INFINITE-DIMENSIONAL CONTINUOUS-TIME ...

()

SIAM J. CONTROL AND OPTIMIZATION Vol. 34, No. 3, pp. 757-812, May 1996

1996 Society for Industrial and Applied Mathematics 001

INFINITE-DIMENSIONAL CONTINUOUS-TIME LINEAR SYSTEMS: STABILITY AND STRUCTURE ANALYSIS* RAIMUND J. OBERt AND YUANYIN WUt Abstract. The question of exponential and asymptotic stability of infinite-dimensional continuous-time statespace systems is investigated. It is shown that every (par)balanced realization is asymptotically stable. Conditions are given for (par)balanced, input-normal, or output-normal realizations to be asymptotically and]or exponentially stable. The boundedness of the system operators is also studied. Examples of delay systems are given to illustrate the theory.

Key words, linear infinite-dimensional systems, balanced realizations, stability, Hankel operators, semigroups of operators

AMS subject classifications. 93B 15, 93B20, 93B28, 93D20

1. Introduction. For a finite-dimensional linear system with transfer function G, there are standard ways to obtain a minimal, i.e., reachable and observable, state-space realization:

k(t) y(t)

Ax(t) + Bu(t), Cx(t) + Du(t).

This realization is unique in the sense that every other minimal realization is equivalent to it. The spectrum of the state propagation operator A is precisely the set of poles of the transfer function G(s) C(sI A)-IB + D, which is proper rational. Hence the realization is exponentially stable if and only if the poles of G are all in the open left half plane. Furthermore, exponential stability of the system is equivalent to asymptotic stability. This paper is concerned with the question of stability for infinite-dimensional systems. If the transfer function G is not rational, then we have an infinite-dimensional system of the above form, where the system operators A, B, and C are usually unbounded operators. In general, it is no longer true that all observable and reachable realizations are equivalent. The correspondence between the spectrum of the realization and the singularities of the transfer function does not necessarily hold. In general the exponential stability of a system cannot be determined by the location of the singularities of the transfer function (see, e.g., [18]). Also asymptotically stable systems are typically not exponentially

stable. There have been attempts to extend the results for finite-dimensional systems mentioned above to the infinite-dimensional case by restricting the transfer functions to a certain class. For example, Curtain [4], Yamamoto [29], and several other authors considered the equivalence between input]output stability and internal stability. We refer to [4] and [29] and the reference therein for the work in this direction. Inevitably, the stronger the results are, the smaller the class of transfer functions is. Here we present another approach. Instead of putting too stringent restrictions on the class of transfer functions to be studied, we restrict the class of realizations to (par)balanced realizations and the closely related input-normal and output-normal realizations. These types of realizations have been advocated by several authors [20], 13], 14], [30]. They were introduced in the finite-dimensional case as a means to perform model reduction in an easy fashion *Received by the editors March 10, 1993; accepted for publication (in revised form) November 9, 1994. This research was supported in part by NSF grant DMS-9304696. The research of the second author was supported in part by Texas Advanced Research Program grant 00974103. Center for Engineering Mathematics, University of Texas at Dallas, Richardson, TX 75083-0688.

757

758

RAIMUND J. OBER AND YUANYIN WU

[20]. Glover, Curtain, and Partington 14] derived infinite-dimensional cominuous-time balanced realizations for a class of transfer functions with nuclear Hankel operators. Young [30] developed a general realization theory of balanced realizations of infinite-dimensional discrete-time systems. The results were generalized to the continuous-time case by Ober and Montgomery-Smith [23]. The results by Young were also used by the authors to conduct an analysis of the stability and structural properties for infinite-dimensional discrete-time systems in [24]. In this paper we extend our analysis in [24] to the continuous-time case. The exponential and asymptotic stability properties of parbalanced, input-normal, and output-normal realizations are studied in detail. It is shown that all parbalanced realizations are asymptotically stable. For a subclass of transfer functions--namely, strictly noncyclic functions--results that are reminiscent of the finite-dimensional case are obtained. For this class of transfer functions the location of the singularities of the transfer function determines the exponential stability properties of parbalanced systems. The stability properties of parbalanced realizations are studied without the explicit presentation of the realizations. Structural properties of the realizations are also analyzed. In particular the boundedness of the system operators of the input- and output-normal realizations is investigated. Most of the results presented in this paper are in terms of the properties of the transfer functions and the Hankel operators with the transfer functions as symbols. This may therefore be regarded as expressing the internal properties of a system in terms of input/output properties. Related topics can be found in Dewilde [7], where systems with strictly noncyclic transfer functions are studied from an input/output point of view. We also refer to Baras, Brockett, and Fuhrmann [2], [3], [11]. For realization theory of nonrational transfer functions, Fuhrmann 11] and Helton 16] reference for transfer provide general references.

Our main tool is a bilinear map that maps discrete-time systems to continuous-time systems. This bilinear map is routinely used for finite-dimensional systems to translate discretetime results to continuous-time results and vice versa. In [23] properties of this bilinear map were studied for infinite-dimensional systems (see also 11]). Some continuous-time questions, however, such as exponential stability, cannot be directly answered by simply applying the bilinear transform to a discrete-time result. In such cases a more detailed study of the problem is necessary. The contents of the paper can be summarized as follows. In 2 we review the settings of infinite-dimensional continuous-time systems we will deal with. We restrict ourselves to so-called admissible systems. We relate continuous-time systems to discrete-time systems in 3, using the above-mentioned bilinear map. As Hankel operators play an important role in our approach, we discuss Hankel operators in 4 in both the discrete- and the continuous-time case. Concrete constructions of the continuous time restricted and *-restricted shift realizations are given in 5. They respectively represent the classes of input-normal and output-normal realizations and are intimately related to Hankel operators and translation semigroups. In 6 we establish the asymptotic stability of all parbalanced continuous-time realizations. Conditions for input-normal or output-normal realizations to be asymptotically stable are also given in terms of the cyclicity of the transfer functions. The topic of 7 is exponential stability. Necessary and sufficient conditions are given for the input- and output-normal realizations to be exponentially stable. These conditions are based on the spectral properties of the transfer functions. They also hold for parbalanced realizations as long as the transfer functions are strictly noncyclic. In 8 we investigate when the system operators are bounded, and finally some examples are given in

9.

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INFINITE-DIMENSIONAL SYSTEMS

The following symbols are used: the open unit disk, D the unit circle, the complement of (0D) U U,Y admissible discrete-time systems (3), DxU,Y admissible continuous-time systems (3), Cx the domain of an operator A on X, D(A) c_ X the space D(A) equipped with norm IIx (D(A), I1’ Ilxll 2 (D(A) (’), II, {fl f’(D(A), I1" Ilz) --> C, antilinear, bounded}, I[Gd(1) Gd(ZX)], Z ]]), for Gd TLD U’r O(z) lim r G(r), Gc(+Oo) the Hankel operator with symbol K, HK {FI F W --+ L(U, Y) analytic, supzew IlF(z)ll

112A

+ IIAxII 2,

or RH P,

(z) L(U,Y)

L(a) LHP

P+

{fl f D --+ Y analytic on D and suP0 Y square integrable on A }, A the open left half plane: {s C Re(s) < 0}, The orthogonal projection of L2y(A) onto Hy(W); A orA=i,W--RHP, the orthogonal projection of Hy(W) onto X or RHP,

RHP S S* S(Q) S(Q)* a(A)

crp(A) or(Q) cr.(G)

TLDU, v TLCU, v XvY (F, G)L = ir (F, G)R IU

_

Hy(W); W

the open right half plane: {s C Re(s) > 0}, the forward shift: (Sf)(z) zf(z) for f Hy(), the backward shift: (S*f)(z) z-[f(z) f(0)] for f H2(), Px SIx, the compression of S to X, where X H2y(D) 0 (QHZy(D)), S*IH()e(QH2(D)), the restriction of S* to Hy2(D) 3 (QHy2(D)), the spectrum of an operator A, the point spectrum of an operator A, the spectrum of an inner function Q 6 H (W) (Lemma 7.3), the set of points in C where G has no analytic continuation (7), {Gd[ Gd De --> L(U, Y) has a reachable and observable admissible realization}, L(U, Y) has a reachable and observable {Gel Gc RHP admissible realization}, closed linear span of subsets X and Y of a Hilbert space, F and G are weakly left coprime (3), F and G are weakly right coprime (3).

-

2. Admissible continuous-time state-space systems. The main aim of this section is to briefly set out the notation and introduce the most important system theoretic concepts for this paper. More details can be found in [11], [23], [27], and [6]. In the first subsection, admissible continuous-time systems are discussed. Input-normal, output-normal, and parbalanced

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realizations are defined in the second subsection. It is these classes of systems that are being analyzed in detail in later sections. What is meant by system equivalence for infinite-dimensional systems is defined in the third subsection.

2.1. Admissible continuous-time systems. It is well known that if A is the generator of a strongly continuous semigroup of operators (e tA)t>_O with domain of definition D(A), then D (A) is a Hilbert space with inner product induced by the graph norm

IIxlI2A --IIxlI2x + IIAxlI2x,

x

D(A).

Since IIx IIa Ilxll for x E D(A), we can embed X in D(A) ’, the set of antilinear continuous functionals on (D (A), I1" a ), by

X --+ D(A) (’), (y (x, y)).

x

Note that D(A) ’) is a Hilbert space with norm Ilfll’ := supllxllA_O on a Hilbert space, then the adjoint (A*, D (A*)) of (A, D (A)) is the generator of the adjoint * (see [26]). Hence, we have similarly that semigroup er-tA)t>O

D(A*) c_ X c_ D(A*) 0. We are now in a position to define admissible continuous-time systems. DEFINITION 2.1. A quadruple of operators (At., Bc, Cc, D.) is called an admissible continuous-time system with state space X, input space U, and output space Y, where X, U, and Y are separable Hilbert spaces, if 1. (Ac, D(A)) is the generator of a strongly continuous semigroup of contractions on X; 2. Bc U -+ (D(A*) (’), I1" I1’) is a bounded linear operator; 3. C. D(Cc) --+ Y is linear with D(Cc) D(Ac) + (I Ac) -1 B,.U and CclD(A (D(Ac), [l’llac) Y is bounded; 4. Cc(I Ac) -1Be L(U, Y); 5. A, B, and C are such that lim s Cc(sI A,.) -1B,. 0 in the norm topology; 6. De L(U, Y). We write ’r for the set of admissible continuous-time systems with input space U, output space Y, and state space X. By the resolvent identity, part 4 of the definition implies that G (s) :- Cc (s I A)L(U, Y) for all s RHP and G is analytic on the RH P. The function G,. is called the transfer function of the system, and (Ac, Bc, C,., Dc) is called a realization of G.

-

CVx

2.2. Duality, observability, reachability, and parbalanced realizations. In order to define observability and reachability for continuous-time systems we need to introduce the notion of the dual system of an admissible continuous-time system. y DEFINITION 2.2. Let (Ac, Bc, Cc, Dc) Then the dual system (A,., C, D,.) of (Ac, Bc, C, De) is given by 1. (Ac, D(Ac)) (A*c, D(A*c)), the adjoint operator of (Ac, D(Ac)); 2. /(y)[.] (y, Cc(.)) Y --+ D(A)(’; y

CUx

t"

,

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3.

(fc D(C-c) -+ (u, CfcXO) ((fcXo, u)

U,

D((fc)

D(A-c) + (I -/c) -1 lcY, where cXo is defined by

Bc(u)[xo], xo e D(A*c), u e U, (Yo, Cc(l Ac)-lBcu), x0 (I Ac)-l/yo, Yo e Y, u e U;

4.16c := D*c

Y-+ U. It can be directly verified that the dual system (Ac,/, 6rc,/5c) of an admissible continuoustime system (Ac, Bt., C, D) is admissible. If the continuous-time transfer function G(s) RH P L(U, Y) has an admissible realization (A, B, C, D), then the dual system (Ac, Bc, Cc, D) is a realization of the transfer function G(s) := (G(g))*, s e RH P, i.e., for all s RHP,

((s)

C(sl )-1 q_

(G(g))*

The definition of observability and reachability of admissible continuous-time systems is now given. DEFINITION 2.3. Let (At,, Bc, C, D,) C xu,Y then the operator

Oc D(Oc)

-->

Ly([0, x)), (CcetAx)t>O

X

is called the observability operator of the system (At,,

D(Oc)

{x

X

CcetAcx exists for almost all

Bc, Ct., D), where [0, oo), and CcetAcx

L2r,([0, oo))}.

We say that (A, Bt., C, Dr.) has a bounded observability operator if D(Ac) c_g_ D(O) and Oc extends to a bounded operator on X. This extension will also be denoted by 0. If (A, B, Cc, D) has a bounded observability operator O such that Ker(Oc) {0}, then the system (A, B, Cc, D) is called observable. Let (a-c, B-c, C, i) be the dual system of (ac, B, C, Dc). Ifthe observability operator 0- of (a-o C, D-) is a bounded operator on X, the adjoint of O is called the reachability operator, denoted by of (Ac, B, Co De), i.e.,

,

,

IfTc exists and range() is dense in X, the system (A, Bc, Cc, Dc) is said to be reach[3 able. The set of all reachable and observable continuous-time systems with input space U, output space Y, and state space X is denoted by L C xu,Y We mainly deal with this set of systems. The reachability Gramian VV and the observability Gramian Jl of a continuous-time system with bounded reachability operator 7 and bounded observability operator Oc are defined to be W,

.=

,a/It, :=

J* x---> x, OOc .x -+ x.

When ]/Yr. AA and the admissible system is observable and reachable, we say that the is A reachable and observable admissible system is said to be balparbalanced. system anced if V and A// /Y has a diagonal representation with respect to an orthonormal basis of the state space. If ]/Vc I, then a reachable and observable admissible system is called input-normal. If A/It. I, then a reachable and observable admissible system is called

output-normal.

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2.3. System equivalence. The concept of an equivalent state-space transformation of an admissible continuous-time system is slightly more complicated than in the discrete time case as the system operators are in general unbounded. Two systems (Zic Bi C., Dic) E C XU’Y 1 2, are called equivalent if there exists a such that invertible V operator boundedly L(X1, X2)

((ZZc, D(AZc)), BZc, (C2c, D(CZc)), DZc) ((VAV -1, VD(A))), VBlc, (Clc V-l, VD(Clc)), Dlc), where is given by

[BZc(u)l(x) (VB)(u)[x]

Bl(u)[V*x],

u

U, x e

D(AZc *) (V*)-ID(A)*).

If V is a unitary operator, then the two systems are said to be unitarily equivalent. We have the following results concerning equivalent systems. U’Y PROPOSITION 2.4. Let (aic B{. ci Di) C x, 1, 2, be two equivalent systems such

that

((A2c, D(A2c)), B2c, (C2c, D(C2c)), D2c) ((VAV -1 VD(A)), VB (Clc v-1 VD(C)), D) with V L(XI, X2) a boundedly invertible operator. Then 1. both (Z, BJ, CJ, DI) and (Z2, B2, C2, D2) realize the same transfer function. 2 if(Z) D) C XlU’Y hasobservabilityoperatorOandreachabilityoperatorT, then the observability and reachability operators of (a2 B2, D2) Cx2v’Y are respectively

BI C

C2

OV

-1

and

[3 The proof is straightforward. Thus equivalent systems have the same transfer function as well as the same observability and reachability properties. Moreover, it can be seen that unitary equivalent systems have the same Gramians. Hence unitary equivalence preserves parbalancing. We point out that for an admissible system ((A C’ D(A))) DI) C Xlv’r and a unitary operator V X -+ X2, the system

Proof

BC

2 D2c) ((A2 D(A2c)) B2c (C2c D( Cc)), ((VA)V -1, VD(A))), VB, (C)V -1, VD(C)), Dc)

is also admissible, where VB) is defined as above Therefore ((A), D(A))), B), Cc, Dlc) and (Ac2, B2, C, D) are unitarily equivalent. The class of continuous-time transfer functions that we are interested in are those that have reachable and observable continuous time realizations on some state space X, where X where U and Y are the is a separable Hilbert space. This class will be denoted by T LC input and output spaces, respectively. We characterize those transfer functions in terms of their Hankel operators in 4.

v,r,

3. Connection between continuous- and discrete-time systems. What is essential in our development is to relate discrete-time systems to continuous-time systems using a generalization of the well-known bilinear transformation for finite-dimensional systems. Thereby it is possible to carry some of the results in [24] for discrete-time systems over to continuoustime systems. It should be noted, however, that not all results of discrete-time systems can be translated to the continuous-time case in this way. For example, under this bilinear map an exponentially stable continuous-time system does not necessarily correspond to a power stable discrete-time system.

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3.1. Admissible discrete-time systems. We recall [24] that an admissible discrete-time system with input space U, output space Y, and state space X, with U, X, and Y being separable Hilbert spaces, is a quadruple of operators (Aa, Ba, Ca, Da) that satisfy the following: 1. Aa L(X) is a contraction and -1 cp(Aa); 2. Ba L(U, X), Ca L(X, Y) and Da L(U, Y); 3. the limit limr>l,rl Ca(rI + Aa) -1Ba exists in the norm topology. The set of all such systems is denoted by D xv,r For (Aa, Ba, Ca, Da) D xU,Y the function

Gd(Z)

Cd(ZI

Ad) -1 Bd -1- Dd

e ---> L(U, Y)

is called the transferfunction of (Ad, Bd, Cd, Dd) and (Ad, Bd, Cd, Dd) is called a realization of Gd. Evidently, the transfer function Gd is analytic on De and at infinity. is defined For (Ad, Bd, Cd, Dd) D xU,Y its observability operator Oa D(Od) -+ as

H2r

((QdX)(Z)--Z(CdAndX)Zn’n>O

xGD((Qd)’={xIZ(CdAndx)ZnGH2y} "n>o

If D(Oa) X, Oa is bounded and Ker(Oa) {0}, then the system (Aa, Ba, Ca, Da) is said to be observable. The system (Aa, Bd, Ca, Da) is said to be reachable if its reachability operator Ta D(d) --> X defined by

N where D(a) {--_,,=0 un Z IN 0, un 6 U} can be extended to a bounded operator with range dense in X. The set of all reachable and observable discrete-time admissible systems with input space U, output space Y, and state space X is denoted by L DxU’r. The set of all discrete-time transfer functions that have realizations (Aa, Ba, Ca, Da) LD xU,Y for some state space X is denoted by T LD U,r. A characterization will be given of this class of transfer functions in the next section. For (Ad, Bd, Ca, Dd) L Dx’r, we define its reachability Gramian kVa X X as

WdX

d*dX,

x

X,

X

X.

-

and its observability Gramian J/ld X --+ X as

:4dX

OOdX,

If Wd Wld and (Ad, Bd, Cd, Dd) is reachable and observable, then (Ad Bd, Cd, Dd) is said to be a parbalanced realization. If the Gramian of a parbalanced realization has a diagonal representation with respect to an orthonormal basis, the realization is said to be balanced. If Wd I, then the reachable and observable admissible system is called input-normal. If .AAd I, then the reachable and observable admissible system is called output-normal.

3.2. Bilinear transform. In the following theorems (see [23]) we introduce the map T D xU,Y --> C xU,Y which transforms discrete-time systems to continuous-time systems. Throughout the rest of this paper T will denote this map. THEOREM 3.1. Let (Ad, Bd, Cd, Dd) D xU,Y ;then T((Ad, Bd, Cd, Dd)) :-- (Ac, Be, C., D.) C xU,Y where the operators Ac, B, C., and D are defined as follows: 1. ac :-- (I + Ad)-l(Ad-- l) (Ad-- I)(I + Ad) -1, D(Ac) :--- D((I + Ad)-I). It a generates strongly continuous semigroup of contractions on X given by pt(Ad), > O, with qgt(z)

e t’+

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2. The operator Be is given by

Be := /(I + Ad) -1 Bd U u

D(A*c)(’) ,/-(I + Ad) -1 Bd(u)[x] / < Bd(u), (I + A})-I(x) >x

b.---->

3. The operator Ce is given by

Ce D(Cc)

Y,

"-+

limx

x

v/Cd(.l "k- Ad)-lx,

>1

where D(Cc)

D(Ac)

+ (I

Ae) -1BeU. On D(Ae) we have

%/Cd( I t_ Ad) -1.

CcID(Ac)

limz--,1 Cd()I -+- Ad)-lBd. Moreover, let the admissible discrete-time system (Ad, Bd, Cd, Dd) be a realization

4.

De "= Dd

of the

transfer function Gd Ie i.e., Gd(z)

Cd(ZI

Ad) -1Bd

+ Dd

---> L(U, Y)

for z

(Ac, Bc, Cc, Dc)

]Ie. Then T((Ad, Bd, Cd, Dd))

is an admissible continuous-time realization of the

Ge(s)

Gd 1

+

s) s

transfer function

RH P --+ L(U, Y).

The inverse map is considered in the next theorem [23]. THEOREM 3.2. Let (Ac, Be, Cc, Dc) C xU,Y ;then T_I ((Ac, Bc, Cc, Dc)) "--(Aa, Ba, Ca, Da) eD xU,Y where the operators Aa, Ba, Ca, and Dd are defined as follows: 1. Aa :-- (I+Ae)(I-Ae)-,andforx D(Ac)wehaveAax (I-Ac)-(I+Ac)x. 2. Ba := ./(I ac)-l Bc. 3. Cd :-- Cc(I Ac) -1. 4. Da :-- Cc(I Ac) Be + De. Moreover, let the admissible continuous time system (Ae, Be, Cc, Dc) be a realization of the transfer function

-

Gc RH P --> L(U, Y) i.e., Gc(s)

Cc(sI

Ac)-lBe + Dcfor s RHP. (Aa, Bd, Ca, Dd)

T -1 ((Ac, Be, Cc, Dc))

is an admissible discrete-time realization of the

Gd(Z)

Then

"=Gc( z-z+ 11)"

transfer function ]]])e

’-->

L(U,Y).

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We recall that two discrete-time systems (Adi Bdi Ccli Ddi) E D X’r (i 1 2) are there is a if bounded equivalent (unitarily equivalent) operator (a unitary operator) V from X1 onto

X2 such that (Adl, Bdl, Cdl, Ddl)

(VAd2V -1, VBd2, Cd2 V-l, Dd2).

In [23] it was shown that T preserves (unitary) equivalence of systems and respects duality of systems.

Note that in the previous two theorems the state spaces for the continuous- and discretetime realizations are the same. As will be seen in later sections for continuous-time systems it is more natural to work on a different yet unitarily equivalent state space that is a subspace of Hy2 (R H P). Here we point out the equivalence of the Hilbert spaces Hy2 (D) and Hy (R H P), where Y is a separable Hilbert space (see [25, Thm. 4.6]). PROPOSITION 3.3. The spaces H2y (D) and H2y (R H P) are unitarily equivalent by the map

The inverse of V is given by

W 1" Hy(RHP)

--->

(Wf.(o = fe(ol :=

(

+ of.

i

-,

The next result shows that observability and reachability properties as well as the Gramians are preserved under T. This implies that the transformation preserves parbalancing of systems. This result is the translation of a result in [23] to the frequency domain. Y THEOREM 3.4. Let (Ac, Bc, Cc, Dc) and (Ad, Bd, Cd, Dd) D xU,Y be such that

cUx

(Ac, Bc, Cc, Dc)

T((Ad, Bd, Cd, Dd)).

Then 1. (At., B., Cc, D.) is observable (reachable)ifandonlyif(Ad, Bd, Cd, Dd) is observable (reachable). In fact, if O (T) and Od (Td) are the observability (teachability) operators of (Ac, Bc, Cc, Dc) and (Ad, Bd, Cd, Dd), respectively, and if either (A, B, Cc, Dc) or (Ad, Bd, Cd, Dd has a bounded observability (reachability) operator, then the following relations hold:

glOdX where V1

H2y ([)

0c x,

X

X

n2y (RH P)

(’.dg-lu "]P.c.,-lu,

and V2

mations as defined in Proposition 3.3:

u

HZu(RHP))

H () --> H2v (RH P) are unitary transfor-

l-s), flHv2(I),

ff--(1 +s) 3 l+s

and

f E H2u(ID),

l 2

is the Laplace transform. 2. If the reachability Gramians W and /d (observability Gramians A/t and ./d) (Ad, Bd, Cd, Dd) and (Ac, Bc, Co, Dc) are defined, then

and

2c

Wd

(Mc

Md).

of

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Proof In [23] a "time domain" version of this result was proven. The presem result follows from the result in [23] by applying the z-transform (respectively, Laplace transform) q and using the unitary transformation of Proposition 3.3. z- then Gg TLD ’v i.e., G has a reachable and observTherefore if Gg(z) G,( z.--), i.e., if and only if G able discrete-time admissible realization, if and only if G. T LC has a reachable and observable continuous-time admissible realization. The combination of Theorems 3.1, 3.2, and 3.4 gives us an effective machinery to transform discrete-time results to the continuous-time case. Before doing this, we need to study Hankel operators which will be important in the analysis of parbalanced, input-normal, or output-normal realizations treated in the sequel.

,,

4. Linear systems and l-lankel operators. In the study of discrete-time systems Hankel operators on H2(]) play an important role 11]. Given a discrete-time transfer function, a Hankel operator can be associated with it in a natural way. The so-called restricted shift realization of the transfer function is constructed by using the range of the Hankel operator as its state space (see 11], [30], [24], and 5 below). When the Hankel operator is compact, a balanced realization can be obtained whose Gramians have diagonal representations with diagonal entries equal to the singular values of the Hankel operator [30]. In the continuoustime situation Hankel operators on H2(RHP) will be of equal importance. We therefore examine the relationship between discrete-time Hankel operators and their continuous-time counterparts.

4.1. l-lankel operators and realizability. Let G be analytic on )e and at infinity so that Gh(z) z-l[Gd(Z -1) Gd(OC)] is analytic on D. We define the operator Ha),

D(H,)

-

Hr2(D) by

(Ha),Df)(z)- P+adJf

(f

D(Ha))),

G-

{f H2 (D) f polynomial, Jf has nontangential limit inD almost everywhere (a.e.) at OD with limit in L2r(OI)} and (Jf)(z) f(1/z). The operator If D(Ha,) is dense in H(D) and Ha_, is called the Hankel operator with symbol 2 extension is also called the Hankel operator this on to a extends bounded operator H (D), and is denoted by Ha; with symbol The following lemma [24] relates te existence of a reachable and observable realization of a discrete-time transfer function G, to the boundedness of the Hankel operator Ha_,. LEMMA 4.1. The transferfunction G, is in T L D :’r i.e., G, has an admissible reachable and observable realization if and only if (i) G, is analytic on De and at infinity, (ii) the is limit lira r6]R G,t(r) exists in the norm topology, and (iii) the Hankel operator r H(RHP) is bounded. ] Proof This follows from Theorem 3.4, Proposition 4.3, and Lemma 4.1.

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4.2. Range spaces of Hankel operators and factorizations of transfer functions. It is known that the orthogonal complement,

H(RHP) ) rangeH

(rangeHc) +/-

c,

of the range of the Hankel operator H is invariam under any multiplication operator with symbol in H. Hence by Beurling’s theorem, the subspace (rangeHc) +/- is either {0} or Q HZr(RHP), where Q H(RHP) is a rigid function. A rigid function is a function Q 0 such that Q(iy) is for a.e. y a ]R a partial isometry with a fixed initial space (see, e.g., 11, p. 186], and [15]). In particular, inner functions are rigid functions. Using the above-defined notions, we introduce the concept of cyclicity of continuous-time transfer functions, which relates Hankel operators with their symbols. The discrete-time case was studied in, e.g., Fuhrmann [11]. A general study of strictly noncyclic transfer functions can also be found in Dewilde [7]. DEFINITION 4.5. Let Gc Hv,r (RH P). Then Gc is called 1. cyclic if(rangeH,Ri4p) +/- {0}; 2. noncyclic if (rangeHc,R/4e) +/QH2r(RHP), where Q H(RHP) is a rigid

function; 3. strictly noncyclic if (rangeHa,li4p) +/inner function.

QH2r(RHP), where Q H(RHP) is an

[3

Evidently in the scalar case G c is strictly noncyclic if and only if it is noncyclic. In the sequel it will be seen that the cyclicity of the transfer functions has much to do with the stability and other properties of their realizations. Here we present more information on cyclicity of H transfer functions. DEFINITION 4.6. Let G be in HL(v,y)(RHP). Then the L(U, Y)-valued function defined on L H P is called a meromorphic pseudocontinuation of bounded type of G if 1. is of bounded type, i.e.,

where F is a L(U, Y)-valued function and h is a scalar-valued function and both functions are bounded and analytic in L H P. 2. G and have the same strong radial limits on iN, i.e., for a.e. y IR lim x0,x--->0

G(x + iy).

U

The following proposition summarizes the connection between discrete- and continuoustime transfer functions in terms of cyclicity, meromorphic pseudocontinuation of bounded type, and factorizations. We refer to 11] for a discussion of these concepts for discrete-time transfer functions, which are analogous to those that have been defined here for continuoustime transfer functions. PROPOSITION 4.7. Let Gc TLC U’, Gd TLD tL’, and set G-(z) Z-I[Gd(Z -1) G a (cx)]. Assume that

Ga(z)

Gc

(Z-+ l

(z

tE

]]])e),

or equivalently

Gc(s) Then

l+s)

Gd 1- s

(s

RHP).

769


G

1. is strictly noncyclic (cyclic, noncyclic) ifand only if Gc is strictly noncyclic (cyclic, noncyclic). 2. Let Qd,1 Hr)(D), Qd,2 Ht)(D), Qc,1 Hr)(RHP), and Qc,2 be inner (RHP) functions. Let Fd, nL(U,y) (]), Fc Hf(v L(Y,U) (]])), Fd 2 and P), (RH Fc,2 Hv,r (RH P). Assume Hr,r

Fd’l(Z)

1-z Fc’l 1 AI-z _Gc(1)*Qc,1

Fd,2(zl

Fc,2

(1-)

Qa,i(z)=Qc,

z

i+

(1-Z)Gc(1),, +z

ac,2

1+

- D,

zi

l

z6II), i=1,2,

+

or equivalently

F,., (s)

F (s) Qc,i(s)

(1-;) (1-s) + Fa (i-;) + ( ) + -I- Ga(o)* Qa,1

Fa,1 1 -t-

1

s

sRHP,

.

RHP,

(ZFd,2(Z))* Qd,2(Z) (Z

0]).

1- s

Qa,

Qa,i

1

Ga(c)*,

l+s

s

s6RHP, i= 1 2.

+

Then Gc can be factored on iN as

Fc*,2Qc,2

Gc-- Qc, lFc*,l

if and only if G- can be factored on 0

G-(z)

as

Qd, I(Z)(ZFd, I(Z))*

3. Assume that Gc H(,r) (RHP) and n(,r) (I). Let Fa (Fc) be a L(U, Y)valued analyticnction in (LHP) and ha (hc) be a scalar-valued analyticnction in (L H P ), both bounded, such that

G

e

Fa(z)

1[ Fc (i-Z)

Gc(1)hc

hc 1

e,

+Z

Z

hd(Z)

z

’+ Z

or equivalently

Fc(s)

hc(s) Then by

e

l+s 1

hd 1

(I-s) + +--s)

Fd

l+s

s

s

-

(l-z)] i+

Gd(o)hd

LHP.

G-) has a meromorphic pseudocontinuation

ha

z

(l-s) l+s

Ze

s

LHP

of bounded type in ]])e, which is given

770


if and only if G

r

has a meromorphic pseudocontinuation

c

of bounded type in L H P

which

is given by

he [q Proof The results can be directly verified. The next theorem provides some convenient ways to determine whether a transfer function is cyclic, noncyclic, or strictly noncyclic Note that Q 6 HL(Z,Y) (RHP) and F 6 HL(U,Y) (R H P) are said to be weakly left coprime if Q Hz2 (R H P) v F Hu2 (R H P) Hy2 (R H P) where v denotes the closed linear span. In this case we write (Q, F)L Iy. If two functions Q1 HL(X,y)(RHP) and F1 H(u,z)(RHP) are such that and/ are weakly left (QI())* and/l(S) coprime, where 01(s) (Fl(ff))* (s RHP), they are said to be weakly right coprime, and we denote this by (Q1, F1)R IV (see Fuhrmann 11]). THEOREM 4.8. Let Gc HL(U,Y) (RHP) with finite-dimensional U and Y. Then the following statements are equivalent: 1. Gc is strictly noncyclic. 2. Gc has a meromorphic pseudocontinuation of bounded type on L H P. 3. On iN the function G can be factored as

Gc- QF- FQ2, where Q1 and Q2 are inner.nctions in HLy)(RH P) and Hv (RH P), respectively. The c (RH P) and HL(U,Y) (RH P), respectively, and the coprimefunctions F1 and F2 are in HL(Y,U) hess

conditions

(Q1, F1)R

Iv,

(Q2, F2)L

Iu

hold. Ifpart3 holds, then QIH2v(RHP) (rangeHac) and 02Hu(RHP) (rangeHc)+/-, where O2(s) (Q2(g))* and c(S) (Gc(5))*. Proof. Analogous results are shown in 11 for discrete-time transfer functions. Thus the theorem follows from Proposition 4.7. The factorization in the theorem is Fuhrmann’s generalization of the Douglas, Shapiro, and Shields factorization [8] to matrix-valued functions. For a given function, part 2 of the theorem may be easy to check. For example, the function e -us R(s) is strictly noncyclic, where c > 0 and R(s) is any rational function in H(u,r)(RH P). This is because e R(s) has a meromorphic pseudocontinuation of bounded type on L H P of the form F (s)leaSh (s), where if al an denote the poles of R(s), then, +/-

(S al) (s --an) (s -k- al)’’’ (s + an)’ and F(s) h(s)R(s). Part 2 of the theorem also gives the following corollary. COROLLARY 4.9. Under the assumption of the theorem, G Hv,r (RH P) is strictly [3 H(r,v (R H P) is strictly noncyclic. noncyclic if and only if

h(s)

r

4.3. Hankel operators with closed range. Similarly to Theorem 4.8, the following theorem (see 11]) gives necessary and sufficient conditions for the Hankel operator to have closed range. THEOREM 4.10. Let Gc HL(U,y)(RHP) with U and Y finite dimensional. Then the Hankel operator Hc has closed range ifand only ifon i thefunction Gc has thefactorization

Gc(s)- Q(s)F(s)*,

771


where Q

Hy)(RHP), F Hy,u)(RHP ), and the equality WQ + VF

Hr

holds for some W (RH P) and V right coprime. In this case

H

Ir

H(u,r) (RH P); that is,

Q and F are strongly

2 2 ( QH,(y)(RHP). (H(u)(RH P)) H(F)(RHP)

[q

This section essentially established that the unitary equivalence of the spaces H: (D) and and H6c, where implies the unitary equivalence of the Hankel operators z-1 Ga(z) Gc( z-)’ z 6 D. Therefore the spaces rangeHa) and rangeHa are unitarily equivalent. As a consequence, the discrete-time transfer function Gel and the continuous-time transfer function Gc have the same cyclicity properties. These results will be repeatedly used in the next section when we obtain the restricted shift realization of a continuous-time system by applying the bilinear map in 3 to the corresponding discrete-time system.

H:(RH P)

H6

5. Continuous-time shift realizations via a bilinear transformation. As a direct application of the bilinear transformation T given in 3 the continuous-time restricted and *-restricted shift realizations can be obtained from the corresponding discrete realizations. These realizations can be further analyzed via the connection between continuous and discretetime transfer functions shown in 3 and 4. Restricted and *-restricted shift realizations are central to the development here since they serve as prototypes of output-normal (respectively, input-normal) realizations. It will be shown in Proposition 6.2 that each output-(input-)normal realization of an admissible transfer function G is unitarily equivalent to the restricted (*-restricted) shift realization. The concrete representations of the continuous-time shift realizations obtained in this section will allow us to analyze input- and output-normal realizations in some detail in later sections.

Another important result of this section is Proposition 5.11, in which the state spaces of the restricted shift realizations for strictly noncyclic transfer functions are characterized through the inner factors in the Douglas-Shapiro-Shields factorizations of the transfer function.

5.1. Discrete-time shift realizations. We first recall the discrete-time restricted and *-restricted shift realizations of a discrete-time transfer function (see 11], [30], and [24]). THEOREM 5.1. Let Gel TLD U’r. Then Gel has two state-space realizations: with state space Xcl and (Acl,,, Bcl,,, Col,,, Dcl.,) with state space Xcl,,, i.e., (Acl, Bcl, Col, Dcl)

for z [e Gcl(Z)

Ccl(zI

Acl) -1Bcl

_

+ Dcl Ccl,,(zI Acl,,) -1Bcl,, 5-

They are given in the following way: 1. The state space Xcl is given by Xcl

rangeHei

Hrz(D), where

is the Hankel operator with symbol G-). The operators Acl, Bcl, Col, and Dcl are given as follows:

and

H6

772


f (z) f (0) feX, zeD,

(Adf)(z) :-- (S* f)(z) (Bdu)(z)

"---G(z)u,

D,

U, z

u

Caf:--f(O), fX,

Odu :--Gd(-[-OO)u,

U

U,

where S is the Oorward) shift operator (Sf)(z) zf (z), f H2r(D), z D. The realization (Ad, Bd, Cd, Dd) is called the restricted shift realization of the transfer function G d. It is admissible, observable, and reachable, and the observability and reachability operators Rd and Od are, respectively,

IX, R= H6).

Od

2. The realization (Ad,,, Bd,,, Cd,,, Dd,,) is given as follows: The state space Xd,, is given by Xd,, rangeH0a with

Cd(Z)

(Gd())* and

5k(z) =1z

(d ()--Gd(OO))

z6D

The operators Ad,,, Bd,,, Cd,,, and Ca,, are defined as

Ad,,

Px., SIx,,,

Bd,,

U

Ca,,

Xd , -+ Y x

Xd,,; u

-

PXd,,U, gTu

.L (Z-(Z))*X(z)dz

PYnGaX (Hax)(O),

D

Gd(+O0), Dd,. where Y is considered a subspace embedded in y} c_ H(D), and Px., and Pr

Hr2(D)"

Y

{Yo

are orthogonal projections from

+ 0z + 0z 2 + Yo H2r(D) onto Xd,, and Y,

respectively. The realization (Ad,,, Bd,,, Ca,,, Dd,,) is called the *-restricted shift realization of the transfer function G d. It is admissible, observable, and reachable, and the observability and reachability operators Od,, and 7P,.d,, are, respectively,

Td,,

ex,, H2v(D) -+ X,

and Od,,

H) Ixa,, H)]x.,.

[3

5.2. Continuous-time restricted shift realization. Now we apply T to the realizations given in the theorem to get the continuous-time realizations. We need some simple lemmas. LEMMA 5.2. For any x H2r(D), liturgy, r>-X, r--l(1 d- r)x(r) 0 in the norm ofr. For any f H2r(RHP), limr, r-++ f (r) 0 in the norm of Y. Proof For x 6 Hrz(D) and z 6 D we have x(z) Yn>_O z Xn where 2n Y and Thus En>_0 II-n 2 IIx

1122().

Iznl IInl[ _
_l, r-+-l(1 q- r)x(r)

O.

[lnll 2

773


Now for f

Hr2 (R H P) and any s

R H P, it is shown in [22, p. 254] that

IIf(s)ll _< 3(Re(s)) -x/2, where 6 is a constant depending on f. Thus the lemma is proven. LEMMA 5.3. With the notation of Theorem 5.1 we have (l + z)h(z) 1. range(I + aa) (z xl x(z) and for x

{

range(/+ Aa) the limit limr,

where x

r>-l, r--*-I X(F) exists;

[()I -t- Ad)-lx](z)

2.

IDe,

Xd, )

[(I + Aa)-lx](z)

+ )(1 + )z) x )GIDe,

X

EXd;

rlR, r>-l, r--I

x(r)

z x(z)

+ Lz

.

Xd and 1

1..q_zX(Z)

(l@xzX(Z)) Xa.

(;I

6

+ L(1 + z) x

Take any y in the invariant space have z-y H]()@ Xa. Therefore,

This shows that P+ operator on Xd for )

(l+z)h(z)-h(O)

+ r)h(r)

h(O)

Xa. By

h(0).

IDe, the element

(1) =e+ (z) +Zz x(z)

H (I3) O Xa. Since

y

x, Z

Hy(]I)) Since

Xa} we

for some h

r

relR, r>-l, r--I

Xa.

P+

(1

lira

2. First we show that for x

1

1

range(/+ Aa) then x(z)

lim

+

-

1_ z+)

Y

6

HOO for )

6

I)e, we

O.

Ad is a contraction, (LI + Aa) -1 is a bounded

13e. Then the equality in 2. follows from the equality

+ Ad) 1 +)Z x(z) + )(1 + ZZ) x

3. Using 2. and the definition of Ca we get 3. 4. If x 6 range(/+ Aa), then by 1. and its proof there exists h (l+z)h(z)-h(O)

Xa}

x e range(/+ x(z) + -x(-1) z +z limre, r>-l,r--I x(r). Since range(/+ Aa) {x + Aaxl x Xa} {x(z)-x(0) + x(z)l x

where x(-1) Proof. 1. have the equality in 1. If x Lemma 5.2,

is in

+ )z

1

Cd(LI-lr-ad)-ax--x(--),

3. 4.

x(z)

;

Z

I3), h

and limr,

r>-l, r----I

x(- 1)

x(r)

h(0). Set lim

rN, r>-l, r---I

x(r)

h(O).

We have Z

l+z

[

x(z)

1

+ -x(--1) z

h(z),

x(z).

Xa such that x(z)

774


which is in Xd. Note that 1 ap(Ad). Thus (I equality in 4. then follows from

(I + Ad)

(+

I

z

1

x(z) +

z

LEMMA 5.4. Let f

x(-1)l) s

lim f-- no n-+cxz n + s n Since

6

s s

f=0.

IIn-f(s)ll2y IIf(s)ll for any s 6 iI andn > 0and 2

s

lim for a.e. s

x(z).

Then in L2y(i) norm,

lim

Proof

(l+z)h(z)-h(O) z

[(I + Ad)h](z)

z

L2y(i).

+ Ad) -1 is defined on range(/+ Ad). The

f(s)

nWs

[:] i, the lemma follows from the Lebesgue dominated convergence theorem. TLC v’r. Set X- rangeHa andD- Px{] u U}. Then

LEMMA 5.5. Let Gc the map

M1 "79

Y,

PXl-

[(c(1) ((cx)lu

is well defined and the map

M2" X

--+

X,

f

w-

Px f l+s

is injective.

Proof Assume Px T-u’

px ]_ 47.uz

Px ul-uz l+s H2y(RHP) 0 X. Therefore, for any f H2y(RHP), 1

+s

Then

H2RHP, v

O. This shows that

1 --{- s

Ul--U2

H2RHP u

(-s)) _ - (Ul +s- -U2 [Gc(s)-Gc(+Cxz)]f(-s)) =( Gc(+cxz)]*ull u2’ f (-s)) ( [dc(s) dc(cx)lull u2 f (s)). U2 ]Ul , P+[Gc l+s

[Gc(s)

Gc(+Cxz)]f

W

s

Hence [((s)

((1)] u’-uz 1--s

0. So we have

[(s) (1)](Ul

on the real line, we get

[c(1) (c(+Cx)l(ul This shows that indeed M1 is well defined.

u2)

0.

u2)

0. Taking the limit

775


To show that M2 is injective, assume Px 0. Hence hi(s) 1

h2(s) hi, h2 E X. ThenPx h(s)-h2(s)l+s Px-i--,

(s)

hl

h2(s)

+s

E

H fg X.

By Lemma 5.4, we have lim

n-

l+shl(s)-h2(s) l +s/n

l

(hi

+s

h2)

H(RHP)

lim (hl --Sn--,x n--S

h2)

0.

1+, hlO.-h2’)_ Hence limn+ l+s/n hi h2 in Hr2 Note that Hv2 O X is an invariant space l+s h(s)-hz(s) LI 2 and 6 H for n > 0. So l+s X, and hence hi h2 G Hr O X. --r 1+./--i--Since h h2 G X, we therefore have h h2 0. This shows that M2 is injective. We will need the following result on the reproducing kernel in HZ(RH P) (see, e.g., 10]). LEMMA 5.6. For f HZ(RHP), u U, andc RHP the following hold:

in

( ) ( s---’ f) u

u

=2rc(f(ot),u),

=27r(u,f(ot))t;.

We are ready to present the continuous-time restricted shift realization using the bilinear transform T. For a continuous-time transfer function G. we first realize the discrete-time transfer function G a defined by

Ga(z)--G(z-z+il)

_

in terms of the restricted shift realization. Applying T to this discrete-time realization we obtain a realization of G with the same state space. Then we use a unitary transformation to get the continuous-time restricted shift realization with state space ran--6H THEOREM 5.7. Let Gc TLC ’v. Then Gc has a state-space realization (A, Bc, Dc, C) CxU,Y which is given in the following way: 1. The state space is given by

X--rangeHc,,/4e

H2(RHP).

(eta)t>_O corresponding to the realization is given by e tAc’. X X, (e tz f)(s) P+e t f(s). f The infinitesimal generator (Ac, D(A)) of the semigroup (eta)t>_O is given by Ac D(Ac) --+ X, (Af)(s) sf(s) lim r rf(r). f 2. The semigroup

-

The domain D(Ac) is dense in X, and we have

D(A,.)

{

f f (s)

1 [h(s)-h(1)]" (sRHP) 1- s

hEX}

776


The domain of the adjoint A*

D(A*c)

of the operator Ac is h(s)

Px

f] f (s)

(s6RHP),

h6X}

and

A*f f

h for f (s)

h(s)

Px-f + s

D(AS).

On/2-1(X) c__ LZv([O, +x)) the semigroup is given by et Ac

/-1 (X), (etAc f)(’C)

PLz([O,+))(f (’c + t))>_O.

3. The input operator is given by

D(A*c)(’) Bc(u), where for u

U and x(s)= [Bc(u)l(x)

D(A*), Px h(AOl+s

1(1-1

[Gc(s)

j

s

A*c)X

Gc(1)lu, (1

)

(u, (Hdch)(1)} U 4. The output operator is given by

Cc D(C)

D(Ac) -t- (I

Ac) -1BcU x

5. The feedthrough operator is given by

Dc

U --+ u

Y, Gc(+cx)u

"=

lim

r6l

lim r rx(r).

Gc(r)u.

The realization (Ac, Bc, Cc, Dc) of G is called the restricted shift realization. These results are obtained by applying the map T of Theorem 3.1 to the restricted shift realization (Ad, Bd, Cd, Dd) of Gd(Z) De), with the state space then Gc( Z--1 y-f ), (z transformed by the unitary operator V Vy defined in Proposition 3.3. 1. Let (Acl, Bcl, Ccl, Dcl) T((Aa, Be, Ce, De)) and

Proof

(Ac, Bc, Cc, Dc)

(VAcl V -1 VBcl CV -1 Dcl)

1-z We usethe following notation: Gd(Z) [Gd()--Gd(OO)]- 2[G.(]-)-Gc(1)](z ]])), Xd rangeHa), and (t (Z) e z+l, > 0. Then by Proposition 4.3 X VXd, and qbt(Ad) is

777


I)(Ad -i- i)-1

the semigroup of contractions on Xd with infinitesimal generator (Ad (see [28, p. 141]). Specifically, for x Xd we have

(Ae)x

P+dt

-

P+ t / x

x

P+e

-z 1.

Acl

x.

Thenit is easyto see that A. VA V generates the semigroup of contractions V (Aa) V on X. If we extend the unita transfoation V H() H(RHP) naturally m

V

:L2(0)

L2(iR),

xe

(Uxe(,l-

(l+,xe

Moreover, by considering zy for n

we still have a unita transfoation. can show that

V

P P

P"

P

VPe

Vt(Ad)V f

1--Z

V

-

Z and y

f

H2(RHP)are the ohogonal 1--a

P+P VetV f- p+et, L

(+a)x.(a)Tx(O)[x Clearly, D(Acl) range(Ad+l), and by Lemma 5.3 range(Ad+I) Since D(Ac) VD(A) and x(0) 2(Vx)(1) for x Xd, we have D(Ac)

V

Vrange(Ad

+ I)

V

(l+z)x(z)-x(0) Z

+ z)x(z)

(1

Y we

v,

where H2()and pttP L2(i) L2(0) projections. From this it follows that for f 6 X,

et& f

-

x(O)

Xd

x

2,f(1)

1--s

+s

Ix Xd

Xd }.

1-s

if X

l+s

_--{f(s)l_s-f(1)lf6X}. For x D(Acl) range(Ad + I) the limit limrz, r>-l, r-,-1 x(r) exists by Lemma 5.3. it x Denoting by (-1) and using Lemma 5.3, part 4, we have

(acx)(z) (Ad

l)

(+[ z

z

From this we obtain, for f

(Acf)(s)

sf (s)

I)(aa

[(ad

1

x(z)+ -x(- 1) Z

D(Ac)

(VAcl V-l f)(s)

,

lim

r- +o

-

])

(1-z)x(z)-2x(-1) l+z

VD(Acl),

V

((1-z)(V-lf)(z)l+z-2(V-If)(-1))

rf (r),

where we have used the fact that for f

(V-If)(-1)

+ l)-x]()

D(Ac)

-

lrim (1 + r)f(r)=

lirm rf(r). q-o

778


Now we show the form of Ac*. Recall that Ac* is the generator of the strongly continuous semigroup (e Ac),. Let

D(A)-

{fl f

h(s)_ for some h 6 X / Px l+s

(s)

/

and

,f

_

X f

,

h

f

Px-14- s

By Lemma 5.5, the operator is well defined. For f and D(A) andg D(Ac) there are v and w inX such that f -0(1) By the definition of A and we have A.g v It then follows w and Af 1-s that

(Acg, f)-- w, This shows that

(I

,) D (A)

D(A)

,,

PXl+

v

D(A*c) and/]

6

,v --(g,

1-s

On the other hand, we clearly have

AIo(A).

X and hence

A*c)D(A

(I Let x

Px

D (A*). Then there exists x

6

X.

D (A) such that

(I-A*c)X --(I-A*c)X. Since A* is the infinitesimal generator of a semigroup of contractions, the number 1 is not in the spectrum of A*. Thus we must have X X. This shows that

.

D(A*c) c_ D(). Therefore D (A c*) D () and hence A c* 2. For the operator B. we first compute BI, following the definition of T"

,qf(I 4- Ad)-iBd U

Bci

u

V -a, (I

Note that V*

+ a)

D(A*c)(’), v-- x/-(I + Ad) -1 Bd(u)[’], "= < Bd(u), (I + a)-l(.) -+

(VBdu)(s)-

1 Gc(s)- G(1)

u (s

h(s) D (A’c) c_ X, we have (I Ac*)X Px T-

Thus for x

(Bcu)(x)

(VBcl)(X)

,v/(VBdu, 1

--xd

1/2(I a*a), and

-1

A*clx

h and

/-(ndu,

(

RHP).

VBdu,

(I 4- A*d)-lV-lx)xd

-V(I A*cl)V-Ix

X

779


(Gc(s)-Gc(1)

x

1-s

Since

* Ac)x

(I

u

x

H&, we have

(He,.)*

(Bcu)(x)

l

+ s’

By Lemma 5.6 the right-hand side is Vc-(u, (H&.h)(1))u. 3. To compute C. we use Lemma 5.3, part 3, to get

Cd(.i _+_ Ad)

So for x e D(Ccl)

D(Acl)

+ (I

-

lx()

AeDe.

Bc U we have lira C(I + Ad)-lx X>I

Ac)

C.x

1

-

1

-x(-)

i>_ x(>.

The existence of the limit for x e D (A.) follows Eom Lemma 5.3, pa 1, because D (Ac) range(Ae + I). For x (I Ac) B U we have that the limit lim Ce(l

also exists by the admissibility of G,, since (I Now it can be verified that VD(C.) D(A) Hence we get, for f e D(Cc),

+

B,u Gu

(see [23]). A.)Bcu + (I Ac)-BU, i.e., VD(C) D(Cc).

Cf _CclV_lf =lim lSX

v/

lim (1 r

Finally, the obvious expression Dc

Dcu

Dcu

Gd(cx)u

Gc(1)u Gc(+ec).

+ r)f(r)

f(1-:)l+ lim rf(r). r

Gc(ec) can also be verified as follows:

Cd(I + A)-IBu Ddu- lim A>I

(1) --’ " , 1}(-1) [ Gc ()+ll)-G(1)lu-limGc(’+l ) lim Cd()l >1

+ Ad)_l Gu

Gc(1)u

lim .>1

>,

V1

l

G

)-i

780


Regarding the expressions for the operator Bc in the theorem we have the following corollary. COROLLARY 5.8. Bcu X, (u U)if and only if[Go- Gc(+Cxz)]u X (u U). In this case

[Gc(s)- Gc(+x)]u (u a U).

(cU)(S)

In particular, if G,. satisfies +cx

sup

IIG,(x

x>O

+ iy) Gc(+Cx)ll2dy

where for s RHP the expression IIGc(s) operator Gc(s) G,(+cx) L(U, Y), then

G.(-+-)II denotes the operator norm of the

Bcu X and

1

(B.u)(s)

_,=--[Gc(s)

Gc(+Oc)]u

for any u U. Proof First we assume that [G,.

G,.(+cxz)]u 6 X (u 6 U). Define F(s) X. It follows from the formula for D(A) that

G.(+). Then Fu

[G(s)-G,(1)]u

[F(s)u- F(1)ul 1-s

1 -s

and hence for x

6

i[Gc(s) A.)

1 l

1-s 1

D(A)

D (A),

[c()](x)

1 ((i

6

G.(s)

s

[G,.(s)

AS)XlHzr(RHP

G.(1)lu, (I

Gc(1)]u x

1-s

([G.(s)- Gc(+oc)lu, x)-

r

1

1--r

(Fu, x).

Here we have used the definition of Ac and the fact that the limit lira r Gc(r)u exists, which follows from the admissibility of Gc. Thus we have shown that Bc(u) X and Fu for any u U. Bc(U) On the other hand, if B,.(u) X, (u U), then there is fu X such that [B(u)](x) (f,, x) for any x 6 X. Therefore 1

This shows that

h

6

(

1

[Gc(s)-Gc(1)]u,

1_-1 [Go(s)- Gc(1)]u e D((I

(I-A*c)X)=(fu, A*)*)

D(I

X such that 1

[Gc(s)-Gc(1)]u -s

h(s)-h(1) 1 -s

x)(xX). A)

D(Ac). So there is

781


Hence G,.(s) G,.(1) 0 (see Lemma 5.2), we have h(s) h(1). Since lim s h(s) G. G.(+cx) h E X. To complete the proof of the corollary, it suffices to show that the condition that G is analytic for Re(s) > 0 and satisfies sup

IlGc(x

x>0

+ iy) Gc(+Cxz)ll2dy < cx

ec

implies that [G. G,.(+oe)]u E X for any u 6 U. Again let F(s) Gc(s) G,.(+oc). We have the equality of Hankel operators:

HF.

HG,.

LZv(iI)

The assumption on G,. implies that Fu LZy (iI) norm

Fu- lim n--+ cx

and hence Fu

6

X

for any u

6

U. Now we show that in

n

HFU n s

+

rangeHF. The proof will then be complete.

Consider

Fu

HF n

S

+ s I (iN) By Lemma 5.4, we have limn_, FullL(iN) lim n--+cx

So we indeed have Fu

S

P+Fu n-s

u

limn__, HF

P+

--s g/

gt

s

Eu

O. Therefore O.

Fu S

ns u, converging in Lzv(iIR) norm.

5.3. Continuous-time ,-restricted shift realization. If we apply the map T in Thez-1 and then transform the orem 3.1 to the *-restricted shift realization of Gd(z) G,.( z-Tf) state space by the unitary operator of Proposition 3.3, we obtain the *-restricted shift realization of G Alternatively, we can find the restricted shift realization of the transfer function (f,, E TLCr, U first, and then the dual system of this restricted shift realization will be the *-restricted shift realization of Gc. THEOREM 5.9. Let Gc TLC U’r. Then Gc has a state-space realization (A,,, Be,,, Cc,,, Dc,,) 6 CxU,Y which is given in the following way: 1. The state space is given by

X,

rangeHd,.,/He

c_C_

HZ(RHP),

where Jc(S) (G(g))* for s RHP. 2. The semigroup (e tac,*)t>_O corresponding to the realization is given by

et Ac,,

f

(e ta,* f)(s)

Px, e -ts f(s),

where the operator Ac,, has domain

D(Ac,,)

Px, i-(s h(s)"

h

X,

782


and for f (s)

Px,-i--sh(S) E D(Ac,,), Ac,,f--f-h.

On -1(X,) c_

L([0, cx)) the semigroup is given by eta

/- (X,)

f

-

/- (X,),

(e taC,* f)(s)

Pc-(x),f(s

t).

3. The input operator Bc , U --+ D(A* (’) is given by

B() with

[Bc,,fu)]fx)

l+s

u’

,,)x

(1-

1(1) l+s

u’h

"v/(u, h(1))t:,

x

h(s) -h(1)

* h ED (ac,,),

X*

4. The output operator has the following form"

D(Cc,)

D(A,,c) + (I

-

h(s)

A ,,c) -1 Bc,,U

=Px, l+s hX, +ex, is

If x

Px, h-l+s then C,,x

and if x

Px, -74-2’ then Cc,,X

HG,.h (1)

-[G(1)

G(+cx)lu.

5. The feedthrough operator is given by

Dc,," U u +

Y, Gc(+Cx:)u

lim r G(r)u.

The realization (Ac,,, Bc,,, C,,, Dc,,) of Gc is called the *- restricted shift realization. Proof Let (A, B, C, D) be the restricted shift realization of the transfer function c(s) (G(g))*. Take (Ac,,, B,,, Cc,,, Dc,,) to be the dual system (,,/, ,/)) of (A, B, C, D). Then (Ac,,, Bc,,, Cc,,, D,,) is a realization of G (see Definition 2.2). We show that (A,, z B,,, C,,, Dc,,) obtained this way has the expressions as given in the theorem. Notice that A A* i.e. A,, A* 1. By Theorem 5.1 the state space of the realization (A, B, C, D) is rangeHc. Thus the dual system (,,/, ,/) has the same state space. That is,

X,

rangeH6c.

783


2. The semigroup generated by A is defined as e

taf P+etsf

(f

E X,).

It is easy to verify

etA* f

(etA)* f

Px, e-t’ f (f e X,).

That is, e tAc,*f

Px, e -tsf (f e X,).

By Theorem 5.1, D(A*)

O(Ac,,)

Ac ,f

f

Px, 1

1

Px,

h for f (s)

X,}

s h(s)lh h(s)

+s

D(Ac,,),

and Ac,, is well defined. 3. By the definition of the dual system (Definition 2.2), we have

/}" U For x(s)

D(A)(’);

--+

1-_ [h(s)- h(1)] 6 D(A) (h /()[x]

(u, Cx)

By Lemma 5.6, /-(u, h(1))v

u w-->

/}(u)[.]

(u, C(.)).

X,), we have

lu,\

limrx(r)l/

/(u, h(1)).

reN

-(-f--u,

h)Hv2(RHP)"

’. Again use Definition 2.2" D((;) D(.) + (I )- [U D(Ac,,) + (I

4. Now we compute C,.,,

Ac,,) -1Bc,,U,

and x0 is defined by

(y, dxo)

B(y)[xo],

(;xo, y)

(uo, C(I

A)-IBy),

xo xo

D(Ac,,) (I

Ac,,) -I Bc,,uo, uo

(

x

Px 1

for x-

Px i +s

Since by Theorem 5.7

B(y)[x]

-(y, (HGch)(1))y

h(s)

+s

ED(Ac,*))

we have

dx

v/-(Hach)(1)

h(s)

E

D(Ac,,).

From Lemma 5.5 it follows that ( is well defined for x D(Ac,,). Note that C(I A) -1 By [(c(1) c(+Cxz)]y. Thus

x0

[Gc(1)

Gc(+Cxz)luo for xo

(I

Ac,,)-lBc,,Uo.

U, y

Y.

784


Now we show that (I Theorem 5.7 (I A*c,, )-Ix

x-x(l

D(A,,).

Since by

we have

1-s

Ac,,)-IBc,,Uo,

((I

uo Let x Px, 7-"

1B.,,uo

At.,,)

x)

Ac,,)-lBc,,Uo](x)

[(I

A,,)-lx)

[B.,,uol((I

--[B"*u]( x-x(1))l-s This shows that (I Ac,,) -1 Bc,,Uo is defined in the following way:

D(C,.,,)=Px,

PX,

X

l+s

l+s

x

u0 Hence, to sum up, the operator C,,, Px, -s"

1+

[hX, +Px,

(

l+s

Px, l+s then

If x

,/(Hc,.h)(1),

Cc,,x and ifx=Px, 1--, then

C,.,,x

-[Gc(1)

Gc(+oe)lu.

Note that by Lemma 5.5 Cc,,X is also well defined for x 4. It is straightforward to get

De,,

D*

((Gc(+O))*)*

6

Px,{ 1-.

u 6

U}.

[-]

Gc(+Cx).

Note that the restricted and *-restricted shift realizations of admissible transfer functions in H are well posed in the sense of Curtain and Weiss [5] and Salamon [27]. Indeed we have the following corollary concerning the reachability and observability of the restricted and *-restricted shift realizations. COROLLARY 5.10. 1. The reachability operator of the restricted shift realization is given

9y

R.c" L2v[O, + cxz

-+ X,

f t- HG

R H P ff_,

- -

-lx.

L

The observability operator of the restricted shift realization is given by

Oclx X

LZr[0, +cx),

x

-

2. The reachability operator of the *-restricted shift real&ation is given by

7-4.c.,

L2u[O, + cxz

-+ X ,

f

-

Px,

f

The observability operator of the *-restricted shift realization is given by

Oc,, X, Here

L2r[0, +cx),

denotes the Laplace transform.

x

ff.,-1HGc,RHpX.

785


This follows from Theorem 3.4 and Theorem 5.7 U We categorize the state spaces of the restricted and *-restricted shift realizations here for

Proof later use.

PROPOSITION 5.11. Let X and X, be, respectively, the state spaces of restricted and *-restricted shift realizations of Gc TL C u, Y. Then 1. if G,. is cyclic, then X H2r(RHP) and X, H(RHP); 2. if G is noncyclic, then X (R H P) 0 O (R H P) and X, (R H P) (3 where and Q1 HLy)(RHP) Q2 HL()(RHP) are rigidfunctions; Q2H2(RHP), 3. if Gc is in Hc(x,r)(RHP), is strictly noncyclic, and has factorization Gc Q1F{ where 01 Hcr)(RHP) and Q2 H(c)(RHP) are inner, al and F are left coprime, and Q2 and Fz are also left coprime, then X H(RHP) 3 QIHr(RHP) and

H2r

1H2r

H2v

P02,

X,

H(RHP) QzH(RHP).

Proof

This follows from Definition 4.5 and Theorems 4.8, 5.7, and 5.9.

6. Continuous-time input-normal, output-normal, parbalanced realizations and their asymptotic stability. Recall that a reachable and observable admissible system I. It is output-normal if A/[ I. (Ac, B., C, D) is said to be input-normal if Vc The reachable and observable admissible systems are said to be parbalanced if

Here W,. and A//c are, respectively, the reachability and observability Gramians of the system. Given a transfer function G. 6 T L CU’ r, by Corollary 5.10 the restricted and *-restricted shift realizations are examples of, respectively, output-normal and input-normal realizations of G,.. Proposition 6.2 shows that up to unitary equivalence all observable input-normal and reachable output-normal realizations of an admissible transfer function G are up to unitary equivalence *-restricted and restricted shift realizations, respectively. In this section we establish the existence of a parbalanced realization for any G 6 T LC u,r and study the stability properties of input-normal, output-normal, and parbalanced realizations.

A parbalanced realization of a continuous-time transfer function Gc TLC u’Y can be obtained from the map T in Theorem 3.1 applied to a discrete-time parbalanced realization of the corresponding discrete-time transfer function Ga. The existence of parbalanced realizations was shown by Young [30]. In [23] Young’s results are cast into the continuous-time situation and the following theorem is proven. THEOREM 6.1. 1. For Gc TLC u’r, there exists a parbalanced realization ’r (Ac, B., C., Dc) of Gc. The state space of this realization is given by the closure of the range of the Hankel operator with symbol G, i.e., X rangeHac. If(A-c, B-c, (f,., 15c) is another parbalanced realization of Gc with state space X, then (Ac, Bc, C, Dc) and

CUx

(c, 1ffc, (c, 1.) are unitarily equivalent. 2. If in addition G(s) is continuous on the extended iN (i.e. on iN tO {icx}) and is a compact operator for each s iN, then there is a basis of X rangeHcc on which the Gramians of the above realization have a diagonal matrix representation with its diagonal consisting of the Hankel singular values of G. We will call this realization a balanced [3 realization of Gc. 6.1. Characterization of the realizations. Concerning the equivalence of different realizations, we have the following proposition. PROPOSITION 6.2. 1. Any two input-normal (output-normal) realizations ofGc T LC U,Y are unitarily equivalent. Hence every input-normal (output-normal) realization of G is unitarily equivalent to the *-restricted (restricted) shift realization of G c.

786


2. An input-normal realization and an output-normal realization of Gc T LC U’Y are equivalent if and only if the Hankel operator HG,, has closed range. 3. All reachable and observable admissible realizations of G, are equivalent if and only the if Hankel operator HG has closed range. Proof Analogous results in the discrete-time case are shown in [24] (see Theorem 3.1, Corollary 3.1, and Proposition 4.1 therein). Applying Theorem 3.1, Theorem 3.4, and Propo71 sition 4.3 to these results we have the proposition. of of the the is A consequence study input-normal (output-normal) realproposition that izations reduces to the study of the *-restricted (restricted) shift realizations. This point will be used repeatedly. Part 2 of the proposition shows when the state-space isomorphism theorem holds. Note that the Hankel operator Hc to have closed range is a very strong condition. This condition can be stated in terms of the Douglas-Shapiro-Shields factorization of the transfer function G,, (see Theorem 4.10 and [11]).

6.2. Asymptotic stability. Now we turn to the study of stability properties of continuoustime systems and use the classes Ci, to describe different asymptotic stability properties of systems [28]. DEFINITION 6.3. Let (etZ")t>_O be a semigroup of contractions on the Hilbert space H. Then 1. (etZc)t> 0 CO. iflimt_ eta"h --O for all h H, 2. (etA")t>_O C.o iflimt-+ etZh --O for all h H, 3. (etZ")t>_O C1. iflimt__, etZ"h O for all h H, 4. (etZ")t>_O C.1 iflimt-+eta*h O for all h H.

We further set

Cij

Ci. A C.j,

i, j

O, 1.

[3

The notions of stability that we consider are the following. DEFINITION 6.4. A continuous-time system (At., Be, C., Dc) 1. asymptotically stable iffor all x X, e

as -+ cx, i.e.,

tA’x

CxU,Y is

0

(eta)t>_O Co.; if

2. exponentially stable co

"-inf{o O such that [[eta"[[ 0)}

[3 The number co is called the growth bound of the semigroup. We comment that the asymptotic and exponential stability of a system is preserved by system equivalence. Moreover, if two systems are unitarily equivalent, they will have the same growth bound. An important result in [28, Prop. 9.1, p. 148] implies that a continuous-time system is asymptotically stable if and only if the corresponding discrete-time system is asymptotically stable. PROPOSITION 6.5. Let (A, B, C, D) D xU,Y and (A, B, C, D) C xU,Y such that

(Ac, Be, Cc, Dc) Then for all x

T((Ad, Bd, Cd, Dd)).

X, lim

n-- (x)

lim I]etA"x]l ]lAx]]- t-o

787


and lira rt---- oQ

II(A*)nxll-

lira

IletA*xll.

Therefore, the study of asymptotic stability of a continuous-time system reduces to the study of the asymptotic stability of the corresponding discrete-time system. Now we state the main result of this section, which asserts that any admissible parbalanced realization of an admissible continuous-time transfer function is asymptotically stable. THEOREM 6.6. Let Gc TC U’r. Let (Ab, Bb, Cb, Db), (Ai, Bi, Ci, Di), and (Ao, Bo, Co, Do) be, respectively, a parbalanced, an input-normaL and an output-normal observable and reachable realization of G Then 1. (a) (etAi)t> 0 C= C.o, (b) (etAi)t>_O Coo if G{ is strictly noncyclic, (C) (etAi)t>_O C10 if G _L is cyclic, 2. (a) (e tAo )t>_o Co., i.e., asymptotically stable, (b) (eta)t>_O Coo if G is strictly noncyclic, (C) (eta)t>_O C01 if Gc is cyclic,

(etZb)t>_O COO. Proof The corresponding asymptotic stability 3.

results for discrete-time systems were obtained in Theorem 3.2 and Theorem 4.2 of [24]. Hence, combining those theorems with q Proposition 6.5 and part of Proposition 4.7, we have the theorem. Since by Proposition 6.2 all reachable and observable realizations of G are equivalent when the Hankel operator HG,. has closed range, and equivalent realizations have the same asymptotic stability properties, the theorem has the following corollary. COROLLARY 6.7. If the Hankel operator HG,. has closed range, then all reachable, ob[3 servable, and admissible realizations of Ge. are asymptotically stable.

7. Spectral minimality and exponential stability of input-normal, output-normal, and parbalanced realizations. This section aims to examine the exponential stability of continuous-time input-normal, output-normal, and parbalanced realizations of certain classes of transfer functions. The results are mainly based upon a detailed spectral analysis of input-normal and output-normal realizations. While the asymptotic stability properties of continuous-time systems can be obtained directly from the discrete-time case as we did in the previous section, exponential stability properties of continuous-time systems do not follow in the same way. However, we can relate the spectrum of the discrete-time system to that of the continuous-time system and thus establish the exponential stability results. Recall that a continuous-time system (A, Be., Cc, D) is exponentially stable if

inf{o

6

1 there exists M >_ 0 such that ][e tA‘‘ _< M et for >_ 0}

< 0.

The following proposition gives an interpretation of the growth bound of a semigroup in terms of the spectral radius of the semigroup (see, e.g., [21, p. 60]). PROPOSITION 7.1. Let co be the growth bound of the semigroup (etZ")t> 0 and r(e tA’) the spectral radius of e A, then

r(etac) for

eCOt

q > O. Note that it follows from this proposition that equivalent systems have the same growth

bound. 7.1. Spectral analysis. Thus we have to investigate the spectral properties of a continuous-time linear system (Ae., B., C, De.) in order to study its exponential stability.

788


The way we do this is to relate the spectral properties of (Ac, Bc, Cc, De) to those of the corresponding discrete-time system (Ad, Bd, Cd, Dd). First we have the following relation between a(Ad) and a(Ac). PROPOSITION 7.2. Let A be the infinitesimal generator of a semigroup of contractions and Ad the co-generator such that A (Ad I)(Ad + i)-1. Then

rp(Ac)

z 4- 1 z 4- 1

z

rp(Ad)

z

c(Ad),

and r(Ad) \ {--1}

:/: --1

Z

s6Crp(A)

1 --s

s6r(Ac)

--s

r(Ac) since e tA‘: is a semigroup of contractions and that by

First note that 1

Proof

and Cp(Ad)

Theorem 3.1,

Acx where D(Ac)

(7.1)

(sI

(Aa

I)(Ad

+ I)-x

I)x for x

D(Ac),

+ 1). Hence the following relations hold: Ac)(Ad + I)x [sl (Ad l)(Ad + l)-X](Ad + I)x range(Ad

[s(Ad 4- I)

(Ad 4- I)(sI

+SI_Ad)

(Ad 4- I)[sI

Ac)x

l)]x

(Ad

1 1--s

(l--s)

(7.2)

+ I)-l(Ad

(Ad

s5

1;

I)(Ad 4- I)-X]x

(Ad

l+SI--Ad)

(l--s)

x Xd,

x

1--s

D(Ac), s # 1.

x

x,

The equations (7.1) and (7.2) show that

l+s

rp(Ad) Now if

l+s

(11

Crp(Ac)

S

S

Ad)-1 exists and is bounded, then

I

cr(Ad), i.e., if (Ad 4- I)

1

+ s I Ad --s

)-1 (114-s)

-1

I

Ad

(Ad 4- I).

Sl

Ad

X

--s

Thus by (7.1) and (7.2)

(7.3)

So (sl

(7.4)

(sI

Ac)-lx

(1

s)-l(Ad 4- I)

(1

s)

-

1

]

+sI s

1--s

Ad)

Ac) -1 is bounded and densely defined, i.e., s

l+s 1

s

s

cr(A.)}

-1

(Ad 4- I)x, x

a(Ac). Hence

c_ o’(Ad).

D(Ac).

789


On the other hand, ifs # ands to verify in this case that 1+s I 1--s

Ad

)-1

(1 -s) 2 (sI

x

Ac) -1 is a bounded operator. It is easy

or(At), then (sl

Ac)

2

-s

121

X

2

D(Ac).

X

In fact from (7.3) we have

(1+

2

)[(l-s) (1-s)22 (ll_s+S Ad) s

1-s

I

Ad

I

(1 s) 2 (1

Ac) -1

(sl 2

S) -1 (Ad

(sI

+ I)x

+

1-Sl] 2

x

(l +Sl--Ad) + 1--S(ll+Sl--Ad) x2 Ac)-lx -+-

1--s

x2 1--s

--s

D(At.).

x, x

Similarly, 2

Thus

l+s

(SI

o" (Ad).

Ac) -1

+

2

x --S

I--Ad x--x,

x e D(Ac).

So we have

(7.5)

s

l+s 1

s

(Ad)]

or(At).

Combining (7.4) and (7.5) we have that

a(At.)-

s"

l+s 1-s

cr(Ad)]-- { Z+I z-1

Z

e cr(A,),z 5/:

-1},

which implies

cr(a) \ {-1}-

l+s 1-s

In our application of the proposition, At. is the state propagation operator of a continuoustime system (A, Bt., Ct., Dr.) C xU,Y and Ad is the state propagation operator of the corresponding discrete-time system (A, Bd, Cd, Dd) D xU,Y which is related to (At., by (Ac, Be, Cc, De)

T((Ad, Bd, Cd, Dd)),

where T is the bilinear mapping in Theorem 3.1. A powerful tool in spectral analysis is the spectral mapping theorem for Co operators (see, e.g., [22, p. 74]). A contraction W L(M), where M is a separable Hilbert space, is called a Co operator, denoted W 6 Co, if there exists no subspace V 6 M such that W Iv V --+ V 0. The least is unitary and if there exists an inner function m 6 H(J) such that m(W) common divisor of all such inner functions is called the minimal function of W, denoted

790


0. Note that if W is a Co operator, m w, which is still an inner function such that m w (W) is W unitarily equivalent to a left shift S* restricted to a left invariant space of the form Hy2 (D) @ Q 2 (D), where Q is inner (see, [22, p. 72]). It can be seen that minimal functions are the generalizations of minimal polynomials of matrices. As in the matrix case, the spectrum of a Co operator is given by the "zeros" of its minimal function in the following sense (see

H

[22, p. 72]). LEMMA 7.3. If W Co, then or(W) c(mw) and rp(W) cr(mw) fq I, where for an inner function Q Hr, (]I)), Y is a Hilbert space, and the spectrum of Q is defined as inf IIQ()ylI=0 or(Q)= .]Di lim inf Ilyll-I 8--+0 I-Xl< ID

Given a Co operator W and a function 4

q(W) :--

yY

6

H (]D), the operator

lim r 0 such that Q2(wn) -1 Z and Q2(s) -1 is bounded by M in a neighborhood of each point wn. As exists for all n to the point spectrum, we have

For IZl

rp(e At’i) \ {0}

{e

s

.

RHP, KerQ2(s)

: {0}}.

2. Under the same assumption as in 1., for the generator Ac,i we have

cr(ac,i) {-)" rp(ac,i)- {-s 3.

G

s

If O,. is noncyclic and range(Hoe)

o(Q2)} cr,(Gc), RHP, KerQz(s) -f: {0}}.

(Q2H2u(RH p))_t_, where Q2 is a non-inner rigid

function, then r (Ac,i

4.

the closed left halfplane.

r(ea,i)

,

cr (A,i

the closed left halfplane.

If G is cyclic, then

crp(e A’,’) \ {0}-

,

The following proposition gives the spectral properties of parbalanced realizations in the case of strictly noncyclic transfer functions.

PROPOSITION 7.8. and Y, then

lf G Ht:,r (RH P) is strictly noncyclic with finite dimensional U

cr(Ac,o) cr(Ac,i) cr(Ac,b) crs(Gc), where (A,.,, B,, Cc,, D.,) is a parbalanced realization of G.

Proof The analogous results in the discrete-time case are proven in [24, Cor. 4.3]. Since z-1 )’ (z {z-1 Ill)e), the statement cr,(G,) 4-f[ z rs(Ga) z -1} where Ga(z) G,( z[3 follows from Propositions 4.7 and 7.2 and Theorem 7.5.

794


7.3. Exponential stability. Before we can give a criterion for the exponential stability of input- and output-normal realizations, we need some results concerning the relation between the spectrum of a semigroup and the spectrum of its generator. The following lemma can be deduced from [9, p. 622] (see also [21, p. 84]). LEMMA 7.9. Let e tA be a strongly continuous semigroup of operators on a Hilbert space with X infinitesimal generator A. Ifty(e tA) {) I,1 0), then r(A) c_ {s < Re(s) Note that in particular if Iletall 0, then r(e ta) 0), if and only if r(A) {s Re(s) < or}. Here is a real number. Proof The necessity part follows from Lemma 7.9. Now assume r(A) {s: Re(s) < o}. Since cr(e tA) {: I)1 _< 1}, we may assume

_

By Lemma 7.3, we have cr(Ad)

Since or(A)

cr (m). On the other hand Proposition 7.2 shows that

l+s

cr(ad) \ 1--1}-

{s

1- s

Re(s) < o}, we have cr (Ad)

{-1} c

/

+s

if(m)

{-1} c

/

+s

Re(s)

0} t_J {cx}. The Blaschke product q is determined by the poles of g. For example if

g(s)

(s

(s 1)(s -i- 2) + 3)(s + 4)(s + 5)’

(s (s

3)(s -4)(s + 3)(s + 4)(s

then the Blaschke product is given by

q(s)

5) + 5)

and

(s + 1)(s 2) (s + 3)(s + 4)(s + 5) It follows from the results in 6 that the state space of the restricted and *-restricted shift realization of the transfer function g is given by

f(s)

X

(qH2(RHP)) +/-.

806


Note that by Kronecker’s theorem (see, e.g., [22]) X is a finite-dimensional space with dimension equal to the number of zeros or poles (counted with multiplicities) of the Blaschke product. From the construction it is clear that the Blaschke product is completely determined by the poles of the transfer function. Hence we have recovered the well-known result that the dimension of a minimal state-space realization equals the number of poles of the transfer function.

Example 2: Delay system with strictly proper rational part. In this example we consider single-input single-output delay systems. We continue with the notation in the above e -asg(S) with c > 0. Let example and let the transfer function have the form gl(s) p(s) e-’q(s). Clearly p is in H(RHP) and inner. Later we will show that in fact p and f are weakly coprime. For now assume that this is true. Thus by Theorem 4.8 gl is strictly noncyclic, and by Proposition 5.11 the state space X of the restricted shift realization (Ac, B., Cc, D.) has the form

X

H2(RHP) 3 pH2(RHP).

x(s)-x(1) x The domain ofA, isD(A) X}. Hence forh 1-s h(s) x(-)-x(1)l_. for some x 6 X, limr,r+ rh(r) x(1) and

(Ah)(s)

lim

sh(s)

rh(r)

D(A,) we will have x(1).

sh(s)

Note that g satisfies the condition in Proposition 8.4. So the operator Bc is defined as

__[g(s)

(Bcu)(s)

and B,. is bounded. Hence (I

O(Cc)

We have, for h

1

-gl(s)u,

gl(W)]u

Ac) -1 B(

u

G_

C,

D(Ac) and

Ac)-IBU

D(Ac) -t- (I

D(Ac).

D(Ac),

C,.h

lim

E,r--- +cx

rh(r).

Note that because c 0, by Corollary 8.8 C is unbounded. The operator D,. is D gl (+cx:)) 0. We can directly verify that this is a realization of g. Let will show that for h D(Ac) h(s) ((I ac)-lh)(s)

6

R H P. An easy calculation

h(). -s

(We remark here that this formula is true in general, not just for this particular example.) Then

(( I

Ac) -1 BcU)(S)

1

Ac) -lg (S)U--

(I

1

gl(S)

gl() S

Hence

Cc(I

Ac)-lBcu

lim

re], r--++cx

r

gl(r)-gl() r

gl().

This realization is exponentially stable by Theorem 7.11 since g is clearly analytic on Re(s) > -3. It also follows from Theorem 7.11 that the degree of stability is -3 max{s s is a pole of g }. Consequently the parbalanced realization will also be exponentially stable

807


with the same degree of stability. Notice that g is continuous in the extended ilK. Hence the Hankel operator Hg is compact. Therefore by Theorem 6.1 there exists a balanced realization. To show that p and f are weakly coprime, consider the closed linear span S :-H2 (R H P). The space S is obp H2 (R H P) v f H2 (R H P). We need to show that S Hence of by Beurling’s theorem [22] there is viously a (right) invariant subspace HZ(RH P). (R) that an inner function 6 H(RH P) such

(R)H2(RHP).

S

Hence pH2(RHP) c (R)H2(RHP) and fH2(RHP) c (R)H2(RHP) Let q(s)(which is the inner part of the inner-outer factorization of f; see [22, p. 11 ]). Then

-

qlHe(RHP)

s-2

s+2

f H2(RHP).

So by [22, Cor. 5, p. 13] we must have that p/(R) and ql/( are both inner functions. Note that (R)(2) 0 since otherwise h(2) 0 for any h pHZ(RHP) c_ (R)HZ(RHP), and this is certainly not true. Thus the inner function ql (s)/(R)(s) has a zero at 2. Hence the function e+zq,(s) will still be in H(RHP) That is, 1/(R) 6 H(RHP). Hence HZ(RHP) s-2 (+(s) c )(1/(9)H2(RHP) (R)H2(RHP)= S. Note that exactly the same argument in this example will apply for any transfer function e g(s), where g is a stable and strictly proper rational function and ot > 0. Also, in a gl similar manner we can obtain the *-restricted shift realization which will have bounded output operator and has the same stability properties as the restricted shift realization. We summarize these as follows. PROPOSITION 9.1. If a scalar transfer function G has the form G(s) e-aS g(s), ot > O, where g is a stable and strictly proper rational function, then 1. G has a balanced realization; 2. all reachable output-normal realizations of G have bounded input operator and unbounded output operator, whereas all observable input-normal realizations have bounded output operator and unbounded input operator; 3. all reachable and observable input- and output-normal realizations and all parbalanced realizations are exponentially stable with growth bound equal to max{Re(s) s is a pole of G }.

Example 3: Delay system with not strictly proper rational part. When the rational transfer function g in the previous example is not strictly proper, the resulting realizations will be different: the input operator of the restricted shift realization is not necessarily bounded, and it is not clear whether there is a balanced realization of g because the Hankel operator Hg is not compact. A parbalanced realization, however, exists by Theorem 6.1. We first consider the simplest case with g(s) 1. This is a simple delay gl (s) e -a* (or > 0). The state space of the restricted shift realization is X H 2 O e H 2, which is the image of the Laplace transform on L2([0, or]). Let (Ac, Be., Cc, D,.) be the restricted shift realization and let (A, B, C, D)

(_,-1Ac, -Bc, Cc, De).

We know that (see Theorem 5.7)

(etAc f)(x) where f(x

+ t)lt0,al

f (x + t)lt0,,l, f(t

+ x) if Af

q- x

f’,

f L 2([0, ot]),

x

[0, or], > 0,

[0, a] and 0 otherwise. Thus

f e D(A),

808


with D(A) {x Theorem 5.7, for x

Le([0, or]) D (A)

,/ (

[.(ul(x

L2([0, or]), x(a)

x is absolutely continuous, x’ and u C,

(

1

-

s

(1-A*c)X

[G.(s)-Gc(1)]u,

e -as

e

1-s

0}. By

)

u,E-l(1-A*)E-x)

(et-ault0,al, ;-1(1 A*c)_..-lx)L2([O,a])

A*)/2-1X)L2(t0,al),

{et-au, (1

where

-.-

(1 c_

et-aul[o, al

shows that for x

D(A*)

[B(u)l(x)

[,-1Bcul(x

le

1-s

)

u (t) is

L2([0, a]),

[ff-.*Bcu](x)

et-au for

[0, oe] and 0 otherwise. This

[Bcul(x)

(et-au, (1- A*)x)L2([0,a] ).

It can be shown that

D(A*)

{x

L2([0, or])

-x’ for x

and A*x

X

D(Cc),

D(C) c_ Cx

Finally, Dc

(e t-a u,x

A*)X)L2([O,a])

-

C.x we have for x

6

L2([0, or]),

x(0)

0},

D(A*). Hence

[B(u)l(x)- (et-au (1Since for x

is absolutely continuous, x’

+

ux(oe).

lim rx(r), 6lR

L2([0, c]), Cc,X

lim r(fx)(r) IR

limx()

x(0).

.>0

g(+ec) 0. This realization is, by Theorem 7.11, exponentially stable. In fact, the spectrum of e tA is {0} (t > 0). The operators B and C are both unbounded. 1. Clearly Now consider the factorization e and f (s) q f*, where q(s) e this is a strongly coprime factorization. Therefore by Proposition 6.2 all reachable and obare equivalent. This shows that all reachable and observable servable realizations of e realizations are exponentially stable and have unbounded input, output, and state propagation operators. As in the previous example, we can generalize this result. PROPOSITION 9.2. If a scalar transfer function G has the form G(s) e g(s), where g is a stable proper rational function and g(ec) 7 0, a > 0, then 1. all reachable and observable admissible realizations of G are equivalent; 2. if (A, B, C, D) is a reachable and observable admissible realization of G, then the operators A, B, and C are all unbounded; 3. every reachable and observable admissible realization of G is exponentially stable with growth bound equal to max{ Re(s) s is a pole of G}.

809


Proof Since g is a stable proper rational function, g has a factorization g qf* such that q and f are stable proper rational and strongly coprime (see Theorem 4.10 for the definition

-

of strong coprimeness). Hence

Since g(ec)

inf [Iq(s)l

sERHP

0, we must have that f(ec)

+ If(s)l] > 0. 0. Therefore

inf [Iq(s)e-’"l

sERHP

+ If(s)l] > 0.

and f are strongly coprime. This, by the Corona theorem (see [22, p. 66]), shows that q e So by Theorem 4.10 the Hankel operator H has closed range and by Proposition 6.2 all reachable and observable realizations of G are equivalent. Thus 1. is proven. Since G is not analytic at infinity, by Theorem 8.2 the state propagation operator of any reachable output-normal realization is unbounded. Note that in the factorization G (qe -’) f* the inner function does not satisfy condition 3 in Theorem 8.6 because now ot 0. Therefore by Corollary 8.8 the output operator of the restricted shift realization and the input operator of the *-restricted shift realization are unbounded. Thus 2. follows from 1. Since G is strictly noncyclic and

inf{ot

G(s) has analytic continuation on Re(s) > max{Re(s) s is a pole of g} 0. This shows that g has a strongly coprime Douglas-Shapiro-Shields factorization. Hence the Hankel operator Hg has closed range. Thus by Proposition 6.2 all reachable and observable admissible realizations of g are equivalent. Therefore all these realizations are asymptotically stable. They are exponentially stable if and only if there exists ot > 0 such that g is analytic on Re(s) > -or. Since R(s) is rational and in H(RHP), we know that g is analytic on Re(s) > -o for some o > 0 if and only if there is ) > 0 such that B(s) is analytic on Re(s) > -). Note that the last condition on B(s) is equivalent to that there is ) > 0 such that Re(fin) > n-- 1, 2 By Corollary 8.8 we know that the input and output operators of any reachable and observable admissible realization of g are bounded if and only if Re(,,) < The second case is that R(s) is strictly proper, no zero of R(s) coincides with any accumulation point of the poles of B(s), and infinity is not an accumulation point of the poles of B (s). In this case B is analytic at infinity and the poles of B have accumulation points on the imaginary line. As in the first case, g has a strongly coprime factorization and hence Hg has closed range. Thus all reachable and observable admissible realizations of g are equivalent and asymptotically stable. However, no reachable and observable realization of g is exponentially stable, since the poles of B have accumulation points on the imaginary line and hence g is not analytic on Re(s) > -or for any ot > 0. Since in this case we have g E HZ(RH P) by Proposition 8.4, the input and output operators of any reachable and observable realization of g are bounded. The third case is that R(s) is strictly proper, no zero of R(s) coincides with any accumulation point of the poles of B(s), and infinity is an accumulation point of the poles of B(s). In this case we can show as was done in Example 2 that the factorization of g in the first case is a weakly coprime factorization. Hence g is strictly noncyclic. Thus all input-normal, output-normal, and parbalanced realizations of g are asymptotically stable. As in the first case, an input-normal, an output-normal, or a parbalanced realization of g is exponentially stable if and only if there exists > 0 such that Re(n) > ,k, (n 1, 2 ). From Corollary 8.8 it follows that the input operator of an input-normal realization or the output operator of an output-normal realization is bounded if and only if Thus by Proposition 8.9 and Corollary 8.10 the input operator and output operator of any parbalanced realization of g are bounded if y Re(fl,,) < x. Since clearly g E HZ(R H P), by Proposition 8.4 the input operator of an output-normal realization and the output operator of an input-normal realization of g are bounded. If in addition no accumulation point of the poles of B (s) is on the imaginary line, then g is continuous in the extended imaginary line and therefore g has a balanced realization. We observe that in this case an output-normal realization cannot have a bounded output operator and still be exponentially stable. An analogous fact holds for an input-normal realization and its input operator. The fourth and final case is that at least one of the zeros of R(s) coincides with an accumulation point of the poles of B(s). Note that this accumulation point must be on the imaginary line. As in the previous case, the factorization of g in the first case is a weakly coprime factorization. Hence g is strictly noncyclic. Thus all input-normal, output-normal, and parbalanced realizations of g are asymptotically stable. They are not exponentially stable because g is not analytic on Re(s) > -or for any ot > 0.

,,

,


811

Again by Corollary 8.8 the input operator of an input-normal realization or the output operator of an output-normal realization is bounded if and only if Re(fin) < z. Thus by Proposition 8.9 and Corollary 8.10 the input operator and output operator of any parbalanced realization of g are bounded if R e(n) < x. If every accumulation point of the poles of B is a zero of R, then g is continuous on the extended imaginary line. Hence g has a balanced realization. We now summarize the results as follows. PROPOSITION 9.3. Consider g(s) R(s)B(s), where R(s) is a proper rational function and B(s) is an infinite Blaschke product, both in H(RH P), and B and R have no pole-zero cancellation. 1. If R(s) is not strictly proper and no zero of R(s) coincides with any accumulation point of the poles of B(s), then (a) all reachable and observable admissible realizations of g are equivalent; (b) all reachable and observable admissible realizations of g are asymptotically stable; (c) all reachable and observable admissible realizations of g are exponentially stable where 1, 2 if and only if there exists > 0 such that Re(n) > or, n are the zeros of B(s); fl, n 1, 2 (d) all reachable and observable admissible realizations of g have bounded input and output operators if and only if Re(n) < x. 2. If R (s) is strictly proper, no zero of R (s) coincides with any accumulation point of the poles of B (s), and infinity is not an accumulation point of the poles of B (s), then (a) all reachable and observable admissible realizations of g are equivalent; (b) all reachable and observable admissible realizations of g are asymptotically stable; (c) no reachable and observable admissible realization of g is exponentially stable; (d) all reachable and observable admissible realizations of g have bounded input and output operators. 3. If R(s) is strictly proper, no zero of R(s) coincides with any accumulation point of the poles of B(s), and infinity is an accumulation point of the poles orB(s), then (a) all input-normal, output-normal, and parbalanced realizations of g are asymptoti-

,

cally stable; (b) all input-normal, output-normal, andparbalanced realizations ofg are exponentially stable if and only if there exists > 0 such that Re() > ot (n 1, 2 ); (c) the input operator ofan input-normal realization or the output operator ofan outputnormal realization of g is bounded if and only if Re() < x. The input operator and output operator of any parbalanced realization of g are bounded if Re(fl) < ; (d) the input operator of an output-normal realization and the output operator an inputnormal realization of g are bounded. If in addition, no accumulation point of the poles of B is on the imaginary line, then g has a balanced realization. 4. If at least one of the zeros of R coincides with an accumulation point of the poles of B, then (a) all input-normaL output-normaL and parbalanced realizations of g are asymptotically stable; (b) no input-normaL output-normaL or parbalanced realization of g is exponentially stable; (c) the input operator ofan input-normal realization or the output operator ofan outputnormal realization of g is bounded if and only if y Re(fin) < x3. The input operator and output operator of any parbalanced realization of g are bounded if Re(fin)