Delay-dependent generalized H2 filtering for uncertain systems with multiple time-varying state delays

ARTICLE IN PRESS

Signal Processing 87 (2007) 709–724 www.elsevier.com/locate/sigpro

Delay-dependent generalized H2 filtering for uncertain systems with multiple time-varying state delays Wen-an Zhanga,, Li Yua, Xiefu Jiangb a

Department of Automation, Zhejiang University of Technology, Hangzhou 310032, PR China Faculty of Business & Informatics, Central Queensland University, Rockhampton, Qld 4702, Australia

b

Received 11 April 2006; received in revised form 5 July 2006; accepted 14 July 2006 Available online 15 August 2006

Abstract A delay- and parameter-dependent approach to generalized H2 filtering is proposed for linear continuous-time uncertain systems with multiple time-varying state delays. The uncertain parameters are assumed to reside in a polytope, and the aim is to design a parameter-dependent or a parameter-independent filter such that the filtering error systems are assured to be asymptotically stable and a prescribed generalized H2 performance is guaranteed. The proposed filter design method possesses many advantages, such as the reduced conservatism of the obtained delay-dependent criteria due to adopting the newly established integral-inequality; the proposed new linearization technique and the parameter-dependent design method for the parameter-dependent filters. Both the conditions for the existence of the parameter-dependent and parameter-independent filters are presented in terms of linear matrix inequalities, and convex optimization problems are formulated to design the desired filters. A numerical example is given to illustrate the validity of the proposed design. r 2006 Elsevier B.V. All rights reserved. Keywords: Generalized H2 filtering; Time-delay systems; Polytopic uncertainty; Delay-dependence; Linear matrix inequality (LMI)

1. Introduction State estimation of dynamic systems is a subject of great practical and theoretical significance which has received considerable attention for decades; see, e.g. [1–5]. When the celebrated Kalman filtering scheme is no longer applicable in the case that a priori information on the external noises is not precisely known, some alternatives, such as the HN filtering, the generalized H2 filtering and the l1 filtering schemes were introduced and have received increasing research efforts in the past two decades [6], see, e.g. [3–8]. On the other hand, time delays are frequently encountered in many practical engineering systems, and are often the sources of instability and performance deterioration, therefore, stability analysis and controller synthesis for time-delay systems have received considerable research efforts. Filtering for dynamic systems is Corresponding author. Tel./fax: +86 571 88320200.

E-mail address: [email protected] (W.-a. Zhang). 0165-1684/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2006.07.004

ARTICLE IN PRESS 710

W.-a. Zhang et al. / Signal Processing 87 (2007) 709–724

therefore desired to take into account the presence of the time delays when they occurred and cannot be neglected, which leads to the filter design problem for time-delay systems. The past few years have witnessed an increasing interest in the filtering for time-delay systems, in particular, with parameter uncertainties, and many results have been reported, see, e.g. [6,9–12]. These results can be roughly classified into two types according to their dependence on the size of the delays, namely, delay independent results and delaydependent results. Since the delay-dependent results contain information on the delay size, they are generally less conservative than the delay-independent ones as might be expected, especially when the delay is small. For time-delay systems with polytopic-type uncertainties, the robust HN filtering for continuous-time systems with multiple time-varying state delays was addressed in [9], and for discrete time state-delayed systems was reported in [10], both provided delay-independent results. In [6], the authors studied the robust l2lN filtering for uncertain continuous time-delay systems, and by using Moon inequality, the delay-dependent l2lN performance criterion was obtained. However, it is noticed that most of the aforementioned results are quite conservative to some extent because they are based on the quadratic stability notion, that is, a same parameter-independent Lyapunov functional was used for the entire uncertainty domain [13]. Very recently, a parameter-dependent Lyapunov functional approach was used in [12] to study the delay-dependent HN filtering for discrete-time state-delayed systems with polytopic type uncertainties, and some improved results were obtained. In addition, motivated by the parameter-dependent stability idea, a parameter-dependent filter was firstly designed in [13] for a class of delay free discrete-time systems, and the conservatism of the obtained results were further reduced. The generalized H2 filtering is the energy-to-peak gain filtering, which was first solved for nominal systems in [14], and was then solved for uncertain systems of both continuous-time and discrete-time in [15]. The generalized H2 filtering for Markovian jump systems and linear repeated nonlinear systems were reported in [16] and in [17], respectively. Recently, generalized H2 filtering for time-delay systems has received increasing attentions, and it was solved for continuous-time delay systems and discrete-time delay systems in [16] and in [18], respectively, and both delay-independent and delay-dependent approaches were adopted in these two results. Besides, the generalized H2 filtering was also addressed for nonlinear time-delay systems and stochastic delay systems in [19] and in [20], respectively. However, for time-delay systems few result concerns the generalized H2 filtering using the delay- and parameter-dependent approach. Besides, the conservatism of the existing delay-dependent criteria can be further reduced by using the newly developed techniques. These motivate the present research. In this paper, we investigate the generalized H2 filtering problem for continuous-time systems with multiple time-varying state delays and polytopic type uncertainties. We first derive delay-dependent representations of generalized H2 performance for the filtering analysis of the systems under consideration. The conditions are derived by using the newly established integral inequality without involving a fixed model transformation and the bounding for certain cross terms, which enables us to obtain much less conservative results. Instead of directly adopting the existing linearization techniques used in [2,9,10] and in [12,13], we propose a new linearization technique that is similar to the variables changing method commonly used in the dynamic output feedback control problem (see e.g. [21]). By introducing some slack matrix variables, the obtained performance conditions are modified to obtain some equivalent matrix inequality representations, and these new representations enable us to use the new linearization technique to design the desired filters. By using the newly established linearization technique and the parameter-dependent Lyapunov function approach proposed recently in [22], the delay- and parameter-dependent conditions for the existence of the parameter-dependent filters and the parameter-independent filters are both formulated in terms of Linear matrix inequalities (LMIs). Convex optimization problems and one-dimensional search are involved to design the desired filters with minimized generalized H2 performances. Illustrative example is finally given to demonstrate the effectiveness of the proposed results. The notation used in the paper is fairly standard: The superscript T stands for matrix transposition, Rn denotes the n-dimensional Euclidean space, Rn  m is the set of all n  m real matrices, and diag {?} stands for a block-diagonal matrix. In symmetric block matrices, we use an asterisk () to represent a term that is induced by symmetry. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

ARTICLE IN PRESS W.-a. Zhang et al. / Signal Processing 87 (2007) 709–724

711

2. Problem formulation and preliminaries Consider the following uncertain linear continuous-time systems with multiple time-varying state delays _ ¼ A0 ðlÞxðtÞ þ S : xðtÞ

q X

Aj ðlÞxðt  hj ðtÞÞ þ BðlÞwðtÞ,

j¼1

yðtÞ ¼ C 0 ðlÞxðtÞ þ

q X

C j ðlÞxðt  hj ðtÞÞ þ DðlÞwðtÞ,

j¼1

zðtÞ ¼ LðlÞxðtÞ, xðtÞ ¼ fðtÞ; 8t 2 ½hmax ; 0Þ,

ð1Þ

n

m

p

where x(t)AR is the state vector; y(t)AR is the measured output; z(t)AR is the signal to be estimated; w(t)ARq is the noise signal vector; f(t) is the given initial vector function that is continuous on the interval [hmax, 0); hj(t) (j ¼ 1, y, q) are time-varying bounded delays satisfying h_j ðtÞphjd o1;

hj ðtÞphjm o1; and

8tX0

hmax ¼ maxfhjm g. j

The system matrices A0(l), Aj(l), B(l), C0(l), Cj(l), D(l), L(l), j ¼ 1, y, q are appropriately dimensioned matrices, and it is assumed that O :¼ ðA0 ðlÞ; . . . ; Aq ðlÞ; BðlÞ; C 0 ðlÞ; . . . ; C q ðlÞ, DðlÞ; LðlÞÞ 2 U,

ð2Þ

where O is a given convex bounded polyhedral domain described by N vertices as follows: ( ) N N X X   U :¼ OðlÞ OðlÞ ¼ li O i ; li ¼ 1; li X0 i¼1

i¼1

and Oi :¼ ðA0i ; . . . ; Aqi ; Bi ; C 0i ; . . . ; C qi ; Di ; Li Þ denotes the ith vertex of the polytope. It is also assumed that l does not depend explicitly on the time variable. Here, according to the properties of the uncertain parameters, that is, whether the uncertain parameters are online measurable or not, we are interested in designing the full order parameter-dependent or parameterindependent filters described by F1 and F2, respectively. Case 1: The uncertain parameters vary slowly enough such that they can be measured online. In this case, attention will be focused on the design of the following parameter-dependent filter _^ ¼ Af ðlÞxðtÞ ^ þ Bf ðlÞyðtÞ, F 1 : xðtÞ ^ z^ðtÞ ¼ C f ðlÞxðtÞ.

ð3aÞ

Case 2: The uncertain parameters cannot be measured online. In this case, the design of the following parameter-independent filter of general structure will be of our interest _^ ¼ Af xðtÞ ^ þ Bf yðtÞ, F 2 : xðtÞ ^ z^ðtÞ ¼ C f xðtÞ.

ð3bÞ

Remark 1. A parameter-dependent filter takes into account the exact values of the online measurable uncertain parameters, and uses different filter gain matrices as the uncertain parameters vary. Therefore, the filtering results obtained by using a parameter-dependent filter are expected to be less conservative than those obtained by using the parameter-independent filters. This approach has recently been used in [13].

ARTICLE IN PRESS W.-a. Zhang et al. / Signal Processing 87 (2007) 709–724

712

Denote

(

F jðAF ; BF ; C F Þ ¼

F 1 jðAf ðlÞ; Bf ðlÞ; C f ðlÞÞ;

case 1

F 2 jðAf ; Bf ; C f Þ;

case 2

Then, both the parameter-dependent and parameter-independent filters to be designed can be written in the following form: _^ ¼ AF xðtÞ ^ þ BF yðtÞ, F : xðtÞ ^ z^ðtÞ ¼ C F xðtÞ.

ð4Þ

Augmenting the model of S to include the states of the filter F, we obtain the filtering error system: _ ¼ A0 ðlÞxðtÞ þ Ad ðlÞxd ðtÞ þ BðlÞwðtÞ I : xðtÞ eðtÞ ¼ C 0 ðlÞxðtÞ, T

ð5Þ T

T

where xðtÞ ¼ ½ xT ðtÞ x^ ðtÞ T , xd ðtÞ ¼ ½ x ðt  h1 ðtÞÞ    x ðt  hq ðtÞÞ T , eðtÞ ¼ zðtÞ  z^ðtÞ, and " # " # 0 Aj ðlÞ A0 ðlÞ 0 A0 ðlÞ ¼ ; Aj ðlÞ ¼ , BF C j ðlÞ 0 BF C 0 ðlÞ AF " # h i BðlÞ BðlÞ ¼ ; Ad ðlÞ ¼ A1 ðlÞ    Aq ðlÞ BF DðlÞ C 0 ðlÞ ¼ ½LðlÞ

 C F .

Before presenting the main objectives of this paper, we introduce the following definition. Definition 1. Given a positive scalar g, the filtering error system I in (5) is said to be asymptotically stable with a generalized H2 disturbance attenuation level g if it is asymptotically stable and under zero initial conditions qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R1 T kek1 ogkwk2 for all nonzero wAl2[0, N), where kek1 :¼ supt eT ðtÞeðtÞ, and kwk2 :¼ 0 w ðtÞwðtÞ dt. Then, the generalized H2 filtering problem addressed in this paper can be expressed as follows. Generalized H2 filtering problem: Given an asymptotically stable system S in (1), and positive scalar g, determine an asymptotically stable linear filter of form (4) such that the filtering error system (5) is asymptotically stable over the entire uncertaint domain U and ensures a generalized H2 disturbance attenuation level g. 3. Generalized H2 filtering analysis Before proceeding further, we introduce the following lemma which plays a key role in the derivation of the delay-dependent conditions. Lemma 1. [23] Let x(t)ARn be a vector-valued function with first-order continuous-derivative entries. Then, the following integral inequality holds for any matrices M1, M2ARn  n and X ¼ XT40, and a scalar function h: ¼ h(t)X0: " #T " T # Z t xðtÞ M 1 þ M 1 M T1 þ M 2 T _ dsp  x_ ðsÞX xðsÞ xðt  hÞ n M T2  M 2 th " # " #T " T # " # xðtÞ xðtÞ xðtÞ M1    þh X 1 M 1 M 2 . ð6Þ xðt  hÞ xðt  hÞ xðt  hÞ M T2 By applying the parameter-dependent Lyapunov functional approach, some generalized H2 performance conditions for the filtering error system (5) are presented in the following theorems.

ARTICLE IN PRESS W.-a. Zhang et al. / Signal Processing 87 (2007) 709–724

713

Theorem 1. Consider filtering error system (5), and let g40, hjm40, hjd40, j ¼ 1, y, q, be given scalars. If there exist symmetric positive-definite matrices P(l), Qj(l), Sj(l)AR2n  2n, and matrices M0j(l), M1j(l)AR2n  2n, j ¼ 1, y, q, such that the following matrix inequalities hold 2 6 Oa ðlÞ ¼ 6 4

" Ob ðlÞ ¼

~ þ A~ T ðlÞP~ T ðlÞ þ MðlÞ ~ ~ AðlÞ PðlÞ n

T A~ ðlÞXðlÞ XðlÞ

n

n

3 M d ðlÞ 7 7o0, 0 5 ¯ m Sd ðlÞ D

# ¯ T ðlÞ PðlÞ C 0 40, n g2 I

(7a)

(7b)

where 2 6 6 ~ MðlÞ ¼6 6 4

~ 11 ðlÞ þ M

q P

~ 12 ðlÞ M

Qj ðlÞ

j¼1

~ 22 ðlÞ  ðI  Dd ÞQd ðlÞ M

n

0

3

7 7 7, 0 7 5

n n I h i T T ~ 12 ðlÞ ¼ M 01 ðlÞ þ M ðlÞ     M 0q ðlÞ þ M ðlÞ , M 11 1q n o ~ 22 ðlÞ ¼ diag M 11 ðlÞ  M T ðlÞ; . . . ; M 1q ðlÞ  M T ðlÞ , M 11 1q

~ 11 ðlÞ ¼ M

q   X M 0j ðlÞ þ M T0j ðlÞ ;

 Dd ¼ diag h1d ; . . . ; hqd ,

j¼1

~ ¯ 0 ðlÞ A ¯ d ðlÞ BðlÞ, ¯ AðlÞ ¼ ½A q n o X ¯ m ¼ diag h1 ; . . . ; h1 , XðlÞ ¼ hjm S j ðlÞ; D 1m qm

Qd ðlÞ ¼ diagfQ1 ðlÞ; . . . ; Qq ðlÞg; ~ PðlÞ ¼ ½PT ðlÞ 0

0T ;

j¼1 T T M d ðlÞ ¼ ½ M 0d ðlÞ M 1d ðlÞ 0 T ,

S d ðlÞ ¼ diagfS1 ðlÞ; . . . ; Sq ðlÞg; M 0d ðlÞ ¼ ½ M 01 ðlÞ

M 0q ðlÞ ;



M 1d ðlÞ ¼ diagfM 11 ðlÞ; . . . ; M 1q ðlÞg.

Then, the filtering error system (5) is asymptotically stable with a generalized H2 disturbance attenuation level g.

Proof. Choose the following parameter-dependent Lyapunov–Krasovskii functional T

V ðxðtÞÞ ¼ x ðtÞPðlÞxðtÞ þ

Z q X j¼1

Z

0

Z

t

þ hjm

T

t

thj ðtÞ

xT ðaÞQj ðlÞxðaÞ da !

_ da db , x_ ðaÞS j ðlÞxðaÞ

ð8Þ

tþb

P where ðPðlÞ; Qj ðlÞ; Sj ðlÞÞ ¼ N PTi ¼ Pi 40, QTji ¼ Qji 40, S Tji ¼ S ji 40, j ¼ 1, y, q, i¼1 li ðPi ; Qji ; Rji Þ, i ¼ 1, y, N. Then, by Lemma 1, the time derivative of V(x(t)) along any trajectory of the filtering error

ARTICLE IN PRESS W.-a. Zhang et al. / Signal Processing 87 (2007) 709–724

714

system (5) is given by q X

_ þ xT ðtÞ V_ ðxðtÞÞ ¼ 2xT ðtÞPðlÞxðtÞ

! Qj ðlÞ xðtÞ

j¼1



q X

ð1  h_j ðtÞÞxT ðt  hj ðtÞÞQj ðlÞxðt  hj ðtÞÞ

j¼1 q X

T þ x_ ðtÞ

! _  hjm S j ðlÞ xðtÞ

j¼1

q Z X j¼1

q X

_ þ xT ðtÞ p2xT ðtÞPðlÞxðtÞ

!

t

T _ da, x_ ðaÞSj ðlÞxðaÞ

thjm

Qj ðlÞ xðtÞ

j¼1



q X

ð1  hjd ÞxT ðt  hj ðtÞÞQj ðlÞxðt  hj ðtÞÞ

j¼1 T

_  þ x_ ðtÞXðlÞxðtÞ

q Z X

t

T _ da x_ ðaÞS j ðlÞxðaÞ

thj ðtÞ

j¼1

q X

_ þ xT ðtÞ p2x ðtÞPðlÞxðtÞ T

!

Qj ðlÞ xðtÞ  xTd ðtÞðI  Dd ÞQd ðlÞxd ðtÞ

j¼1

T _ þ Z¯ T ðtÞ þ x_ ðtÞXðlÞxðtÞ

"

~ 12 ðlÞ ~ 11 ðlÞ M M

þ

"

#T "

xðtÞ xðt  hj ðtÞÞ

j¼1

M 0j ðlÞ

Z¯ ðtÞ

~ 22 ðlÞ M

n q X

#

# hjm S 1 j ðlÞ

M 1j ðlÞ

h

M T0j ðlÞ

M T1j ðlÞ

i

"

xðtÞ xðt  hj ðtÞÞ

# ,

ð9Þ

T T where Z¯ T ðtÞ ¼ ½ x ðtÞ xd ðtÞ . Therefore, when assuming the zero disturbance input, from (9) and (5) we can obtain that

V_ ðxðtÞÞp¯ZT ðtÞOa ðlÞ¯ZðtÞ, where Oa ðlÞ ¼

"

# PðlÞ h 0 "



PðlÞ

i h A0 ðlÞ Ad ðlÞ þ A0 ðlÞ #T

0 h

 A0 ðlÞ

" þ

~ 11 ðlÞ M n

i

Ad ðlÞ þ

"

~ 12 ðlÞ M

#

~ 22 ðlÞ M # M 0d ðlÞ

M 1d ðlÞ

Ad ðlÞ

iT

h iT þ A0 ðlÞ Ad ðlÞ XðlÞ

¯ m Sd ðlÞ D

" #T 1 M 0d ðlÞ M 1d ðlÞ

.

By the Schur complement, matrix inequality (7a) guarantees Oa ðlÞo0, which implies V_ ðxðtÞÞo0 for all nonzero Z¯ ðtÞ. Then, we conclude from the Lyapunov–Krasovskii stability theorem that the filtering error system (5) is asymptotically stable. & Now, to establish the generalized H2 performance for the filtering error system, we consider the following performance index: Z t wT ðaÞwðaÞ da. (10) J ¼ V ðxðtÞÞ  0

ARTICLE IN PRESS W.-a. Zhang et al. / Signal Processing 87 (2007) 709–724

715

 Under zero initial condition, i.e. V ðxðtÞÞt¼0 ¼ 0, we have Z t Z t   J¼ V_ ðxðaÞÞ  wT ðaÞwðaÞ dap ZT ðaÞOa ðlÞZðaÞ da, 0 T

0 T

xTd ðaÞ

T

w ðaÞ . Then, (7a) guarantees Jo0 for all nonzero Z(t), which implies that where Z ðaÞ ¼ ½ x ðaÞ Z t V ðxðtÞÞo wT ðaÞwðaÞ da. 0

On the other hand, using Schur complement, it follows from (7b) that ¯ T0 ðlÞC ¯ 0 ðlÞxðtÞog2 xT ðtÞPðlÞxðtÞ eT ðtÞeðtÞ ¼ xT ðtÞC Z t og2 V ðxðtÞÞog2 wT ðaÞwðaÞ da 0 Z 1 2 og wT ðaÞwðaÞ da. 0 Taking the supremum over tX0 yields eðtÞ 1 og wðtÞ 2 for all nonzero wAL2[0, N), then the filtering error system (5) has a generalized H2 disturbance attenuation level g according to the Definition 1. This completes the proof. Remark 2. Instead of using model transformations and bounding techniques for cross terms, we use the integral inequality proposed in [23] to derive the delay-dependent generalized H2 performance conditions. Therefore, the conditions presented in Theorem 1 are expected to be less conservative than some existing ones. By introducing additional slack matrix variables the performance conditions in Theorem 1 will be modified to obtain some equivalent matrix inequality representations, which enable us to use the new linearization technique introduced in this paper to design the desired filters. This is the following theorem. Theorem 2. Consider error system (5), and let g40, hjm40, hjd40, j ¼ 1, y, q, be given scalars. If there exist symmetric positive-definite matrices P(l), Qj(l), Sj(l)AR2n  2n, and matrices E(l), F(l), M0j(l), M1j(l)AR2n  2n, j ¼ 1, y, q, such that the following matrix inequalities hold 2 3 ~ þ A~ T ðlÞE~ T ðlÞ þ MðlÞ ~ AðlÞ ~ ~ þ A~ T ðlÞF T ðlÞ  EðlÞ ~ EðlÞ PðlÞ M d ðlÞ 6 7 6 7o0, (11a) n F T ðlÞ  F ðlÞ þ XðlÞ 0 4 5 ¯ n n Dm Sd ðlÞ "

# T PðlÞ C¯ 0 ðlÞ 40, n g2 I

(11b)

~ where EðlÞ ¼ ½ E T ðlÞ 0 0 T , then the filtering error system (5) is asymptotically stable with a generalized H2 disturbance attenuation level g. Proof. We will prove the theorem by showing the equivalence between (7a) and (11a). If (7a) is true, (11a) is readily established by choosing ET(l) ¼ E(l) ¼ P(l), FT(l) ¼ F(l) ¼ X(l). On the other hand, by Schur complement, (11a) is equivalent to 2 3 ~ þ A~ T ðlÞE~ T ðlÞ þ MðlÞ ~ AðlÞ ~ ~ þ A~ T ðlÞF T ðlÞ  EðlÞ ~ EðlÞ PðlÞ 4 5 n F T ðlÞ  F ðlÞ þ XðlÞ " # " #T M d ðlÞ

1 M d ðlÞ Dm Sd ðlÞ o0. ð12Þ þ 0 0

ARTICLE IN PRESS W.-a. Zhang et al. / Signal Processing 87 (2007) 709–724

716 T

Since the matrix ½ I A~ ðlÞ  has full rank, (15) implies that 8" 3 T T < EðlÞ ~ T ðlÞF T ðlÞ  EðlÞ ~ ~ ~ ~ ~ ~ ~ PðlÞ þ A AðlÞ þ A ðlÞ E ðlÞ þ MðlÞ T 5 ½ I A~ ðlÞ  T : F ðlÞ  F ðlÞ þ XðlÞ n # # " #T 9" M d ðlÞ

1 M d ðlÞ = I o0, Dm Sd ðlÞ þ ~ ; AðlÞ 0 0 "

which is T ~ þ A~ T ðlÞP~ T ðlÞ þ MðlÞ ~ ~ AðlÞ ~ PðlÞ þ A~ ðlÞXðlÞAðlÞ

1 þ M d ðlÞ Dm S d ðlÞ M Td ðlÞo0.

ð13Þ

By Schur complement, (13) is equivalent to (7a), therefore, if (11a) holds, then, (7a) is true. This shows the equivalence between (7a) and (11a), the proof is thus completed. & 4. Generalized H2 filter design By introducing some parameter-dependent slack matrix variables, the coupling between the error system matrices and the Lyapunov matrices is eliminated. However, new coupling between the slack matrices and the filter matrices are introduced, since the filter matrices are to be determined, the above conditions are actually nonlinear matrix inequalities. Therefore, in order to solve the filter design problem some linearization procedures have to be adopted. In the following, we will introduce a new linearization technique to transform the performance conditions in Theorem 2 into LMI conditions. Then, we have the following theorem that provides sufficient LMI conditions for the existence of the generalized H2 filter of the form (4). Theorem 3. Consider system (1) with polytopic uncertainties (2), and let g40, 0oap1, hjm40, hjd40, j ¼ 1, y, q, be given scalars. Then, an admissible generalized H2 filter of the form (4) exists if there exist ¯ j11 ðlÞ; Q ¯ j22 ðlÞ; S¯ j11 ðlÞ; S¯ j22 ðlÞ 2
12

21

22

11

12

21

Y ðlÞ, C(l), AF ; BF ; C F , P12 ðlÞ; Qj12 ðlÞ; S j12 ðlÞ, M 0j ðlÞ; M 0j ðlÞ; M 0j ðlÞ; M 0j ðlÞ, M 1j ðlÞ; M 1j ðlÞ; M 1j ðlÞ; 22

M 1j ðlÞ 2
" Gb ðlÞ ¼

n

PðlÞ

PT ðlÞ

n

g2 I

n

n

0 Dm S d ðlÞ

7 7 7 7o0, 7 7 5

(14a)

(14b)

40,

F11 ðlÞ ¼ jT11 ðlÞ þ j11 ðlÞ þ M 11 ðlÞ þ j11 ðlÞ ¼ 4

M 1d ðlÞ 0

3

#

where

2

M 0d ðlÞ

q X

Qj ðlÞ,

j¼1 T

T

Y ðlÞA0 ðlÞ

Y ðlÞA0 ðlÞ

X T ðlÞA0 ðlÞ þ BF C 0 ðlÞ þ AF

¯ F C 0 ðlÞ X T ðlÞA0 ðlÞ þ B

3 5,

ARTICLE IN PRESS W.-a. Zhang et al. / Signal Processing 87 (2007) 709–724

82 3 2 3T 9 11 12 11 12 q > M 0j ðlÞ M 0j ðlÞ > = < M 0j ðlÞ M 0j ðlÞ X 6 7 6 7 , þ M 11 ðlÞ ¼ 4 21 5 4 5 22 21 22 > ; M 0j ðlÞ M 0j ðlÞ > j¼1 : M 0j ðlÞ M 0j ðlÞ 2 3 Qj11 ðlÞ Qj12 ðlÞ 5, Qj ðlÞ ¼ 4 n Qj22 ðlÞ 2 2 3 2 11 3T 11 12 12 M ðlÞ M 01 ðlÞ M 11 ðlÞ M 11 ðlÞ 6 4 01 5þ4 5 M 12 ðlÞ ¼ 4 21 22 21 22 M 01 ðlÞ M 01 ðlÞ M 11 ðlÞ M 11 ðlÞ 2 3 2 3T 3 11 12 11 12 M 0q ðlÞ M 0q ðlÞ M 1q ðlÞ M 1q ðlÞ 6 7 6 7 7     4 21 5 þ 4 21 5 5, 22 22 M 0q ðlÞ M 0q ðlÞ M 1q ðlÞ M 1q ðlÞ j12d ðlÞ ¼ ½ j121 ðlÞ    j12q ðlÞ , 2 3 T T Y ðlÞAj ðlÞ Y ðlÞAj ðlÞ 5, j12j ðlÞ ¼ 4 T X ðlÞAj ðlÞ þ BF C j ðlÞ X T ðlÞAj ðlÞ þ BF C j ðlÞ 2 3 T Y ðlÞBðlÞ 5; j22 ðlÞ ¼ M 22 ðlÞ  ðI  Dd ÞQd ðlÞ j13 ðlÞ ¼ 4 X T ðlÞBðlÞ þ BF DðlÞ n o T Qd ðlÞ ¼ diag Q1 ðlÞ; . . . ; Qq ðlÞ ; j44 ðlÞ ¼ aEðlÞ  aE ðlÞ þ XðlÞ, XðlÞ ¼

q X

 hjm S j ðlÞ; Sd ðlÞ ¼ diag S 1 ðlÞ; . . . ; Sq ðlÞ ,

j¼1

2 EðlÞ ¼ 4

T

Y ðlÞ

T

Y ðlÞ

X T ðlÞ þ CðlÞ X T ðlÞ

3

2

5; S j ðlÞ ¼ 4

S j11 ðlÞ S j12 ðlÞ n

S j22 ðlÞ

3 5,

8 2 3 2 11 3T 11 12 12 > < M 11 ðlÞ M 11 ðlÞ M 11 ðlÞ M 11 ðlÞ 54 5 ; ; M 22 ðlÞ ¼ diag 4 21 22 21 22 > : M 11 ðlÞ M 11 ðlÞ M 11 ðlÞ M 11 ðlÞ 2 3 2 3T 9 11 12 11 12 M 1q ðlÞ M 1q ðlÞ M 1q ðlÞ M 1q ðlÞ > = 6 7 6 7 , 4 21  5 4 5 22 21 22 ; M 1q ðlÞ M 1q ðlÞ M 1q ðlÞ M 1q ðlÞ > 2 33 22 3 11 12 11 12 M ðlÞ M ðlÞ M 01 ðlÞ M 01 ðlÞ 0q 0q 77 6 5  6 4 21 5 5, M 0d ðlÞ ¼ 4 4 21 22 22 M 01 ðlÞ M 01 ðlÞ M 0q ðlÞ M 0q ðlÞ 82 2 39 3 11 12 12 > > M ðlÞ M ðlÞ = < M 11 1q 1q 11 ðlÞ M 11 ðlÞ 7 5; . . . ; 6 , M 1d ðlÞ ¼ diag 4 21 4 5 22 21 22 > ; : M 11 ðlÞ M 11 ðlÞ M 1q ðlÞ M 1q ðlÞ > 2 3 " # T T P11 ðlÞ P12 ðlÞ L ðlÞ  C F 5. PðlÞ ¼ ; PT ðlÞ ¼ 4 n P22 ðlÞ LT ðlÞ

717

ARTICLE IN PRESS W.-a. Zhang et al. / Signal Processing 87 (2007) 709–724

718

Moreover, under the above conditions, an admissible generalized H2 filter of form (4) is given by #" " # " # AF BF AF BF C1 ðlÞ 0 ¼ . CF 0 CF 0 0 I

(15)

Proof. For filters design purpose, we specialize the matrices F(l) as aE(l), where a is a positive scalar satisfying 0oap1. In this case, it can be observed from (11a) that E(l) is invertible, in addition, if (14a) holds, T we can explore the fact that aEðlÞ þ aE ðlÞ4XðlÞ40, which means " T # T Y ðlÞ þ Y ðlÞ Y ðlÞ þ X ðlÞ þ CT ðlÞ 40. (16) n X ðlÞ þ X T ðlÞ Therefore, X(l) and Y ðlÞ are nonsingular. Denote Y ðlÞ ¼ Y as

" T

E ðlÞ ¼

# X ðlÞ H 0 ðlÞ ; H 1 ðlÞ UðlÞ

" E

T

ðlÞ ¼

1

ðlÞ and partition ET(l) and its inverse ET(l)

# Y ðlÞ F 0 ðlÞ . F 1 ðlÞ V ðlÞ

(17)

By (17), we have I ¼ Y T ðlÞX T ðlÞ þ F T1 ðlÞH T0 ðlÞ, 0 ¼ Y T ðlÞH T1 ðlÞ þ F T1 ðlÞU T ðlÞ. Denote

"

Y ðlÞ

I

#

"

ð18Þ

Y ðlÞ 0

#

; Y~ ðlÞ ¼ F 1 ðlÞ 0 0 I 9 9 8 8 > > > > = = < < J d ðlÞ ¼ diag JðlÞ; . . . ; JðlÞ ; Y~ d ðlÞ ¼ diag Y~ ðlÞ; . . . ; Y~ ðlÞ > > ; ; :|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}> :|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}> JðlÞ ¼

q

q

~ ¼ diagfJðlÞY~ ðlÞ; J d ðlÞY~ d ðlÞ; Ig, JðlÞ CðlÞ ¼ H T1 ðlÞF 1 ðlÞY ðlÞ. Since matrix ½ Y T ðlÞ Y T ðlÞ  has full rank, (16) implies that " T #" # T Y ðlÞ þ Y ðlÞ Y ðlÞ þ X ðlÞ þ CT ðlÞ Y ðlÞ T T ½ Y ðlÞ Y ðlÞ  o0, Y ðlÞ n X ðlÞ þ X T ðlÞ which is YT(l)(CT(l)C(l))Y(l)40; therefore, C(l) is nonsingular. Hence, one can always find square and nonsingular matrices H1(l) and F 1 ðlÞY ðlÞ satisfying CðlÞ ¼ H T1 ðlÞF 1 ðlÞY ðlÞ. Denote

T PðlÞ ¼ JðlÞY~ ðlÞ PðlÞJðlÞY~ ðlÞ,

T Qj ðlÞ ¼ JðlÞY~ ðlÞ Qj ðlÞJðlÞY~ ðlÞ,

T S j ðlÞ ¼ JðlÞY~ ðlÞ S j ðlÞJðlÞY~ ðlÞ,

T M 0j ðlÞ ¼ JðlÞY~ ðlÞ M 0j ðlÞJðlÞY~ ðlÞ,

T M 1j ðlÞ ¼ JðlÞY~ ðlÞ M 1j ðlÞJðlÞY~ ðlÞ, AF ¼ H T1 ðlÞAF F 1 ðlÞY ðlÞ; BF ¼ H T1 ðlÞBF , C F ¼ C F ðlÞF 1 Y ðlÞ; j ¼ 1; . . . ; q.

ð19Þ

ARTICLE IN PRESS W.-a. Zhang et al. / Signal Processing 87 (2007) 709–724

719

T ~ Multiplying from the left and the right of (11a) by diagfJ~ ðlÞ; ðJ d ðlÞY~ d ðlÞÞT ; Ig and diagfJðlÞ; J d ðlÞY~ d ðlÞ; Ig, respectively, and applying (5) and (18), we obtain (14a). Multiplying (7b) by diagfðJðlÞY~ ðlÞÞT ; Ig and diagfJðlÞY~ ðlÞ; Ig on the left and on the right, respectively, it follows that (7b) is equivalent to (11b). Now, denote the filter transfer function from y(t) to z^ðtÞ by T(s) ¼ CF(sIAF)1BF. Substituting the filter matrices with (19) and considering the relationship CðlÞ ¼ H T1 ðlÞF 1 ðlÞY ðlÞ yield TðsÞ ¼ C F C1 ðlÞ ðsI  AF C1 ðlÞÞ1 BF . Therefore, the state-space realization of the filter in (15) is established, and by Theorem 2, the filter guarantees that the filtering error system (5) is asymptotically stable with a generalized H2 disturbance attenuation level g. The proof is completed. &

Remark 3. Theorem 3 is a preliminary result for solving the generalized H2 filtering problem, which presents LMI conditions for the existence of the generalized H2 filter of form (4). Based on these obtained LMI conditions we are now ready to separately establish the sufficient conditions for the existence of the parameterdependent filter F1 in the case 1 and the parameter-independent filter F2 in the case 2. The main filtering results are shown as follows. Case 1 (Parameter-dependent filter): To further reduce the conservatism of the design results, we make full use of the parameter-dependent stability idea to design a parameter-dependent filter of the form (3a) in the case that the uncertain parameters can be measured online. In this case, the slack matrix variable E(l) is introduced to be a parameter-dependent matrix, which results in the parameter-dependent matrices X(l) and Y ðlÞ. Then, it can be observed that the LMI conditions in Theorem 3 are not convex in the parameter l due to the coupling between the system matrices and the filter matrices, and, the coupling between the system matrices and the introduced slack matrices X(l), Y ðlÞ. Besides, these conditions are infinite-dimensional ones. Therefore, the LMI conditions in Theorem 3 still cannot be directly implemented. In the following, we will use a technique recently proposed in [22] to convexify the matrix inequalities in Theorem 3, and thus, to obtain convex LMI conditions depend only on the vertices of the polytope. Before presenting the main results, we first make the following assumptions and denotations. 11 12 21 22 Assume that the matrix functions Y ðlÞ, X(l), C(l), AF ; BF ; C F , M lj ðlÞ; M lj ðlÞ; M lj ðlÞ; M lj ðlÞ, j ¼ 1, y, q, l ¼ 0, 1 in (14) have the following forms: Y ðlÞ ¼

N X

li Y i ; X ðlÞ ¼

i¼1

N X

li X i ; CðlÞ ¼

N X

i¼1

AF ¼ Af ðlÞ ¼

N X

li Afi ; BF ¼ Bf ðlÞ ¼

N X

i¼1

C F ¼ C f ðlÞ ¼

li C i ,

i¼1

li Bfi ,

i¼1

N X

li C fi ,

i¼1 11

M lj ðlÞ ¼

N X

11

12

li M lji ; M lj ðlÞ ¼

i¼1 21

M lj ðlÞ ¼

N X

N X

12

li M lji ,

i¼1 21

22

li M lji ; M lj ðlÞ ¼

i¼1

N X

22

li M lji .

i¼1 11

12

21

22

Replace the matrices M lj ðlÞ; M lj ðlÞ; M lj ðlÞ; M lj ðlÞ, j ¼ 1, y, q, l ¼ 0, 1 in M 11 ðlÞ; M 12 ðlÞ; M 22 ðlÞ; 11

12

21

22

M 0d ðlÞ; M 1d ðlÞ by M lji ðlÞ; M lji ðlÞ; M lji ðlÞ; M lji ðlÞ, respectively, and denote the obtained matrices by M 11i ; M 12i ; M 22i ; M 0di ; M 1di , respectively. Denote 2 3 T T Y i A0i Y i A0i 5, j11i ¼ 4 T X i A0i þ Bfi C 0i þ Afi X Ti A0i þ Bfi C 0i

ARTICLE IN PRESS W.-a. Zhang et al. / Signal Processing 87 (2007) 709–724

720

2 j12ji ¼ 4 2

T

3

T

Y i Aji

Y i Aji

X Ti Aji þ Bfi C ji

X Ti Aji þ Bfi C ji T

5,

T

Y i A0k þ Y k A0i

j11ik ¼ 4

X Ti A0k þ X Tk A0i þ Bfi C 0k þ Bfk C 0i þ Afi þ Afk 3 T T Y i A0k þ Y k A0i 5, X Ti A0k þ X Tk A0i þ Bfi C 0k þ Bfk C 0i 2 T T Y i Ajk þ Y k Aji 4 j12jik ¼ X Ti Ajk þ X Tk Aji þ Bfi C jk þ Bfk C ji 3 T T Y i Ajk þ Y k Aji 5, X Ti Ajk þ X Tk Aji þ Bfi C jk þ Bfk C ji 2 3 2 3 T T T Y i Bi Y i Bk þ Y k Bi 5; j13ik ¼ 4 5, j13i ¼ 4 T X i Bi þ Bfi Di X Ti Bk þ X Tk Bi þ Bfi Dk þ Bfk Di 2 3 2 3 T T T T T T Yi Yi Yi þ Yk Yi þ Yk 5; E ik ¼ 4 5, Ei ¼ 4 T X i þ Ci X Ti X Ti þ X Tk þ Ci þ Ck X Ti þ X Tk

"

P11i

P12i

#

2

Qj11i

; Qji ¼ 4 n P22i 2 3 T LTi  C fi T 5; j12di ¼ ½ j121i Pi ¼ 4 LTi Pi ¼

n

j12dik ¼ ½ j121ik



j12qik ; Xi ¼

Qj12i Qj22i 

q X

3

2

5; S ji ¼ 4

S j11i

Sj12i

n

Sj22i

3 5

j12qi ,

hjm S¯ ji ,

j¼1

n o  Qdi ¼ diag Q1i ; . . . ; Qqi ; Sdi ¼ diag S1i ; . . . ; S qi , Pik ¼ Pi þ Pk ; Qjik ¼ Qji þ Qjk ; Sjik ¼ S ji þ S jk , Qdik ¼ Qdi þ Qdk ; Sdik ¼ Sdi þ Sdk ; Xik ¼ Xi þ Xk , M 11ik ¼ M 11i þ M 11k ; M 12ik ¼ M 12i þ M 12k , M 22ik ¼ M 22i þ M 22k ; M 22ik ¼ M 22i þ M 22k , M 1dik ¼ M 1di þ M 1dk ; i; k 2 W ¼ ð1; . . . ; NÞ. Then, replace the matrices j11(l), j12d(l), j13(l), XðlÞ; PðlÞ; EðlÞ, Qj ðlÞ; Qd ðlÞ; S d ðlÞ, M 11 ðlÞ; M 12 ðlÞ; M 22 ðlÞ; M 0d ðlÞ; M 1d ðlÞ in Ga(l) by matrices j11i, j12di, j13i, Xi ; Pi ; E i , Qji ; Qdi ; S di , M 11i ; M 12i ; M 22i ; M 0di ; M 1di , respectively, and denotes the obtained matrix by Gai; replace the matrices j11(l), j12d(l), j13(l), XðlÞ; PðlÞ; EðlÞ, Qj ðlÞ; Qd ðlÞ; S d ðlÞ, M 11 ðlÞ; M 12 ðlÞ; M 22 ðlÞ; M 0d ðlÞ; M 1d ðlÞ, I in Ga(l) by matrices j11ik, j12dik, j13ik, Xik ; Pik ; E ik , Qjik ; Qdik ; S dik , M 11ik ; M 12ik ; M 22ik ; M 0dik ; M 1dik , 2I, respectively, and denotes the obtained matrix by Gaik.

ARTICLE IN PRESS W.-a. Zhang et al. / Signal Processing 87 (2007) 709–724

721

The sufficient LMI conditions for the existence of the parameter-dependent filter of the form (3a) are finally formulated in the following theorem. Theorem 4. Consider system (1) with polytopic uncertainties (2), and let g40, 0oap1, hjm40, hjd40, j ¼ 1, y, q, be given scalars. Then, an admissible generalized H2 filter of the form (3a) exists if there exist symmetric positive-definite matrices P11i ; P22i ; Qj11i ; Qj22i ; Sj11i ; S j22i 2
12

21

22

11

12

21

22

Afi ; Bfi ; C fi , P12i ; Qj12i ; S 12i , M 0ji ; M 0ji ; M 0ji ; M 0ji , M 1ji ; M 1ji ; M 1ji ; M 1ji 2
i ¼ 1; . . . ; N,

(20a)

Gaik o0; i ¼ 1; . . . ; N  1; k ¼ i þ 1; . . . ; N, " # Pi PTi Gbi ¼ 40; i ¼ 1; . . . ; N, n g2 I

(20b) (20c)

Moreover, under the above conditions, the matrix functions for an admissible generalized H2 filter of the form (3a) are given by 3 " # " PN #2  P 1 PN N Af ðlÞ Bf ðlÞ i¼1 li Afi i¼1 li Bfi l C 0 i¼1 i i 4 5. (21) ¼ PN C f ðlÞ 0 0 i¼1 li C fi 0 I Proof. Note that Ga(l) in (14a) and Gb(l) in (14b) can be, respectively, written as Ga ðlÞ ¼

N X

l2i Gai þ

i¼1

Gb ðlÞ ¼

N X

N X N X

li lk Gaik ,

(22a)

i¼1 k¼iþ1

li Gbi .

(22b)

i¼1

P Imposing (20a)–(20c) and taking into account that liX0, N i¼1 li ¼ 1 one has that (14a) and (14b) hold for all admissible l. Therefore, by Theorem 3, the parameter-dependent filter with a state-space realization (Af(l), Bf(l), Cf(l)) defined in (21) assures that the filtering error system (5) has a generalized H2 disturbance attenuation level g. This completes the proof. & Case 2: Parameter-independent filter In the case that the uncertain parameters cannot be measured online, the proposed parameter-dependent filter of the form (3a) cannot be realized in practice, in such case, an alternative is to design a parameterindependent filter of the form (3b). In this case, the matrices Y ðlÞ, X(l), C(l) in (14a) and (14b) are assumed to be fixed, and take the forms

 Y ðlÞ ¼ Y , X(l) ¼ X, C(l) ¼ C, respectively. Besides, the filter matrices A ; B ; C in (14a) and (14b) take F F F  the forms Af ; Bf ; C f . Then, the LMI conditions in Theorem 3 are convex in the parameter l, therefore, one can easily use the convex combination of the vertex matrices to cast the infinite-dimensional LMI conditions into finite-dimensional conditions. Now, replace the matrices Y i , Xi, Ci, Afi ; Bfi ; C fi , i ¼ 1, y, N in Gai by the matrices Y , X, C, Af ; Bf ; C f , respectively, and define the obtained matrix by G~ ai ; Replace the matrices C fi , i ¼ 1, y, N in Gbi by the matrix ~ bi . The sufficient LMI conditions for the existence of the parameterC f , and define the obtained matrix by G independent filter of form (3b) are finally formulated in the following theorem. Theorem 5. Consider system (1) with polytopic uncertainties (2), and let g40, 0oap1, hjm40, hjd40, j ¼ 1, y, q, be given scalars. Then, an admissible robust generalized H2 filter of the form (3b) exists and

ARTICLE IN PRESS W.-a. Zhang et al. / Signal Processing 87 (2007) 709–724

722

guarantees that the filtering error system (5) is asymptotically stable with a generalized H2 disturbance attenuation level g if there exist symmetric positive-definite matrices P11i ; P22i ; Qj11i ; Qj22i ; Sj11i ; S j22i 2
12

21

22

11

12

21

22

and any matrices X, Y , C, Af ; Bf ; C f , P12i ; Qj12i ; Sj12i , M 0ji ; M 0ji ; M 0ji ; M 0ji , M 1ji ; M 1ji ; M 1ji ; M 1ji 2
(23a)

~ bi 40. G

(23b)

Proof. By the virtue of the properties of convex combination, and using the characterization (14) in Theorem 3, it can be concluded that the theorem is true under conditions (23a) and (23b). The proof is completed. & Remark 4. With the results of Theorem 4 and 5, the parameter-dependent and parameter-independent robust generalized H2 filter can be obtained by solving the feasibility problems subject to LMIs in Theorem 4 and 5, respectively. Further more, it is noted that the LMI conditions in Theorem 4 and 5 are convex in the scalar g2, therefore, convex optimization problems can be further formulated to obtain the desired filters of either parameter-dependent one or parameter-independent one with minimized generalized H2 performance g. For example, the global minimum to the optimization problem (24) can be found by applying a simple onedimensional search over the variable a, and thus the parameter-dependent filter of form (3a) with minimized generalized H2 performance g can be obtained. Minimize r subject to ð20aÞ2ð20cÞ; with r ¼ g2 .

(24)

5. Illustrative examples Consider system (1) with the following matrices borrowed from [6]: " # " # 0:1 0 0 3þr ~ ; A1 ¼ , S1 : A0 ¼ 0:2 0:2 þ s 4 5 " # " # 0 0:1 0:4545   ; B¼ ; C 0 ¼ 0 100 , A2 ¼ 0:2 0:3 þ s 0:9090     C 1 ¼ C 2 ¼ 0 0 ; L ¼ 0 100 ; D ¼ 1; h1d ¼ 0:3; h2d ¼ 0:5,

ð25Þ

  where r and s are uncertain real parameters satisfying rpr, ¯ jsjps, ¯ s¯ are known scalars. In addition, ¯ and r; we assume h1m ¼ 0.2 and h2m ¼ 0.3. First, we consider the case that the uncertain parameters r and s cannot be measured online, then, assuming r¯ ¼ 0:3, s¯ ¼ 0:1 and in order to illustrate the advantage of the delaydependent approach proposed in this paper, we will compare our results with those in [6], where l2lN filters (in the some meaning as the generalized H2 filters defined in this paper) are designed using the parameterindependent Lyapunov functional method. The parameter-independent and delay-dependent approach proposed in [6] yields the minimum generalized H2 disturbance attenuation level given by g ¼ 1.3180. For comparison, we first adopt the parameter-independent Lyapunov functional method (simply set P11i ; P22i ; Qj11i ; Qj22i ; S j11i ; Sj22i in (23a) and (23b) to be P11 ; P22 ; Qj11 ; Qj22 ; S j11 ; S j22 , respectively, for all i ¼ 1, y, N) to design the parameter-independent filters, and by the delay-dependent approach used in this paper, the minimum generalized H2 disturbance attenuation level obtained is g ¼ 1.1173, which occurs at a ¼ 0.6, showing the less conservatism of the adopted delay-dependent approach. Furthermore, choosing a ¼ 0.6 and by adopting the parameter-dependent Lyapunov functional approach a smaller generalized H2 disturbance attenuation level g ¼ 1.0036 can be obtained with the filter matrices shown as follows:  Af ¼

2:1230

1:1368

13:7686 94:9965



 ; Bf 1 ¼

4:3759 158:0951

 ;

 C f 1 ¼ 0:0628

 0:5806 .

ARTICLE IN PRESS W.-a. Zhang et al. / Signal Processing 87 (2007) 709–724

723

Now, we consider the case that the uncertain parameters r and s vary slowly and can be measured online, in this case, a less conservative result can be achieved by designing a parameter-dependent filter, By Theorem 4, the minimum generalized H2 performance of admissible parameter-dependent generalized H2 filters can be achieved to be g ¼ 0.9965, which occurs at a ¼ 0.6. The associated matrices needed for the calculation of the parameter-dependent filters are given by 

473:3535

242:8901





586:7300

304:2246



; C2 ¼ 17:0084 167:8750 35:9107 200:2949     448:8743 232:0820 553:0430 289:9854 C3 ¼ ; C4 ¼ ; 32:7627 154:3715 38:9138 192:7657     0:1000 0:0730 0:1218 0:1133 ; Af 2 ¼ 104  ; Af 1 ¼ 104  0:0592 1:7103 0:0763 2:1048     0:0849 0:0474 0:1169 0:1039 4 4 Af 3 ¼ 10  ; Af 4 ¼ 10  ; 0:0495 1:6595 0:0738 2:0505       11:9015 11:9015 7:4930 ; Bf 2 ¼ ; Bf 3 ¼ ; Bf 1 ¼ 200:1184 200:1184 157:7932     13:7973 ; C f 1 ¼ 0:0492 99:6570 ; Bf 4 ¼ 194:6629     C f 2 ¼ 0:0182 99:7121 ; C f 3 ¼ 0:1439 99:7026 ;   C f 4 ¼ 0:0128 99:7221 :

C1 ¼

Then, the filter matrix functions with respect to the values of r and s can be given by

Af ðr; sÞ ¼

4 X i¼1

Bf ðr; sÞ ¼

4 X i¼1

! li Afi

4 X

!1 li C i

,

i¼1

li Bfi ; C f ðr; sÞ ¼

4 X i¼1

! li C fi

4 X

!1 li C i

,

i¼1

where     1 r  s 1 r  s 1þ 1þ 1 þ ; l2 ¼ 1 , 4 r¯ 4 r¯ s¯ s¯      1 r s 1 r s 1 1 1 þ ; l4 ¼ 1 . l3 ¼ 4 r¯ 4 r¯ s¯ s¯ l1 ¼

6. Conclusions This paper has presented a delay- and parameter-dependent approach to design the generalized H2 filters for a class of continuous-time systems with multiple time-varying state delays and polytopic bounded parameters. Sufficient conditions for the existence of the parameter-dependent filters and the parameter-independent filters are presented in terms of LMIs, and design procedures for the desired filters were also proposed. For the delay-dependent aspect, a newly developed integral-inequality approach was adopted to derive the delay-dependent performance conditions without involving some model transformations and the bounding for cross terms, therefore, the criteria obtained are of less conservatism than some existing ones. For the parameter-dependent aspect, the design method proposed possesses the advantages by making full use of the parameter-dependent stability idea: parameter-dependent Lyapunov

ARTICLE IN PRESS 724

W.-a. Zhang et al. / Signal Processing 87 (2007) 709–724

functionals have been used to design the parameter-dependent filters in the case that the uncertain parameter can be measured online. Acknowledgments The first and the second author’s research work were supported by the National Natural Science Funds for Distinguished Young Scholar under Grant 60525304. The third author’s work was supported by Central Queensland University for the 2005 small grand Scheme ‘‘T-S Model-based Fuzzy Control for Nonlinear Networked Control Systems’’. References [1] B.D.O. Anderson, J.B. Moore, Optimal Filtering, Prentice-Hall, Englewood Cliffs, NJ, 1979. [2] J.C. Geromel, M.C. De Oliveira, H2 and HN robust filtering for convex bounded uncertain systems, IEEE Trans. Automat. Control. 46 (1) (January 2001) 100–107. [3] P.P. Khargonekar, M.A. Rotea, E. Baeyens, Mixed H2/HN filtering, Int. J. Robust Nonlinear Control 6 (6) (1996) 313–330. [4] L. Xie, C.E. De Souza, M. Fu, HN estimation for discrete-time linear uncertain systems, Int. J. Robust Nonlinear Control 1 (1991) 111–123. [5] A. Pila, U. Shaked, C. de Souza, HN filtering for continuous-time linear systems with delay, IEEE Trans. Automat. Control 44 (7) (October 1999) 1412–1417. [6] H.J. Gao, C.H. Wang, Robust L2LN filtering for uncertain systems with multiple time-varying delays, IEEE Trans. Circuits Syst. I 50 (4) (April 2003) 594–599. [7] Z.S. Duan, J.X. Zhang, C.S. Zhang, E. Mosca, A simple design method of reduced-order filters and its applications to multirate filter bank design, Signal Processing 86 (5) (May 2006) 1061–1075. [8] R.M. Palhares, P.L.D. Peres, LMI approach to the mixed H2/HN filtering design for discrete-time systems, IEEE Trans. Aerosp. Electron. Syst. 37 (1) (January 2001) 292–296. [9] C.E. De Souza, R.M. Palhares, P.L.D. Peres, Robust HN filtering design for uncertain linear systems with multiple time-varying state delays, IEEE Trans. Signal Processing 49 (3) (March 2001) 569–576. [10] R.M. Palhares, C.E. De Souza, P.L.D. Peres, Robust HN filtering for uncertain discrete-time state-delayed systems, IEEE Trans. Signal Process. 49 (8) (August 2001) 1696–1703. [11] S.Y. Xu, J. Lam, T.W. Chen, Y. Zou, A delay-dependent approach to robust HN filtering for uncertain distributed delay systems, IEEE Trans. Signal Process. 53 (10) (October 2005) 3764–3772. [12] H.J. Gao, C.H. Wang, A delay-dependent approach to robust HN filtering for uncertain discrete-time state-delayed systems, IEEE Trans. Signal Process. 52 (6) (June 2004) 1631–1640. [13] H.J. Gao, J. Lam, L.H. Xie, C.H. Wang, New approach to mixed H2/HN filtering for polytopic discrete-time systems, IEEE Trans. Signal Process. 53 (8) (August 2005) 3183–3192. [14] K.M. Grigoriadis, J.T. Watson, Reduced order HN and l2lN filtering via linear matrix inequalities, IEEE Trans. Aerosp. Electron. Syst. 33 (4) (1997) 1326–1338. [15] R.M. Palhares, P.L.D. Peres, Robust filtering with guaranteed energy-to-peak performance—an LMI approach, Automatica 36 (6) (June 2000) 851–858. [16] H. Liu, F. Sun, Z. Sun, Reduced-order filtering with energy-to-peak performance for discrete-time Markovian jumping systems, IMAJ. Math. Control Inform. 21 (20) (2004) 143–158. [17] H. Gao, J. Lam, C. Wang, Induced l2 and generalized H2 filtering for systems with repeated scalar nonlinearities, IEEE Trans. Signal Process. 53 (11) (November 2005) 4215–4226. [18] H. Gao, C. Wang, X.Z. Gao, Robust energy-to-peak filtering for uncertain discrete time state-delayed systems: delay independent and dependent approaches, Intell. Automat. Soft Comput. 11 (4) (2005) 245–257. [19] H. Gao, C. Wang, Delay-dependent robust HN and l2lN filtering for a class of uncertain nonlinear time-delay systems, IEEE Trans. Automat. Control 48 (9) (September 2003) 1661–1666. [20] H. Gao, J. Lam, C. Wang, Robust energy-to-peak filter design for stochastic time-delay systems, Systems Control Lett. 55 (2) (February 2006) 101–111. [21] W.H. Chen, Z.H. Guan, X.M. Lu, Delay-dependent output feedback guaranteed cost control for uncertain time-delay systems, Automatica 40 (7) (July 2004) 1263–1268. [22] P.J. de. Oliveira, R.C.L.F. Oliveira, V.J.S. Leite, V.F. Montagner, P.L.D. Peres, HN guaranteed cost computation by means of parameter-dependent Lyapunov functions, Automatica 40 (6) (June 2004) 1053–1061. [23] X.M. Zhang, M. Wu, J.H. She, Y. H, Delay-dependent stabilization of linear systems with time-varying state and input delays, Automatica 41 (8) (August 2005) 1405–1412.