Automatica 38 (2002) 1381–1387www.elsevier.com/locate/automatica

Brief Paper

H2 and H∞ norm computations of linear continuous-time periodicsystems via the skew analysis of frequency response operators�

Jun Zhou ∗, Tomomichi HagiwaraDepartment of Electrical Engineering, Kyoto University, Yoshida, Sakyo-ku, Kyoto 606-8501, Japan

Received 7 September 2000; received in revised form 20 January 2002; accepted 7 February 2002

Abstract

The H2 and H∞ norm computations of 0nite-dimensional linear continuous-time periodic (FDLCP) systems through the frequencyresponse operators de0ned by steady-state analysis are discussed. By the skew truncation, the H2 norm can be reached to any degree ofaccuracy by that of an asymptotically equivalent linear time-invariant (LTI) continuous-time system. The H∞ norm can be approximatedby the maximum singular value of the frequency response of an asymptotically equivalent LTI continuous-time system over a certainfrequency range via the modi0ed skew truncation. By the latter result, a Hamiltonian test is proved for FDLCP systems in an LTI fashion,based on which a modi0ed bisection algorithm is developed. ? 2002 Elsevier Science Ltd. All rights reserved.

Keywords: H2 and H∞ norms; Continuous-time periodic systems; Frequency response operator; Skew truncation; Convergence

1. Introduction

The H2 and H∞ norms are used to quantify performancein system analysis and synthesis (Green & Limebeer, 1995;Zhou, 1998). Their computations in linear time-invariant(LTI) systems have been solved, respectively, by the traceformula via solutions of algebraic Lyapunov equations,and by solutions of algebraic Riccati equations or theHamiltonian test (Green & Limebeer, 1995). However, in0nite-dimensional linear continuous-time periodic (FDLCP)systems, the computations are much more di;cult. Thelifting technique (Bamieh & Pearson, 1992a,b; Bittanti andColaneri, 2000; Yamamoto & Khargonekar, 1996), di?eren-tial equation analysis (Colaneri, 2000a; Bolzern & Colaneri,1988), fast sampling=fast hold approximation (Keller & An-derson, 1992; Yamamoto, Madieviski, & Anderson, 1999)and truncations on frequency response operators de0nedvia steady-state analysis (Wereley & Hall, 1990; Wereley,1990) are frequently adopted, each of which has its ad-vantages and drawbacks. In particular, in the H2 and H∞norm computations by the frequency response operators

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

∗ Corresponding author. Tel.: +81-75-753-5334;fax: +81-75-753-3338.

E-mail address: zhouj@jaguar.kuee.kyoto-u.ac.jp (J. Zhou).

(Wereley & Hall, 1990; Wereley, 1990), the numerical im-plementation is nontrivial since the frequency response op-erators are in0nite dimensional. To solve this problem, thesquare truncation is proposed (Wereley, 1990). However,its convergence has not been veri0ed, which is nontrivialespecially when the operator is noncompact.

In this paper we establish formulas for the H2 andH∞ norm computations, via skew and staircase trun-cations on the frequency response operator de0nedvia the steady-state analysis. These truncations lead to‘asymptotically equivalent’ LTI continuous-time systemsfor the norm computations as opposed to the ‘equivalent’operated-valued linear shift-invariant (LSI) discrete-timesystems derived via the lifting approach. The implicationis two-fold: on the one hand, these truncations do giveways to compute the H2 and H∞ norms, by extending thetrace formula and the Hamiltonian test to FDLCP systemsin an LTI continuous-time fashion (as opposed to an LSIdiscrete-time fashion via the lifting approach); on the otherhand, these truncations bridge a gap between the harmonicanalysis of FDLCP systems and its implementation. In the-ory, harmonic analysis leads to the (in0nite-dimensional)harmonic Lyapunov equation (Zhou & Hagiwara, 2002) forFDLCP systems, from whose solution an ‘exact’ trace for-mula can be obtained for the H2 norm of FDLCP systems(Zhou, Hagiwara, & Araki, 2001). It should be remarkablethat the limit of the ‘asymptotic’ trace formula provided

0005-1098/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved.PII: S 0005 -1098(02)00038 -9

1382 J. Zhou, T. Hagiwara / Automatica 38 (2002) 1381–1387

in this paper as the truncation parameter tends to in0nitycoincides with this ‘exact’ trace formula (Zhou, Hagiwara,& Araki, 2001). This coincidence reveals the e?ectivenessof the truncations in dealing with the in0nite dimension-ality of the frequency response operators of FDLCP sys-tems. The skew truncation is 0rst introduced in Zhou andHagiwara (2001) for the frequency response gains compu-tation in FDLCP systems. In this paper we elaborate on it toderive a Lyapunov-equation-based method for the H2 norm.The staircase truncation is introduced to compute the H∞norm as well as to extend the Hamiltonian test into FDLCPsystems.

In Section 2, the frequency response operator of FDLCPsystems and its properties are quickly reviewed. As prepara-tions, the skew and staircase truncations are also explainedin Section 2. The H2 norm computation is the focus of Sec-tion 3, which only relies on the skew truncation. Section 4tackles the H∞ norm computation via the staircase trun-cation. Numerical examples are given in Section 5. Somesketched proofs are provided in the appendix.

Notations. The Euclidean norm for 0nite-dimensional vec-tors as well as the matrix norm induced by this vector normis denoted by ‖·‖. ‖·‖l2=l2 is the induced norm on l2. With alittle abuse; we say F(t)∈L2[0; h] to mean that F is a matrixfunction; each element of which is h-periodic and belongsto L2[0; h] when its domain is restricted to [0; h]; similarlyfor other function sets de0ned over [0; h]. Z is the set of allintegers.

2. Stability and frequency response of FDLCP systems

Consider the FDLCP system

x = A(t)x + B(t)u;

y = C(t)x + D(t)u;(1)

where the state matrix A(t) belongs to LPCD[0; h], andB(t); C(t); D(t) belong to LCAC[0; h]. Here, LPCD[0; h] is theset of h-periodic functions that are piecewise continuousand di?erentiable at a.e. t ∈ [0; h], while LCAC[0; h] is theset of h-periodic functions that are continuous and Fourierseries are absolutely convergent. Clearly, PCD stands forpiecewise continuous and di?erentiable and CAC standsfor continuous and absolutely convergent. System (1) isstrictly proper if D(t) = 0;∀t ∈ [0; h]. The Floquet theorem(Lukes, 1982) says that the transition matrix �(t; t0) (t0 isthe initial time) can be expressed as �(t; t0)=P(t; t0)eQ(t−t0)

where P(t; t0) is a nonsingular h-periodic matrix and Q isa constant matrix. Moreover, the system is asymptoticallystable if and only if the eigenvalues of the matrix Q are inthe open left-half plane.

Next we quickly review the frequency response operatorsof FDLCP systems de0ned via steady-state analysis. Underthe stability assumption, it is shown (Wereley & Hall, 1990;

Zhou & Hagiwara, 2002) that the frequency response oper-ator of system (1) is given by

G(j’) :=C(E(j’) − A)−1B+ D

= C(E(j’) − Q)−1B+ D: (2)

Here, E(j’):=diag[ : : : ; j’−1I; j’0I; j’1I; : : : ] with !h:=2�=h; ’i:=’+ i!h; i∈Z and ’∈I0:=[−!h=2; !h=2) andA:=T{A(t)} is the Toeplitz transformation of A(t) given by

A=

. . ....

...... . .

.

: : : A0 A−1 A−2 : : :: : : A1 A0 A−1 : : :: : : A2 A1 A0 : : :

. .. ...

......

. . .

:

Here {Am}+∞m=−∞ is the Fourier coe;cients sequence of A(t).

The operators B, C and D are de0ned similar to A but interms of the Fourier coe;cients of B(t), C(t) and D(t), re-spectively. In the second equation of (2), similarity trans-formation formulas (3) of FDLCP systems (Wereley, 1990;Zhou, Hagiwara, & Araki, 2002) have been used, whereQ; P, B and C are the Toeplitz transformations ofQ, P(t; 0),B(t):=P−1(t; 0)B(t) and C(t):=C(t)P(t; 0), respectively:

(E(j0) − A)−1 = P(E(j0) − Q)−1P−1;

B= P−1B; C = C P (3)

The reformation of G(j’) given in (2) plays a key role inavoiding the inverse computation of the in0nite-dimensionaloperator E(j’)−A so that this reformation makes it possibleto implement truncations on G(j’).

Now we describe the skew and staircase truncations onG(j’). For the purposes of this paper, it is assumed that thematrix D(t) is constant.

(i) Skew truncation: Let us approximate G(j’) by

G[N ](j’) = C[N ](E(j’) − Q)−1B[N ] + D; (4)

where B[N ] is formed by skew truncating B, i.e.,

B[N ]:=T{∑Nm=−N Bme

jm!ht} where {Bm} is the Fouriercoe;cients sequence of B(t). The operator C[N ] is formedsimilarly but in terms of C. No truncation is done on Dsince it already has the skew-truncated form. The structureof G[N ](j’) is helpful in trace computation. Hence it willbe used in the H2 norm computation.

(ii) Staircase truncation: To compute the H∞ norm,only the skew truncation cannot reduce the problem to0nite-dimensional computations. The staircase truncation isintroduced as

G[N;M ](j’) = C[N;M ](EM (j’) − QM

)−1B[N;M ] + DM : (5)

J. Zhou, T. Hagiwara / Automatica 38 (2002) 1381–1387 1383

Here, the matrix B[N;M ]:=diag[ : : : ; BNM ; BNM ; BNM ; : : : ] with

BNM =

B0 : : : B−N 0...

. . .. . .

BN. . . B−N

. . .. . .

...0 BN : : : B0

︸ ︷︷ ︸(2M+1)-blocks

:

In (5), C[N;M ] is de0ned similar to B[N;M ] but in termsof the Fourier coe;cients of C(t). To conform to theblock-diagonal form of B[N;M ] and C[N;M ], E( j’); Q and Dare rewritten as EM ( j’); Q

Mand DM . That is,

QM

:= diag[ : : : ; QM ; QM ;QM ; : : : ](=Q);

DM := diag[ : : : ; DM ; DM ;DM ; : : : ](=D);

EM ( j’) := diag[ : : : ; EM;−1( j’); EM0( j’); EM1( j’); : : : ](=E( j’))

with DM :=diag[D;D; : : : ; D]︸ ︷︷ ︸(2M+1)

and QM being de0ned simi-

larly. Also,

EMm( j’) = diag[ j’m(2M+1)−MI; : : : ; j’m(2M+1)I; : : : ;

j’m(2M+1)+MI ] (6)

for m∈Z. In our discussions, we take M¿N + 1. Toguarantee convergence of the truncations, properties of theFourier series expansions of B(t) and C(t) are needed. Itis shown (Zhou et al., 2002) that under the assumptionson (1), B(t) and C(t) belong to LCAC[0; h].

3. H2 norm computation of FDLCP systems

The H2 norm of G( j’) is given by the square root of thequantity

‖G‖22 =

12�

∫ !h=2−!h=2

trace(G( j’)∗G( j’)) d’: (7)

By Colaneri (2000a), Zhang and Zhang (1997) and Zhouet al. (2002), this de0nition is equivalent to the time-domainone. However, since G( j’) is in0nite dimensional, ‖G‖2

cannot be computed directly. To solve this problem, letus de0ne the following quantity and state the convergencelemma:

‖GN‖22 =

12�

∫ !h=2−!h=2

trace(G[N ]( j’)∗G[N ]( j’)) d’: (8)

Lemma 1. Under the assumptions of system (1); ifthe system is asymptotically stable and strictly proper;then for any �¿ 0; there exists an integer N0(�) such that|‖G‖2 − ‖GN‖2|¡� (∀N¿N0(�)).

Lemma 1 implies that the H2 norm ‖G‖2 can be approx-imated to any degree of accuracy by ‖GN‖2. Even thoughthe latter still involves the in0nite-dimensional operatorG[N ]( j’) as in (8), it is actually possible to compute ‖GN‖2

exactly, only through 0nite-dimensional computations.To show this, we de0ne the LTI continuous-time systemgiven by

GN (s):=[QN BNCN 0

];

CN :=

C−N 0...

. . .

C0 C−N...

. . ....

. . .

CN : : : C0 : : : C−N. . .

.... . .

...

CN C0

. . ....

0 CN

(9)

with QN = QN − EN0( j0) and

BN :=[BT−N ; : : : ; B

T−1; B

T0 ; B

T1 ; : : : ; B

TN ]T:

It is clear that in G[N ](s), (E( j’)−Q)−1 is block diagonaland B[N ] and C[N ] have 2N + 1 (sub)diagonal strips alongthe diagonal, and the mth entry of G[N ]( j’)∗G[N ]( j’) onthe diagonal is B∗

N (EN0( j’m)−QN )−∗C∗NCN (EN0( j’m)−

QN )−1BN = GN ( j’m)∗GN ( j’m). Then it follows that

‖GN‖22 =

12�

∫ !h=2−!h=2

+∞∑m=−∞

trace(GN ( j’m)∗GN ( j’m)) d’

=12�

∫ ∞

−∞trace(GN ( j!)∗GN ( j!)) d!

= ‖GN‖22; (10)

where ‖GN‖2 is the H2 norm of GN (s). In (10), the in-0nite summation and integral are interchanged, whose va-lidity is guaranteed by the Levy theorem (Naylor & Sell,1982) and the fact that

∑km=−k trace(GN ( j’m)∗GN ( j’m))

is absolutely convergent over k. Now the following result isimmediate.

Theorem 2. Suppose that system (1) is asymptotically sta-ble and strictly proper. Then

limN→∞

trace(B∗NVNBN ) = lim

N→∞trace(CNWNC

∗N )

= ‖G‖22;

where VN and WN are; respectively; the solutions of the3nite-dimensional Lyapunov equations

Q∗NVN + VNQN + C∗

NCN = 0;

QNWN + WNQ∗N + BNB

∗N = 0:

1384 J. Zhou, T. Hagiwara / Automatica 38 (2002) 1381–1387

4. H∞ norm computation of FDLCP systems

In the H∞ norm computation, the staircase truncationis useful. In particular, we derive a Hamiltonian test withthis truncation. By Colaneri (2000a,b), Wereley (1990) andZhou et al. (2002), the H∞ norm of system (1) is

‖G‖∞:=max’∈I0

‖G( j’)‖l2=l2 :

The following lemma (Zhou et al., 2001) is useful in provingthe convergence of the staircase truncation on G( j’).

Lemma 3. Suppose that system (1) is asymptoticallystable. Then for any �¿ 0; there exists an integerN0(�)¿ 0 such that ‖G( j’) − G[N ]( j’)‖l2=l2¡� (∀N¿N0(�);∀’∈I0).

By Lemma 3, to show that G[N;M ]( j’) converges toG( j’) uniformly over ’∈I0 in the l2-induced norm asN;M → ∞, it su;ces to show that G[N;M ]( j’) convergesto G[N ]( j’) as M → ∞ in the same sense. Indeed one canclaim the following convergence lemma.

Lemma 4. Suppose that system (1) is asymptoticallystable. Then; for each 3xed N and for any �¿ 0; thereexists an integer M0(N; �)¿ 0 such that ‖G[N ]( j’) −G[N;M ]( j’)‖l2=l2¡� (∀M¿M0(N; �);∀’∈I0).

Now, let us further de0ne the LTI continuous-time system

GNM (s) :=[QM BNMCNM DM

]; (11)

where QM is as before and BNM :=BNM ; CNM :=CNM ,DM :=DM . The following theorem is helpful in establishinga Hamiltonian test for the H∞ norm computation of FDLCPsystems.

Theorem 5. Suppose that system (1) is asymptoticallystable. Then

‖G‖∞ = limN→∞

limM→∞

‖GNM‖∞;IM ;

where ‖GNM‖∞;IM :=sup!∈IM‖GNM ( j!)‖ with IM

being the union of the angular frequency intervalsIMm:=[ − !h=2 + m(2M + 1)!h; !h=2 + m(2M + 1)!h);m∈Z; i.e.; IM =

⋃+∞m=−∞ IMm.

Proof. By the stability assumption; QM has no eigenvalueson the imaginary axis. Then; by the fact that G[N;M ]( j’) isblock diagonal; it follows that

max’∈I0

‖G[N;M ]( j’)‖l2=l2 = ‖GNM‖∞;IM : (12)

On the other hand; by Lemmas 3 and 4; we obtainthat for all ’∈I0; limM→∞ limN→∞‖G[N;M ]( j’)‖l2=l2=

‖G( j’)‖l2=l2 . Using this in (12) yields

‖G‖∞ = max’∈I0

limM→∞

limN→∞

‖G[N;M ]( j’)‖l2=l2

= limM→∞

limN→∞

‖GNM‖∞;IM :Note that the order exchanges involved are valid since the

convergence of Lemmas 3 and 4 is uniform with respect to’∈I0.

Theorem 5 shows that if N and M are large enough,we can get tight estimations of ‖G‖∞ by computing‖GNM‖∞;IM . The following theorem gives a basis by which‖GNM‖∞;IM can be determined for any 0xed N and Mthrough a modi0ed bisection algorithm we will describeshortly.

Theorem 6. Suppose that system (1) is asymptotically sta-ble. Then; for any 3xed large enough N and M satisfy-ing M¿N +1; ‖GNM‖∞;IM ¡# only if S$(D)¡# and theHamiltonian matrix

HNM :=

[QM + BNMR

−1M D∗

MCNM

−C∗NM (I + DMR

−1M D∗

M )CNM

BNMR−1M B∗

NM

−(QM + BNMR−1M D∗

MCNM )∗

]

has no eigenvalues on the jIM portion of the imaginaryaxis. In the above RM :=#2I −D∗

MDM .

Theorem 6 can be shown by following the necessity proofof Lemma 3.7.2 of Green and Limebeer (1995). Theorem 6gives only a necessary condition because the range of jIMis not connected and hence we cannot employ a simple con-tinuity argument (Green & Limebeer, 1995). Thus, for some#, the knowledge on the eigenvalues of the Hamiltonian ma-trix HNM corresponding to # alone fails to tell whether # isan upper bound or a lower bound of ‖GNM‖∞;IM . To copewith such situations, we propose the following modi0ed bi-section algorithm for the computation of ‖GNM‖∞;IM for0xed N and M . Now we introduce

SIMm :=[!h

2+ (m− 1)(2M + 1)!h;

− !h2

+ m(2M + 1)!h); m∈Z

and SIM :=⋃+∞m=−∞ SIMm. Note that IM and SIM are dis-

joint and that the whole imaginary axis is the union of jIMand j SIM .

Modi&ed Bisection Algorithm for Computing ‖GNM‖∞;IMStep 1: Select an upper bound #u and a lower bound #l

such that ‖D‖6 #l6 ‖GNM‖∞;IM 6 #u. We may simplytake #l = ‖D‖ and #u = ‖GNM‖∞.Step 2. If (#u − #l)=#l ≤ speci0ed error tolerance level,

then stop, and let ‖GNM‖∞;IM = (#u + #l)=2; otherwise, set#= (#u + #l)=2 and go to the next step.

J. Zhou, T. Hagiwara / Automatica 38 (2002) 1381–1387 1385

Step 3: Compute HNM with given #, and determine theeigenvalues of HNM . If there exists an eigenvalue of HNM onjIM , it is concluded by Theorem 6 that ‖GNM‖∞;IM ¿ #.This leads to the operation #l = # and go back to Step 2;otherwise go to Step 4.Step 4: There are two cases: (i) If HNM has no

eigenvalues on the imaginary axis, it follows by theusual Hamiltonian test and IM ⊂ (−∞;+∞) that#¿ ‖GNM‖∞¿ ‖GNM‖∞;IM . This leads to the operation#u = # and go back to Step 2. (ii) If HNM has eigenval-ues on the imaginary axis (i.e., on j SIM ), it remains un-known whether # is an upper bound or a lower bound of‖GNM‖∞;IM . To resolve such uncertainty, go to the nextstep.Step 5: Let j SIMm1 ; : : : ; j SIMmp be the intervals containing

the eigenvalues of HNM on j SIM . For each i = 1; 2; : : : ; p,check if ‖GNM ( j!−

i )‖¡# and ‖GNM ( j!+i )‖¡#, where

!−i := inf SIMmi =

!h2

+ (i − 1)(2M + 1)!h;

!+i := sup SIMmi = −!h

2+ i(2M + 1)!h:

If one of these tests fails, then # is a lower bound of‖GNM‖∞;IM . Hence, set #l = # and go back to Step 2;otherwise, ‖GNM ( j!)‖¡# (∀!∈IM ) by the simplecontinuity argument. Hence, set #u = # and go back toStep 2.

5. Numerical examples

We consider the H2 and H∞ norm computations for thefollowing �-periodic system when the input weighting

Table 1H2 norms: skew truncation approach

N 1 2 3 5 10 15 30

' = 0 0.7323 0.7323 0.7323 0.7323 0.7323 0.7323 0.73230.1 0.6825 0.6839 0.6836 0.6836 0.6836 0.6836 0.68360.2 0.6352 0.6409 0.6408 0.6408 0.6408 0.6408 0.64080.3 0.5907 0.6046 0.6052 0.6052 0.6053 0.6053 0.60530.4 0.5500 0.5761 0.5781 0.5781 0.5782 0.5782 0.57820.5 0.5137 0.5566 0.5608 0.5608 0.5610 0.5611 0.5611

Table 2H∞ norms: staircase truncation approach

(N;M) (1; 6) (2; 6) (3; 6) (5; 6) (10; 12) (15; 17) (30; 52)

' = 0 1.5024 1.5024 1.5024 1.5024 1.5024 1.5024 1.50240.1 1.4574 1.4582 1.4618 1.4618 1.4618 1.4618 1.46180.2 1.4158 1.4202 1.4264 1.4273 1.4273 1.4273 1.42730.3 1.3778 1.3893 1.3981 1.3981 1.3981 1.3981 1.39810.4 1.3460 1.3654 1.3761 1.3769 1.3769 1.3769 1.37690.5 1.3230 1.3478 1.3601 1.3601 1.3601 1.3601 1.3601

parameter ' varies from 0 to 0:5:

x =

[ −1 − sin2(2t) 2 − 12 sin(4t)

−2 − 12 sin(4t) −1 − cos2(2t)

]x

+

[0

1 − 2'((t)

]u;

y = [1 1]x:

Here, the �-periodic function ( is given by ((t) = sin(2t)when 06 t6 �=2 and ((t)=0 when �=2¡t6 �. The tran-sition matrix of the system has a Floquet factorization of theform

P(t; 0) =

[cos(2t) sin(2t)

−sin(2t) cos(2t)

]; Q =

[−1 0

0 −2

]:

Noting that the transition matrix is available, the Flo-quet transformed form, i.e., the second equality of (2), ofG( j’) can be determined explicitly and thus all the re-sults apply. The H2 norm computation results are listed inTable 1.

Now introduce a feedthrough D=1 into the above systemand compute the H∞ norm of the corresponding �-periodicsystem (which is not strictly proper) by the modi0ed bisec-tion algorithm. The results are given in Table 2. The initialupper and lower bounds for the H∞ norm are #u = 4:6200and #l = 1(=‖D‖) while the tolerance error is 0:0001. Here,#u is estimated by working on ‖GNM ( j!)‖∞, which is takenuniformly over '∈{0; 0:1; 0:2; 0:3; 0:4; 0:5}; N=1; 2; : : : ; 30and M = 6; 7; : : : ; 52 for simplicity. The Fourier coe;cientsused are computed by numerical integrations.

1386 J. Zhou, T. Hagiwara / Automatica 38 (2002) 1381–1387

6. Conclusion

By truncations, the H2 and H∞ norm computations ofFDLCP systems are converted to those of asymptoticallyequivalent LTI continuous-time systems. Hence, the resultsfor the H2 and H∞ norms in LTI systems are extendedto FDLCP systems. In other words, the skew analysis isan alternative approach to get insight into the behavior ofFDLCP systems. In addition, we believe this skew analysiscan also convert the H2 problem into that of an asymptot-ically equivalent discrete-time system by the well-knownimpulse modulation formula and the factorization tech-nique (Hagiwara & Araki, 1995), though this idea is notpursued here. The staircase truncation is also useful for thefrequency response gains computation of FDLCP systems,as an alternative to the skew-rectangular truncation method(Zhou et al., 2001).

In the proposed methods, the Floquet transforma-tion is introduced to avoid the invertibility problem ofin0nite-dimensional operators and guarantee the conver-gence, which is standard in FDLCP system analysis. There-fore, it is necessary to have the transition matrix knowledgebefore applying our results. Fortunately, our discussionsestablish in particular that it is enough to have the numer-ical description of the transition matrix in the sense thatthe convergence is guaranteed theoretically without anyassumptions on the analytical properties of the transitionmatrix.

Acknowledgements

The 0rst author is grateful to Professor M. Araki for hisadvice and comments on this paper.

Appendix A. The sketched proofs of lemmas

Proof of Lemma 1. Note that; by the stability assumption;G( j’) is bounded for any ’∈I0. Thus; by Naylor and Sell(1982); we have

trace{G( j’)∗G( j’)}6 ‖C‖2l2=l2

∞∑m=−∞

K2Qf(m)2

∞∑n=−∞

‖Bm−n‖2: (A.1)

To get (A.1); we used the fact that there exists KQ¿ 0 suchthat

‖( j’mI − Q)−1‖6KQf(m); (A.2)

where the function f(n) of an integer n is de0ned by f(n)=1; n = 0 and f(n) = |n|−1; n = 0. Then; it holds (Araki;Ito;& Hagiwara; 1996) that

∑∞n=−∞ f(n)2¡ 5 and for any

N + 1¿ 2;∑∞n=N+1 f(n)2¡ 1=N . In (A.2); since Q is

stable; KQ can be chosen to be independent of ’∈I0. Notealso that C is bounded on l2 and

∑∞m=−∞ ‖Bm‖2¡∞.

Noting that G[N ]( j’) is uniformly bounded on l2 over’∈I0 (Zhou et al., 2001), similar procedure as in (A.1)yields trace(G[N ]( j’)∗G[N ]( j’))6K,¡∞ for someK,¿ 0, which is independent of N . Furthermore, for anorthonormal basis {xn}∞n=−∞ of l2, we obtain that

|trace(G( j’)∗G( j’)) − trace(G[N ]( j’)∗G[N ]( j’))|

6√

2[trace(G( j’)∗G( j))

+ trace(G[N ]( j’)∗G[N ]( j’))]1=2

·[ ∞∑n=−∞

‖(G( j’) − G[N ]( j’))xn‖2l2

]1=2

; (A.3)

where {xn}∞n=−∞ is an orthonormal basis of l2, withxn:=[ : : : ; 0T; uTn ; 0

T; : : : ]T and |un| = 1;∀n∈Z. Inequality(A.3) says that to show the convergence assertion, it su;cesto show that

∞∑n=−∞

‖(G( j’) − G[N ]( j’))xn‖2l2

¡ 5K2Q‖B‖2

l2=l2

∑|m|¿N

‖Cm‖2

+ 5K2Q‖C[N ]‖2

l2=l2

∑|m|¿N

‖Bm‖2 → 0 (A.4)

as N → ∞. However, this is obvious since B is bounded onl2 and ‖C[N ]‖2

l2=l2 is uniformly bounded over N .

Proof of Lemma 4. To prove this lemma; we focus on theinequality

‖G[N ]( j’) − G[N;M ]( j’)‖l2=l26 ‖ TC[N;M ](E( j’) − Q)−1‖l2=l2‖B[N;M ]‖l2=l2

+ ‖C[N ]‖l2=l2‖(E( j’) − Q)−1 TB[N;M ]‖l2=l2 ; (A.5)

where TB[N;M ]:=B[N ] − B[N;M ]. More explicitly; it is given by

TB[N;M ] :=

. . .. . .

. . . 0TBNM l 0 TBNMu

TBNM l 0 TBNMu

0. . .

. . .. . .

=: TBl[N;M ] + TBu[N;M ];

where TBl[N;M ] and TBu[N;M ]are the lower and upper triangle

portions of TB[N;M ]; respectively. The matrix TC[N;M ] is de0ned

J. Zhou, T. Hagiwara / Automatica 38 (2002) 1381–1387 1387

similarly but in terms of {Cm}Nm=−N . Also; it is easy tosee that ‖B[N;M ]‖l2=l2 ; ‖C[N ]‖l2=l2 and ‖ TBNM l‖ are uniformlybounded with respect to N and M .

Now we prove the main result of Lemma 4. It su;ces toshow that

‖(E( j’) − Q)−1 TB[N;M ]‖l2=l26 ‖(E( j’) − Q)−1 TBl[N;M ]‖l2=l2

+‖(E( j’) − Q)−1 TBu[N;M ]‖l2=l2 → 0 (A.6)

as M → ∞. By the structures of EM ( j’); QM

, and TBl[N;M ]

and the fact that the entries of TBNM l are zeros except itsright-upper blocks, we have

‖(E( j’) − Q)−1 TBl[N;M ]‖l2=l26 ‖ TBNM l‖KQf(M − N + 1) (A.7)

under the assumption M¿N + 1. From (A.7), onecan conclude that for each 0xed N and for any �¿ 0,there exists an integer M0(N; �)¿ 0 such that ‖(E( j’) −Q)−1 TBl[N;M ]‖l2=l2¡�, ∀M¿M0(N; �);∀’∈I0 becausef(n) is monotonically decreasing. These arguments canbe repeated on the second term of the right-hand sideof inequality (A.6). Then, the convergence assertionfollows.

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Jun Zhou received the B.Sc. degree inelectroics from Sichuan University, China,in 1984. He received the M.Sc. degreein electrical and information science fromLanzhou University, China, in 1987. Cur-rently he is a Ph.D. candidate in thedepartment of electrical engineering ofKyoto University, Kyoto, Japan.

From 1987 to 1995, he was a lec-turer with the Department of Electronicsand Information Science at Lanzhou Uni-versity, China. Since 1995, he has been

an associated professor with the same department. His research inter-est includes system structure analysis, robust stability and periodicallytime-varying systems control and their applications.

Tomomichi Hagiwara was born in Osaka,Japan, on March 28, 1962. He receivedthe B.E., M.E. and Ph.D. degrees in elec-trical engineering from Kyoto University,Kyoto, Japan, in 1984, 1986 and 1990,respectively.

Since 1986 he has been with theDepartment of Electrical Engineering,Kyoto University, where he is a Professorsince 2001. His research interests includesampled-data control, nonlinear=robuststability, two-degree-of-freedom control

systems, and dynamical system theory.