuousand monotonically increasing, it follows that for jK(0)jXm, there exists a 0
>0 such that
F
( 0);
and thus :S!S with = 0.
Proof of (ii): Let c
Ki = (Ki) for Ki 2 S; i = 1;2. It follows from (4.1) that if we dene (t) =x
1( t)?x
2(
t), then
(t) =? Z t
0
S(t?s)B[(K 1(
s)?K 2(
s))x 1(
s) +K 2(
s)(s)]ds
Thus, we have as above
j(t)jX c 2 ec 2 Z t 0 jK 1( s)?K
2(
s)jXmjx 1(
s)jXdsc 2 c 4 ec 2 jK 1 ?K 2 jY;
where we assumed that jx 1(
t)jX c 4 on [0
;]. Hence, using (4.2) and the above we have
j c
K
1( t)?
c
K
2(
t)jXm c Z t
0
(jx 1(
s)?x 2(
s)jXjx 2(
s)jX+je 1(
s)jXjx 1(
s)?x 2(
s)jX)ds
cc 2 c 4 2 ec 2[
c
4+ ( c 4+ c 3)] jK 1 ?K 2 jY:
We may choose 0<
0such that = 1 cc 2 c 4 2 ec 2[
c 4+( c 4+ c 3)]
<1 and thus (4.3) is satised.
By the Banach xed point theorem, there exits a unique xed pointK() inS which denes a local
Suppose (2.6) is satised, then it follows from (3.9) that
jK(t)jXmjK(0)jXm+ const
jx
0
jX+jK(0)?KmjXm+ Z t
0
jxm(s)j 2
Xds
; t2[0;]
which implies that a locally dened solution K() is a unique global solution. 2
5 Examples and Numerical Results
Here we present two examples to demonstrate the feasibility of our proposed adaptive scheme.
Remark 5.1
The examples presented below cannot be treated within our theoretical framework in Sections 3 and 4, because A 2= L(X ;X). However, the plant generator in these two examplesgenerates an analytic semigroup in X. Thus, we can restate Theorem 3.2 for the case of analytic
semigroups as follows.
Theorem 5.1
We assume that A 0 andAp generate an analytic semigroup on X. Furthermore we
assume that (A1) (A)
0+ 0
A+Q>I, for some >0
(A2) Axm()2L 2
loc(0;1;X).
Then the closed loop system is well posed. Moreover, if Axm()2L 2(0
;1;X), then
lim
t!1
e(t) = 0;
and u()2L 2(0
;1;RIm).
Proof:
It follows from (3.5) - (3.9)d
dt
V(t) = ?2hAe(t); 0
e(t)i?hQe(t);e(t)i?h( 0
BB
0+ 1
2 00)
e(t);e(t)i
+2h 0
e(t);Axm(t)i:
Thus, from (A1)
d
dt
V(t)?min
;
1 2 2
je(t)j 2
X+j 0
e(t)j 2
jX
+ 2 2
jAxm(t)j 2
X;
which implies that
V(t) + min
;
1 2 2
Z t
0
je()j 2
X+j 0
e()j 2
X
d V(0) + 2 2
Z t
0
jAxm()j 2
Xd:
Using (A2) and similar arguments as in the proof of Theorem 3.2, we can show that Z
1
0
je()j 2
Xd <1
and p h
0
Remark 5.2
A detailed discussion concerning assumption(A1) will appear in a forthcoming paper.In nite dimensional systems it resembles the denition of quadratic stability, [8]. The second assumption of Theorem 5.1 is satised ifxm(0)2D(Ap) andr()2L
2
loc(0;1;RIm).
Example 1:
We consider the heat diusion equation in the interval [0;1] given by @xp(t;)@t
=ap @
2
xp(t;) @
2 + bp
@xp(t;) @
+cpxp(t;) +B()u(t);
with boundary and initial conditions
xp(t;0) = 0 =xp(t;1); xp(0;) =x 0
2L 2
(0;1):
In this case we have that the operator Ap is given by
Ap'() = d
d
apq() d'()
d
+bp d'()
d
+cp'();
with D(Ap) =H 2(0
;1)\H 1
0(0
;1), where we choose
q() = 1:0?
1
2 sin [2(?0:25)]; 0 1:
The nominal plant operator A
0 is chosen to be
A
0
'() =a 0
d 2
'() d
2 + c
0 '():
In our calculations we have chosen the following numerical values for the coecients of Ap and A 0
to be
ap = 1:510 ?2
bp = 1:010 ?2
cp = 2:010 ?2
and a
0 = 0 :510
?2
c
0 = 2 :010
?3
:
The input operator B is taken to be
B() = [0:3;0:7](
); 0 1;
and the reference signal is given by
r(t) =
3:0 + 0:1sin
t
50
e ?0:01t
:
All computations are carried out by a numerical approximation method using nite element methods with the linear spline elements and the Fehlberg fourth-fth order Runge-Kutta method for time integration. The numerical implementation of the proposed adaptive law and its nite dimensional approximation and convergence proofs will appear in greater detail in a forthcoming paper.
0 10 20 30 40 50 60 70 80 90 100 0
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01
Time (sec)
|e(t)|
non−adaptive
adaptive
Figure 1: State error,e(t) =xp(t)?xm(t): adaptive and non-adaptive case.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Kp Km
K(100)
ξ
converges to zero faster that the non-adaptive case where the constant feedback law (1.9) was used. This agrees with the theoretical estimates ofx2 andx3 concerning the constants
1 and ~
1 that are
related to the tracking of the state error. Our calculations were carried out by choosing the initial values of the plant and reference model states and the initial guess of the estimate K(t) at t= 0
as xp(0;) = 1:010 ?2sin(
), xm(0;) = 0 andK(0)0, respectively.
In Figure 2, we plot the nominal feedback gain Km (solid line), and the nal value of the
estimate K(100) (dashed line). In addition, we plot the feedback gain Kp (dotted line) which is
the feedback gain that corresponds to the \optimal" gain based on the actual plant operator Ap,
i.e. Kp=B
p where p solves
A
pp+ pAp?p
BB
?
1
2
I
p+Q
x= 0; 8x2 D(Ap):
This shows that the adaptive lawK(t) is adjusted to some optimal feedback gain for the uncertain
Example 2:
As a second example, we consider a one dimensional Euler-Bernoulli beam with Kelvin-Voigt viscoelastic damping. It is assumed that a single piezoceramic patch located in the center of the beam is used for actuation, see [2]. The underlying equation is given by@ 2
xp(t;) @t 2 + @ 2 @ 2 EIp @ 2
xp(t;) @
2 + cDIp
@ 3
xp(t;) @ 2 @t ! = @ 2 @ 2 KB [
1;2 ](
)u(t)
+f(t;) (5.1)
with boundary and initial conditions
xp(t;0) =
@xp(t;0) @
=xp(t;l) =
@xp(t;l) @
= 0;
xp(0;) =w 0(
) and
@xp(0;) @t
=w 1(
); 0 l :
If we write (5.1) in a rst order form (see [1]) on the state space X=H 2
0(0 ;l)L
2(0
;l), then the
plant generator is given by
Ap = "
0 I Kp Dp
#
;
with D(Ap) = f(; )2 X : 2 H 2
0(0
;l) and Kp+Dp 2 L 2(0
;l)g, where the plant stiness
and damping operators are given by
Kp'() = d 2 d 2 EIp d 2 '() d 2 !
Dp'() = d
2
d 2
cDIp d 3 '() d 2 @t ! :
Similarly, the nominal plant generator is given by
A 0 = " 0 I K 0 D 0 # ;
with the corresponding stiness and damping operators given by
K 0 '() = d 2 d 2 EI 0 d 2 '() d 2 ! D 0 '() = d 2 d 2
cDI 0 d 3 '() d 2 @t ! :
andD(A 0) =
f(; )2X : 2H 2
0(0
;l) and K 0
+D 0
2L 2(0
;l)g. We have chosen the following
numerical values for the coecients of Ap and A 0 to be
EIp= 25 cDIp = 1:010
?3 and
EI
0= 20 cDI
0 = 2 :010
?3
:
The (unbounded) input operator B 2L(RI;H ?2(0
;l)) is given by
Bu(t) = @ 2 @ 2 KB [
1;2 ](
)u(t)
where [
1; 2](
) is the characteristic function on the interval [ 1
;
2] for 0
l = 0:6 with
[ 1
;
2] = [0
:15;0:45], and the constant KB is a piezoceramic constant that is given by KB =
2:33165510
?3. The reference signal
r(t) is given by
r(t) = 2000
In Figure 3 we plot the norm of the displacement error for the adaptive (solid line) and non-adaptive cases (dashed line). Once again we observe that the state error using the non-adaptive law (3.1) converges to zero faster than the non-adaptive case with the constant feedback law (1.9). The displacement feedback gainsKm (solid line),Kp(dotted line) and the nal value K(1) (dashed line)
are plotted in Figure 4. It is observed, like the previous example, that the displacement feedback gain K(t) converges to some optimal feedback gain but not to the nominal displacement gain Km
or the optimal displacement gain for the uncertain plant Kp. The 3-D plots of the displacement
errors for the adaptive and non-adaptive cases are plotted in Figures 5 & 6 respectively. Inspection of the last two gures reveals that the state error e(t;) converges to zero faster in the adaptive
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0
0.02 0.04 0.06 0.08 0.1 0.12
Time (sec)
|e(t)|
non−adaptive
adaptive
Figure 3: State error,e(t) =xp(t)?xm(t): adaptive and non-adaptive case.
0 0.1 0.2 0.3 0.4 0.5
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5
3x 10 7
solid: K m
dotted: K p
dashed: K(1)
ξ
0
0.2
0.4
0.6 0
0.2 0.4
0.6 0.8
1 0
0.5 1 1.5 2 2.5 3
x 10−3
Time (sec)
ξ
Figure 5: State error,e(t;) =xp(t;)?xm(t;): adaptive case.
0
0.2
0.4
0.6 0
0.2 0.4
0.6 0.8
1 0
0.5 1 1.5 2 2.5 3
x 10−3
Time (sec)
ξ
Figure 6: State error,e(t;) =xp(t;)?xm(t;): non-adaptive case.
6 Conclusions and Future Research
exten-sions of the above adaptive scheme would involve systems with unbounded input operators and uncertainties in input operators as well as a more general class of system operators in order to include a wider class of innite dimensional systems. This is the subject of our current research. The proposed adaptive control law (1.2), (1.4) assumes full state observation. We will extend our study to the construction of adaptive compensator dynamics based on a partial state observation. Moreover, the convergence properties of the approximation of the adaptive law that uses a nite dimensional approximation of the Riccati solution 0 and an approximated reference model
xm(t)
will be studied.
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