Disturbance attenuation for a class of distributed parameter systems

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, VOL. 7, 759—775 (1997)

DISTURBANCE ATTENUATION FOR A CLASS OFDISTRIBUTED PARAMETER SYSTEMS

ALBERTO DE SANTIS* AND LEONARDO LANARI

Dipartimento di Informatica e Sistemistica, Universita di Roma ‘La Sapienza’, Via Eudossiana 18, 00184 Roma, Italy

SUMMARY

This paper deals with the problem of robust stabilization and disturbance attenuation via measuredfeedback, for a class of dissipative collocated distributed systems with disturbances affecting both the inputand the measured output. The proposed solution is based on a direct L

2-gain characterization which avoids

the usual Riccati equation argument. For this purpose only strong stabilizability of the semigroup isrequired. A flexible slewing link is chosen as an application example. ( 1997 by John Wiley & Sons, Ltd.

Int. J. Robust Nonlinear Control, 7, 759—775 (1997)No. of Figures: 1 No. of Tables: 0 No. of References: 22

Key words: H= control; distributed parameter system; strong stability; flexible beam

1. INTRODUCTION

In the last decade much research effort has been devoted to the study of linear H= control.The problem consists of finding a feedback control law ensuring closed-loop stability anddisturbance attenuation. This last requirement is achieved by preventing the energy transferbetween the undesired inputs (disturbances) and suitable penalty variables from exceeding anassigned bound. This is also known as the suboptimal H= control problem, which consists ofrendering the L

2-gain from the disturbances to the penalty variables less than a given value c2.1

Penalty variables definition usually involves the energy of either regulation or tracking errorsof the variables (state or output) to be controlled, as well as the energy related to the control inputs.

Since the relevant literature is quite vast, we mention only a few significant contributionsreferring to the references therein. Initially this problem has been studied in a frequency domainsetting.2,3 State-space formulations were then presented in Reference 4, based on non-standardsteady-state Riccati equation (SSRE) arguments, and in Reference 5 where differential gametheory was used. In the infinite-dimensional setting, see e.g., Reference 6 for a survey, thestate-space approach for the full information case (i.e., full state available for feedback) wasrecently introduced in References 7 and 8, thus extending known results from the finite-dimen-sional setting.5,9 In References 7 and 8 algebraic operator Riccati equations are encountered asopposite to the integral version dealt with in Reference 10, where a very general problem is

This paper was recommended for publication by editor M. J. Grimble

*Correspondence to: A. de Santis, Dipartimento di Informatica e Sistemistica, Universita di Roma ‘La Sapienza’, ViaEndossiana 18, 00184 Roma, Italy

CCC 1049-8923/97/080759—17 $17.50 Received 8 August 1994( 1997 by John Wiley & Sons, Ltd. Revised 4 December 1995

studied, including also time-varying systems. It is worthwhile noting that the exponentialstabilizability of the semigroup is always assumed in these works.

Robust stabilization of distributed positive real systems is addressed in Reference 11 based ona transfer function coprime factorization. In Reference 12 such a factorization is expressed interms of the strong stabilizing solution of the SSRE.

For practical purposes a great deal of work has been devoted to designing output feedbackcompensators. In References 13 and 14 the existence of a static output controller is studied for thefinite dimensional case, while in Reference 15 the measurement feedback H= control problem issolved for the class of Pritchard—Salomon distributed systems.

In this paper we consider a class of infinite-dimensional dissipative systems in the case ofcollocated input/output, with additive disturbances corrupting both the input and the measuredoutput. This class can be adopted in modelling flexible structures with collocated sen-sors/actuators.16 We show that an output static control law can be designed in order toasymptotically stabilize the system and attenuate the effect of the disturbances on some well-defined penalty variable. Moreover, only strong stabilizability is required and the L

2-gain

characterization is achieved without the SSRE argument.In Section 2 basic definitions and assumptions are given and the problem formulation is stated.

The main result is presented in Section 3, and some interesting particular cases are discussed. Theflexible slewing link is chosen as an application example in Section 4.

2. PRELIMINARIES AND PROBLEM FORMULATION

Consider the following infinite-dimensional linear system on the separable Hilbert space H

xR (t)"Ax(t)#Bu(t) (1a)

y(t)"B*x(t) (1b)

with u (t)3Hu, H

ubeing another separable Hilbert space, and y(t)3H

y"H

u, under the follow-

ing assumptions:

(P1) A is a closed densely operator on D (A)LH generating a strongly continuous semi-group (or C

0-semigroup) S (t). Moreover the resolvent R(j,A)"(jI!A)1, j'0, is

compact, i.e., A has a pure point spectrum.(P2) A and A* are dissipative operators i.e.,17

[Ax, x]#[x,Ax])0 x3D (A)

[A*z, z]#[z, A*z])0 z3D (A*)

(P3) B :HuPH is a linear bounded operator and the pair (A,B) is controllable i.e.,17

B*S* (t)v"0 Nv"0, v3H, t*0

Several systems arising in applications belong to this class. Appealing examples are continuummodels of flexible space structures with collocated sensors/actuators.16

The mild solution17 of (1) is

x (t)"S (t)x (0)#Pt

0

S(t!q)Bu(q) dq, x (0)3H (2)

y(t)"B*x (t) (3)

760 A. DE SANTIS AND L. LANARI

Int. J. Robust Nonlinear Control, 7, 759—775 (1997) ( 1997 by John Wiley & Sons, Ltd.

where S (t)x(0)"xh(t) is the solution of the homogeneous version of (1). The behaviour of x

h(t), as

t goes to infinity, gives rise to different types of stability listed below:

(i) the semigroup S (t) is exponentially stable if

ES(t)E)Me~at, M, a'0

(ii) it is strongly stable if, for every z3H,

limt?=

ES (t)zE"0

(iii) it is weakly stable if, for every z, v3H,

limt?=

[S (t)z, v]"0

For the links between these different kinds of stability the reader is referred to Reference 17.Prior to any control problem is the feedback stabilization issue. It consists of finding a control

law us(t)"!Px(t), with P linear bounded, such that the closed-loop operator (A—BP) generates

a stable semigroup according to any one of (i)—(iii). Note that a semigroup cannot be, in general,exponentially stabilized when the input operator B is compact,18 as it often occurs in practicalapplications where B is even finite-dimensional.16 Therefore, weak and strong stability representa more general framework for distributed parameter systems control design. Moreover, sucha control law u

s(t) requires full-state measurement, whereas only the measured output is available

in most situations.For the considered class of systems, it can be easily verified that the output feedback

u(t)"!ky (t), for any k'0, strongly stabilizes system (1). This is a direct consequence of theresults in Reference 19. Indeed by assumption (P2), S (t) is a contraction semigroup,17 i.e.,ES(t)E)1, t*0. Because of (P3), it is also weakly stabilizable by the feedback !kB*x(t),k'0, i.e., the closed-loop operator A

k"(A!kBB*) generates a weakly stable semigroup

Sk(t). Since the resolvent of A is compact, so is the resolvent of A

k, and therefore S

k(t) is strongly

stable.17Besides stabilizability, another interesting concept for control design purposes is the system’s

L2-gain, i.e., the ratio between the L

2-norms of both the forced system’s response and the input.

For these quantities to be well defined let u3¼u"¸

2[[0,R),H

u]. If S (t) is exponentially stable,

then the solution of (1) belongs to ¼x"¸

2[[0,R),H] and consequently

y3¼y"¸

2[[0,R),H

y].17 Note that in this case the whole system’s response y belongs to ¼

y.

Then, the forced component yf

of the output y, obtained from equations (2), (3) by settingx(0)"0, is given by

yf(t)"B* P

t

0

S(t!q)Bu(q) dq

and establishes a linear bounded input—output map

G :¼uP¼

y, y

f"Gu

with

EGE" supEu E

L2/1EGuE

L2*0,=)with EuE2

L2*0,=)"P

=

0

[u (t), u (t)]Hudt

The quantity EGE is defined as the L2-gain of system (1).8

DISTURBANCE ATTENUATION 761

( 1997 by John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control, 7, 759—775 (1997)

If the semigroup S (t) is only strongly stable, the state evolution (2) no longer belongs to ¼x, so

that the quantity EGE cannot be generally defined. However, Lemma 2 in the next section willshow that for the considered class of systems the input—output mapping G, involving theclosed-loop semigroup S

k(t), is bounded. Therefore the L

2-gain can still be defined without the

exponential stability property.Having introduced the concepts of stabilization and L

2-gain, we can now state the problem to

be solved. We want to achieve closed-loop stability via output feedback and attenuate theinfluence of an exogenous input d(t)"(w1 (t)

w2 (t)) on a suitably defined penalty variable z(t). More

precisely, given the system

xR (t)"Ax(t)#Bu(t)#Bw1(t) (4a)

y(t)"B*x(t)#w2(t) (4b)

satisfying (P1)—(P3), with w13¼

uand w

23¼

y, find a static output control law

u(t)"!ky (t)

such that, given the penalty variable

z (t)"Ay (t)

u (t)Bthe closed-loop system is strongly stable and the L

2-gain from the signal d to z is less than a

given value c'0, i.e., EzE2L2*0,=)

)c2EdE2L2*0,=)

. This results in designing a finite energycontrol u(t) to minimize the energy transfer from the disturbances to the output regulationerror.

The solution can be achieved through the direct L2-gain characterization as shown in the next

section.

3. MAIN RESULT

In this section we find an upper bound for the L2-gain from the disturbances to the penalty

variable for the given class of systems, via direct computation. We first state two lemmas to beused in the next theorem.

Lemma 1

Consider the system

xR (t)"(A!kBB* )x (t)#Bu (t) (5)

with u3¼uand the pair (A,B) satisfying (P1)—(P3) so that (A!kBB*) generates a strongly

stable contraction semigroup Sk(t). For some finite ¹'0 choose a sequence

MunN3C1[[0, ¹ ],H

u] (i.e., the space of functions strongly continuously differentiable in (0, ¹ ),

with derivative continuous in [0, ¹ ]) converging to u in the ¼u-norm. Let

v(t)"Pt

0

Sk(t!q)Bu(q) dq, t3[0, ¹ ]



be the mild solution of (5) with x (0)"0, and vn(t) be the analogous quantity with u"u

n. Then

limn?=

E v(t)!vn(t)E

L2*0,T +"0 (6a)

limn?=

Ev(t)!vn(t)EH"0 (6b)

limn?=

D[u, v]L2*0,T +

![un, v

n]L2*0,T +

D"0 (6c)

Proof. See Appendix.

We can now show that the input—output map of the closed-loop system has a well-definedL

2-gain.

Lemma 2

Consider the system

xR (t)"(A!kBB* )x (t)#Bu(t)(7)

y(t)"B*x (t), x (0)"0

with u3¼u, k'0, and the pair (A,B) satisfying (P1)—(P3); then the system’s forced response

y belongs to ¼y"¸

2[[0, R), H

y].

Proof. Choose a sequence MunN3C1[[0, ¹ ],H

u] converging to u in the ¼

u-norm. Define

v(t)"Pt

0

Sk(t!q)Bu(q) dq, v

n(t)"P

t

0

Sk(t!q)Bu

n(q) dq (8)

By Theorem 4.8.2 in Reference 17, function vn(t) is strongly continuously differentiable in (0, ¹ ) so

that we can write, using also property (P2),

d

dtEv

n(t)E2H"[A

kvn(t), v

n(t)]H#[v

n(t),A

kvn(t)]H#2[Bu

n(t), v

n(t)]H

)!2k[BB*vn(t), v

n(t)]H#2[Bu

n(t), v

n(t)]H (9)

Integrating equation (9) from 0 to ¹, one obtains

Evn(¹ )E2H)2[Bu

n(t), v

n(t)]

L2*0,T +!2kEB*v

n(t)E2

L2*0,T +

and therefore

EB*vn(t)E2

L2*0,T +)

1

k[Bu

n(t), v

n(t)]

L2*0,T +)

1

kEu

n(t)E

L2*0,T +EB*v

n(t)E

L2*0,T +

from which

EB*vn(t)E

L2*0,T +)

1

kEu

n(t)E

L2*0,T +)

1

kEu

n(t)E

L2*0,=)

implying that B*vn3¼

y. Now note that

EB*v(t)EL2*0,T +

)EB*(v (t)!vn(t) )E

L2*0,T +#

1

kEu

n(t)E

L2*0,=)



where the first r.h.s. term tends to zero as nPR by Lemma 1, while the second one is in ¼ufor

any n. Therefore, we can now let ¹ go to infinity which leads to B*v3¼y. K

Lemma 2 states that system (7) has a well defined L2-gain.

Remark 1

As argued in Section 2, the state trajectories of system (7) do not belong to ¼x

for any x (0),since this would require

P=

0

ESk(t)x (0)E2 dt(R, ∀x (0)3H

which, according to Datko’s theorem,20 is true if and only if Sk(t) is exponentially stable.

However, it can be shown that not only the systems’s forced response but the whole outputtrajectory belongs to ¼

yfor any x (0)3H. We indeed have

Ey (t)E2L2*0,=)

)2EB*Sk(t)x (0)E2

L2*0,=)#2EB*v(t)E2

L2*0,=)

and only the boundedness of the first term in the r.h.s. need be shown. Owing to the denseness ofD(A

k), for any x (0)3H we can find a sequence Mx

nN3D (A

k) converging to x (0). Now, being

Sk(t)x

ndifferentiable for any n, we can use the same argument as in Lemma 2, obtaining

d

dtES

k(t)x

nE2H)!2kEB*S

k(t)x

nE2H

from which

ESk(t)x

nE2H!Ex

nE2H)!2kEB*S

k(t)x

nE2L2*0,T+

By the strong stability of Sk(t) we eventually obtain that

EB*Sk(t)x

nE2L2*0,=)

)ExnE2H

A limiting argument can now be used to substitute x (0) to xnin the above inequality.

We now state the following theorem.

Theorem 1

Given the system

xR (t)"Ax (t)#Bu(t)#Bw1

(t) (10a)

y(t)"B*x (t)#w2(t) (10b)

satisfying (P1)—(P3) with w13¼

u"¸

2[[0, R), H

u] and w

23¼

y"¸

2[[0, R), H

y], the output

feedback law

u (t)"!ky(t), k'0

strongly stabilizes the system and the L2-gain from d (t)"(w1 (t)

w2 (t)) to z (t)"(y(t)

u(t)) is bounded by

c2"(1#k2)maxG1,1

k2H (11)



Proof. The closed-loop system

xR (t)"(A!kBB*)x (t)#B (w1(t)!kw

2(t))

is strongly stable, as already discussed in Section 2. For the L2-gain characterization, we need to

find an upper bound for

EzE2L2*0,=)

"EyE2L2*0,=)

#EuE2L2*0,=)

"(1#k2)EyE2L2*0,=)

(12)

Define the intermediate variables v1(t) and v

2(t) as follows

v1(t)"P

t

0

Sk(t!q)Bw

1(q) dq, v

2(t)"P

t

0

Sk(t!q)Bw

2(q) dq

The closed-loop system output, with initial conditions x (0)"0, is given by

y (t)"B* Pt

0

Sk(t!q)B(w

1(q)!kw

2(q)) dq#w

2(t)

therefore

y(t)"B* (v1(t)!kv

2(t))#w

2(t)

and

EyE2L2*0,=)

"EB* (v1!kv

2)E2

L2*0,=)#Ew

2E2L2*0,=)

#2[w2,B* (v

1!kv

2)]

L2*0,=)(13)

where all the terms are well defined by Lemma 2.Consider a sequence of compact time intervals MI

jN"M[t

j~1, t

j]N such that Z=

j/1"[0, R) and

let sIj(t) be the indicating function of I

j(i.e., s

Ij(t)"1 if t3I

j, and s

Ij(t)"0 elsewhere), so that we

can consider the restrictions

wj1(t)"w

1(t)s

Ij(t), wj

2(t)"w

2(t)s

Ij(t)

of w1(t) and w

2(t) to the generic interval I

j. The function

wj (t)"[w1(t)!kw

2(t)]s

Ij(t)"w(t)s

Ij(t)"wj

1(t)!kwj

2(t)

of course belongs to ¼u

for any j. Moreover, with

v(t)"v1(t)!kv

2(t)"P

t

0

Sk(t!q)B(w

1(q)!kw

2(q)) dq

we have

vj (t)"v (t)sIj(t)"P

t

tj~1

Sk(t!q)B(w

1(q)!kw

2(q)) dq

"Pt

tj~1

Sk(t!q)Bwj (q) dq, t3I

j

On any Ijwe can find a sequence Mwj

n(t)N3C1 (I

j) converging to wj(t) in the ¼

u-norm, so that the

function

vjn(t)"P

t

tj~1

Sk(t!q)Bwj

n(q) dq



is strongly continuously differentiable for any t3Ij, obtaining

vR jn(t)"A

kvjn(t)#Bwj

n(t), t3I

j

Let us compute the quantity

d

dtEvj

n(t)E2H"[A

kvjn(t)#Bwj

n(t), vj

n(t)]H#[vj

n(t),A

kvjn(t)#Bwj

n(t)]H

)2[wjn(t),B*vj

n(t)]H!2kEB*vj

n(t)E2H , t3I

j(14)

where we used assumption (P2). Intergrating (14) from tj~1

to tjwe obtain

Evjn(tj)E2H)2[wj

n(t),B*vj

n(t)]

L2*tj~1, tj+!2kEB*vj

n(t)E2

L2*tj~1, tj+and therefore

Evjn(tj)E2H#2kEB*vj

n(t)]2

L2*tj~1, tj+)2[wj

n(t) ,B*vj

n(t)]

L2*tj~1, tj+(15)

Now, by Lemma 1, as nPR both sides in equation (15) converge and it eventually results that

Evj (tj)E2H)2[wj (t),B*vj (t)]

L2*tj~1, tj+!2kEB*vj(t)E2

L2*tj~1, tj+

Now, for any N, we can add up the terms of each interval Ij, j)N, obtaining

N+j/1

Evj (tj)E2H)2[w(t),B*v (t)]

L2 *0, tN+!2kEB*v(t) E2

L2*0, tN+(16)

Here, we used the fact that

Evj(t)E2L2*tj~1, tj+

#Evj~1(t)E2L2*tj~2, tj~1+

"Ev(t)sIjXIj~1

E2L2*tj~2, tj+

since vl(t) and vm (t) are orthogonal in the ¸2-sense for lOm. By Lemma 2 both terms in the r.h.s.

of (16) are bounded as tNPR, so that the l.h.s is bounded for any N; we finally have

2kEB*v(t)E2L2*0,=)

)2[w (t), B*v (t)]L2*0,=)

(17)

In order to find an explicit bound for EyE2L2*0,=)

, we now expand inequality (17) to obtain

kMEB*v1E2L2*0,=)

#k2EB*v2E2L2*0,=)

!2k[B*v2, w

2]L2*0,=)

!2k[B*v1,B*v

2]L2*0,=)

#2[B*v1, w

2]L2*0,=)

N

)!k EB*v1E2L2*0,=)

#2k2[B*v1,B*v

2]L2*0,=)

!k3EB*v2E2L2*0,=)

#2[w1,B*v

1]L2*0,=)

!2k[w1,B*v

2]L2*0,=)

(18)

which used in (13) yields

EyE2L2*0,=)

)!EB*v1E2L2*0,=)

#2k[B*v1,B*v

2]L2*0,=)

!k2EB*v2E2L2*0,=)

#

2

k[w

1,B*v

1]L2*0,=)

!2[w1,B*v

2]L2*0,=)

#Ew2E2L2*0,=)

)!EB* (v1!kv

2)E2

L2*0,=)#

2

k[w

1,B* (v

1!kv

2)]

L2*0,=)#Ew

2E2L2*0,=)



)!KKB* (v1!kv

2)!

1

kw1 KK

2

L2*0,=)

#

1

k2Ew

1E2L2*0,=)

#Ew2E2L2*0,=)

)

1

k2Ew

1E2L2*0,=)

#Ew2E2L2*0,=)

)maxG1,1

k2H EdE2L2*0,=)

Therefore, by substituting in (12), one obtains

EzE2L2*0,=)

)(1#k2) maxG1,1

k2H EdE2L2*0,=)

Hence the L2-gain is bounded above by (1#k2)maxM1, 1

k2N. K

Note that the L2-gain upper bound attains its minimum at c*2"2 for k"1.

We can derive some interesting particular cases from the previous analysis.

Corollary 1

In the case w1"w

2"w, and of course d"w, the L

2-gain can be bounded as

EzE2L2*0,=)

EdE2L2*0,=)

)(1#k2) A1#D1!k D

k B2

(19)

Proof. From (13) we have

EyE2L2*0,=)

"(1!k)2EB*vE2L2*0,=)

#EwE2L2*0,=)

#2(1!k) [w,B*v]L2*0,=)

(20)

with obvious meaning of the terms involved. From (17), using the Scwartz inequality, we obtain

EB*vE2L2*0,=)

)

1

k[w, B*v]

L2*0,=))

1

kEwE

L2*0,=)EB*vE

L2*0,=)

from which

EB*vE2L2*0,=)

)

1

kEwE2

L2*0,=)(21)

By using (21) in (20), after a little computation, one has

EzE2L2*0,=)

EdE2L2*0,=)

)(1#k2) A1#D1!k D

k B2

which attains its minimum at c*2"2 for k"1. K

Remark 2

Corollary 1 has an interesting interpretation. Consider the closed-loop system withu(t)"!ky (t)

xR (t)"(A!kBB*)x (t)#(1!k)Bw(t)



Setting k"1 the output feedback exactly cancels the effect of the disturbances in the stateequation. Therefore the effect of the disturbances on the output is given only by the directfeedthrough between the disturbances and the output.

Corollary 2

In the case w1"0 (disturbances affecting only the output) the L


EzE2L2*0,=)

EdE2L2*0,=)

)(1#k2) (22)

(with d"w"w2).

Proof. From (13) we have

EyE2L2*0,=)

"k2EB*vE2L2*0,=)

#EwE2L2*0,=)

!2k[w,B*v]L2*0,=)

By setting w1"v

1"0 in (18) we obtain

0)2k2[w,B*v]L2*0,=)

!2k3EB*vE2L2*0,=)

0)k (2k[w,B*v]L2*0,=)

!k2EB*vE2L2*0,=)

)!k3 EB*vE2L2*0,=)

and therefore

k2EB*vE2L2*0,=)

!2k[w, B*v]L2*0,=)

)!k2EB*vE2L2*0,=)

)0

which implies

EyE2L2*0,=)

)EwE2L2*0,=)

, EzE2L2*0,=)

)(1#k2)EdE2L2*0,=) K

Remark 3

Corollary 2 states that the minimum L2-gain upper bound achievable with the above

computations is c*2"1 attainable for kP0. In other words the output feedback used to stabilizethe system cannot attenuate the effect of the direct feedthrough from the disturbances to theoutput. A trade-off between two different objectives (stabilization and disturbance attenuation)has to be done.

Corollary 3

In the case w2"0 (disturbances affecting only the input) the L


EzE2L2*0,=)

EdE2L2*0,=)

)A1#1

k2B (23)

(with d"w"w1).

Proof. We have

EyE2L2*0,=)

"EB*vE2L2*0,=)



Arguing as in Corollary 1, one obtains the same inequality as (21) which implies

EyE2L2*0,=)

)

1

k2EwE2

L2*0,=), EzE2

L2*0,=))A1#

1

k2B EdE2L2*0,=)

K

Remark 4

The minimum L2-gain upper bound achievable is c*2"1 for kPR. In other words since the

stabilizing controller does not depend upon the disturbances directly, the more the system isstabilized, the better bound can be achieved.

In order to evaluate the obtained results we consider an example referring to a standard H=

control technique, for the chosen class of systems under the stronger assumption of exponentialstabilizability. In order to have an easy solution of the SSRE involved, it will be further assumedthat D(A)"D (A*) and that property (P2) holds with the equality sign

[Ax, x]#[x,Ax]"0, ∀x3D(A)

Then Theorem 4.3.1 in Reference 17 applies and A generates a semigroup S (t) of isometries(ES (t)E"1, t*0) on H. Note that in this case, Corollary 3 can be seen as a special case ofCorollary 4.2 in Reference 8 with D

11"0 which would lead to

c2"k2

k2!1

i.e., the minimum c*2"1 is attained for kPR.

Example

As in Reference 8 consider the case

xR (t)"Ax(t)#Bu(t)#Bw(t) (24a)

y(t)"B*x(t) (24b)

with penalty variable z1(t)"(y(t)`w(t)

u(t)) . Applying Corollary 4.2 of Reference 8, one obtains the

following Riccati equation

CPx, AA#

1

c2!1BB*BxD#CAA#

1

c2!1BB*Bx, PxD

#A1

c2!1!1B [PBB*Px, x]#

c2c2!1

[BB*x, x]"0

Then setting P"kI it results that

c2"2k2

k2!2

Therefore the minimum c*2"2 is attained for kPR. Note that the resulting control lawu(t)"!kB*x (t) belongs to the class of feedback laws considered here. The same result can berecovered, under the loosened assumption of strong stabilizability of A (the context in which all



the presented results are derived), from Corollary 3. The two penalty variables are actually relatedas follows

Ez1E2L2*0,=)

"EzE2L2*0,=)

#EwE2L2*0,=)

#2[y, w]L2*0,=)

)A1#1

k2B EwE2L2*0,=)

#EwE2L2*0,=)

#

1

kEwE2

L2*0,=)(25)

The minimum L2-gain upper bound attains its minimum at c*2"2 for kPR.

This example shows how, for the considered class of systems and under hypotheses(P1)— (P3), the obtained upper bounds for the L

2-gain are quite satisfactory considering that

there is need for neither exponential stability nor full-state measurement (nor dynamic outputcompensator).

4. FLEXIBLE SLEWING LINK EXAMPLE

In this section we briefly introduce a significative case study which belongs to the class of systemsaddressed in this paper. Let us consider the motion of a flexible beam rotating in the horizontalplane where bending occurs.

The rigid-body motion is defined as the rotation of the non-inertial frame (X, ½) in which thebeam deflection d (t,s) is described, with respect to the inertial frame (X

0, ½

0); t and s denote

respectively time and spatial variable. The X-axis is tangent to the beam at the base. Note that therigid body motion h (t) corresponds to the effectively measured joint angular position (Fig. 1).

We will briefly state the beam model, referring the interested reader to Reference 21 fora detailed derivation. Let EI denote the beam flexural rigidity, l its length, q(t) the applied torqueat the hub, M

pand J

pthe mass and inertia of the payload, J

hthe hub’s inertia and, assuming

a uniform beam with linear mass density o, Jbthe beam inertia

Jb"o P

l

0

s2 ds

Furthermore let J be

J"Jh#J

b#J

p#M

pl2

Figure 1. The flexible slewing link



In terms of g (t, s)"d (t, s)#sh(t) and under the small deflection hypothesis, the system behaviouris described by the following PDE

EIg(IV) (t, s)#og(t, s)"0 (26)

with boundary conditions

g (t, 0)"0

Jhg @ (t, 0)!EIgA (t, 0)"q(t)

(27)EIgA(t, l)#J

pg@(t, l)"0

EIg@@@(t, l )!Mpg(t, l)"0

where fQ and f @ denote respectively the time and spatial derivative of a function f. Note that therigid-body motion h (t) is equal to g@(t, 0).

Following the approach in Reference 16, the previous PDE can be put in a state-space form byincluding the boundary variables as part of the state in addition to the functional part describingthe beam deformation. The state vector p can therefore be chosen as

pT"(g( · ) g(t, l) g@(t, l ) g@ (t, 0))T

As a consequence we can write in an abstract setting the following set of equations

Mp(t)#Ap (t)#Bu(t)"0 (28)with u (t)"!q (t), A, M and B such that

Ap"AEIg(IV) ( · )

!EIg@@@ (t, l )

EIgA (t, l )

!EIgA (t, 0) B , p3D(A)

M"Ao 0 0 0

0 Mp

0 0

0 0 Jp

0

0 0 0 JhB , B"A

0

0

0

1 Band D(A) defined as

D(A)"Mg : g@, gA, g@@@, g(IV)3¸2[0, l], and g(0)"0N]R3

On this domain we consider the inner product [ · , · ]

[a, b]"Pl

0

a (s)b(s) ds#a (l )b (l)#a @(l )b @ (l)#a @ (0)b @(0)

Equation (28) is defined on the Hilbert space H obtained as completion of D (A) in the norminduced by the chosen inner product.

M is a bounded self-adjoint positive definite operator with bounded inverse, A is a self-adjointnon-negative definite operator with compact resolvent and B is a bounded operator. The



eigenvalues uk

and corresponding eigenvectors /k

can be obtained by solving the followingeigenvalue problem

A/k!u2

kM/

k"0, [M/

k, /

k]"1

Note that j"0 is in the spectrum of A due to the presence of the rigid body motion g@(t, 0), andthe /

k’s constitute a basis on H.

In order to cast equation (28) as a well-posed first order abstract differential equation on theHilbert space H]H in terms of p and pR , we need to define a suitable norm. Unfortunately theusual ‘energy norm’ [Ap, p]#[MpR , pR ], i.e., the kinetic plus the potential energy, does not definea norm on H]H due to the zero eigenvalue in A. To avoid this difficulty (see Reference 22) let usfirst close the motor position feedback

u(t)"KpB*p(t)#v(t) , K

p'0

The resulting closed-loop equation is

Mp (t)#App (t)#Bv (t)"0 (29)

with Ap

defined as

App"A

EIg(IV) ( · )

!EIg@@@(t, l )

EIgA (t, l )

!EIgA (t, 0)#Kpg@(t, 0) B

Note that Ap

is a positive definite self-adjoint operator with compact resolvent and boundedinverse, i.e., equation A

pp"q can be solved uniquely for every q3H. In other words A

phas no

zero eigenvalue. Defining

x"Ap

pR B3H]H

we can write the following first order equation

xR (t)"Apx (t)#Bv (t) (30)

with

Ap"A

0

!M~1Ap

I

0B , B"A0

!M~1BBwhich is now well-posed by endowing H]H with the following energy inner product

[½, Z]E"[A

p½

1, Z

1]#[M½

2, Z

2]

½"A½

1½

2B , Z"A

Z1

Z2B

We denote this space by HE. As output y(t) we consider the joint angular velocity gR @(t, 0) which,

according to the state vector definition, can be expressed as

y (t)"Cx(t)"B*x (t) (31)



where the adjoint is computed according to the inner product adopted. We obtain the so called‘collocated case’, i.e., C"B*.

In this setting, Ap

is positive definite with a compact resolvent and zero is in the resolvent set.Moreover

A*p"!A

p(32)

where the adjoint operator is again intended in the energy inner product. Therefore the followingdissipativity equality holds

[Apx, x]#[x, A

px]"0

and, because of (32), the same is true for A*p

. Thus both Ap

and A*p

are dissipative operators sothat A

pgenerates a strongly continuous semigroup S (t) of isometries on H

E. It can also be easily

verified that the pair (Ap,B) is controllable.16 Hence hypotheses (P1)—(P3) are fulfilled; therefore

if disturbances d (t)"(w1(t)w2(t)

) affect system (30), (31) as in (10), Theorem 1 can be applied and thefeedback

v(t)"!KvB*x (t), K

v'0 (33)

can be chosen to strongly stabilize the closed-loop system (30)—(31)—(33) and render the L2-gain

from d(t) to z (t)"(y(t)v(t)

) less than the prescribed value given in (11) with k"Kv. In conclusion, the

proportional plus derivative feedback

u(t)"KpB*p (t)#K

vM~1B*pR (t)

"Kph(t)#

Kv

Jh

h0 (t)

provides both strongly stability and disturbance attenuation for the flexible slewing link (28).

5. CONCLUSIONS

For a class of dissipative distributed parameter systems with finite-dimensional collocatedinput/output we solved the stabilization and disturbance attenuation control problem viameasured feedback under the loosened condition of strong stabilizability. The main point consistof showing that the closed-loop system has a well defined L

2-gain despite of lack of exponential

stability. The solution goes through the direct characterization of the system’s L2-gain, without

invoking the commonly used Riccati equation argument. The obtained control law is a staticoutput feedback. A flexible showing link has been chosen as an application example.

APPENDIX

Since C1[[0, ¹ ],Hu], is dense in ¸

2[[0, R),H

u], there always exists such a sequence Mu

nN mentioned in

Lemma 1. Then we can write

Ev(t)!vn(t)E

L2*0,T+"KK P

t

0

Sk(t!q)B(u(q)!u

n(q)) dq KK

L2*0,T+

)KK Pt

0

ESk(t!q)E EBE Eu(q)!u

n(q)Edq KK

L2*0,T+



Now, by applying the Schwartz inequality to the integral above, we obtain

Pt

0

ESk(t!q)E EBE Eu (q)!u

n(q)Edq)EBECP

t

0

ESk(q)E2 dqD

1@2

CPt

0

Eu(q)!un(q)E2 dqD

1@2

)EBEJ¹Eu!unEL2*0,T+

(34)

where the last step holds because Sk(t) is a contraction. Finally we have

Ev(t)!vn(t)E

L2*0,T+)¹ EBE Eu!u

nEL2*0,T+

)¹ EBE Eu!unEL2*0,=)

and the r.h.s. goes to zero as nPR, which proves (6a).Moreover

Ev (t)!vn(t)EH)P

t

0

ESk(t!q)E EBE Eu(q)!u

n(q)Edq

and by (34) we have

limn?=

Ev (t)!vn(t)EHP0

i.e., we also have pointwise convergence (6b). Finally we can write

D[u, v]L2*0,T +

![un, v

n]L2*0,T +

D"D[u!un, v]

L2*0,T +#[u

n, v!v

n]L2*0,T +

D

)Eu!unEL2*0,T +

EvEL2*0,T +

#EunEL2*0,T +

Ev!vnEL2*0,T +

where we applied the Schwartz inequality. Both terms in the r.h.s. converge to zero as nPR thus proving(6c). K

ACKNOWLEDGEMENTS

The authors would like to thank Professor S. Monaco for his helpful suggestions on an earlyversion of this paper. This work is partially supported by M.P.I. 60% and A.S.I. 94RS17 funds.

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Disturbance attenuation for a class of distributed parameter systems

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