internal model based tracking and disturbance rejection for stable well-posed systems

Available online at www.sciencedirect.com

Automatica 39 (2003) 1555–1569

www.elsevier.com/locate/automatica

Internal model based tracking and disturbance rejectionfor stable well-posed systems�

Richard Rebarbera ;∗, George Weissb

aDepartment of Mathematics and Statistics, University of Nebraska - Lincoln, Lincoln, NE 68588-0323, USAbDepartment of Electrical & Electronic Eng., Imperial College London, Exhibition Rd, London SW7 2BT, UK

Received 24 September 2001; received in revised form 26 March 2003; accepted 27 April 2003

Abstract

In this paper we solve the tracking and disturbance rejection problem for in0nite-dimensional linear systems, with reference anddisturbance signals that are 0nite superpositions of sinusoids. We explore two approaches, both based on the internal model principle. Inthe 0rst approach, we use a low gain controller, and here our results are a partial extension of results by H3am3al3ainen and Pohjolainen. Intheir papers, the plant is required to have an exponentially stable transfer function in the Callier–Desoer algebra, while in this paper weonly require the plant to be well-posed and exponentially stable. These conditions are su9ciently unrestrictive to be veri0able for manypartial di:erential equations in more than one space variable. Our second approach concerns the case when the second component of theplant transfer function (from control input to tracking error) is positive. In this case, we identify a very simple stabilizing controller whichis again an internal model, but which does not require low gain. We apply our results to two problems involving systems modeled bypartial di:erential equations: the problem of rejecting external noise in a model for structure/acoustics interactions, and a similar problemfor two coupled beams.? 2003 Elsevier Ltd. All rights reserved.

Keywords: Well-posed linear system; Tracking; Internal model principle; Input–output stability; Exponential stability; Dynamic stabilization; Positivetransfer function; Optimizability; Structural acoustics; Coupled beams

1. Introduction and statement of the main results

In this paper we solve a tracking and disturbance rejectionproblem for stable well-posed linear systems, using a lowgain controller suggested by the internal model principle.These results are a partial extension of those in H3am3al3ainenand Pohjolainen (2000) (detailed comments on this connec-tion will be given). Our results may also be regarded as ageneralization of certain results from Logemann and Town-ley (1997a), who consider controllers with one pole on theimaginary axis (at zero), while we allow several such poles,corresponding to several relevant frequencies.We assume that the plant �p is a well-posed linear system

and it is exponentially stable (see Section 2 for the concepts).

� This paper was not presented at any IFAC meeting. This paper wasrecommended for publication in revised form by Associate Editor FaryarJabbari under the direction of Editor Roberto Tempo.

∗ Corresponding author.E-mail addresses: [email protected] (R. Rebarber),

[email protected] (G. Weiss).

The plant has two inputs,w and u. The inputw consists of theexternal signals (references and disturbances) and u is thecontrol input. These signals take values in the Hilbert spacesW and U , respectively. The output signal of �p, denoted byz, which represents the tracking error, takes values in theHilbert space Y . The transfer function of the plant is

P= [P1 P2];

where P1(s)∈L(W; Y ) and P2(s)∈L(U; Y ). Let �c bethe well-posed controller which is to be determined. Theclosed-loop system obtained by interconnecting these sys-tems is shown in Fig. 1.We denote the state space of �p by X (this is a Hilbert

space). The state trajectories x of this system satisfy thedi:erential equation

ddt

x(t) = Ax(t) + B1w(t) + B2u(t); (1.1)

where A is the generator of a strongly continuous semigroupT on X and B1 and B2 are admissible control operators

0005-1098/03/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0005-1098(03)00192-4

mailto:[email protected]

mailto:[email protected]

1556 R. Rebarber, G. Weiss / Automatica 39 (2003) 1555–1569

Fig. 1. The closed-loop system built by interconnecting the stablewell-posed plant �p with the well-posed controller �c. The signal w con-tains disturbances and references, and z is the error signal which should bemade small. The transfer functions of �p and �c are P= [P1 P2] and C.

Fig. 2. The closed-loop system from Fig. 1, with two additional inputsignals uin and zin, which could be interpreted as noise or as measurementerrors.

for T. This follows from the general representation theoryof well-posed linear systems, which will be recalled brieIyin Section 2. The exponential stability of �p means that‖Tt‖ decays at an exponential rate. If �p is regular (regularsystems are a subclass of well-posed systems, see Section2), then z (the output function of �p) is given by

z(t) = C�x(t) + D1w(t) + D2u(t); (1.2)

for almost every t¿ 0. Here, C� is an unbounded opera-tor from X to Y (de0ned in Section 2), D1 ∈L(W; Y ) andD2 ∈L(U; Y ). In this case,

P1(s) = C�(sI − A)−1B1 + D1;

P2(s) = C�(sI − A)−1B2 + D2:

If �p is not regular, then (1.2) has to be replaced by amore complicated formula, see (2.8) in Section 2. A similardescription applies to �c.

For technical reasons, we consider two additional arti0cialinput signals injected into the feedback system in Fig. 1, uinand zin, which are added to u and z, as shown in Fig. 2. Thesesignals could be interpreted as additional disturbances, noise,measurement errors or quantization errors. We consider theoutput signals to be zout and uout (the output signals of �p

and �c). Thus, from the three inputs to the two outputs, wehave six relevant transfer functions. These, arranged in a2 × 3 matrix, form the transfer function of the closed-loopsystem.The controller �c will be chosen such that the closed-loop

system in Fig. 2 (with inputs w, uin and zin and outputs zoutand uout) is well-posed. In this case, the closed-loop systemis described by equations similar to (1.1) and (1.2), and itsstate space is the Cartesian product of the state spaces of �p

and �c (see Sta:ans, 1998; Weiss & Curtain, 1997).Let J be a 0nite index set of integers. We assume that w

is of the form

w(t) =∑j∈J

wjei!jt ; wj ∈W; !j ∈R: (1.3)

Thus, w is a superposition of constant and sinusoidal signals.The frequencies !j are assumed to be known (for designpurposes), but the vectors wj (which determine the ampli-tudes and the phases) are not known in advance.We introduce some terminology and notation. For any

a∈R, we putCa := {s∈C |Re s¿a}:For any Banach space Z , we denote by H∞

a (Z) the space ofbounded analytic Z-valued functions on Ca. For a = 0 wealso use the notation H∞(Z). In the sequel, when the spaceZ is clear from the context, we write H∞

a for H∞a (Z) and

H∞ for H∞(Z). A transfer function is called stable if it isin H∞, and exponentially stable if it is in H∞

a for somea¡ 0. If a well-posed system is exponentially stable, thenits transfer function is also exponentially stable, see Section2. In particular, for the system in Fig. 1, there exists a¡ 0such that

P1 ∈H∞a (L(W; Y )); P2 ∈H∞

a (L(U; Y )):

For any �∈R we denote

L2�[0;∞) =

{f∈L2

loc[0;∞)∣∣∣∣∫ ∞

0e−2�t |f(t)|2 dt ¡∞

}:

The corresponding space of Y -valued functions is denotedby L2

�([0;∞); Y ) (or also L2([0;∞); Y ) if �=0). However,by some abuse of notation, we sometimes just write L2

�[0;∞)when the range space Y is clear from the context.The objective of this paper is to 0nd a controller �c

with transfer function C so that the closed-loop system inFig. 1 is exponentially stable, and the output z (the trackingerror) decays exponentially to zero, by which we mean thatz ∈L2

�[0;∞) for some �¡ 0. From an engineering point ofview, it would look more convincing if we would requirethat limt→∞ z(t) = 0. However, such an objective is unre-alistic due to the very general class of plants that we con-sider. Indeed, the output of �p resulting from a non-smoothinitial state may be in L2

� without any further continuityproperties, so that point-evaluations of z do not make sense.However, the condition z ∈L2

�[0;∞) with �¡ 0 actually

R. Rebarber, G. Weiss / Automatica 39 (2003) 1555–1569 1557

expresses a very rapid decay of the output, since the se-quence En =

∫ n+1n ‖z(t)‖2 dt decays exponentially.

We now recall the main result of H3am3al3ainen and Pohjo-lainen (2000). Their block diagram is slightly di:erent, itcorresponds to taking P1 = [ − I P2] in Fig. 1 (see furthercomments on this after Fig. 2), and they consider U and Yto be 0nite-dimensional, but these are not essential restric-tions. They consider P to be an exponentially stable transferfunction in the Callier–Desoer algebra. The internal modelprinciple of Davison, Wonham and Francis (see Davison,1976a,b; Francis & Wonham, 1975) suggests that C shouldhave poles at {i!j | j ∈J}. Following this principle, the fol-lowing controller transfer function has been proposed andanalyzed in H3am3al3ainen and Pohjolainen (2000).

C(s) = − ∑j∈J

Kj

s − i!j(1.4)

with

Kj ∈L(Y; U ); "(P2(i!j)Kj) ⊂ C0: (1.5)

Note that (1.5) implies that P2(i!j)Kj is invertible, hencethe range of P2(i!j) is Y (i.e., the matrix P2(i!) is ontoat the relevant frequencies). It was shown in H3am3al3ainenand Pohjolainen (2000) that for all ¿ 0 su9ciently small,the feedback system in Fig. 1 (with the assumptions ofH3am3al3ainen and Pohjolainen (2000) just described) has ex-ponentially stable transfer functions and moreover, if w isas in (1.3), then the error z tends to zero. The approach ofH3am3al3ainen and Pohjolainen (2000) is algebraic and theyconsider also multiple poles in C, which are needed if wewant to allow the coe9cients wj in (1.3) to be polynomialsin t (we have not reproduced the formulas corresponding tothe multiple poles, since we only consider constant wj).

This result is important because it allows tracking and dis-turbance rejection for external signals that are superpositionsof a constant and several sinusoidal signals, with very littleinformation about the plant. Indeed, all we have to know isthat the plant is stable and we must have some knowledgeabout P2(i!j), which can be measured, but we do not needto be very precise (since the conditions (1.5) are robust withrespect to small changes of P2(i!j)). By contrast, the in-ternal model theory developed in many references requiresa detailed knowledge of the plant equations. We feel thatthis is an important and beautiful result, and we wanted tounderstand it from a di:erent perspective, using an analyticapproach.How is our setup di:erent from the one in H3am3al3ainen

and Pohjolainen (2000): the small di:erences in the blockdiagram and in the spacesU; Y have already been mentioned(in both respects, our framework is slightly more general).We have constant coe9cients wj in (1.3), so we consideronly simple poles for C, and in this respect we are lessgeneral. Most importantly, we replace the class of plantsin H3am3al3ainen and Pohjolainen (2000) with exponentially

stable well-posed plants, which is substantially more generalthan the class of stable plants with transfer functions in theCallier–Desoer algebra. The conditions for a plant to bewell-posed are su9ciently unrestrictive to be veri0able formany partial di:erential equations in more than one spacevariable; this is probably not true if we have to verify thatthe transfer function is in the Callier–Desoer algebra.Our 0rst main results follows. The concepts of exact con-

trollability and exact observablity, which appear in the the-orem, will be recalled in Section 2.

Theorem 1.1. Suppose that �p is an exponentially stablewell-posed linear system with transfer function P=[P1 P2],where P1(s)∈L(W; Y ), P2(s)∈L(U; Y ). Let �c be an ex-actly controllable and exactly observable realization of atransfer function C of the form

C(s) = −

C0(s) +

∑j∈J

Kj

s − i!j

; (1.6)

where C0 ∈H∞� (L(Y; U )) with �¡ 0, Kj ∈L(Y; U ) and

"(P2(i!j)Kj) ⊂ C0.Then the closed-loop system shown in Fig. 2 is well-posed

and exponentially stable for all su<ciently small ¿ 0. Forany such there exists #¡ 0 such that, if w is of the form(1.3), then z ∈L2

#[0;∞).

It follows from Theorem 1.1 that (for su9ciently small ¿ 0) the six transfer functions of the closed-loop systemin Fig. 2 are in H∞

# for some #¡ 0.Note that, in (1.6), we have added the extra term C0

(when compared to (1.4)). The theorem would of courseremain valid without this extra term, but C0 may be neededto satisfy some other design requirements, possibly derivedfrom robustness considerations, or to shorten the transientresponse.This theorem has not been stated in the strongest pos-

sible form: it is a consequence of a more general theoremstated and proved in Section 3. In the more general version,the conditions “exactly controllable” and “exactly observ-able” are replaced with the less restrictive conditions “opti-mizable” and “estimatable”. The reason for not stating thisversion here is that the concepts of optimizability and esti-matability would probably be less familiar to most readers.If U and Y are 0nite-dimensional and C is rational (as itwould be in most applications), then it is natural to take �c

to be a minimal realization of C, so that �c would be con-trollable and observable, as required in the theorem.We make some comments on the generality of the block

diagram in Fig. 1 and the signal w in (1.3). Typically wwill contain both the reference r and the disturbance d, i.e.,w=[r d]�. In this case r(t)∈Y and d(t) is in another Hilbertspace V , we have W = Y × V and P1 can be decomposedas P1 = [ − I P0

1], so that, neglecting the inIuence of theinitial states of �p and �c,

z = P1w + P2u= −r + P01d+ P2u= y − r:


Fig. 3. The stable well-posed plant �0p interconnected with the well-posed

controller �c. The transfer functions of these systems are P0 = [P01 P2]

and C. If we lump together the disturbance d and the reference r to formthe signal w, then this block diagram reduces to the one in Fig. 1, withP1 = [− I P0

1].

Here, y=P01d+P2u is the Laplace transform of the output

signal y of the plant �0p with transfer function P0 = [P0

1 P2],corresponding to the initial state zero. The system �0

p is al-most the same as �p (it has the same state trajectories) butits inputs are only d and u (instead of r; d and u), as shownin Fig. 3. The paper (H3am3al3ainen & Pohjolainen, 2000) usesa block diagram which corresponds to this decomposition,with the additional restriction P0

1 = P2. In the theory thatwe develop, such a decomposition of w and of P1 is notneeded. We have assumed that the plant is exponentially sta-ble, but in applications �p might be obtained by stabilizingan initially unstable plant. This is illustrated in our Section5 (disturbance rejection for a coupled beams system).Usually, W is the complexi0cation of a real Hilbert

space W0, so that any w∈W has a unique decomposi-tion w = w0 + iw1 with w0;w1 ∈W0 and so the complexconjugate Qw = w0 − iw1 is well de0ned. Moreover, thesignal w takes values in W0, which (with proper indexing)implies that

!−j = −!j and w−j = Qwj:

Similarly, U and Y are usually the complexi0cations of realHilbert spaces U0 and Y0 and the transfer function P is real,which means that

P1(−i!)w0 = P1(i!)w0 ∀w0 ∈W0;

and a similar condition is satis0ed by P2. In this case, thecontroller transfer function C can be chosen to be real aswell, by choosing

K−jy0 = Kjy0 ∀y0 ∈Y0;

and by choosingC0 to be real. These restrictions on w, P andC had to be mentioned because they would probably arise inany engineering application. However, these restrictions arenot necessary for our theory, so that we will not make them.Taking them into account would not modify the theory at all.

The results of H3am3al3ainen and Pohjolainen (2000) buildon earlier work by the same authors, notably Pohjolainen(1982) and H3am3al3ainen and Pohjolainen (1996). Low-gaintracking results for in0nite-dimensional systems canbe found in Logemann, Bontsema, and Owens (1988),Logemann and Curtain (2000), Logemann and Mawby(2001), Logemann and Owens (1989), Logemann andTownley (1997a,b), Logemann, Ryan, and Townley (1998),Pohjolainen (1982, 1985), and Pohjolainen and L3atti (1983).In Logemann and Curtain (2000), Logemann and Mawby(2001), Logemann et al. (1998), and Logemann and Town-ley (1997a,b) the reference and disturbance signals areconstant and the class of plants considered is the classof regular systems, which is a large subset of well-posedsystems. In Logemann and Curtain (2000), Logemannand Mawby (2001), and Logemann et al. (1998), the sys-tems are allowed to have nonlinearities. In Logemann andTownley (1997a), the plants are uncertain and it is shownhow the gain can be chosen adaptively. In Logemannand Townley (1997b), analogous results are obtained foruncertain discrete-time systems, with the gain obtainedadaptively; these results are applied to continuous-timeregular systems to obtain adaptive sampled-data low-gaintracking controllers. The works by Logemann et al. (1988),Logemann and Owens (1989), Pohjolainen (1982, 1985),and Pohjolainen and L3atti (1983) are earlier works wherethe classes of systems considered are substantially morerestrictive than here, and the references and disturbancesare constant.Let P be an L(U )-valued transfer function de0ned on (a

set containing) the half-plane C0. We say that P is a positivetransfer function if

ReP(s) := 12 [P(s) + P(s)∗]¿ 0 ∀s∈C0:

When the second component of the plant transfer func-tion (from control input to error) is positive, the followingtheorem states that certain simple controllers will stabilizethe system in the sense of Theorem 1.1 and also achievetracking. Moreover, in this situation, there is no need toadjust an unknown small gain.

Theorem 1.2. Suppose that �p is an exponentially stablewell-posed linear system with transfer function P=[P1 P2],where P1(s)∈L(W;U ), P2(s)∈L(U ), P2 is a positivetransfer function, and ReP2(i!j) is invertible for all j ∈J.Let �c be an exactly controllable and exactly observablerealization of a transfer function C of the form

C(s) = −C0(s) +

∑j∈J

Kj

s − i!j

; (1.7)

where Kj∈L(U ); Kj¿0; K−1j ∈L(U ); C0 ∈H∞

� (L(U ))with �¡ 0 and

ReC0(s)¿ 12 I ∀s∈C0:


Then the feedback system in Fig. 2 is well-posed andexponentially stable. If w is of the form (1.3), then z ∈L2#[0;∞) for some #¡ 0.

Concerning the conditions “exactly controllable” and “ex-actly observable” that appear in this theorem, the same com-ments apply as those made after Theorem 1.1. The moregeneral version of Theorem 1.2 is stated and proved in Sec-tion 3.We remark that a simple choice for Kj and C0 is to take

them scalar multiples of the identity operator. Then C isessentially a single input-single output system repeated inidentical copies as many times as the dimension of U . Thischoice still allows C0 to be a function of s, but of course thesimplest choice for C0 would be to take it a real constantlarger than 0.5.In Section 4 we apply Theorem 1.1 to a model for struc-

ture/acoustics interaction, showing that external noise at0xed frequencies can be rejected. In Section 5 we applyTheorem 1.2 to a model for coupled Euler–Bernoulli beams.

2. Well-posed linear systems and related concepts

In this section we recall some basic facts about admis-sible control and observation operators, well-posed linearsystems, their transfer functions, static output feedback,exponential stability and its relation to optimizability andestimatability.Throughout this section, X is a Hilbert space and

A :D(A) → X is the generator of a strongly continuoussemigroup T on X . The Hilbert space X1 is D(A) withthe norm ‖z‖1 = ‖(#I − A)z‖, where #∈ ((A) is 0xed(this norm is equivalent to the graph norm). The Hilbertspace X−1 is the completion of X with respect to the norm‖z‖−1=‖(#I−A)−1z‖. This space is isomorphic toD(A∗)∗,and we have

X1 ⊂ X ⊂ X−1;

densely and with continuous embeddings. T extends to asemigroup on X−1, denoted by the same symbol. The gener-ator of this extended semigroup is an extension of A, whosedomain is X , so that A :X → X−1.Let U be a Hilbert space and B∈L(U; X−1). Then B

is called an admissible control operator for T if it has thefollowing property: If x is the solution of

x(t) = Ax(t) + Bu(t) (2.1)

with x(0) = x0 ∈X and u∈L2([0;∞); U ), then x(t)∈X forall t¿ 0 (see Weiss (1989a) for details). In this case, x is acontinuous X -valued function of t. We have x(t) = Ttx0 +)tu, where )t ∈L(L2([0;∞); U ); X ) is de0ned by

)tu=∫ t

0Tt−"Bu(") d": (2.2)

The above integration is done in X−1, but the result is in X .The Laplace transform of x is

x(s) = (sI − A)−1[x0 + Bu(s)]:

B is called bounded if B∈L(U; X ) (and unbounded other-wise).Let Y be a Hilbert space and C ∈L(X1; Y ). Then C is

called an admissible observation operator for T if it hasthe following property: For some (hence, for every) T ¿ 0there exists a kT ¿ 0 such that∫ T

0‖CTtx0‖2 dt6 k2T‖x0‖2 ∀x0 ∈D(A); (2.3)

see Weiss (1989b). C is called bounded if it can beextended such that C ∈L(X; Y ). If C ∈L(X1; Y ), thenC∗ ∈L(Y; X d

−1), where X d−1 is the completion of X with re-

spect to the norm ‖z‖d−1 = ‖(#I − A∗)−1z‖, with #∈ ((A∗)

(we have identi0ed Y with its dual). C is an admissible ob-servation operator for T if and only if C∗ is an admissiblecontrol operator for the dual semigroup T∗.

We regard L2loc([0;∞); Y ) as a FrSechet space with the

seminorms being the L2 norms on the intervals [0; n]; n∈N.Then the admissibility of C means that there is a continuousoperator , :X → L2

loc([0;∞); Y ) such that

(,x0)(t) = CTtx0 ∀x0 ∈D(A): (2.4)

The operator , is completely determined by (2.4), becauseD(A) is dense in X . We de0ne the �-extension of C by

C�z = lim-→+∞

C-(-I − A)−1z; (2.5)

where - is real. The domain D(C�) consists of all z ∈X forwhich the above limit exists. If we replace C by C�, thenformula (2.4) becomes true for all x0 ∈X and for almostevery t¿ 0. If y =,x0, then its Laplace transform is

y(s) = C(sI − A)−1x0: (2.6)

By a well-posed linear system we mean a linear time-invariant system such that on any 0nite time interval, theoperator from the initial state and the input function to the0nal state and the output function is bounded. The input,state and output spaces are Hilbert spaces, and the input andoutput functions are of class L2

loc. For the detailed de0ni-tion, background and examples we refer to Salamon (1987,1989), Sta:ans (1997), Sta:ans and Weiss (2002) or Weiss(1994a,b).We recall some facts about well-posed linear systems. Let

� be such a system, with input space U , state space X andoutput space Y . We consider positive time, t¿ 0. The statetrajectories of � satisfy the equation (2.1) and the commentsfrom the beginning of this section apply. T is called thesemigroup of � and B is called the control operator of �.If u is the input function of �, x0 is its initial state and y is


the corresponding output function, then

y =,x0 + Fu: (2.7)

Here, , is given by (2.4) and continuous extension to X ,and C appearing in (2.4) is called the observation operatorof �. Clearly, both B and C are admissible.The operator F :L2

loc([0;∞); U ) → L2loc([0;∞); Y ) is eas-

iest to represent using Laplace transforms. We do not dis-tinguish between two analytic functions de0ned on di:erentright half-planes if one is a restriction of the other. For somea∈R, there exists a unique L(U; Y )-valued function G inH∞

a , called the transfer function of �, which determines Fas follows: if u∈L2([0;∞); U ) and y = Fu, then y has aLaplace transform y and, for Re s su9ciently large,

y(s) =G(s)u(s):

This determines F, since L2 is dense in L2loc. G is in H∞

afor any a∈R which is larger than the growth bound of thesemigroup T, see Weiss (1994a, Section 4). In particular, ifT is exponentially stable, then G is in H∞

a for some a¡ 0(recall that such a transfer function is called exponentiallystable). An operator-valued analytic function de0ned on adomain containing a right half-plane and which is boundedon this half-plane is called a well-posed transfer function.We have

G(s) − G(#) = C[(sI − A)−1 − (#I − A)−1]B;

for any s; # in the open right half-plane determined by thegrowth bound of T. This shows that G is determined byA; B and C up to an additive constant operator. In additionto (2.7), there is also a formula which expresses y locallyin time: for any complex # with Re # su9ciently large,

y(t) = C�[x(t) − (#I − A)−1Bu(t)] +G(#)u(t); (2.8)

valid for almost every t¿ 0, where x is the state trajectoryof �, see Sta:ans and Weiss (2002).Any operator-valued well-posed transfer function G has

a realization, i.e., a well-posed linear system with transferfunction G, see Salamon (1989) and Sta:ans (1999). More-over, it follows from the material in Salamon (1989) andSta:ans (1999) that if G is exponentially stable, then therealization can be chosen to be exponentially stable.An operator K ∈L(Y; U ) is called an admissible feed-

back operator for � (or for G) if I −GK has a well-posedinverse (equivalently, if I−KG has a well-posed inverse). Ifthis is the case, then the system with output feedback shownin Fig. 4 is well-posed (its input is v, its state and output arethe same as for �). This new system is called the closed-loopsystem corresponding to � and K , and it is denoted by �K .Its transfer function is GK =G(I −KG)−1 =(I −GK)−1G.We have that −K is an admissible feedback operator for �K

and the corresponding closed-loop system is �. For moredetails on closed-loop systems we refer to Weiss (1994b).

Fig. 4. A well-posed linear system � with output feedback via K . If Kis admissible, then this is a new well-posed linear system �K , called theclosed-loop system.

The system � is called regular if the limit

lims→+∞G(s)v = Dv

exists for every v ∈U , where s is real (see Weiss, 1994b). Inthis case, the operatorD∈L(U; Y ) is called the feedthroughoperator of �. Regularity is equivalent to the fact that theproduct C�(sI − A)−1B makes sense. In this case,

G(s) = C�(sI − A)−1B+ D; (2.9)

like in 0nite dimensions. This, together with (2.8), impliesthat the function y from (2.7) satis0es, for almost everyt¿ 0,

y(t) = C�x(t) + Du(t); (2.10)

where x is the state trajectory of the system. The oper-ators A; B; C; D are called the generating operators of �,because they determine � via (2.1) and (2.2). (A; B; C) iscalled a regular triple if A; B; C; 0 are the generating oper-ators of a regular linear system. Equivalently, A generatesa semigroup, B and C are admissible, the product C�(sI −A)−1B exists and it is bounded on some right half-plane (seeSection 2 of Weiss and Curtain (1997)). In particular, if Ais a generator, one of B and C is admissible and the other isbounded, then (A; B; C) is a regular triple.

If, in the feedback system shown in Fig. 4, � is regularand its feedthrough operator is zero, then the semigroupgenerator of �K is

AK = A+ BKC�;

D(AK) = {x∈D(C�) |Ax + BKC�x∈X }: (2.11)

In this case, �K is regular and its other generating operatorsare

BK = B; CK = C�|D(AK ); DK = 0; (2.12)

where C�|D(AK ) denotes the restriction of C� to D(AK). Itis interesting to note that in this case CK

� = C�, see Weiss(1994b, Section 7) for details.

De�nition 2.1. The well-posed system � (or the pair(A; B)) is optimizable if for every x0 ∈X there exists a


u∈L2([0;∞); U ) such that x∈L2([0;∞); X ), where x is thestate trajectory de0ned by x(t) =Ttx0 +)tu. The system �(or the pair (A; B)) is exactly controllable if for some t ¿ 0,)t is onto.

Optimizability is the most natural extension of the con-cept of stabilizability from 0nite-dimensional systems to thecontext of well-posed systems.Motivated by linear quadraticoptimal control theory, this property is sometimes called the=nite cost condition. For details concerning optimizabilitywe refer to Weiss and Rebarber (2001). It is easy to see thatexact controllability implies optimizability.

De�nition 2.2. The well-posed system� (or the pair (A; C))is estimatable if (A∗; C∗) is optimizable. The system � (orthe pair (A; C)) is exactly observable if (A∗; C∗) is exactlycontrollable.

Estimatability is equivalent to the solvability of a cer-tain 0nal state estimation problem, see Weiss and Rebarber(2001, Section 3). Since it is the dual concept of optimiz-ability, estimatability is an extension of the concept of de-tectability from 0nite-dimensional systems to the context ofwell-posed systems. Exact observability is equivalent to thefact that there exist T ¿ 0 and KT ¿ 0 such that∫ T

0‖CTtx0‖2 dt¿KT‖x0‖2 ∀x0 ∈D(A):

Exact observability implies estimatability. We refer again toWeiss and Rebarber (2001) for further details.

It is easy to see that if a well-posed system is exponentiallystable, then it is optimizable and estimatable. Optimizabilityand estimatability are invariant under static output feedback.One of the main results of Weiss and Rebarber (2001) isthe following:

Theorem 2.3. A well-posed linear system is exponentiallystable if and only if it is optimizable, estimatable and input–output stable.

The following theorem is a consequence of the last the-orem (related results for regular linear systems were givenin Weiss and Curtain (1997)).

Theorem 2.4. Suppose that�p and�c are well-posed linearsystems with transfer functions denoted P and C (whereP = [P1 P2]), connected in feedback as shown in Fig. 2.Suppose that �p is optimizable via u, �c is optimizable,and both systems are estimatable. Suppose that the fourtransfer functions

(I − P2C)−1; C(I − P2C)−1; (I − P2C)−1P2;

C(I − P2C)−1P2 (2.13)

are all stable (i.e., in H∞). Then the closed-loop system�p;c shown in Fig. 2 is an exponentially stable well-posedlinear system.

Proof. Under the given optimizability and estimatabilityhypotheses, it follows fromWeiss and Rebarber (2001, The-orem 6.4) that the closed-loop system �p;c is well-posed andexponentially stable if (and only if) (I−L)−1 ∈H∞(U×Y ),where

L=

[0 C

P2 0

]:

It is easy to verify that (I −L)−1 ∈H∞ if (and only if) thetransfer functions listed in (2.13) are all in H∞.

The transfer functions appearing in (2.13) are closely re-lated to the transfer functions of the system in Fig. 2, withw = 0. For example, (I − P2C)−1 is the transfer functionfrom zin to z, hence (I − P2C)−1 − I maps zin to zout.

Remark 2.5. With the assumptions and the notation ofTheorem 2.4, let / denote the growth bound of the semi-group of �p;c, so that /¡ 0 according to the theorem.Then according to the theory recalled earlier, all of thetransfer functions associated with this system (such as thefour transfer functions listed in (2.13)) are in H∞

# for any#¿/. Similarly, G= (I −P2C)−1P1, which is the transferfunction from w to z for �p;c, is in H∞

# for any #¿/:

3. Proof of the main results

To prove the stability parts of Theorems 1.1 and 1.2 weshow that an appropriate set of closed-loop transfer functionsis stable, and then we apply Theorem 2.4 to conclude thatthe closed-loop system is exponentially stable.To be able to apply Theorem 2.4, a key step is to show

that the 0rst of the four transfer functions listed in (2.13),called the sensitivity and denoted by S, is stable (i.e., inH∞) for all su9ciently small ¿ 0.

Lemma 3.1. Suppose that P2 ∈H∞a (L(U; Y )) with a¡ 0

and C is given by (1.6) and satis=es the conditions listedafter (1.6). Then there exists ∗ ¿ 0 such that for all ∈ (0; ∗],

S := (I − P2C)−1 ∈H∞:

Proof. Let r0 ¿ 0 be such that |!j − !k |¿ 2r0 for allj; k ∈J with j �= k. We de0ne the half-disks

Dj := {s∈C |Re s¿ 0; |s − i!j|6 r0}:

Since P2 and C0 are bounded on C0 and J is 0nite set, itis clear that

sups∈G

∥∥∥∥∥∥P2(s)

C0(s) +

∑j∈J

Kj

s − i!j

∥∥∥∥∥∥¡∞;


where

G= C0\⋃j∈J

Dj:

Therefore, there exists ∞ ¿ 0 such that

S(s) =

I + P2(s)

C0(s) +

∑j∈J

Kj

s − i!j

−1

is uniformly bounded for all s∈G and all ∈ (0; ∞].Fix j ∈J. To analyze S on Dj, we de0ne

Sj(s) :=(I +

P2(i!j)Kj

s − i!j

)−1

:

(this is a good approximation of S(s) near the point s=i!j)and

Qj(s) =(P2(s) − P2(i!j))Kj

s − i!j+ P2(s)C0(s)

+∑

k∈J; k �=j

P2(s)Kk

s − i!k:

It is clear that the last two terms in the de0nition of Qj arebounded on Dj. Since P2 is analytic at i!j, the 0rst term inthe de0nition of Qj is also bounded on Dj. Therefore Qj

is bounded on Dj, with a bound that is independent of (because Qj is independent of ).We have S−1 − S−1

j = Qj, so that we can write

S(s) = Sj(s)(I + Qj(s)Sj(s))−1: (3.1)

If we can show that Sj is bounded on Dj, with the boundindependent of ¿ 0, then from (3.1) we see that there exists j ¿ 0 such that S is bounded on Dj for all ∈ (0; j]. SinceJ has 0nitely many elements, this will 0nish the proof, with

∗ =min{ j | j ∈J} ∪ { ∞}:To show that Sj is bounded on Dj, with the bound indepen-dent of ¿ 0, denote Pj = P2(i!j)Kj, z = (s − i!j)= , andnote that

sups∈Dj

‖Sj(s)‖ = supRe z¿0; |z|6r=

∥∥∥∥∥(I +

Pj

z

)−1∥∥∥∥∥

6 supRe z¿0

‖z(zI + Pj)−1‖:

Since "(Pj) ⊂ C0, z(zI+Pj)−1 is analytic onC0. Moreover,

lim|z|→∞

z(zI + Pj)−1 = I;

so z(zI + Pj)−1 is continuous on the one-point compacti-0cation of C0, hence it is bounded there. Therefore Sj isbounded on Dj with a bound independent of , which (asexplained earlier) 0nishes the proof of the lemma.

Theorem 1.1 is an immediate consequence of the follow-ing, more general theorem.

Theorem 3.2. Let �p be an exponentially stable well-posedlinear system with transfer function P = [P1 P2], whereP1(s)∈L(W; Y ), P2(s)∈L(U; Y ). Let �c be an optimiz-able and estimatable realization of a transfer function Cof the form (1.6), where C0 ∈H∞

� (L(Y; U )) with �¡ 0,Kj ∈L(Y; U ) and "(P2(i!j)Kj) ⊂ C0.

Then there exists an ∗ ¿ 0 such that for every ∈ (0; ∗],the feedback system in Fig. 2 is well-posed and exponen-tially stable. For such , there exists a #¡ 0 such that ifw is of the form (1.3), then z ∈L2

#[0;∞).

The simplest way to obtain an optimizable and estimatablerealization of C is to take an exponentially stable realizationof − C0 and connect it in parallel with a minimal (i.e.,exactly observable and exactly controllable) realization ofthe rational transfer functionC+ C0. Note that this minimalrealization is not 0nite-dimensional in general, because thecoe9cients Kj are operators.

Proof. Since �p is exponentially stable, we have P∈H∞a

with a¡ 0. According to Lemma 3.1, for ∈ (0; ∗] we have(I − P2C)−1 ∈H∞. We need to show that for such , theother transfer functions listed in (2.13) are also in H∞. Dueto the stability of P2, this follows once we have shown that

C(I − P2C)−1 ∈H∞:

To prove this, we notice that by (1.6), lims→i!j

(s − i!j)C(s) = − Kj for any j ∈J. This implies that

lims→i!j

1s − i!j

[I − P2(s)C(s)]−1

=[lim

s→i!j

P2(s)(s − i!j)C(s)]−1

= − 1 [P2(i!j)Kj]−1: (3.2)

Note that P2(i!j)Kj is invertible according to (1.5). Now,using again the decomposition (1.6), we conclude thatC(I −P2C)−1 has a 0nite limit at i!j, for all j ∈J, so thatC(I − P2C)−1 is bounded on a neighborhood N of the set{i!j | j ∈J}. Since (I −P2C)−1 ∈H∞ and C is uniformlybounded on C0 \ N, it follows that C(I − P2C)−1 ∈H∞.As mentioned earlier, this implies that all of the transferfunctions listed in (2.13) are in H∞.Because �p is exponentially stable, it is optimizable and

estimatable. �c is optimizable and estimatable by hypothe-sis. Now we apply Theorem 2.4 to conclude that �p;c is awell-posed system and it is exponentially stable, so that thegrowth bound / of its semigroup satis0es /¡ 0.Now we choose #∈ (/; 0) and we prove that z ∈L2

#[0;∞).First we recall from Remark 2.5 that the transfer function G


from w to z is in H∞# . From (3.2),

lims→i!j

1s − i!j

G(s) = −1 [P2(i!j)Kj]−1P1(i!j): (3.3)

According to (1.3), we have

w(s) =∑j∈J

wj

s − i!j:

This, together with (3.3), shows that Gw has removable sin-gularities at the points i!j; j ∈J. SinceG is bounded onC#

and w decays like 1=s for large |s|, we see that Gw∈H 2# (Y ).

Here H 2# (Y ) denotes the space of those Y -valued analytic

functions on C# which can be obtained by shifting a func-tion in the Hardy space H 2(Y ) (on C0) to the left by |#|.Equivalently, by the Paley–Wiener theorem, H 2

# (Y ) is thespace of Laplace transforms of functions in L2

#([0;∞); Y ).The output function z of �p;c can be decomposed as z =

z1 + z2, where z1 is the component due to the initial stateand z2 is the component due to the input w. According toWeiss (1994a, formula (4.2)), z1 ∈L2

#[0;∞). Since z2 =Gw∈H 2

# , we have that z2 ∈L2#[0;∞). Thus we have shown

that z ∈L2#[0;∞).

We now turn our attention to the proof of Theorem 1.2.We will need the following simple lemma, which is re-lated to the computations usually found in the theory ofthe Cayley transform, but we are not aware of a referencefor this lemma. Recall from Section 1 that for any operatorT∈L(H), we denote ReT= 1

2(T+ T∗).

Lemma 3.3. Suppose that H is a Hilbert space andT∈L(H). Then

ReT¿ 12 I

if and only if there exists Q∈L(H) such that

T= (I − Q)−1 with ‖Q‖6 1: (3.4)

Furthermore, if the above conditions are satis=ed and wedenote M = ‖T‖, then

ReQ6(1 − 1

2M 2

)I:

Proof. Suppose that ReT¿ 12 I . Then for any u∈H with

‖u‖ = 1,

‖Tu‖¿ |〈Tu; u〉|¿Re 〈Tu; u〉 = 〈(ReT)u; u〉¿ 12 ;

so that T is bounded from below. By a similar argument,T∗ is also bounded from below, hence T is onto, so that Tis invertible. We have for any v∈H

‖(T − I)v‖2 = 〈(T − I)v; (T − I)v〉= ‖Tv‖2 − 2Re 〈Tv; v〉 + ‖v‖2

6 ‖Tv‖2 − ‖v‖2 + ‖v‖2 = ‖Tv‖2:

Denoting z = Tv, we have shown that

‖(T − I)T−1z‖6 ‖z‖ ∀z ∈H; (3.5)

so that Q = (T − I)T−1 = I − T−1 is a contraction. It isclear that T= (I − Q)−1.Conversely, suppose that (3.4) holds. Then obviously T is

invertible andQ=(T−I)T−1, so that (3.5) holds. Denotingv = T−1z, we get

‖(T − I)v‖6 ‖Tv‖ ∀v∈H:

Developing ‖(T − I)v‖2 as in the 0rst part of the proof,we obtain that 2Re 〈Tv; v〉¿ ‖v‖2, which is the same asReT¿ 1

2 I .The 0nal part of the lemma follows from

Q∗ +Q = 2I − T−∗ − T−1 = 2I − T−∗(T∗ + T)T−1

6 2I − T−∗T−1:

If ‖T‖ =M then T−∗T−1¿ I=M 2, so that

Q∗ +Q6 2I − IM 2 :

Theorem 1.2 is an immediate consequence of the follow-ing, more general theorem.

Theorem 3.4. Suppose that �p is an exponentially stablewell-posed linear system with transfer function P=[P1 P2],where P1(s)∈L(W;U ), P2(s)∈L(U ), P2 is a positivetransfer function, and ReP2(i!j) is invertible for all j ∈J.Let �c be an optimizable and estimatable realization of atransfer function C of the form (1.7), where Kj ∈L(U ),Kj ¿ 0, K−1

j ∈L(U ), C0 ∈H∞� (L(U )) with �¡ 0 and

ReC0(s)¿ 12 I ∀s∈C0:

Then the feedback system in Fig. 2 is well-posed andexponentially stable. If w is if the form (1.3), then z ∈L2#[0;∞) for some #¡ 0.

Proof. For everyKj ∈L(U ) withKj¿ 0 and every!j ∈R,

Re(

Kj

s − i!j

)¿ 0 ∀s∈C0:

Therefore, from formula (1.7) we see that

− ReC(s)¿ 12 I; (3.6)

which implies that for all s∈C0

‖C(s)v‖¿ 12‖v‖ ∀v∈U: (3.7)

By Lemma 3.3, (3.6) implies that there exists Q∈L(H)such that

−C= (I − Q)−1; ‖Q‖6 1:


Since S= (I − P2C)−1, we obtain that

S−1 = I + P2(I − Q)−1 = [I − Q+ P2](I − Q)−1

=−[I − Q+ P2]C: (3.8)

Since by our assumptions ReP2(i!j) is positive and invert-ible for each j ∈J, there exists ¿ 0 and 5j ¿ 0 such thatfor all j ∈J,

ReP2(s)¿ I for every s∈Dj := {s∈C0| |s − i!j|¡5j}:Since ReQ6 I and ReP2¿ I for s∈Dj, we see thatRe [I − Q(s) + P2(s)]¿ I , so that

‖[I − Q(s) + P2(s)]u‖¿ ‖u‖ ∀u∈U:

Using this, together with (3.7) and (3.8), we obtain

‖S−1(s)v‖¿ 2‖v‖ ∀v∈U; s∈

⋃j∈J

Dj: (3.9)

Furthermore, C is bounded on C0 \ ⋃j∈J Dj, so from

Lemma 3.3 there exists 1 ¿ 0 such that ReQ6 (1 − 1)Iin C0 \ ⋃

j∈J Dj. Since by hypothesis ReP2(s)¿ 0 fors∈C0, Re (I − Q(s) + P2(s))¿ 1I for s∈C0 \ ⋃

j∈J Dj.Therefore, using again the formulas (3.7) and (3.8), weobtain that

‖S−1(s)v‖¿ 12

‖v‖ ∀v∈U; s∈C0

∖ ⋃j∈J

Dj: (3.10)

Combining (3.9) and (3.10), we see that S−1 is uniformlybounded from below on C0, i.e., S(s) is uniformly boundedon C0. Now we can follow the rest of the proof of Theorem3.2, taking = 1.

4. Noise reduction in a stuctural acoustics model

We consider a model for the following situation: A rect-angle 6 represents a two-dimensional cross section of anacoustic cavity. An acoustic velocity potential z satis0es thewave equation in 6. 70 is one side of the boundary of 6, andrepresents a cross section of the active wall of the cavity. Wedenote the displacement of this wall by v, which satis0es abeam equation with Kelvin–Voigt damping. The boundaryconditions for the wave equation include boundary dampingwhich causes the system energy to decay exponentially. Thecontrol action is realized by the placement of a piezoelectricceramic patch on 70; a voltage is applied on this patch andthe resulting bending moments can be interpreted as deriva-tives of delta functions. Furthermore, there is a noise sourced(t) applied to the active wall, which we assume to be thesum of 0nitely many sinusoids; in an application the acous-tic cavity might be an airplane cockpit and the noise mightbe engine noise. We refer to zt as the acoustic pressure, al-though it is actually proportional to the acoustic pressure.We are interested in reducing the acoustic pressure near a

point 80 ∈6, so we take the observation y(t) to be an integralof the acoustic pressure over a small neighborhood 60 ⊂6 of 80. We wish to design a controller to maintain theexponential stability of the system, and reject the e:ect ofthe noise d(t) from the output. We will apply Theorem 1.1 toobtain this controller. This model has been studied in detailin Lasiecka (2002, Chapters 4–7). A tracking problem fora related structural acoustics model has been numericallystudied in Banks, Demetrion, and Smith (1996).Let z = z(8; t) for t ∈ [0;∞) and 8∈6, let v = v(9; t)

for t ∈ [0;∞) and 9∈70, and let @=@; denote the outwardnormal derivative on 7. Let U = Y = R. In our motivatingexample b2 will represent the action of one piezoelectricceramic patch, and hence it will be of the form b2 = �5′=1 −�5′=2 , where �∈R, =1; =2 ∈70, and 5′= is the (distributional)derivative of the Dirac delta function concentrated at =. Werefer to Banks and Smith (1995, Eq. (2.3)) for a derivationof this input operator.Let the disturbance d be of the form considered in

Section 1,

d(t) =∑j∈J

djei!jt ; dj ∈R; !j ∈R; (4.1)

where J is a 0nite index set. Let b1(9) be a function on 70

which describes the spatial distribution on which the distur-bance term d(t) acts; for instance, if the noise is uniformlydistributed throughout 70, then b1(9) = 1. We will assumefor simplicity that b1 ∈L2(70), although the same procedurewill work also for some distributions b1 not in L2(70). Weconsider the following model:

ztt =Vz on [0;∞) × 6;

@z@;

+ #zt = vt on [0;∞) × 70;

@z@;

+ #zt = 0 on [0;∞) × 7 \ 70;

vtt = −D4v − D4vt − zt + b1d+ b2u on [0;∞) × 70;

v(a; t) = v(b; t) = 0; (Dv)(a; t) = (Dv)(b; t) = 0; (4.2)

where D is the derivative w.r.t. 9 operator, and t ∈ [0;∞).The corresponding output signal is

y(t) =∫60

zt(8; t) d8: (4.3)

The damping, parameterized by #¿ 0, is either a naturalpart of the system, such as an energy-absorbing coating,or it would be introduced as a stabilizing feedback beforethe low-gain tracking is applied. This system 0ts into theframework described by Fig. 3 (which is a special case ofthe system in Fig. 1), where the reference signal r is takento be zero, P2 is the transfer function from u to y, and P0

1 isthe transfer function from d to y. For d=0, this system is aspecial case of the one described by Eqs. (5.2.1) and (5.3.23)


in Lasiecka (2002), and the results in Lasiecka (2002) willallow us to verify the hypotheses of Theorem 1.1.The state of this system is x(t) = [z(t) zt(t) v(t) vt(t)]�,

and the state space with the appropriate compatabilityconditions is

X :={[z1 z2 v1 v2]� ∈H 1(6) × L2(6) × H 2

0 (70)

×L2(70)∣∣∣∣∫6z2 d8

=∫70

[v1 − #z1] d" − #∫7\70

z1 d"

}:

From Lasiecka (2002, Theorem 5.4.1), Eqs. (4.2) with d=0and u = 0 determine an exponentially stable semigroup Ton X with generator A. (For an explicit description of A werefer to Lasiecka (2002).)The output signal y in (4.3) is obtained via a bounded

observation functional C de0ned on X . If we denote B1 =[0 0 0 b1]� and B2 = [0 0 0 b2]�, then (4.2) and (4.3) canbe rewritten in the form

x(t) = Ax(t) + B1d(t) + B2u(t); y(t) = Cx(t); (4.4)

and B1 is bounded (i.e., B1 ∈X ). Hence, these equations de-termine a well-posed and regular system �p if and only ifthe control operator B2 is admissible for T (as explainedin Section 2, after (2.10)). The admissibility of B2 is es-tablished in Lasiecka (2002, Proposition 6.6.1), when thedamping parameter #=0 and, as pointed out after (Lasiecka,2002, Remark 6.6.1), the addition of damping simpli0es thearguments for proving (Lasiecka, 2002, Proposition 6.6.1).Thus, �p is a well-posed, exponentially stable system, so wecan apply Theorem 1.1 to it (we now have d and y in placeof w and z).We comment on condition (1.5), which in our case re-

duces to P2(i!j)Kj ∈C0. Clearly it will be possible tochoose the gains {Kj}j∈J so that this is satis0ed if and onlyif P2(i!j) �= 0 at all of the frequencies !j. Since P2 is an-alytic and not identically equal to zero in C� for �¡ 0, weimmediately conclude that the zeros of P2 on the imaginaryaxis have no 0nite accumulation point. Unfortunately, it isvery di9cult to compute P2(i!j) directly from (4.2) and(4.3); for this example the zeros of P2 would have to be es-timated numerically. We will now prove that P2(0)=0 (thisimplies that for this model constant disturbances cannot berejected). If the initial state is zero, then (4.3) implies that

y(s) = s∫60

z(8; s) d8: (4.5)

Let us denote the growth bound of T by /, so that /¡ 0.Choose #∈ (/; 0), so that P2 ∈H∞

# . Assume that d = 0,u∈L2

#[0;∞) and x(0) = 0. Then the state trajectory x,

given by x(t) =∫ t0 Tt−"B2u(") d", satis0es x∈H 2

# (X ) (the

de0nition of H 2# was given after (3.3)). Since z is the 0rst

component of x, we have z ∈H 2# (H

1(6)). Now (4.5) showsthat y is analytic in a neighborhood of 0 and y(0)=0. Sincey = P2u, we conclude that P2(0) = 0.

5. Disturbance rejection in a coupled beam

In this section we consider two coupled Euler–Bernoullibeams, not necessarily of equal length, with the disturbanceacting through a term distributed along the beams. The dis-turbance has a 0nite number of sinusoidal components withknown frequencies. There are two control inputs, acting atthe joint. Our goal is to design a controller which exponen-tially stabilizes the system and rejects the disturbance. Wemention the related work of Morgul (2001), who considereda disturbance rejection problem for a single undamped beamwith force input at one end. He showed that his closed-loopsystem is stable (strongly or exponentially, depending on atechnical condition), but the decay rate of the error has notbeen considered.We assume that 9∈ (0; 1), the 0rst beam has spatial ex-

tent [0; 9], the second beam has spatial extent [9; 1], andthe beams are hinged at 0 and 1. Both are uniform Euler–Bernoulli beams with the same mass density per unit lengthm and the same Iexural rigidity EI . We normalize so thatEI=m = 1. Let w(8; t) be the displacement of the coupledbeams at position 8∈ [0; 1] and time t ¿ 0. The notationw(8; t) denotes the derivative of w(8; t) with respect to time,and D denotes the spatial di:erentiation operator. Then wsatis0es the following equations:

3w(8; t) + D4w(8; t) = f(8)d(t); 8∈ (0; 9) ∪ (9; 1);

w(0; t) = w(1; t) = 0; D2w(0; t) = D2w(1; t) = 0;

w(9−; t) = w(9+; t); Dw(9−; t) = Dw(9+; t) (5.1)

where the disturbance d(t) is of the form (4.1) and thespatial distribution of the disturbance is given by the functionf(8). We assume for simplicity that f∈L2[0; 1], but thedevelopment will work for some distributions f as well. Seethe Introduction of Chen et al. (1989) for a discussion ofthe physical meaning of these joint conditions.In order to solve the disturbance rejection problem for the

coupled beams (5.1), we apply a stabilizing feedback, andthen apply the controller described in Theorem 1.2, whichdoes not require the gain to be small. It is possible to ex-ponentially stabilize the coupled beams with either a rigidsupport joint (a pointwise shear force feedback) or an an-gle guide joint (a pointwise bending moment feedback); seeChen et al. (1989) for a discussion of the physical meaningof these feedback controls, and Rebarber (1995) for expo-nential stability and well-posedness results. For instance, theclosed-loop system with the shear force feedback is expo-nentially stable when 9= 1

2 , and the closed-loop system withbending moment feedback is exponentially stable with 9= 1

3 ,


Fig. 5. The damped coupled beams system �p, obtained from the un-damped coupled beams system �0 by negative output feedback. Thetransfer function Q from u to z, and hence the transfer function P2 fromu to z, are positive.

and either feedback leads to a well-posed closed-loop sys-tem for any joint placement. Hence we can apply Theorem1.1 to obtain a tracking and disturbance rejection to eitherof these closed-loop systems. Unfortunately, the exponentialstability of this system with either of these feedbacks is verynon-robust with respect to the joint position (see Rebarber,1995), hence it is not likely to be practical. Therefore, wewill apply a feedback which has been recently shown inAmmari, Liu, and Tucsnak (2002) to exponentially stabilizein a way which is robust with respect to joint position. InAmmari et al. (2002) it is shown that if both types of dissi-pative feedback mechanisms are incorporated into the joint,then the system is exponentially stable independent of thejoint position 9. Thus, Ammari et al. (2002) considers theequations (5.1) together with the joint conditions

D3w(9+; t) − D3w(9−; t) = −w(9; t) + u1;

D2w(9−; t) − D2w(9+; t) = −Dw(9; t) + u2:(5.2)

It is shown in Ammari et al. (2002) that for any 9∈ (0; 1),(5.1)–(5.2) (with u0) determine an exponentially stablesemigroup describing the evolution of [w(·; t)wt(·; t)]� inthe state space

X = {[x1 x2]� ∈H 2(0; 1) × L2(0; 1)| x1(0) = x1(1) = 0};

with inner product 〈[x1 x2]�; [y1 y2]�〉=∫ 10 {D2x1(?)D2y1(?) + x2(?)y2(?)} d?.We need to reformulate (5.1) with (5.2) as the equations

describing the state trajectories of an appropriate well-posedsystem �p. The 0rst step is to show that a related undampedsystem �0 is well-posed and described by

x(t) = A0x(t) + Bdd(t) + Bu(t); z(t) = B∗�x(t); (5.3)

with the same state space X , input spaces W =R, U =R2,and output space Y = R2, with the operators A0, Bd and Bto be determined shortly. In (5.3), B∗

� is the �-extension ofB∗, de0ned similarly to (2.5). We will then show that �p isobtained from �0 by negative output feedback, as shown inFig. 5.

We choose Bd =[0 f(·)]�. We use the notation u for thecontrol input of �0. The state [w w]� of �0 satis0es (5.1)and

D3w(9+; t) − D3w(9−; t) = u 1(t); Dw(9; t) = u 2(t);

z1(t) = w(9; t); z2(t) = D2w(9−; t) − D2w(9+; t): (5.4)

Let u= [u 1 u 2]� and z = [z1 z2]�. We need to identify thecontrol operator B that corresponds to (5.1) with (5.4). Wewill use the approach in Section 3 of Ho and Russell (1983).As usual, (5.3) will be true as an equation in X−1 := D(A∗

0)′,

so A0 is extended to an operator from X to X−1 by

〈A0x; v〉 = 〈x; A∗0v〉 for all v∈D(A∗

0) =D(A0):

We use the same symbol A0 for this standard extension.For (5.1) with (5.4) A0 (and its domain) is given by

A0 :=

[0 I

−D4 0

]; (5.5)

D(A0) = {[x1 x2]� ∈X | x1 ∈H 4(0; 9) ∪ H 4(9; 1);

x2 ∈H 2(0; 1);

D2x1(0) = D2x1(1) = 0; x2(0) = x2(1) = 0;

Dx2(9) = 0; D3x1(9+) = D3x1(9−)}:The operator D4 is applied separately on the intervals (0; 9)and (9; 1). It is shown in Rebarber (1995, Section 3) thatA0 is a skew adjoint operator (which has a Riesz basis ofeigenvectors) on X , hence it generates a unitary semigroupS on X .

Proposition 5.1. For [v1 v2]� ∈D(A∗0) =D(A0), de=ne

B∗1 [v1 v2]� = v2(9); B∗

2 [v1 v2]� = D2v1(9−) − D2v1(9+);

and B= [B1 B2]. Then B is an admissible control operatorfor S, B∗ is an admissible observation operator for S, andthe system �0 described by (5.1) with (5.4) is regular andit can be written in the form (5.3).

Proof. It is obvious that with the above choice of B∗1 and B∗

2 ,the observations in (5.4) can be written for x(t)∈D(A0) asz(t)=B∗x(t). Let A1 be the extension of A0 given by the samematrix of operators (5.5), with the following domain, whichno longer requires Dx2(9) = 0 and D3x1(9+) = D3x1(9−),since these are boundary conditions in which the controlsare incorporated:

D(A1) = {[x1x2]� ∈X |x1 ∈H 4(0; 9) ∪ H 4(9; 1);

x2 ∈H 2(0; 1);

D2x1(0) = D2x1(1) = 0; x2(0) = x2(1) = 0}:Then (5.1) is equivalent to

x(t) = A1x(t); x(t) = [w(t) w(t)]�:


Using integration by parts, we obtain for x=[x1 x2]� ∈D(A1)and v= [v1 v2]T ∈D(A∗

0) =D(A0),

〈A1x; v〉= 〈x;−A0v〉 + [D3x1(9+) − D3x1(9−)]v2(9)

+Dx2(9)[D2v1(9−) − D2v1(9+)]: (5.6)

Since A∗0 = −A0, (5.6) implies that the following equation

holds in X−1:

A1x = A0x + [D3x1(9+) − D3x1(9−)]B1 + Dx2(9)B2;

for all x∈D(A1). Since D3w(9+; t) − D3w(9−; t) = u 1(t)and Dw(9; t)= u 2(t), (5.1) and the 0rst part of (5.4) can bewritten as the 0rst part of (5.3).It is shown in Rebarber (1995, Lemma 3.7) that B2 is an

admissible control operator for S. The admissibility of B1 re-quires a standard argument: we will show that the expansioncoe9cients for B1 satisfy the Carleson measure criterion,found in Ho and Russell (1983, Corollary 2.5). In Rebarber(1995, Proposition 3.3) it is shown that the multiplicity ofeach eigenvalue of A0 is one. In the proof of Rebarber (1995,Lemma 3.7) it is shown that for any h¿ 1, the number Nh ofeigenvalues in {z ∈C | 06Re z6 h; a−h6 Im z6 a+h}satis0es Nh6Mh for some M ¡∞ that is independent ofa∈R.Let {,±!} be a complete set of independent eigenvectors

of A0, corresponding to the eigenvalues ±i!2 (describedbelow). Let C±! ¿ 0 be chosen so that ‖C±!,±!‖ = 1.A detailed description of ,±! is given in Rebarber (1995,Lemma 3.2), and C±! is given by equation (3.38) inRebarber (1995). Let b±! = C±!B∗

1,±!. If (b±!) is abounded sequence, then according to the Carleson measurecriterion B1 is admissible. The eigenvectors associated withnonzero eigenvalues ±i!2 are of the form

,±! =

[ != ± i!2

!

](5.7)

with ! ∈H 2(0; 1), so b±! = C±! !(9).There are two types of eigenvectors of A0 associated with

non-zero eigenvalues. If cos(Cn9)=0 for some n∈Z+, then- = ±i(Cn)2 are eigenvalues with associated eigenvectorsas in (5.7), with != Cn and !(s) = sin(Cns). In this caseC±!=1, so b±!=sin(Cn9), and these coe9cients are clearlybounded.Let ! be a positive real solution of

tanh(!9) − tan(!9) − tanh(!(1 − 9))

+tan(!(1 − 9)) = 0:

Then - = ±i!2 are eigenvalues of A0 with the associatedeigenvectors given in (5.7), with ! given in part (c) ofRebarber (1995, Lemma 3.2). We compute that

!(9) = tanh(!9) − tan(!9): (5.8)

Furthermore, rewriting (3.38) in Rebarber (1995),1

C2±!= 9[tan2(!9) + tanh2(!9)] + (1 − 9)[tan2(!(9 − 1))

+ tanh2(!(9 − 1))]:

Since 9∈ (0; 1), it is clear from this formula and (5.8) that(b±!)=(C±! !(9)) is a bounded sequence for these eigen-vectors. This shows that B1 is an admissible control operatorfor S, hence B is an admissible control operator for S. By astandard duality argument, B∗ is an admissible observationoperator for S.In Rebarber (1995), it is shown that (5.1) and (5.4) with

u1 = 0 (i.e., with B1 replaced by the zero vector) determinea well-posed and regular system with feedthrough operator0. The proof follows directly from the eigenvalue placementand the boundedness of the sequence of expansion coe9-cients for B2 (see Rebarber (1995, Theorem 2.7)), whichis also true for B1, so our system �0 is well-posed andregular.

Now we show that the damped system �p with inputs d,u1 and u2 and outputs z1 and z2, described by (5.1), (5.2)and

z1(t) = w(9; t); z2(t) = D2w(9−; t) − D2w(9+; t); (5.9)

is well-posed. We do this by showing that �p is obtainedfrom �0 by static output feedback. Let Q be the transferfunction from u to z for �0. We claim thatQ is positive. Thedegree of unboundedness of any B∈L(U; X−1), denoted�(B), is the in0mum of those �¿ 0 for which there existpositive constants 5; such that

‖(-I − A)−1B‖L(U;X )65

-1−� ∀-∈ (!;∞)

(this was introduced in Rebarber and Weiss (2000,Section 1)). It is shown in Curtain and Weiss (submitted,Remark 5.4) that if a well-posed system has a skew-adjointgenerator A0, a control operator B with �(B)¡ 1

2 , and itsobservation operator is B∗, then this system is regular andthe transfer function Q(s)=B∗

�(sI −A0)−1B is positive. Toapply this result, we need to show that �(B)¡ 1

2 . For C±!

and ,±! de0ned above, it is shown in Rebarber (1995,Section 3) that ()±!) := (C±!,±!) is a Riesz basis ofeigenvectors of A0 for X . We have shown above that theexpansion coe9ents for B1 in this basis are bounded, and inRebarber (1995, Lemma 3.7) the same is shown for B2. It isshown in Rebarber (1995, Lemma 3.6) that the eigenvaluesof A0 can be enumerated so that -±k = ±i!2

k , where thereexists a; b∈R; m¿ 0 such that mk + b6wk 6mk + a fork ∈N. Therefore, it is easy to see that for any ¿ 0, wecan write B= A1=4+

0 B, where B is bounded. It then followsfrom Krein (1971, Eq. (5.15)) that �(B)6 1

4 , which showsthat Q is positive. Hence Re (I + Q(s))¿ I for s∈C0

and this implies that K = [0 − I ]� is an admissible feed-back operator for �0 (see Section 2 for this concept). Letu = [u1 u2]�. Thus, the feedback u(t) = −z(t) + u(t) for


�0 yields the well-posed and regular damped system �p.According to (2.11) and (2.12), �p can be written (usingthat [Bd B]K = −B) in the form

x(t) = (A0 − BB∗�)x(t) + Bdd(t) + Bu(t);

z(t) = B∗�x(t):

(5.10)

Thus, the semigroup generator of �p is A = A0 − BB∗�.

Using the feedback relation u(t) = −z(t) + u(t) in (5.4),we obtain (5.2) and (5.9), so that (5.1), (5.2) and (5.9)are represented by (5.10). As mentioned earlier, it followsfrom Ammari et al. (2002) that A generates an exponen-tially stable semigroup.The transfer function of �p is P = [P1 P2], where

P2(s)=Q(s)(I+Q(s))−1, which is easily seen from this for-mula to be positive. We will apply Theorem 1.2 to this sys-tem.We note that since Bd is a bounded operator, the transferfunction from d to z is well-posed. Therefore, to reject thedisturbance while maintaining exponential stability, we canapply Theorem 1.2 to the system described by (5.1),(5.2) and (5.9) as long as {!j}j∈J ⊂ R is such thatReP2(i!j) is invertible for all j ∈J. Let us examinethis condition in terms of Q(i!j). Since it is easy to ver-ify that Q∗(i!j) = −Q(i!j), we see that ReP2(i!j) =−Q(i!j)2[I − Q(i!j)2]−1. Since ReQ(i!j)26 0, wesee that [I − Q(i!j)2]−1 is invertible for all j ∈J, soReP2(i!j) is invertible if and only if Q(i!j) is invertible.As an example, we consider the case when 9= 1

2 . Writes= i=2. In this case

Q(s) =

[Q11(s) 0

0 Q22(s)

];

where

Q11(s) =4i=

{tan

=2

− tanh=2

};

Q22(s) =4i=

sinh(==2) sin(==2)sin(==2) cosh(==2) − sinh(==2) cos(==2)

:

Thus,Q(s) is nonsingular everywhere except at points whereQ11(s) = 0 or Q22(s) = 0. We have that Q11(s) = 0 whens=±i=21; k , where {=1; k}∞k=0 is asymptotic to {C=2+2Ck}∞k=0.Furthermore, we can see that Q22(s) = 0 when s = ±i=22; k ,where =2; k = 2C+ 2Ck for k ∈Z+. Both Q11 and Q22 haveremovable singularities at 0, and their analytic extensionshave a zero at 0. Thus, we can track signals containing com-ponents at all frequencies ! except 0 and {±=21; k ;±=22; k}∞k=0.

Acknowledgements

The 0rst author was supported in part by National Sci-ence Foundation (USA) grant DMS-0206951, and by a vis-iting fellowship awarded by the EPSRC (UK) in 2002. Thesecond author was supported in part by the EPSRC plat-form grant “Analysis and Control of Lagrangian Systems”,GR/R05048, awarded in 2001.

References

Ammari, K., Liu, Z., & Tucsnak, M. (2002). Decay rates for a beamwith pointwise force and moment feedback. Mathematics of Control,Signals, and Systems, 15, 229–255.

Banks, H. T., Demetriou, M. A., & Smith, R. C. (1996). Robustnessstudies for H∞ feedback control in a structural acoustic model withperiodic excitation. International Journal of Robust and NonlinearControl, 6, 453–478.

Banks, H. T., & Smith, R. C. (1995). Feedback control of noise in a2-D nonlinear structural acoustics model. Discrete and ContinuousDynamical Systems, 1, 119–149.

Chen, G., Krantz, S. G., Russell, D. L., Wayne, C. E., West, H. H., &Coleman, M. P. (1989). Analysis, designs, and behavior of dissipativejoints for coupled beams. SIAM Journal on Control and Optimization,49, 1665–1693.

Curtain, R., & Weiss, G. Stabilization of essentially skew-adjoint systemsby collocated feedback. Submitted, available at www.ee.ic.ac.uk/CAP/

Davison, E. J. (1976a). The robust control of a servomechanism problemfor linear time invariant multivariable systems. IEEE Transactions onAutomatic Control, 21, 25–34.

Davison, E. J. (1976b). Multivariable tuning regulators: the feedforwardand robust control of a general servomechanism problem. IEEETransactions on Automatic Control, 21, 35–47.

Francis, B. A., & Wonham, W. M. (1975). The internal modelprinciple for linear multivariable regulators. Applied Mathematics andOptimization, 2, 170–194.

H3am3al3ainen, T., & Pohjolainen, S. (1996). Robust control and tuningproblem for distributed parameter systems. International Journal ofRobust and Nonlinear Control, 6, 479–500.

H3am3al3ainen, T., & Pohjolainen, S. (2000). A 0nite dimensional robustcontroller for systems in the CD-algebra. IEEE Transactions onAutomatic Control, 45, 421–431.

Ho, L. F., & Russell, D. L. (1983). Admissible input elements for systemsin Hilbert space and a Carleson measure criterion. SIAM Journal onControl and Optimization, 21, 614–640.

Krein, S. G. (1971). In J. M. Danskin (Trans.), Linear diBerentialequations in Banach space. Translations of Mathematical Monographs,Providence, RI: American Mathematical Society (in Russian).

Lasiecka, I. (2002). Mathematical control theory of coupled PDEs.CBMS-NSF Regional Conference Series in Applied Mathematics.Philadelphia: SIAM.

Logemann, H., Bontsema, J., & Owens, D. H. (1988). Low-gain control ofdistributed parameter systems with unbounded control and observation.Control Theory and Advanced Technology, 4, 429–446.

Logemann, H., & Curtain, R. F. (2000). Absolute stability results forin0nite-dimensional well-posed systems with applications to low-gaincontrol. ESAIM: Control, Optimisation and Calculus of Variations,5, 395–424, http://www.edpsciences.com/articles/cocv/abs/2000/01/cocvVol5-16.

Logemann, H., & Mawby, A. D. (2001). Low–gain integral control ofin0nite dimensional regular linear systems subject to input hysteresis.In F. Colonius et al. (Eds.), Advances in math. systems theory (pp.255–293). Boston: Birkh3auser.

Logemann, H., & Owens, D. H. (1989). Low–gain control of unknownin0nite-dimensional systems: A frequency-domain approach.Dynamicsand Stability of Systems, 4, 13–29.

Logemann, H., Ryan, E. P., & Townley, S. (1998). Integral control ofin0nite-dimensional linear systems subject to input saturation. SIAMJournal on Control and Optimization, 36, 1940–1961.

Logemann, H., & Townley, S. (1997a). Low–gain control of uncertainregular linear systems. SIAM Journal on Control and Optimization,35, 78–116.

Logemann, H., & Townley, S. (1997b). Discrete-time low-gain controlof uncertain in0nite-dimensional system. IEEE Transactions onAutomatic Control, 42, 22–37.

http://www.ee.ic.ac.uk/CAP/

http://www.edpsciences.com/articles/cocv/abs/2000/01/cocvVol5-16

http://www.edpsciences.com/articles/cocv/abs/2000/01/cocvVol5-16


Morgul, O. (2001). Stabilization and disturbance rejection for the beamequation. IEEE Transactions on Automatic Control, 46, 1913–1918.

Pohjolainen, S. (1982). Robust multivariable PI-controllers for in0nitedimensional systems. IEEE Transactions on Automatic Control, 27,17–30.

Pohjolainen, S. (1985). Robust controllers for systems with exponentiallystable strongly continuous semigroups. Journal of MathematicalAnalysis and Applications, 111, 622–636.

Pohjolainen, S., & L3atti, I. (1983). Robust controller for boundary controlsystems. International Journal of Control, 38, 1189–1197.

Rebarber, R. (1995). Exponential stability of coupled beams withdissipative joints: a frequency domain approach. SIAM Journal onControl and Optimization, 33, 1–28.

Rebarber, R., & Weiss, G. (2000). Necessary conditions for exactcontrollability with a 0nite-dimensional input space. Systems &Control Letters, 40, 217–227.

Salamon, D. (1987). In0nite dimensional systems with unbounded controland observation: a functional analytic approach. Transactions of theAmerican Mathematical Society, 300, 383–431.

Salamon, D. (1989). Realization theory in Hilbert space. MathematicalSystems Theory, 21, 147–164.

Sta:ans, O. J. (1997). Quadratic optimal control of stable well-posedlinear systems. Transactions of the American Mathematical Society,349, 3679–3715.

Sta:ans, O. J. (1998). Coprime factorizations and well-posed linearsystems. SIAM Journal on Control and Optimization, 38, 1268–1292.

Sta:ans, O. J. (1999). Admissible factorizations of Hankel operatorsinduce well-posed linear systems. Systems and Control Letters, 37,301–307.

Sta:ans, O. J., & Weiss, G. (2002). Transfer functions of regular linearsystems, Part II: The system operator and the Lax-Phillips semigroup.Transactions of the American Mathematical Society, 354, 3229–3262.

Weiss, G. (1989a). Admissibility of unbounded control operators. SIAMJournal on Control and Optimization, 27, 527–545.

Weiss, G. (1989b). Admissible observation operators for linearsemigroups. Israel J. Math., 65, 17–43.

Weiss, G. (1994a). Transfer functions of regular linear systems, PartI: Characterizations of regularity. Transactions of the AmericanMathematical Society, 342, 827–854.

Weiss, G. (1994b). Regular linear systems with feedback. Mathematicsof Control, Signals, and Systems, 7, 23–57.

Weiss, G., & Curtain, R. F. (1997). Dynamic stabilization of regularlinear systems. IEEE Transactions on Automatic Control, 42, 4–21.

Weiss, G., & Rebarber, R. (2001). Optimizability and estimatability forin0nite-dimensional linear systems. SIAM Journal on Control andOptimization, 39, 1204–1232.

Richard Rebarber received his BA fromOberlin College and his MA and Ph.D. inMathematics from University of Wisconsin,Madison. He is currently a Professor in theDepartment of Mathematics and Statisticsat University of Nebraksa, Lincoln. His re-search interests are in control theory of in0-nite dimensional systems, and in partial dif-ferential equations.

George Weiss received the Control Engi-neer degree from the Polytechnic Instituteof Bucharest, Romania, in 1981 and thePh.D. degree in Applied Mathematics fromthe Weizmann Institute, Rehovot, Israel, in1989. He has been working at Brown Uni-versity (Providence), Virginia Tech (Blacks-burg), the Weizmann Institute (Rehovot),Ben-Gurion University (Beer Sheva) andthe University of Exeter (UK). Currently heis with Imperial College London, UK. Hisresearch interests are distributed parameter

systems, operator semigroups, power electronics, repetitive control andperiodic (in particular, sampled-data) linear systems.

internal model based tracking and disturbance rejection for stable well-posed systems

Documents