Filtering for uncertain 2-D discrete systems with state delays

ARTICLE IN PRESS

Signal Processing 87 (2007) 2213–2230 www.elsevier.com/locate/sigpro

Filtering for uncertain 2-D discrete systems with state delays Ligang Wua, Zidong Wangb,, Huijun Gaoa, Changhong Wanga a

Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin 150001, PR China b Department of Information Systems and Computing, Brunel University, Uxbridge, Middlesex UB8 3PH, UK Received 20 July 2006; received in revised form 2 March 2007; accepted 6 March 2007 Available online 13 March 2007

Abstract This paper is concerned with the problem of robust H1 filtering for two-dimensional (2-D) discrete systems with timedelays in states. The 2-D systems under consideration are described in terms of the well-known Fornasini–Marchesini local state-space (FMLSS) models with time-delays. Our attention is focused on the design of a full-order filter such that the filtering error system is guaranteed to be asymptotically stable with a prescribed H1 disturbance attenuation performance. Sufficient conditions for the existence of desired filters are established by using a linear matrix inequality (LMI) approach, and the corresponding filter design problem is then cast into a convex optimization problem that can be efficiently solved by resorting to some standard numerical software. Furthermore, the obtained results are extended to more general cases where the system matrices contain either polytopic or norm-bounded parameter uncertainties. A simulation example is provided to illustrate the effectiveness of the proposed design method. r 2007 Elsevier B.V. All rights reserved. Keywords: Filtering; H1 norm; Linear matrix inequality (LMI); Time-delay; Two-dimensional (2-D) systems

1. Introduction In the past few decades, two-dimensional (2-D) discrete-time systems have received considerable research attention since 2-D systems have extensive applications in image processing, seismographic data processing, thermal processes, water stream heating, etc., see [1]. So far, many important results have been reported in the literature. For example, the stability analysis problem for 2-D systems has been investigated in [2,3], the controller and filter design problems have been considered in [4–7], and the model approximation problem for 2-D digital filters has been studied in [8]. On the other hand, in the signal processing and control communities, the H1 filtering problem has recently drawn a great deal of research interests. The aim of H1 filtering problem is basically to find a full-order (or reduced-order) filter such that the associated filtering error system satisfies a prescribed H1 norm-bound constraint. Much work has been done for H1 filtering problem, see e.g., [9–15] and references therein. It has also been well recognized that time-delay exists commonly in dynamic systems and is frequently a source of Corresponding author.

E-mail addresses: [email protected] (L. Wu), [email protected] (Z. Wang), [email protected] (H. Gao), [email protected] (C. Wang). 0165-1684/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2007.03.002

ARTICLE IN PRESS L. Wu et al. / Signal Processing 87 (2007) 2213–2230

2214

instability and poor performance. Therefore, the last 10 years have witnessed significant advances in dealing with analysis and design problems for time-delay systems. In particular, the H1 filtering problem has been thoroughly studied for various time-delay systems, see e.g., [16–22] for some recent papers. However, the aforementioned results are only concerned with one-dimensional (1-D) time-delay systems. When it comes to the 2-D systems, most published results have been restricted to the 2-D discrete-time-delay free systems, see [4,23,24]. In the simultaneous presence of time-delays and parameter uncertainties, unfortunately, the robust H1 filtering problem for 2-D discrete-time systems has not gained enough research attention mainly due to the complexity in the stability analysis, despite its potential in engineering applications. This situation motivates our current investigation. It is, therefore, our intention in this paper to investigate the problem of H1 filtering for 2-D systems with time-delay in states. The mathematical model of the 2-D systems is established in terms of the well-known Fornasini–Marchesini local state-space (FMLSS) model incorporating time-delays. We aim at designing a fullorder filter that guarantees the asymptotic stability of the filtering error system while keeping the prescribed H1 disturbance attenuation performance. By using a linear matrix inequality (LMI) approach, we derive the existence conditions of the desired filters, and convert the corresponding filter design problem into a convex optimization one that can then be efficiently handled with help from available numerical software [25]. Furthermore, the obtained results are extended to some more general cases where the system matrices also contain uncertain parameters. Most frequently used descriptions for the parameter uncertainties, including polytopic and norm-bounded characterizations, are taken into consideration within the unified LMI framework. A numerical example is provided to demonstrate the effectiveness of the proposed filter design procedures. The remainder of this paper is organized as follows. The problems of H1 filtering for 2-D discrete statedelayed systems is formulated in Section 2. Section 3 presents our main results of filtering for 2-D discrete-time systems with state delays, and the results obtained are further extended in Section 4 to more general cases where the parameter uncertainties are considered. Section 5 provides an illustrative example and we conclude this paper in Section 6. Notations. The notations used throughout the paper are fairly standard. The superscript ‘‘T’’ stands for matrix transposition; Rn denotes the n-dimensional Euclidean space; Rmn is the set of all real matrices of dimension m  n and the notation P40 means that P is real symmetric and positive definite; I and 0 represent identity matrix and zero matrix; j  j refers to the Euclidean vector norm; and lmin ðÞ; lmax ðÞ denote the minimum and the maximum eigenvalues of a real symmetric matrix, respectively. In symmetric block matrices or long matrix expressions, we use an asterisk ðÞ to represent a term that is induced by symmetry, and diagf. . .g stands for a block-diagonal matrix. l n2 f½0; 1Þ; ½0; 1Þg denotes the space of square summable sequences on f½0; 1Þ; ½0; 1Þg with values on Rn . Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. 2. Problem formulation Consider the following state-delayed 2-D system SFM described by the FMLSS model [26] with delays in the states: SFM : xði þ 1; j þ 1Þ ¼ A1 xði; j þ 1Þ þ A2 xði þ 1; jÞ þ Ad1 xði  d 1 ; j þ 1Þ þ Ad2 xði þ 1; j  d 2 Þ þ B1 oði; j þ 1Þ þ B2 oði þ 1; jÞ, yði; jÞ ¼ Cxði; jÞ þ Doði; jÞ, zði; jÞ ¼ Exði; jÞ,

(1) l l2 f½0; 1Þ; ½0; 1Þg

where xði; jÞ 2 Rn is the state; oði; jÞ 2 is the disturbance input; yði; jÞ 2 Rm is the p measured output; zði; jÞ 2 R is the signal to be estimated with i; j 2 Zþ ; and d 1 and d 2 are constant positive integers representing delays along vertical and horizontal directions, respectively. A1 , A2 , Ad1 , Ad2 , B1 , B2 , C, D and E are constant matrices with compatible dimensions. The boundary conditions

ARTICLE IN PRESS L. Wu et al. / Signal Processing 87 (2007) 2213–2230

2215

are given by fxðf; jÞ ¼ 0g;

8jX0;

f ¼ d 1 ; d 1 þ 1; . . . ; 0;

fxði; jÞ ¼ 0g;

8iX0;

j ¼ d 2 ; d 2 þ 1; . . . ; 0.

ð2Þ

Throughout this paper, the following assumptions are made. Assumption 1. System SFM in (1) is asymptotically stable. Assumption 2. The boundary condition is assumed to satisfy lim

N!1

N X

ðjx0;k j2 þ jxk;0 j2 Þo1.

(3)

k¼0

The aim of the robust H1 filtering problem addressed in this paper is to estimate the signal zði; jÞ by a linear, full-order, dynamic filter of the structure described by S^ FM : ^ FM : xf ði þ 1; j þ 1Þ ¼ A1f xf ði; j þ 1Þ þ A2f xf ði þ 1; jÞ þ B1f yði; j þ 1Þ þ B2f yði þ 1; jÞ, S zf ði; jÞ ¼ C f xf ði; jÞ, xf ði; jÞ ¼ 0

for i ¼ 0 or j ¼ 0,

ð4Þ

where xf ði; jÞ 2 Rn is the filter state vector, and A1f , A2f , B1f , B2f and C f are appropriately dimensioned constant matrices to be determined. ^ FM , we obtain the following filtering error Now, augmenting the model of SFM to include the states of filter S ~ FM : system S ~ FM : xði þ 1; j þ 1Þ ¼ A~ 1 xði; j þ 1Þ þ A~ 2 xði þ 1; jÞ þ A~ d1 xði  d 1 ; j þ 1Þ þ A~ d2 xði þ 1; j  d 2 Þ S þ B~ 1 oði; j þ 1Þ þ B~ 2 oði þ 1; jÞ, ~ eði; jÞ ¼ Cxði; jÞ, T T where xði; jÞ9½ x ði; jÞ xf ði; jÞ T , eði; jÞ9zði; jÞ  zf ði; jÞ and " # " # " # 0 0 A1 A2 Ad1 0 ~ ~ ~ A1 9 ; A2 9 ; Ad1 9 , B1f C A1f B2f C A2f 0 0 " # " # " # h i B1 B2 Ad2 0 ~ ~ ~ ~ E C f . ; B1 9 ; B2 9 ; C9 Ad2 9 B1f D B2f D 0 0

(5)

ð6Þ

Before problem formulating, we give the following definitions. Definition 1. Consider the filtering error system S~ FM in (5). Given a scalar g40 and constant weighting ~ ~ ~ FM is said to have an H1 performance level g if it is matrices P40, Q40, Q~ 1 40 and Q~ 2 40, the system S asymptotically stable and satisfies k¯eði; jÞk22 og2 P1 T T ~ ~ j¼0 x01 ð0; jÞPx01 ð0; jÞ þ i¼0 x10 ði; 0ÞQx10 ði; 0Þ P1 P1 xT ð0; jÞQ~ 1 xk1 ð0; jÞ þ xT ði; 0ÞQ~ 2 x1k ði; 0Þ;

P1

½koði; ¯ jÞk22 þ P1 P1 þ j¼0 k¼d

1

k1

i¼0

k¼d 2

(7)

1k

where xab ði; jÞ9xði þ a; j þ bÞ. In the case of the zero boundary conditions as in (2), the above H1 performance measure (7) reduces to k¯eði; jÞk2 ogkoði; ¯ jÞk2

ðg40Þ,

(8)

ARTICLE IN PRESS L. Wu et al. / Signal Processing 87 (2007) 2213–2230

2216

where e¯ ði; jÞ9½ eT ði; j þ 1Þ eT ði þ 1; jÞ T , oði; ¯ jÞ9½ oT ði; j þ 1Þ k¯eði; jÞk22 9

oT ði þ 1; jÞ T and k  k2 is l 2 norm defined by

1 X 1 X ½eT ði; j þ 1Þeði; j þ 1Þ þ eT ði þ 1; jÞeði þ 1; jÞ, i¼0

koði; ¯ jÞk22 9

j¼0

1 X 1 X i¼0

½oT ði; j þ 1Þoði; j þ 1Þ þ oT ði þ 1; jÞoði þ 1; jÞ.

j¼0

^ FM in (4) is said to be an H1 filter if the filtering error system S~ FM in (5) is Definition 2. The filter S asymptotically stable and satisfies H1 performance in (8) with zero boundary conditions as in (2). The objective of this paper is to find the matrices A1f , A2f , B1f , B2f and C f of the full-order H1 filter S^ FM in (4) for the 2-D state-delayed system SFM in (1), such that for any nonzero oði; jÞ 2 l 2 f½0; Þ; ½0; 1Þg the filtering error system S~ FM in (5) is asymptotically stable and satisfies (8). 3. Main results 3.1. Filter analysis In this subsection, we shall analyze the stability and H1 performance for the filtering error system S~ FM . The following lemma is essential in establishing our stability results. ~ FM in (5) with oi;j  0 is Lemma 1 (Theorem 3 of Paszke et al. [6]). The 2-D state-delayed system S asymptotically stable if there exist matrices P40, Q40, Q1 40 and Q2 40 such that the following LMI holds: 2 T3 2 3 A~ 0 0 0 P  Q  Q1 6 1T 7 6 ~ 7 h 7 i 6  Q  Q2 0 0 7 6 A2 7 6 6 T 7P A~ 1 A~ 2 A~ d1 A~ d2  6 7o0. 6 ~ 7 6   Q1 0 7 6 Ad1 7 4 5 4 T 5    Q 2 A~ d2

Next, the following theorem provides a sufficient condition under which the filtering error system S~ FM in (5) is asymptotically stable and the performance constraint (8) is satisfied. ~ FM in (5) is asymptotically stable with an H1 disturbance attenuation Theorem 1. The filtering error system S level bound g if there exist matrices P40, Q40, Q1 40 and Q2 40 such that the following LMI holds: 3 2 PB~ 1 PA~ d1 PA~ 2 PB~ 2 PA~ d2 P 0 0 PA~ 1 7 6 6  I 0 C~ 0 0 0 0 0 7 7 6 6   I 0 0 0 C~ 0 0 7 7 6 7 6 7 6    Q þ Q  P 0 0 0 0 0 1 7 6 6  2    g I 0 0 0 0 7 (9) 7o0. 6 7 6 7 6      Q1 0 0 0 7 6 7 6       Q  Q 0 0 7 6 2 7 6 6        g2 I 0 7 5 4         Q2 Proof. See the Appendix.

&

For the delay free case, i.e., A~ d1 ¼ 0 and A~ d2 ¼ 0, according to the procedure of the proof of Theorem 1, it is clear that setting Q1 ¼ 0 and Q2 ¼ 0 in Theorem 1 would yield the following corollary.

ARTICLE IN PRESS L. Wu et al. / Signal Processing 87 (2007) 2213–2230

2217

~ FM in (5) is asymptotically stable with an H1 disturbance attenuation Corollary 1. The filtering error system S level bound g if there exist matrices P40 and Q40 such that the following LMI holds: 2 ¯ 2 PB ¯1 ¯ 1 PA ¯2 3 PB P 0 0 PA 6  I 0 ¯ C 0 0 0 7 7 6 7 6 ¯ 7 6   I 0 0 C 0 7 6 6    QP 0 0 0 7 7o0. 6 7 6 2 6     g I 0 0 7 7 6 6      Q 0 7 5 4 











g2 I

Remark 1. It should be pointed out that the result in Corollary 1 is actually the main result in [8]. In other words, Theorem 1 in this paper is an extension of the main result of [8]. 3.2. Filter synthesis We are now ready to deal with the H1 filter design problem in the following theorem. Theorem 2. Consider the 2-D state-delayed system SFM in (1) and let g40 be a prescribed constant scalar. Then ^ FM in the form of (4) such that the filtering error system S ~ FM is asymptotically there exists a full-order filter S stable and (8) is satisfied if there exist matrices U40, V40, Q1 40, Q3 40, Q11 40, Q13 40, Q21 40, Q23 40, Q2 , Q12 , Q22 , A1f , A2f , B1f , B2f and Cf such that the following LMIs hold: 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

U  

0

0

UA1 þ B1f C

A1f

UB1 þ B1f D

UAd1

0

UA2 þ B2f C

A2f

UB2 þ B2f D

UAd2

V 0  I

0 0

VA1 þ B1f C E

A1f Cf

VB1 þ B1f D 0

VAd1 0

0 0

VA2 þ B2f C 0

A2f 0

VB2 þ B2f D 0

VAd2 0

0 0

0 0

0 0

E 0

Cf 0

0 0

0 0

0 g2 I

0 0

0 0

0 0

0 0

0 0

0 0

V

 

 

 

I 

 

 

 

 

0 0 Q1 þ Q11  U Q2 þ Q12  V  

Q3 þ Q13  V 















Q11

Q12

0

0

0

0

 

 

 

 

 

 

 

 

Q13 

0 Q21  Q1

0 Q22  Q2

0 0

0 0

 

 

 

 

 

 

 

 

 

 

Q23  Q3 

0 g2 I

0 0

 

 

 

 

 

 

 

 

 

 

 

 

Q21 

0

3

7 7 7 7 7 7 0 7 7 0 7 7 7 0 7 7 0 7 7 7o0, 0 7 7 0 7 7 7 0 7 7 0 7 7 7 0 7 7 Q22 7 5 Q23 0 0

(10) "

"

"

Q1

Q2



Q3

# 40,

Q11 

# Q12 40, Q13

Q21

Q22



Q23

(11)

(12)

# 40.

Moreover, the parameters of a desired H1 filter of the form (4) can be computed from 32 2 3 2 1 3 A1f B1f A1f B1f V 0 0 76 6A 7 6 7 V1 0 54 A2f B2f 5. 4 2f B2f 5 ¼ 4  Cf 0 Cf 0   I

(13)

(14)

ARTICLE IN PRESS L. Wu et al. / Signal Processing 87 (2007) 2213–2230

2218

Proof. According to Theorem 1, if (9) holds P is nonsingular since P40. Now, partition P as " # P1 P 2 P9 T . P2 P3

(15)

Without loss of generality, we assume that P2 is nonsingular (if not, P2 may be perturbed by a matrix DP2 with sufficiently small norm such that P2 þ DP2 is nonsingular and satisfies (9)). Define the following matrices: " # I 0 T G9 U9P1 40; V9P2 P1 3 P2 40, T ; 0 P1 3 P2 " # " # Q1 Q2 Q11 Q12 T T Q9G QG9 40; Q1 9G Q1 G9 40,  Q3  Q13 " # Q21 Q22 T 40 ð16Þ Q2 9G Q2 G9  Q23 and 2

A1f 6A 4 2f Cf

3 2 B1f P2 6 7 B2f 594  0 

0 P2 

32 A1f 0 76 A 0 54 2f Cf I

3 B1f " T P1 3 P2 B2f 7 5 0 0

# 0 . I

Performing congruence transformations to (9) by matrix diagfG; I; I; G; I; G; G; I; Gg, we have 3 2 GT PG 0 0 GT PA~ 1 G GT PB~ 1 GT PA~ d1 G GT PA~ 2 G GT PB~ 2 GT PA~ d2 G 7 6 ~  I 0 CG 0 0 0 0 0 7 6 7 6 ~ 7 6   I 0 0 0 CG 0 0 7 6 7 6 T    Q þ Q1  G PG 0 0 0 0 0 7 6 7 6 2 7o0 6     g I 0 0 0 0 7 6 7 6      Q1 0 0 0 7 6 7 6 7 6       Q2  Q 0 0 7 6 7 6        g2 I 0 5 4         Q2

(17)

(18)

in which " G P A~ j G ¼ T

T

P1 Aj þ P2 Bjf C

T P2 Ajf P1 3 P2

#

; ðj ¼ 1; 2Þ, T T P2 PT P2 Ajf P1 3 P2 Aj þ P2 Bjf C 3 P2 " # " # 0 P1 Adj P1 Bj þ P2 Bjf D T T ~ T T ~ G P Adj G ¼ ; G P Bj ¼ , T T P2 PT P2 PT 0 3 P2 Adj 3 P2 Bj þ P2 Bjf D " T# h i P2 P1 P1 3 P2 T T T ~ ¼ E C f P1 P ; CG G P G¼ 3 2 . T T P2 PT P2 P1 3 P2 3 P2

ð19Þ

Substituting (15)–(17) and (19) into (18), we can obtain (10). Also, from (16), we can obtain (11)–(13). On the other hand, (17) is equivalent to 32 2 3 2 1 3 # A1f B1f A1f B1f " P2 0 0 T 76 A 6A 7 6 7 P2 P3 0 1 B B 2f 5 ¼ 4  2f 5 (20) P2 0 54 2f 4 2f 0 I Cf 0 C 0 f   I

ARTICLE IN PRESS L. Wu et al. / Signal Processing 87 (2007) 2213–2230

2219

^ FM in (4) can be described by and it follows from (4) that the transfer function of filter S Tðz1 ; z2 Þ ¼ C f ½z1 z2 I  z1 A1f  z2 A2f 1 ½z1 B1f þ z2 B2f .

(21)

Substituting (20) into (21) results in 1 1 T 1 T 1 1 Tðz1 ; z2 Þ ¼ Cf PT 2 P3 ½z1 z2 I  z1 P2 A1f P2 P3  z2 P2 A2f P2 P3  ½z1 P2 B1f þ z2 P2 B2f 

¼ Cf ½z1 z2 I  z1 V1 A1f  z2 V1 A2f 1 ½z1 V1 B1f þ z2 V1 B2f . Then, the realization of the filter in (14) can be readily established, which completes the proof.

&

Remark 2. Note that Theorem 2 provides a sufficient condition for the solvability of H1 filter design problem for the 2-D state-delayed system. Since the obtained conditions are expressed by strict LMIs, the desired filter can be determined by solving the following convex optimization problem: min d

s:t: ð10Þ2ð13Þ with d9g2 .

(22)

4. Further extensions In this section, we further extend the results obtained so far to 2-D state-delayed systems with uncertain model data, that is, the uncertain parameters are present in the system matrices A1 , A2 , Ad1 , Ad2 , B1 , B2 , C, D and E. In the following, we will consider two types of parameter uncertainties: polytopic uncertainty and norm-bounded uncertainty, which have been extensively used for studying robust control and filtering problems in the literature (see, for instance, [27] and the references therein). 4.1. Polytopic uncertain case Theorem 2 addresses the H1 filtering problem for system SFM in (1) where the system matrices are all known. However, since LMIs (10)–(13) are affine in the system matrices, Theorem 2 can be directly used to solve the H1 filtering problem for the case where the system matrices are not exactly known but reside within a given polytope. Assumption 3. The matrices A1 , A2 , Ad1 , Ad2 , B1 , B2 , C, D and E of system SFM in (1) contain partially unknown parameters. Assume that O9ðA1 ; A2 ; Ad1 ; Ad2 ; B1 ; B2 ; C; D; EÞ 2 w, where w is a given convexbounded polyhedral domain described by s vertices:  ( )  s s X X  w9 wðlÞwðlÞ ¼ lj wj ; lj ¼ 1; lj X0 ,  i¼1 i¼1 where wj 9ðA1j ; A2j ; Ad1j ; Ad2j ; B1j ; B2j ; C j ; Dj ; E j Þ denotes the jth vertex of the polytope w. We state the following theorem without proof, since the proof can be obtained along the same line of the derivation of Theorem 2. Theorem 3. Consider the 2-D state-delayed system SFM in (1) with Assumption 3 and let g40 be a prescribed constant scalar. Then there exists a full-order filter S^ FM in the form of (4) such that the filtering error system S~ FM is asymptotically stable and (8) is satisfied if there exist matrices U40, V40, Q1j 40, Q3j 40, Q11j 40, Q13j 40, Q21j 40, Q23j 40, Q2j , Q12j , Q22j , A1f , A2f , B1f , B2f and Cf such that, for j ¼ 1; 2; . . . ; s, the following LMIs

ARTICLE IN PRESS L. Wu et al. / Signal Processing 87 (2007) 2213–2230

2220

(23)–(26) hold: 2

U

V

0

0

UA1j þ B1f C j

A1f

UB1j þ B1f Dj

UAd1j

0

UA2j þ B2f C j

A2f

UB2j þ B2f Dj

UAd2j



V

0

0

VA1j þ B1f C j

A1f

VB1j þ B1f Dj

VAd1j

0

VA2j þ B2f C j

A2f

VB2j þ B2f Dj

VAd2j





I

0

Ej

Cf

0

0

0

0

0

0

0

0

0

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4







I

























0

0

0

Ej

Cf

0

0

Q1j þ Q11j  U Q2j þ Q12j  V

0

0

0

0

0

0

0



Q3j þ Q13j  V

0

0

0

0

0

0

0





g2 I

0

0

0

0

0

0 0















Q11j

Q12j

0

0

0

















Q13j

0

0

0

0

















Q21j  Q1j

Q22j  Q2j

0

0





















Q23j  Q3j

0

0























g2 I

0

























Q21j



























0

3

7 7 7 7 7 0 7 7 7 0 7 7 0 7 7 7 0 7 7 7o0, 0 7 7 0 7 7 7 0 7 7 0 7 7 7 0 7 7 Q22j 7 5 Q23j 0

0

(23) "

"

"

Q1j

Q2j



Q3j

#

Q11j

Q12j



Q13j

Q21j

Q22j



Q23j

(24)

40, # 40,

(25)

40.

(26)

#

Moreover, a desired H1 filter is given in the form of (4) with parameters can be computed from (14). 4.2. Norm-bounded uncertain case An alternative way of dealing with uncertain systems is to assume that the deviation of the system parameters from their nominal values is norm-bounded, which has also been widely used in the robust control and filtering problems. Assumption 4. The matrices A1 , A2 , Ad1 , Ad2 , B1 , B2 , C, D and E of system SFM in (1) are assumed to have the following form: A1 ¼ A^ 1 þ DA1 ; A2 ¼ A^ 2 þ DA2 ; C ¼ C^ þ DC;

B1 ¼ B^ 1 þ DB1 ; B2 ¼ B^ 2 þ DB2 ; D ¼ D^ þ DD;

Ad1 ¼ A^ d1 þ DAd1 , Ad2 ¼ A^ d2 þ DAd2 ,

E ¼ E^ þ DE,

ð27Þ

^ D^ and E^ are known constant matrices with appropriate dimensions. DA1 , where A^ 1 , A^ 2 , A^ d1 , A^ d2 , B^ 1 , B^ 2 , C, DA2 , DAd1 , DAd2 , DB1 , DB2 , DC, DD and DE are real-valued time-varying matrix functions representing normbounded parameter uncertainties satisfying 2 3 2 3 M1 DA1 DB1 DAd1  6 7 6 7  4 DA2 DB2 DAd2 5 ¼ 4 M 2 5D N 1 N 2 N 3 , M3 DC DD DE where Di;j is a real uncertain matrix function with Lebesgue measurable elements satisfying DT DpI, and M 1 , M 2 , M 3 , N 1 , N 2 and N 3 are known real constant matrices of appropriate dimensions. Before proceeding further, we give the following lemma which will be used in the proof of this subsection (see, for instance, [7]).

ARTICLE IN PRESS L. Wu et al. / Signal Processing 87 (2007) 2213–2230

2221

Lemma 2. Given appropriately dimensioned matrices S1 , S2 and S3 with ST1 ¼ S1 . Then S1 þ S2 OS3 þ ST3 OT ST2 o0

(28)

holds for all O satisfying OT OpI if and only if for some 40, S1 þ 1 S2 ST2 þ ST3 S3 o0. We now present the robust H1 filtering result for the system SFM in (1) with norm-bounded uncertainties in the following theorem.

Theorem 4. Consider the 2-D state-delayed system SFM in (1) with Assumption 4 and let g40 be a prescribed constant scalar. Then there exists a full-order filter S^ FM in the form of (4) such that the filtering error system S~ FM is asymptotically stable and (8) is satisfied if there exist matrices U40, V40, Q1 40, Q3 40, Q11 40, Q13 40, Q21 40, Q23 40, Q2 , Q12 , Q22 , A1f , A2f , B1f , B2f and Cf , scalars j 40 ðj ¼ 1; 2; . . . ; 6Þ such that the LMIs (11)–(13) and LMI (29) (shown below) hold. In (29), somenotations are defined as follows: C55 9Q1 þ Q11  U þ ð1 þ 3 ÞN T1 N 1 þ 5 N T3 N 3 ; C56 9Q2 þ Q12  V, C57 9ð1 þ 3 ÞN T1 N 2 ; C66 9Q3 þ Q13  V; C77 9ð1 þ 3 ÞN T2 N 2  g2 I, C88 91 N T3 N 3  Q11 ;

C1011 9Q22  Q2 ;

C1111 9Q23  Q3 ,

C1010 9Q21  Q1 þ ð2 þ 4 ÞN T1 N 1 þ 6 N T3 N 3 ; C1012 9ð2 þ 4 ÞN T1 N 2 , C1212 9ð2 þ 4 ÞN T2 N 2  g2 I; C1313 92 N T3 N 3  Q21 . Moreover, a desired H1 filter is given in the form of (4) with parameters can be computed from (14),

(29)

ARTICLE IN PRESS L. Wu et al. / Signal Processing 87 (2007) 2213–2230

2222

Proof. With the result of Theorem 2, we substitute the norm-bounded uncertain matrices A1 , A2 , Ad1 , Ad2 , B1 , B2 , C, D and E defined in (27) into (10) and obtain (28) where 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 S1 96 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

A1f

0

UA^ 1 þ B1f C^ VA^ 1 þ B1f C^

I

0

  

I  

   

   

   

   

0

A1f

UA^ d1 VA^ d1

E^

Cf

0

0

0

0

0

0 Q2 þ Q12  V Q3 þ Q13  V

0 0 0

0 0 0

0 0 0

0 E^ 0 0

0

0 Q1 þ Q11  U 

Cf 0 0

0 0 0

0 0 0

   

   

   

g2 I   

0 Q11  

0 Q12 Q13 

0 0 0 Q21  Q1

0 0 0 Q22  Q2

0 0 0 0

0 0 0 0

   

   

   

   

   

   

   

Q23  Q3   

0 g2 I  

0 0 Q21 

V

0

0



V

0





  

  

       

2

M T1 U 6 M TU 6 2 6 T T 6 M 3 B1f 6 S2 96 T T 6 M 3 B2f 6 6 0 4 0 2

0 6 60 6 60 6 S3 96 60 6 60 4 0

UA^ 2 þ B2f C^ VA^ 2 þ B2f C^

UB^ 1 þ B1f D^ VB^ 1 þ B1f D^

U

0

M T1 V M T2 V

0 0

0 0

0 0

0 0

0 0 0 0

0 0

0 0

0 0

0 0 0 0

M T3 BT1f

0

0

0

0

0 0

0

0

0

0 0

M T3 BT2f

0

0

0

0

0 0

0

0

0

0 0

0 0

M T3

0 M T3

0 0

0 0

0 0 0 0

0 0

0 0

0 0

0 0 0 0

0

0 0

0 0

0 0

N1 0

0 N2 0 0

N3 0

0 0

0 N1

0 0

0 N2

0 N3

0

0

0

N1

0 N2

0

0

0

0

0

0

0 0

0 0

0 0

0 N3

0 0

0 0

0 0

0 0

N1 0

0 0

N2 0

0 0

0

0

0

0

0

0

0

0

N3

0

0

0

3 0 7 07 7 07 7 7; 07 7 07 5 0

UA^ d2 VA^ d2

UB^ 2 þ B2f D^ VB^ 2 þ B2f D^

A2f A2f

0

3

7 0 7 7 7 0 7 7 7 0 7 7 0 7 7 7 0 7 7 0 7 7, 7 0 7 7 0 7 7 7 0 7 7 0 7 7 7 0 7 7 Q22 7 5 Q23

3T 0 07 7 7 07 7 7 , 07 7 07 5 0

D 60 6 6 60 O96 60 6 6 40

0 D

0 0

0 0

0 0

0

D

0

0

0 0

0 0

D 0

0 D

3 0 07 7 7 07 7. 07 7 7 05

0

0

0

0

0

D

2

By invoking Lemma 1 together with a Schur complement operation, (28) holds if and only if (29) holds, which completes the proof. & 5. Illustrative example In a real world, some dynamical processes in gas absorption, water stream heating and air drying can be described by the Darboux equation with time-delays [28]: q2 sðx; tÞ qsðx; tÞ qsðx; t  t1 Þ qsðx; tÞ qsðx; t  t1 Þ ¼ a11 þ a12 þ a21 þ a22 þ a0 sðx; tÞ þ bf ðx; tÞ, qxqt qt qt qx qx  qsðx; tÞ  a21 sðx; tÞ , yðx; tÞ ¼ c1 sðx; tÞ þ c2 qt

(30)



(31)

where sðx; tÞ is an unknown function at x(space) 2 ½0; xf  and t(time) 2 ½0; 1Þ, t1 is the time-delay, a0 , a11 , a12 , a21 , a22 , b, c1 and c2 are real coefficients, f ðx; tÞ is the input function, and yðx; tÞ is the measured output.

ARTICLE IN PRESS L. Wu et al. / Signal Processing 87 (2007) 2213–2230

2223

Note that (30)–(31) is a partial differential equation (PDE) and, in practice, it is often desired to predict the unknown state function sðx; tÞ through the available measurement yðx; tÞ, which renders the filtering problem. Similar to the technique used in [5], we define rðx; tÞ:¼

qsðx; tÞ qsðx; t  t1 Þ  a21 sðx; tÞ þ  a22 sðx; t  t1 Þ, qt qt

x1 ði; jÞ:¼rði; jÞ:¼rðiDx; jDtÞ;

x2 ði; jÞ:¼sði; jÞ:¼sðiDx; jDtÞ,

and then the PDE model (30)–(31) can be converted into the form of a state-delayed 2-D system SFM in (1). As discussed in [5], the discrepancy between the PDE model and its 2-D difference approximation depends on the step sizes Dx and Dt which may be treated as uncertainty in the difference model. Obviously, the smaller the step sizes Dx and Dt, the closer between the PDE model and the difference model. Now, subject to the selection of the parameters a0 , a11 , a12 , a21 , a22 , b, c1 and c2 , we let the system matrices be given as follows: " A1 ¼

0:3

#

0

" ;

B1 ¼

0:3

#

" ;

0:2

Ad1 ¼

0

#

, 0:1 þ 0:02d " # " # " # 0:1 0 0:2 0 0:1 ; B2 ¼ ; Ad2 ¼ , A2 ¼ 0:2 0:2 þ 0:02d 0:4 þ 0:01d 0 0:2 þ 0:02d " # " # " # 1:0 0 0 1:0 1:0 C¼ ; D¼ ; E¼ . 1:0 0:6 þ 0:02d 0:3 þ 0:01d 0 0:8 þ 0:02d 0:2

0:1 þ 0:02d

0:5 þ 0:01d

0

First, we assume that the system matrices are perfectly known, that is, d ¼ 0. Solving the LMIs condition obtained in Theorem 2 by applying the well-developed LMI-Toolbox in the MATLAB environment directly, we obtain that the minimum g is g ¼ 3:8207 and " A1f ¼ " A2f ¼

0:0117

0:0086

0:0086

0:0063

0:0101

0:0074

0:0072

0:0053

#

" ;

B1f ¼

#

" ;

B2f ¼

2:1209

1:0000

1:5539

0:7343

1:5607

1:9339

1:1429

1:4179

# , #

" ;

Cf ¼

1:3918

1:0222

1:1895

0:8761

# .

Now, we assume jdjp1, that is, the system considered has parameter uncertainties. As mentioned in the previous section, there are two types of parameter uncertainties, namely, polytopic uncertainties and normbounded uncertainties. In the following, firstly, we consider the polytopic uncertainties case. In this case, according to Assumption 3, the parameter uncertainties can be represented by a two-vertex polytope. Using Theorem 3, the minimum g obtained is g ¼ 5:4379, and the obtained filter parameter matrices are given as follows: " A1f ¼ " A2f ¼

0:0661

0:0378

0:0375

0:0215

0:0428

0:0245

0:0237

0:0136

#

" ;

B1f ¼

#

" ;

B2f ¼

6:3037

2:3277

3:6136

1:3374

4:6924

4:1119

2:6911

2:3608

# , #

" ;

Cf ¼

1:8108

1:0445

1:5350

0:8889

# .

ARTICLE IN PRESS L. Wu et al. / Signal Processing 87 (2007) 2213–2230

2224

Finally, we consider the norm-bounded uncertainties case, and the uncertainties are characterized as follows according to Assumption 4: " A^ 1 ¼ "

#

0:3

0

0:2

0:1

0:3

#

" A^ 2 ¼

;

"

0:2

#

#

0:1

0

0:2

0:2 "

" ;

A^ d1 ¼

1:0

0

#

0:2

0

0

0:1 "

#

" A^ d2 ¼

;

; ; D^ ¼ ; C^ ¼ 0:3 1:0 0:6 0:4 " # 0   M1 ¼ M2 ¼ M3 ¼ ; N 1 ¼ N 3 ¼ 0 0:02 ; N 2 ¼ 0:02. 1 B^ 1 ¼

0:5

;

B^ 2 ¼

#

, 0 0:2 " # 1:0 1:0 , E^ ¼ 0 0:8

#

0

0 0:1

Using Theorem 4, the minimum g is obtained as g ¼ 5:2074, and the obtained filter parameter matrices are given as follows:

" A2f ¼

0:4439

0:4496

0:2917

1:8433

1:2020

1:2065

0:7864

#

" ;

B1f ¼

#

State response of H−infinity filter

A1f ¼

0:6830

" ;

B2f ¼

2:2071

0:4063

1:4350

0:2651

4:4037

0:6876

2:8895

0:4487

# , # ;

" Cf ¼

1:6334

1:0734

1:4913

0:9842

1 0.5 0 −0.5 −1 20

15

10 ,2.. .

j=1

5

0

0

5

15

10

20

...

i=1,2

Fig. 1. State response of the H1 filter xf 1 ði; jÞ.

State response of H−infinity filter

"

1 0.5 0 −0.5 −1 20

15

10 ,2.. .

j=1

5

0

0

5

15

10

...

i=1,2

Fig. 2. State response of the H1 filter xf 2 ði; jÞ.

20

# .

ð32Þ

ARTICLE IN PRESS

H−infinity filtering error response

L. Wu et al. / Signal Processing 87 (2007) 2213–2230

2225

2 1 0 −1 −2 −3 −4 20

15

10 j=1 ,2.. .

5

0

0

5

15

10

20

...

i=1,2

H−infinity filtering error response

Fig. 3. Error response e1 ði; jÞ.

2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 20

15

10 j=1 ,2.. .

5

0

0

5

15

10

20

...

i=1,2

Fig. 4. Error response e2 ði; jÞ.

Let the disturbance input oði; jÞ be  0:05; 3pi; jp19; oði; jÞ ¼ 0 otherwise: In the following, we shall show the usefulness of the designed H1 filters by presenting simulation results. To show the asymptotic stability of the filtering error system, let the initial and boundary conditions be ( ½ 1 1:5 T ; 0pip15; xð0; iÞ ¼ xði; 0Þ ¼ ½ 0 0 T ; i415: The state response of the designed H1 filter in (4) with (32) are given in Figs. 1 and 2, and Figs. 3 and 4 are the error response for eði; jÞ. It can be seen from Figs. 3 and 4 that the designed H1 filter guarantees that eði; jÞ converges to zero under the above conditions. 6. Concluding remarks In this paper, the problem of robust H1 filtering for a class of 2-D delayed systems has been studied. Some sufficient conditions have been proposed for the existences of robust H1 filter in terms of LMI. The designed robust H1 filter guarantees robust asymptotic stability and a prescribed H1 performance of the filtering error system, and the desired filter can be found by solving a convex optimization problem.

ARTICLE IN PRESS L. Wu et al. / Signal Processing 87 (2007) 2213–2230

2226

In addition, the obtained results have been further extended to more general cases where the system matrices also contain uncertain parameters. The most frequently used methods of dealing with parameter uncertainties, including polytopic and norm-bounded characterizations, have been taken into consideration. An illustrative example has been presented to demonstrate the effectiveness of the proposed methods. One of the future research topics would be the further investigation on the time-varying time-delays case [29]. Acknowledgements The authors are grateful to the anonymous reviewers for their valuable comments and suggestions that helped improve the presentation of the paper. This work was partially supported by the National Natural Science Foundation of China (60504008), Program for New Century Excellent Talents in University of China and the Postdoctoral Science Foundation of China (20060390231). Appendix ~ FM in (5) with oi;j  0. Proof of Theorem 1. First, let us establish the asymptotic stability of the error system S Denote V 11 ði; jÞ9xT11 ði; jÞPx11 ði; jÞ þ

1 X

xTk1 ði þ 1; jÞQ1 xk1 ði þ 1; jÞ þ

k¼d 1

V d1 ði; jÞ9xT01 ði; jÞðP  QÞx01 ði; jÞ þ

1 X

xT1k ði; j þ 1ÞQ2 x1k ði; j þ 1Þ,

k¼d 2

1 X

xTk1 ði; jÞQ1 xk1 ði; jÞ,

k¼d 1

V d2 ði; jÞ9xT10 ði; jÞQx10 ði; jÞ þ

1 X

xT1k ði; jÞQ2 x1k ði; jÞ.

(33)

k¼d 2

Consider the increment DV ði; jÞ given by DV ði; jÞ9V 11 ði; jÞ  V 01 ði; jÞ  V 10 ði; jÞ.

(34)

~ FM , we have Then, along the solution of the filtering error system S DV ði; jÞ ¼ ½A~ 1 xði; j þ 1Þ þ A~ 2 xði þ 1; jÞ þ A~ d1 xði  d 1 ; j þ 1Þ þ A~ d2 xði þ 1; j  d 2 ÞT P  ½A~ 1 xði; j þ 1Þ þ A~ 2 xði þ 1; jÞ þ A~ d1 xði  d 1 ; j þ 1Þ þ A~ d2 xði þ 1; j  d 2 Þ  xT ði; j þ 1ÞðP  Q  Q1 Þxði; j þ 1Þ  xT ði þ 1; jÞðQ  Q2 Þxði þ 1; jÞ  xT ði  d 1 ; j þ 1ÞQ1 xði  d 1 ; j þ 1Þ  xT ði þ 1; j  d 2 ÞQ2 xði þ 1; j  d 2 Þ 9ZT ði; jÞCZði; jÞ, where Zði; jÞ9½xT ði; j þ 1Þ; xT ði þ 1; jÞ; xT ði  d 1 ; j þ 1Þ; xT ði þ 1; j  d 2 ÞT and 2 T3 2 3 A~ 0 0 0 P  Q  Q1 6 1T 7 6 ~ 7 6 7  Q  Q2 0 0 7 6 A2 7 6 ~ 1 A~ 2 A~ d1 A~ d2   6 7 7. A P½ C96 6 ~T 7 6   Q1 0 7 6 Ad1 7 4 5 4 T 5    Q2 ~ Ad2

ð35Þ

ARTICLE IN PRESS L. Wu et al. / Signal Processing 87 (2007) 2213–2230

2227

By Schur complement [25], LMI (9) implies Co0. It follows from Lemma 1 (Theorem 3 of [6]) that the 2-D filtering error system S~ FM in (5) with oði; jÞ  0 is asymptotically stable. ~ FM in (5), introduce the Now, to establish the H1 performance for the filtering error system S following index: J9DV ði; jÞ þ e¯ T ði; jÞ¯eði; jÞ  g2 o ¯ T ði; jÞoði; ¯ jÞ,

(36)

where DV ði; jÞ ¼ ½A~ 1 xði; j þ 1Þ þ A~ 2 xði þ 1; jÞ þ A~ d1 xði  d 1 ; j þ 1Þ þ A~ d2 xði þ 1; j  d 2 Þ þ B~ 1 oði; j þ 1Þ þ B~ 2 oði þ 1; jÞT P½A~ 1 xði; j þ 1Þ þ A~ 2 xði þ 1; jÞ þ A~ d1 xði  d 1 ; j þ 1Þ þ A~ d2 xði þ 1; j  d 2 Þ þ B~ 1 oði; j þ 1Þ þ B~ 2 oði þ 1; jÞ  xT ði; j þ 1ÞðP  Q  Q1 Þxði; j þ 1Þ  xT ði þ 1; jÞðQ  Q2 Þxði þ 1; jÞ  xT ði  d 1 ; j þ 1ÞQ1 xði  d 1 ; j þ 1Þ  xT ði þ 1; j  d 2 ÞQ2 xði þ 1; j  d 2 Þ.

ð37Þ

According to the stability of the system, we have J ¼ ½A~ 1 xði; j þ 1Þ þ A~ 2 xði þ 1; jÞ þ A~ d1 xði  d 1 ; j þ 1Þ þ A~ d2 xði þ 1; j  d 2 Þ þ B~ 1 oði; j þ 1Þ þ B~ 2 oði þ 1; jÞT P½A~ 1 xði; j þ 1Þ þ A~ 2 xði þ 1; jÞ þ A~ d1 xði  d 1 ; j þ 1Þ þ A~ d2 xði þ 1; j  d 2 Þ þ B~ 1 oði; j þ 1Þ þ B~ 2 oði þ 1; jÞ  xT ði; j þ 1ÞðP  Q  Q1 Þxði; j þ 1Þ  xT ði þ 1; jÞðQ  Q2 Þxði þ 1; jÞ  xT ði  d 1 ; j þ 1ÞQ1 xði  d 1 ; j þ 1Þ  xT ði þ 1; j  d 2 ÞQ2 xði þ 1; j  d 2 Þ T T ~ ~ þ 1; jÞ þ xT ði; j þ 1ÞC~ Cxði; j þ 1Þ þ xT ði þ 1; jÞC~ Cxði 2 T 2 T  g o ði; j þ 1Þoði; j þ 1Þ  g o ði þ 1; jÞoði þ 1; jÞ ¼ ½A~ 1 xði; j þ 1Þ þ A~ 2 xði þ 1; jÞ þ A~ d1 xði  d 1 ; j þ 1Þ þ A~ d2 xði þ 1; j  d 2 ÞT P

 ½A~ 1 xði; j þ 1Þ þ A~ 2 xði þ 1; jÞ þ A~ d1 xði  d 1 ; j þ 1Þ þ A~ d2 xði þ 1; j  d 2 Þ  xT ði; j þ 1ÞðP  Q  Q1 Þxði; j þ 1Þ  xT ði þ 1; jÞðQ  Q2 Þxði þ 1; jÞ  xT ði  d 1 ; j þ 1ÞQ1 xði  d 1 ; j þ 1Þ  xT ði þ 1; j  d 2 ÞQ2 xði þ 1; j  d 2 Þ T T ~ ~ þ 1; jÞ þ xT ði; j þ 1ÞC~ Cxði; j þ 1Þ þ xT ði þ 1; jÞC~ Cxði þ 2½A~ 1 xði; j þ 1Þ þ A~ 2 xði þ 1; jÞ þ A~ d1 xði  d 1 ; j þ 1Þ þ A~ d2 xði þ 1; j  d 2 ÞT P

 ½B~ 1 oði; j þ 1Þ þ B~ 2 oði þ 1; jÞ  fg2 oT ði; j þ 1Þoði; j þ 1Þ þ g2 oT ði þ 1; jÞoði þ 1; jÞ  ½B~ 1 oði; j þ 1Þ þ B~ 2 oði þ 1; jÞT P½B~ 1 oði; j þ 1Þ þ B~ 2 oði þ 1; jÞg 9ZT ði; jÞPZði; jÞ þ 2ZT ði; jÞOoði; ¯ jÞ  o ¯ T ði; jÞFoði; ¯ jÞ ¼ ZT ði; jÞPZði; jÞ þ ZT ði; jÞOF1 OT Zði; jÞ  ZT ði; jÞOF1 OT Zði; jÞ þ 2ZT ði; jÞOoði; ¯ jÞ  o ¯ T ði; jÞFoði; ¯ jÞ ¼ ZT ði; jÞðP þ OF1 OT ÞZði; jÞ  ½ZT ði; jÞOF1 OT Zði; jÞ  2ZT ði; jÞOoði; ¯ jÞ þ o ¯ T ði; jÞFoði; ¯ jÞ ¼ ZT ði; jÞSZði; jÞ  ½oði; ¯ jÞ  F1 OT Zði; jÞT F½oði; ¯ jÞ  F1 OT Zði; jÞ 9ZT ði; jÞSZði; jÞ  mT ði; jÞmði; jÞ,

ð38Þ

ARTICLE IN PRESS L. Wu et al. / Signal Processing 87 (2007) 2213–2230

2228

where Zði; jÞ is defined in (35), mði; jÞ9F1=2 ½oði; ¯ jÞ  F1 OT Zði; jÞ and 2

T A~ 1

3 2

T A~ 1

3T

2

T Q þ Q1  P þ C~ C~

0

0

0

3

7 6 7 6 7 6 6 ~T 7 6 ~T 7 7 6 T 6 A2 7 6 A2 7 6  Q2  Q þ C~ C~ 0 0 7 7 6 7 6 7, 6 P96 T 7P6 T 7 þ 6 7 6 A~ 7 6 A~ 7 7 6 0   Q 6 d1 7 6 d1 7 1 5 4 5 4 5 4 T T    Q2 A~ d2 A~ d2 2 T3 A~ 1 7 6 2 T3 6 ~T 7 6 A2 7 B~ 1 7 6 O96 T 7P½ B~ 1 B~ 2 ; F9g2 I  4 T 5P½ B~ 1 B~ 2 ; S9P þ OF1 OT . 6 A~ 7 B~ 2 6 d1 7 5 4 T A~ d2

By Schur complement, LMI (9) implies So0. This together with (36) and (38) yields DV ði; jÞ þ e¯ T ði; jÞ¯eði; jÞ  g2 o ¯ T ði; jÞoði; ¯ jÞo  mT ði; jÞmði; jÞ.

(39)

Therefore we can sum both sides of (39) to obtain 1 X 1 1 X 1 X X ½DV ði; jÞ þ e¯ T ði; jÞ¯eði; jÞ  g2 o mT ði; jÞmði; jÞ ¼ kmði; jÞk22 . ¯ T ði; jÞoði; ¯ jÞo  i¼0

j¼0

i¼0

(40)

j¼0

For any integers p, q40, it follows from (34) that p X q X i¼0

j¼0

DV ði; jÞ ¼

q X ½xT11 ðp; jÞðP  QÞx11 ðp; jÞ  xT01 ð0; jÞðP  QÞx01 ð0; jÞ j¼0

þ

p X

½xT11 ði; qÞQx11 ði; qÞ  xT10 ði; 0ÞQx10 ði; 0Þ

i¼0

2 3 q 1 1 X X X 4 xTk1 ðp þ 1; jÞQ1 xk1 ðp þ 1; jÞ  xTk1 ð0; jÞQ1 xk1 ð0; jÞ5 þ j¼0

k¼d 1

k¼d 1

k¼d 2

k¼d 2

2

3 p 1 1 X X X 4 xT1k ði; q þ 1ÞQ2 x1k ði; q þ 1Þ  xT1k ði; 0ÞQ2 x1k ði; 0Þ5. þ i¼0

ð41Þ

Thus, together with (40) implies that k¯eði; jÞk22  g2 koði; ¯ jÞk22 þ kmði; jÞk22 o

1 X

xT01 ð0; jÞðP  QÞx01 ð0; jÞ þ

j¼0

þ

1 X

xT10 ði; 0ÞQx10 ði; 0Þ

i¼0

1 1 X X j¼0 k¼d 1

xTk1 ð0; jÞQ1 xk1 ð0; jÞ þ

1 1 X X i¼0 k¼d 2

xT1k ði; 0ÞQ2 x1k ði; 0Þ

ð42Þ

ARTICLE IN PRESS L. Wu et al. / Signal Processing 87 (2007) 2213–2230

2229

which implies that " k¯eði; jÞk22 þ kmði; jÞk22 og2 koði; ¯ jÞk22 þ

1 X

~ 01 ð0; jÞ þ xT01 ð0; jÞPx

j¼0

þ

1 1 X X

1 X

~ 10 ði; 0Þ xT10 ði; 0ÞQx

i¼0

xTk1 ð0; jÞQ~ 1 xk1 ð0; jÞ þ

j¼0 k¼d 1

1 1 X X

3 xT1k ði; 0ÞQ~ 2 x1k ði; 0Þ5,

ð43Þ

i¼0 k¼d 2

~ Q1 og2 Q~ 1 and Q2 og2 Q~ 2 . Now, to establish the H1 performance, we show that ~ Qog2 Q, where P  Qog2 P, there exists a scalar a40 such that " 1 1 X X ~ 10 ði; 0Þ ~ 01 ð0; jÞ þ kmði; jÞk22 Xa2 koði; xT01 ð0; jÞPx xT10 ði; 0ÞQx ¯ jÞk22 þ j¼0

þ

1 1 X X

i¼0

xTk1 ð0; jÞQ~ 1 xk1 ð0; jÞ þ

1 1 X X

j¼0 k¼d 1

3 xT1k ði; 0ÞQ~ 2 x1k ði; 0Þ5.

ð44Þ

i¼0 k¼d 2

Consider the inverse system of (5): xði þ 1; j þ 1Þ ¼ A~ 1 xði; j þ 1Þ þ A~ 2 xði þ 1; jÞ þ A~ d1 xði  d 1 ; j þ 1Þ þ A~ d2 xði þ 1; j  d 2 Þ þ B~ 1 oði; j þ 1Þ þ B~ 2 oði þ 1; jÞ 9AZði; jÞ þ Boði; ¯ jÞ ¼ ðA þ BF1 OT ÞZði; jÞ þ BF1=2 mði; jÞ,

ð45Þ

oði; ¯ jÞ ¼ F1 OT Zði; jÞ þ F1=2 mði; jÞ,

(46)

where A9½ A~ 1 A~ 2 A~ d1 A~ d2 , B9½ B~ 1 B~ 2  and Zði; jÞ has been defined before. It can be verified from (9) that the system in (45) is asymptotically stable, thus there exists a bounded b40 such that " 1 1 X X 2 ~ 01 ð0; jÞ þ ~ 10 ði; 0Þ koði; xT01 ð0; jÞðP~  QÞx xT10 ði; 0ÞQx ¯ jÞk2 þ j¼0

þ

1 1 X X j¼0 k¼d 1

xTk1 ð0; jÞQ~ 1 xk1 ð0; jÞ þ

i¼0 1 1 X X

3

xT1k ði; 0ÞQ~ 2 x1k ði; 0Þ5pb2 kmði; jÞk22 .

ð47Þ

i¼0 k¼d 2

This implies (44) with b ¼ 1=a. With zero boundary conditions as in (2), we can easily obtain (8), hence the proof is completed. & References [1] T. Kaczorek, Two-Dimensional Linear Systems, Springer, Berlin, Germany, 1985. [2] H. Gao, J. Lam, S. Xu, C. Wang, Stabilization and H 1 control of two-dimensional Markovian jump systems, IMA J. Math. Control Inf. 21 (2004) 377–392. [3] W.S. Lu, A. Antoniou, Two-Dimensional Digital Filters, Marcel Dekker, New York, 1992. [4] C. Du, L. Xie, Y.C. Soh, H 1 filtering of 2-D discrete systems, IEEE Trans. Signal Process. 48 (6) (2000) 1760–1768. [5] C. Du, L. Xie, C. Zhang, H 1 control and robust stabilization of two-dimensional systems in Roesser models, Automatica 37 (2001) 205–211. [6] W. Paszke, J. Lam, K. Galkowski, S. Xu, Z. Lin, Robust stability and stabilisation of 2D discrete state-delayed systems, Syst. Control Lett. 51 (3–4) (2004) 277–291.

ARTICLE IN PRESS 2230

L. Wu et al. / Signal Processing 87 (2007) 2213–2230

[7] L. Xie, C. Du, Y.C. Soh, C. Zhang, H 1 and robust control of 2-D systems in FM second model, Multidimensional Syst. Signal Process. 13 (2002) 256–287. [8] H. Gao, J. Lam, C. Wang, S. Xu, H 1 model reduction for uncertain two-dimensional discrete systems, Optim. Control Appl. Methods 26 (2005) 199–227. [9] W. Assawinchaichote, S.K. Nguang, H 1 filtering for nonlinear singularly perturbed systems with pole placement constraints: an LMI approach, IEEE Trans. Signal Process. 34 (6) (June 2004) 1659–1667. [10] C.E. de Souza, A. Trofino, An LMI approach to the design of robust H 2 filters, in: Recent Advances on Linear Matrix Inequality Methods in Control, Philadelphia, PA, 1999. [11] J.C. Geromel, M.C. De Oliveira, H 2 and H 1 robust filtering for convex bounded uncertain systems, IEEE Trans. Automat. Control 46 (1) (2001) 100–107. [12] E. Gershon, D.J.N. Limebeer, U. Shaked, I. Yaesh, Robust H 1 filtering of stationary continuous time linear systems with stochastic uncertainties, IEEE Trans. Automat. Control 46 (11) (2001) 1788–1793. [13] Z. Wang, F. Yang, D.W.C. Ho, X. Liu, Robust H 1 filtering for stochastic time-delay systems with missing measurements, IEEE Trans. Signal Process. 54 (7) (2006) 2579–2587. [14] Z. Wang, F. Yang, X. Liu, Robust filtering for systems with stochastic nonlinearities and deterministic uncertainties, Proc. IMechE— J. Syst. Control Eng. 220 (3) (2006) 171–182. [15] Z. Wang, F. Yang, D.W.C. Ho, X. Liu, Robust finite-horizon filtering for stochastic systems with missing measurements, IEEE Signal Process. Lett. 12 (6) (2005) 437–440. [16] H. Gao, C. Wang, Delay-dependent robust H 1 and L2 2L1 filtering for a class of uncertain nonlinear time-delay systems, IEEE Trans. Automat. Control 48 (9) (2003) 1661–1666. [17] H. Liu, F. Sun, K. He, Z. Sun, Design of reduced-order H 1 filter for Markovian jumping systems with time delay, IEEE Trans. Circuits Syst.—II 51 (11) (November 2004) 607–612. [18] R.M. Palhares, C.E. de Souza, P.L.D. Peres, Robust H 1 filtering for uncertain discrete-time state-delayed systems, IEEE Trans. Signal Process. 49 (8) (2001) 1696–1703. [19] L. Wu, P. Shi, C. Wang, H. Gao, Delay-dependent robust H 1 and L2 –L1 filtering for LPV systems with both discrete and distributed delays, IEE Proc. Part D: Control Theory Appl. 153 (4) (2006) 483–492. [20] S. Xu, P.V. Dooren, Robust H 1 filtering for a class of nonlinear systems with state delay and parameter uncertainty, Int. J. Control 75 (10) (2002) 766–774. [21] D. Yue, Q.-L. Han, Robust H 1 filter design of uncertain descriptor systems with discrete and distributed delays, IEEE Trans. Signal Process. 52 (11) (November 2004) 3200–3212. [22] H. Zhang, D. Zhang, L. Xie, An innovation approach to H 1 prediction with application to systems with delayed measurements, Automatica 40 (7) (2004) 1253–1261. [23] T. Hinamoto, 2-D Lyapunov equation and filter design based on Fornasini–Marchesini second model, IEEE Trans. Circuits Syst.: Part I 40 (2) (1993) 102–110. [24] H.D. Tuan, P. Apkarian, T.Q. Nguyen, Robust mixed H 2 =H 1 filtering of 2-D systems, IEEE Trans. Signal Process. 50 (7) (2002) 1759–1771. [25] S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, SIAM, Philadelphia, PA, 1994. [26] E. Fornasini, G. Marchesini, Doubly indexed dynamical systems: state-space models and structural properties, Math. Syst. Theory 12 (1978) 59–72. [27] Z. Wang, B. Huang, H. Unbehauen, Robust H 1 observer design of linear state delayed systems with parametric uncertainty: the discrete-time case, Automatica 35 (1999) 1161–1167. [28] W. Marszalek, Two-dimensional state-space discrete models for hyperbolic partial differential equations, Appl. Math. Modeling 8 (1984) 11–14. [29] A. Megretski, A. Rantzer, System analysis via integral quadratic constraints, IEEE Trans. Automat. Control 42 (6) (1997) 819–830.