Deconvolution filtering for stochastic systems via homogeneous polynomial Lyapunov functions

ARTICLE IN PRESS Signal Processing 89 (2009) 605–614

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Deconvolution filtering for stochastic systems via homogeneous polynomial Lyapunov functions$ Baoyong Zhang a,, James Lam b, Shengyuan Xu a a b

School of Automation, Nanjing University of Science and Technology, Nanjing, 210094 Jiangsu, PR China Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong

a r t i c l e in fo

abstract

Article history: Received 3 July 2008 Received in revised form 8 October 2008 Accepted 10 October 2008 Available online 21 October 2008

This paper deals with the robust H1 and L2 2L1 deconvolution filtering problems for stochastic systems with polytopic uncertainties. The purpose is to design a full-order deconvolution filter such that (i) the deconvolution error system is robustly exponentially mean-square stable with a prescribed decay rate and (ii) an H1 or L2 2L1 performance of the deconvolution error system is guaranteed. Based on a homogeneous polynomial parameter-dependent matrix (HPPDM) approach, sufficient conditions for the solvability of these problems are given in terms of linear matrix inequalities (LMIs). Such conditions are dependent on the decay rate, which enables one to design robust deconvolution filters by selecting the decay rates according to different practical conditions. In addition, when these LMIs are feasible, a design procedure of the desired filters is developed and an exponential estimate for the deconvolution error system is given. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed design methods. & 2007 Elsevier B.V. All rights reserved.

Keywords: Deconvolution filtering Exponential estimates Robust H1 filtering Robust L2 2L1 filtering Uncertain stochastic systems

1. Introduction Over the past years, considerable attention has been devoted to the study of the filtering problem for polytopic uncertain systems. The purpose of this problem is to design a stable filter such that, for all parameter uncertainties residing in a polytope, the error system is asymptotically stable and ensures a certain performance constraint. Generally, there are two approaches to solving this problem: the quadratic (parameter-independent) $ This work was supported in part by the Program for the National Science Foundation for Distinguished Young Scholars of PR China under Grant 60625303, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20060288021, the RGC HKU 7029/05P, the Natural Science Foundation of Jiangsu Province under Grant BK2008047, and the Research Innovation Program for Graduate Students in Jiangsu Province under Grant CX07B_114z.  Corresponding author. E-mail addresses: [email protected] (B. Zhang), [email protected] (J. Lam).

0165-1684/$ - see front matter & 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2008.10.008

Lyapunov function (QLF) approach and the parameterdependent Lyapunov function (PDLF) approach. The QLF approach is relatively simple and easily used for the filter design [1–3]. In addition, such an approach has special advantages to the study of robust and gain-scheduled filtering problems for systems with time-varying uncertainties [4,5]. However, the QLF approach may produce conservative results because the fixed Lyapunov matrix must be used for the entire uncertainty domain [6,7]. In order to reduce such conservatism, the PDLF approach has been used and a number of important results have been reported in the literature; see, for example, [6,8–12]. We note that the parameter-dependent matrices used in these works have the same structures as the system matrices; that is, they are linearly dependent on the uncertain parameters. Very recently, another approach to constructing the parameter-dependent matrices, known as the homogeneous polynomial parameter-dependent matrix (HPPDM) approach, has been proposed in [13–15]. By using this approach, the parameter-dependent matrices

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B. Zhang et al. / Signal Processing 89 (2009) 605–614

are represented in the form of homogeneous polynomials with respect to the uncertain parameters, which are more general than the linear parameter-dependent matrices. It has been shown in [13–15] that, as the degree of the polynomials increases, less conservative robust stability conditions are obtained. It is also worth noting that the HPPDM approach has been generalized to the robust and parameter-dependent filtering problems for polytopic uncertain systems [16–18]. The deconvolution problem is another important research topic that has extensive applications in data transmission, equalization, reverberation cancellation, seismic deconvolution, image restoration, speech processing, and other areas [19–24]. The objective of the deconvolution filtering problem is to design a filter to estimate the unknown input signal of a system using available measurements, which is much different from the aforementioned filtering problems where we estimate the system states. Over the past 10 years, the deconvolution filtering problem for linear systems has received much attention. For example, a game-theory approach was proposed in [25,26] for the H1 deconvolution filter design. In [23,27], the full-order and reduced-order H1 deconvolution filters were designed, respectively. In [28], linear matrix inequality (LMI)-based results on the H2 deconvolution filtering problem were presented. In [29], the problem of finite horizon H1 deconvolution filtering for linear time-varying systems was studied. In [22], 2-D H1 deconvolution filters were designed for 2-D digital systems based on the LMI approach. On the other hand, the stochastic systems have been extensively studied and a lot of results have been reported for different research topics, such as stability and stabilization [30–33], sliding mode control [34,35], proportional-integral-derivative (PID) control [36], H1 model reduction [37], H1 control and filtering [38–42], and L2 2L1 filtering [43]. It is worth noting that the HPPDM approach has been applied in [42] to study the robust H1 filter design problem for a class of discrete-time stochastic systems with polytopic uncertainties. Very recently, the H1 deconvolution filtering problem for stochastic systems with interval uncertainties has been investigated in [44], where some valuable conditions for the design of the H1 deconvolution filters were presented via LMIs. However, to the best of our knowledge, the problems of robust H1 and L2 2L1 deconvolution filtering for stochastic systems with polytopic uncertainties have not been addressed in the literature, which still remain open and unsolved. In addition, in many practical applications, one is concerned not only with the stability of a system but also with the decay rate (also called convergence rate). Therefore, how to design a deconvolution filter for a system such that the deconvolution error system converges with a specified decay rate is also an interesting research topic. For the first time, this paper investigates the robust H1 and L2 2L1 deconvolution filtering problems for stochastic systems with time-invariant polytopic uncertainties. First, parameter-dependent conditions, which are also dependent on a specified decay rate, for the existence of the robust H1 and L2 2L1 deconvolution filters are derived.

Second, the parameter-dependent conditions are transformed into LMIs by using the HPPDM approach and the combinatoric mathematics techniques. Finally, based on the proposed LMIs, a design procedure of the desired filters is developed and an exponential estimate for the deconvolution error system is given. We also provide two examples to demonstrate the effectiveness of the proposed design methods. The contribution of this paper is threefold: (i) a design procedure of the robust H1 and L2 2L1 deconvolution filters for polytopic uncertain stochastic systems is proposed for the first time; (ii) the exponential estimate is considered in the filter design; (iii) the recently developed HPPDM approach is applied, and thus the presented results are generally less conservative. n Notations: Throughout this paper, R denotes the nmn dimensional Euclidean space, and R denotes the set of all m  n real matrices. A real symmetric matrix P40ðX0Þ denotes that P is a positive definite (or positive semidefinite) matrix, and A4ðXÞB means A  B4ðXÞ0. I denotes an identity matrix of appropriate dimension. The superscript T represents the transpose.  is used as an ellipsis for terms that are induced by symmetry. L2 ½0; 1Þ is the space of square-integrable vector functions over ½0; 1Þ. j  j denotes the Euclidean norm for vectors, k  k denotes the spectral norm for matrices, and k  k2 stands for the usual L2 ½0; 1Þ norm. We use diagf. . .g to represent a blockdiagonal matrix. Let ðO; F; fFt gtX0 ; PÞ be a complete probability space with a filtration fFt gtX0 satisfying the usual conditions (i.e., the filtration contains all P-null sets and is right continuous). Efg denotes the expectation operator with respect to the probability measure P. To use the HPPDM approach, we let KðrÞ denote the set of s-tuples obtained as all possible combinations of nonPs negative integers ki , i ¼ 1; 2; . . . ; s, such that i¼1 ki ¼ r; Ps that is, KðrÞ ¼ fk ¼ k1 k2 . . . ks jki 2 Zþ ; i¼1 ki ¼ rg, where Zþ is the set of nonnegative integers. In addition, r! Q denotes the factorial, pðkÞ ¼ si¼1 ðki !Þ, and Ii denotes a combination of 0 and 1 with the i-th element being 1 and the other elements being 0; i.e., Ii ¼ 0 . . . 010 . . . 0. Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations. 2. Problem formulation Consider the following uncertain stochastic system: ðSÞ : dxðtÞ ¼ ½AðaÞxðtÞ þ BðaÞuðtÞ dt þ f ðt; xðtÞ; uðtÞÞ doðtÞ;

xð0Þ ¼ x0 ,

yðtÞ ¼ CðaÞxðtÞ þ DðaÞuðtÞ, n

(1) (2)

m

where xðtÞ 2 R is the state, uðtÞ 2 R is the unknown p input that belongs to L2 ½0; 1Þ, yðtÞ 2 R is the measured l output, and oðtÞ 2 R is an l-dimensional Brownian motion defined on a complete probability space ðO; F; PÞ relative to an increasing family ðFt Þt40 of s-algebras Ft  F, where O is the sample space, F is the s-algebra of subsets of the sample space and P is the probability measure on F. In system ðSÞ, AðaÞ, BðaÞ, CðaÞ and DðaÞ are

ARTICLE IN PRESS B. Zhang et al. / Signal Processing 89 (2009) 605–614

607

uncertain matrices of appropriate dimensions and take the polytopic form " # " # s X AðaÞ BðaÞ Ai Bi , ¼ ai CðaÞ DðaÞ C i Di

Definition 1. The uncertain stochastic system ðSe Þ with uðtÞ ¼ 0 is said to be robustly exponentially mean-square 2n stable if, for all x0 ¼ ½xT0 xTf 0 T 2 R , there exist scalars r40 and s40 such that the following exponential estimate:

where Ai , Bi , C i and Di are known real matrices, while ai , i ¼ 1; 2; . . . ; s, are time-invariant uncertainties satisfying

EfjxðtÞj2 gprest jx0 j2

i¼1

ai X0;

s X

ai ¼ 1.

(3)

i¼1 n

m

such that trace½f ðt; xðtÞ; uðtÞÞT f ðt; xðtÞ; uðtÞÞ " # " #" # xðtÞ T F 1 F 2 xðtÞ p . F T2 F 3 uðtÞ uðtÞ

(4)

For system ðSÞ, we are interested in designing a fullorder filter in the form of ðSf Þ : dxf ðtÞ ¼ ½Af xf ðtÞ þ Bf yðtÞ dt;

xf ð0Þ ¼ xf 0 ,

(5)

uf ðtÞ ¼ C f xf ðtÞ þ Df yðtÞ,

(6)

n

m

where xf ðtÞ 2 R is the filter state, uf ðtÞ 2 R is the filter output that is the estimate of uðtÞ, and Af , Bf , C f and Df are constant real matrices to be determined. Let eðtÞ ¼ uf ðtÞ  uðtÞ and xðtÞ ¼ ½xðtÞT xf ðtÞT T . Then, the deconvolution error system of system ðSÞ with filter ðSf Þ is obtained as ˜ aÞxðtÞ þ Bð ˜ aÞuðtÞ dt ðSe Þ : dxðtÞ ¼ ½Að þ gðt; xðtÞ; uðtÞÞ doðtÞ,

(7)

˜ aÞxðtÞ þ Dð ˜ aÞuðtÞ, eðtÞ ¼ Cð

(8)

where "

AðaÞ Bf CðaÞ

gðt; xðtÞ; uðtÞÞ ¼

0 "

Af

#

" ˜ aÞ ¼ Bð

;

f ðt; xðtÞ; uðtÞÞ

˜ aÞ ¼ ½Df CðaÞ C f ; Cð

BðaÞ Bf DðaÞ

#

#

 0 ; 0

F˜ 2 ¼

 F2 ; 0

keðtÞkE2 pgkuðtÞk2 , (11) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R1 2 where keðtÞkE2 ¼ 0 EfjeðtÞj g dt . (II) Robust L2 2L1 deconvolution filtering problem: For system ðSÞ and prescribed scalars s40 and g40, design a filter in the form of ðSf Þ such that the deconvolution error system ðSe Þ with uðtÞ ¼ 0 is robustly exponentially mean-square stable with a decay rate s and, under zero initial conditions and all nonzero uðtÞ 2 L2 ½0; 1Þ, the following condition is satisfied: keðtÞkE1 pgkuðtÞk2 , where keðtÞkE1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ supt EfjeðtÞj2 g.

(12)

The purpose of this paper is to give decay-ratedependent conditions for the solvability of these two problems.

3.1. Robust H1 deconvolution filter design

0 ˜ aÞ ¼ Df DðaÞ  I. Dð



(I) Robust H1 deconvolution filtering problem: For system ðSÞ and prescribed scalars s40 and g40, design a filter in the form of ðSf Þ such that the deconvolution error system ðSe Þ with uðtÞ ¼ 0 is robustly exponentially mean-square stable with a decay rate s and, under zero initial conditions and all nonzero uðtÞ 2 L2 ½0; 1Þ, the following condition is satisfied:

3. Main results

,

trace½gðt; xðtÞ; uðtÞÞT gðt; xðtÞ; uðtÞÞ 3" " # 2 # xðtÞ T F˜ 1 F˜ 2 xðtÞ 4 5 , p T uðtÞ uðtÞ F˜ 2 F˜ 3

The problems considered in this paper are formulated as follows:

,

It follows from (4) that

where  F1 F˜ 1 ¼ 0

holds for any uncertainties fa satisfying (3). In this case, r and s are called the decay coefficient and the decay rate, respectively.

nl

f ðt; xðtÞ; uðtÞÞ : Rþ  R  R ! R denotes nonlinear stochastic disturbances with f ðt; 0; 0Þ ¼ 0. It is assumed that there exists a real symmetric matrix " # F1 F2 F¼ X0 T F2 F3

˜ aÞ ¼ Að

(10) s i gi¼1

(9)

F˜ 3 ¼ F 3 .

Throughout this paper, we shall adopt the following definition.

Here we study the robust H1 deconvolution filtering problem for system ðSÞ. We first present some parameterdependent conditions for the solvability of this problem, and then transform them into LMIs by using the HPPDM approach. First, for system ðSe Þ we give the following result. Lemma 1. For prescribed scalars s40 and g40, system ðSe Þ is robustly exponentially mean-square stable with a decay rate s and condition (11) is satisfied under zero initial conditions and all nonzero uðtÞ 2 L2 ½0; 1Þ, if there exist parameter-dependent matrices PðaÞ ¼ PðaÞT , Q ðaÞ, RðaÞ and constant scalars m40 and 40, such that the

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B. Zhang et al. / Signal Processing 89 (2009) 605–614

following conditions hold:

mIpPðaÞpI, ˜ aÞT 3 ˜ aÞ þ F˜ 2 Cð F1 ðaÞ F2 ðaÞ Q ðaÞBð 6 7 ˜ aÞ 6  RðaÞ  RðaÞT RðaÞBð 0 7 6 7 6 7o0, 6  ˜ aÞT 7  gI þ F˜ 3 Dð 4 5    gI

(13)

2

˜ aÞxðtÞ þ Bð ˜ aÞuðtÞ. Then, by fol˜ ðtÞ ¼ Að xðtÞT PðaÞxðtÞ and j lowing a similar line as in the above discussion, we obtain 1

g (14)

where ˜ aÞ þ Að ˜ aÞT Q ðaÞT , F1 ðaÞ ¼ sPðaÞ þ F˜ 1 þ Q ðaÞAð

(15)

˜ aÞT RðaÞT . F2 ðaÞ ¼ PðaÞ  Q ðaÞ þ Að

(16)

In this case, a corresponding decay coefficient is given by r ¼ =m. Proof. Under the conditions in (13) and (14), we first prove the robust exponential mean-square stability of system ðSe Þ with uðtÞ ¼ 0. To this end, we use the following Lyapunov function: VðxðtÞÞ ¼ est xðtÞT PðaÞxðtÞ.

(17)

˜ aÞxðtÞ. System (7) with uðtÞ ¼ 0 can be Let jðtÞ ¼ Að rewritten as dxðtÞ ¼ jðtÞ dt þ gðt; xðtÞ; 0Þ doðtÞ.

(18)

Then, by using the Itoˆ formula, we have

LVðxðtÞÞ ¼ sest xðtÞT PðaÞxðtÞ þ 2est xðtÞT PðaÞjðtÞ þ est trace½gðt; xðtÞ; 0ÞT PðaÞgðt; xðtÞ; 0Þ.

T

xðtÞ PðaÞjðtÞ ¼

xðtÞ

#T "

jðtÞ

PðaÞ 0

Q ðaÞ RðaÞ

#"

jðtÞ

(19)

˜ aÞxðtÞ  jðtÞ , Að

: jðtÞ

RðaÞ  RðaÞT



By using the Schur complement equivalence to (14) and noticing that sPðaÞ40, we obtain 2 ˜ 3 ˜ aÞ þ F˜ 2 F1 ðaÞ F2 ðaÞ Q ðaÞBð 6 7 ˜ aÞ 6  7 RðaÞ  RðaÞT RðaÞBð 4 5 ˜   gI þ F 3 2 3 2 T 3T ˜ ˜ aÞT Cð   CðaÞ 6 7 1 6 7 7 7 I 6 þ6 4 0 5 4 0 5 o0,

#9

xðtÞ =

jðtÞ ;

g

˜ aÞT Dð

which, together with (21), implies eðtÞT eðtÞ  guðtÞT uðtÞ þ LV˜ ðxðtÞÞp0.

0

#"

(22)

Then, we have that, for 8t40, Z t    1 E eðsÞT eðsÞ  guðsÞT uðsÞ þ LV˜ ðxðsÞÞ ds p0.

We obtain

F2 ðaÞ

˜ 1 ðaÞ ¼ F˜ 1 þ Q ðaÞAð ˜ aÞ þ Að ˜ aÞT Q ðaÞT . F

g

trace½gðt; xðtÞ; 0ÞT PðaÞgðt; xðtÞ; 0ÞpxðtÞT ðF˜ 1 ÞxðtÞ. 8" # " < xðtÞ T F1 ðaÞ

where

1

#

and, by (9) and (13),

LVðxðtÞÞpest

3 82 ˜ 3 ˜ aÞ þ F˜ 2 xðtÞ T > F2 ðaÞ Q ðaÞBð > F1 ðaÞ 6 7 <6 7 ˜ aÞ 7 6 7 ˜ RðaÞ  RðaÞT RðaÞBð p6 4 jðtÞ 5 >4  5 > : uðtÞ   gI þ F˜ 3 9 2 3 2 T 3T >2 xðtÞ 3 ˜ aÞT ˜ Cð > >   CðaÞ 7 6 7 1 6 7 =6 7 7 6 0 7 6j ˜ 0 (21) þ6 I 4 5 g 4 5 >4 ðtÞ 5, > > uðtÞ T T ; ˜ ˜ DðaÞ DðaÞ 2

˜ aÞT Dð

Note that "

eðtÞT eðtÞ  guðtÞT uðtÞ þ LV˜ ðxðtÞÞ

.

(20) It follows from (14) that " # F1 ðaÞ F2 ðaÞ o0.  RðaÞ  RðaÞT

g

This yields Z t    1 E eðsÞT eðsÞ  guðsÞT uðsÞ ds p0. 0

g

When t ! 1, we obtain the condition in (11). The proof is completed here. & Remark 1. In the proof of Lemma 1, we have found that ˜ ðtÞ can be rewritten as xðtÞT PðaÞj "

This, together with (20), implies LVðxðtÞÞo0. By considering this and recalling (13), we have

mest EfjxðtÞj2 gpEfVðxðtÞÞgpjx j2 , 0

which implies that

 EfjxðtÞj2 gp est jx0 j2 . m Therefore, by Definition 1, system ðSe Þ with uðtÞ ¼ 0 is robustly exponentially mean-square stable with a decay rate s and a decay coefficient r ¼ =m. Now we are in a position to prove that condition (11) holds under zero initial conditions and for all nonzero uðtÞ 2 L2 ½0; 1Þ. For the proof, we define V˜ ðxðtÞÞ ¼

˜ ðtÞ ¼ xðtÞT PðaÞj

xðtÞ

j˜ ðtÞ " 

#T "

PðaÞ

Q ðaÞ

0

RðaÞ

#

j˜ ðtÞ ˜ aÞxðtÞ þ Bð ˜ aÞuðtÞ  j ˜ ðtÞ Að

# ,

˜ aÞxðtÞ þ Bð ˜ aÞuðtÞ, and Q ðaÞ and RðaÞ are any ˜ ðtÞ ¼ Að where j matrices with appropriate dimensions. With the help of this fact, we have derived the conditions in (13) and (14) that do not contain any products of the Lyapunov matrix PðaÞ and the system matrices in ðSe Þ. Thus, conditions (13)–(14) are applicable for the design of robust deconvolution filters when a parameter-dependent Lyapunov matrix is used. Based on Lemma 1, we obtain the following theorem.

ARTICLE IN PRESS B. Zhang et al. / Signal Processing 89 (2009) 605–614

Theorem 1. Consider the uncertain stochastic system ðSÞ. For prescribed scalars s40, g40, d1 and d2 with d2 a0, the robust H1 deconvolution filtering problem is solvable, if there exist parameter-dependent matrices P 1 ðaÞ ¼ P 1 ðaÞT , P 2 ðaÞ, P 3 ðaÞ ¼ P 3 ðaÞT , Q 1 ðaÞ, Q 2 ðaÞ, R1 ðaÞ, R2 ðaÞ, constant ^ f , and constant scalars real matrices S40, W, A^ f , B^ f , C^ f , D m40 and 40, such that the following conditions hold: " # P 1 ðaÞ P 2 ðaÞ mIp (23) pI,  P 3 ðaÞ 2 6 6 6 6 6 6 CðaÞ:¼6 6 6 6 6 6 6 4



7 7 7 7 7 7 7 0 7 7 7 0 7 7 7 T ^T DðaÞ Df  I 7 5 gI T C^ f

C22 ðaÞ C23 ðaÞ C24 ðaÞ C25 ðaÞ C33 ðaÞ C34 ðaÞ C35 ðaÞ



















C55 ðaÞ











3

T

^ CðaÞT D f

6 C11 ðaÞ C12 ðaÞ C13 ðaÞ C14 ðaÞ C15 ðaÞ

C44 ðaÞ C45 ðaÞ

o0,

(24)

where

C11 ðaÞ ¼ sP 1 ðaÞ þ F 1 þ Q 1 ðaÞAðaÞ þ AðaÞT Q 1 ðaÞT T þ B^ f CðaÞ þ CðaÞT B^ f , T

C12 ðaÞ ¼ sP2 ðaÞ þ A^ f þ AðaÞT Q 2 ðaÞT þ CðaÞT B^ f , T

C13 ðaÞ ¼ P 1 ðaÞ  Q 1 ðaÞ þ AðaÞT R1 ðaÞT þ d1 CðaÞT B^ f , C14 ðaÞ ¼ P 2 ðaÞ  W þ AðaÞT R2 ðaÞT þ d2 Cða

T ÞT B^ f ,

C15 ðaÞ ¼ Q 1 ðaÞBðaÞ þ B^ f DðaÞ þ F 2 , T

C22 ðaÞ ¼ sP3 ðaÞ þ A^ f þ A^ f ; T

C24 ðaÞ ¼ P 3 ðaÞ  W þ d2 A^ f ; C33 ðaÞ ¼ R1 ðaÞ  R1 ðaÞT ;

T

C23 ðaÞ ¼ P2 ðaÞT  Q 2 ðaÞ þ d1 A^ f , C25 ðaÞ ¼ Q 2 ðaÞBðaÞ þ B^ f DðaÞ, C34 ðaÞ ¼ d1 W  R2 ðaÞT ,

C35 ðaÞ ¼ R1 ðaÞBðaÞ þ d1 B^ f DðaÞ;

C44 ðaÞ ¼ d2 W  d2 W T ,

C45 ðaÞ ¼ R2 ðaÞBðaÞ þ d2 B^ f DðaÞ;

C55 ðaÞ ¼ gI þ F 3 .

In this case, the parameters of a desired robust H1 deconvolution filter in ðSf Þ are given by 3 " # " #2 ^ ^ Af Bf W 1 0 4 Af Bf 5 . (25) ¼ ^f C f Df C^ f D 0 I Moreover, a corresponding decay coefficient is given by r ¼ =m. Proof. Suppose the conditions of the theorem hold and the parameters of filter ðSf Þ are given by (25). Denote " # " # P1 ðaÞ P2 ðaÞ Q 1 ðaÞ W ; Q ð , a Þ ¼ PðaÞ ¼ P2 ðaÞT P3 ðaÞ Q 2 ðaÞ W " # R1 ðaÞ d1 W RðaÞ ¼ . R2 ðaÞ d2 W Then, it is easy to see that condition (13) is satisfied. Moreover, (24) is equivalent to (14). Therefore, by Lemma 1, we have that the deconvolution error system ðSe Þ is robustly exponentially mean-square stable with a decay rate s and condition (11) is satisfied under zero initial

609

conditions and for all nonzero uðtÞ 2 L2 ½0; 1Þ. The proof is completed here. & Remark 2. Theorem 1 provides sufficient conditions for the solvability of the robust H1 deconvolution filter design problem for uncertain stochastic system ðSÞ. We note that these conditions are still dependent on the uncertain parameters and thus should be transformed into LMIs. One possible way to do that is to set matrices P 1 ðaÞ, etc. to depend linearly on the uncertain parameters. P For example, P 1 ðaÞ ¼ sk¼1 P 1k , where P 1k is a constant real matrix. Such an approach has been largely used in the past; see, for example, [6,8–12]. Another method to transforming conditions (23)–(24) into LMIs is the HPPDM approach [17,18,42]. In this approach, P 1 ðaÞ, etc. can be constructed as homogeneous matrix polynomials with respect to the uncertain parameters. For example, P P 1 ðaÞ ¼ k2KðrÞ ak11 ak22 . . . aks s P 1;k . It is not difficult to see P Ps k1 k2 ks that, when r ¼ 1, k¼1 P 1k . k2KðrÞ a1 a2 . . . as P 1;k ¼ Thus, the HPPDM approach is more general than the methods in [6,8–12]. In the following, we use the HPPDM approach to study the robust deconvolution filtering problems for system ðSÞ. Theorem 2. Consider the uncertain stochastic system ðSÞ. For prescribed scalars s40, g40, d1 and d2 with d2 a0, the robust H1 deconvolution filtering problem is solvable, if there exist a constant scalar 40 and constant real matrices ^ f ,P1;k , P 2;k , P 3;k , Q 1;k , Q 2;k , R1;k , R2;k , S40, W, A^ f , B^ f , C^ f , D k 2 KðrÞ, such that the following LMIs hold for k 2 Kðr þ 1Þ: " # P 1;ðkIi Þ P2;ðkIi Þ X X p ck;i I, (26) 0o  P3;ðkIi Þ i2½1;2;...;s i2½1;2;...;s ki 40

2

6 6 6 6 6 X 6 6 6 Gk : ¼ 6 i2½1;2;...;s6 6 ki 40 6 6 6 4

ki 40

Gk;11

Gk;12

Gk;13

Gk;14

Gk;15



Gk;22

Gk;23

Gk;24

Gk;25





Gk;33

Gk;34

Gk;35







Gk;44

Gk;45









Gk;55











3

^T ck;i C Ti D f

7 7 7 7 7 7 7 0 7 7 7 0 7 7 7 T^T ck;i Di Df  ck;i I 7 5 ck;i gI T ck;i C^ f

o0,

(27)

where ck;i ¼ ðr!Þki =pðkÞ, ki 40, and

Gk;11 ¼ sP1;ðkIi Þ þ ck;i F 1 þ Q 1;ðkIi Þ Ai þ ATi Q T1;ðkIi Þ T þ ck;i B^ f C i þ ck;i C Ti B^ f , T

Gk;12 ¼ sP 2;ðkIi Þ þ ck;i A^ f þ ATi Q T2;ðkIi Þ þ ck;i C Ti B^ f , T

Gk;13 ¼ P 1;ðkIi Þ  Q 1;ðkIi Þ þ ATi RT1;ðkIi Þ þ ck;i d1 C Ti B^ f , T

Gk;14 ¼ P 2;ðkIi Þ  ck;i W þ ATi RT2;ðkIi Þ þ ck;i d2 C Ti B^ f , Gk;15 ¼ Q 1;ðkIi Þ Bi þ ck;i B^ f Di þ ck;i F 2 , T

Gk;22 ¼ sP 3;ðkIi Þ þ ck;i A^ f þ ck;i A^ f , T

Gk;23 ¼ P T2;ðkIi Þ  Q 2;ðkIi Þ þ ck;i d1 A^ f , T

Gk;24 ¼ P 3;ðkIi Þ  ck;i W þ ck;i d2 A^ f , Gk;25 ¼ Q 2;ðkIi Þ Bi þ ck;i B^ f Di ;

Gk;33 ¼ R1;ðkIi Þ  RT1;ðkIi Þ ,

ARTICLE IN PRESS 610

B. Zhang et al. / Signal Processing 89 (2009) 605–614

Gk;35 ¼ R1;ðkIi Þ Bi þ ck;i d1 B^ f Di ,

Gk;34 ¼ ck;i d1 W  RT2;ðkIi Þ ;

Gk;45 ¼ R2;ðkIi Þ Bi þ ck;i d2 B^ f Di ,

T

Gk;44 ¼ ck;i d2 W  ck;i d2 W ; Gk;55 ¼ ck;i gI þ ck;i F 3 .

In this case, the parameters of a desired robust H1 deconvolution filter in ðSf Þ are given by (25). Moreover, a corresponding decay coefficient is given by

 r¼ , m

(28)

where 8 0 1 > > < B X C m ¼ min B ck;i C @ A k2Kðrþ1Þ> > : i2½1;2;...;s k 40 i

0

"

B X B @



i2½1;2;...;s ki 40

k2KðrÞ

X

ak11 ak22 . . . aks s P2;k ,

P3 ðaÞ ¼

ak11 ak22 . . . aks s P3;k ,

k2KðrÞ

Q 1 ðaÞ ¼

X

k1 1

Assumption 1. DðaÞ in (2) is independent of the uncertain parameters; that is, DðaÞ  D, where D is a known real ^ where D ^ is matrix and has full column rank. In (6), Df ¼ D, the Moore–Penrose inverse of D. Remark 3. For the L2 2L1 performance constraint in (12) ˜ aÞ ¼ 0. This is to be well defined, it is required that Dð because input signals with unit L2 norm may have arbitrarily large value under the L1 measure. Note that ˜ aÞ ¼ 0. Assumption 1 ensures that Dð Under Assumption 1, system ðSe Þ is equivalent to the form

k2KðrÞ

X

where Gk is given in (27). Note that Gk o0 for any k 2 Kðr þ 1Þ. Hence, CðaÞo0; that is, (24) is satisfied. Therefore, by Theorem 1, the robust H1 deconvolution filtering problem is solvable and a decay coefficient is given by (28). The proof is completed here. &

In this subsection we consider the robust L2 2L1 deconvolution filtering problem for system ðSÞ. For this problem, we shall make the following assumption.

Proof. We prove this theorem by using Theorem 1. First, the parameter-dependent matrix variables in Theorem 1 are constructed in the homogeneous polynomial form as follows: X k k P1 ðaÞ ¼ a11 a22 . . . aks s P1;k , P2 ðaÞ ¼

k2Kðrþ1Þ

3.2. Robust L2 2L1 deconvolution filter design

11 1 9 > > # > P2;ðkIi Þ C = C . P3;ðkIi Þ A > > ; >

P 1;ðkIi Þ

Therefore, condition (23) is satisfied. In addition, CðaÞ in (24) can be rewritten as X CðaÞ ¼ ak11 ak22 . . . aks s Gk , (29)

k2 2

a a . . . aks s Q 1;k ,

˜ aÞxðtÞ þ Bð ˜ aÞuðtÞ dt ðS0e Þ : dxðtÞ ¼ ½Að þ gðt; xðtÞ; uðtÞÞ doðtÞ,

(30)

˜ aÞxðtÞ, eðtÞ ¼ Cð

(31)

k2KðrÞ

Q 2 ðaÞ ¼

X

ak11 ak22 . . . aks s Q 2;k ,

k2KðrÞ

R1 ðaÞ ¼

X

For this system, we present the following result.

ak11 ak22 . . . aks s R1;k ,

Lemma 2. Under Assumption 1 and for prescribed scalars s40 and g40, system ðS0e Þ is robustly exponentially meansquare stable with a decay rate s and the condition (12) is satisfied under zero initial conditions and all nonzero uðtÞ 2 L2 ½0; 1Þ, if there exist parameter-dependent matrices PðaÞ ¼ PðaÞT , Q ðaÞ, RðaÞ and constant scalars m40 and 40, such that the following conditions hold:

k2KðrÞ

R2 ðaÞ ¼

X

k1 1

k2 2

a a ...a

ks s R2;k .

k2KðrÞ

It can be verified that " # P1 ðaÞ P 2 ðaÞ P 3 ðaÞ



X

¼

k2Kðrþ1Þ

2 6 X

"

ak11 ak22 . . . aks s 6 4

i2½1;2;...;s ki 40

0 X

Xm

k1 1

k2 2

B X

ks B s @

a a ...a

k2Kðrþ1Þ

i2½1;2;...;s ki 40

P1;ðkIi Þ 

3 # P 2;ðkIi Þ 7 7 5 P 3;ðkIi Þ

1 C ck;i C AI



On the other hand, it follows from (26) that

P 1 ðaÞ

P 2 ðaÞ



P 3 ðaÞ

# p

0 X k2Kðrþ1Þ

1

B X

ak11 ak22 . . . aks s B @

i2½1;2;...;s ki 40



(32) (33) ˜ aÞ þ F˜ 2 Q ðaÞBð ˜ aÞ RðaÞBð gI þ F˜ 3

3 7 7o0, 5

(34)

where F1 and F2 are given in (15) and (16), respectively. In this case, a corresponding decay coefficient is given by r ¼ =m.

¼ mI.

"

mIpPðaÞpI, 2 3 ˜ aÞT PðaÞ Cð 4 540,  gI 2 F1 ðaÞ F2 ðaÞ 6 6  RðaÞ  RðaÞT 4

C ck;i C AI ¼ I.

Proof. Using the same Lyapunov function as that in (17) and following a similar line as in the proof of Lemma 1, we can prove that, under the conditions of Lemma 2, system ðS0e Þ is robustly exponentially mean-square stable with a decay rate s and a decay coefficient r ¼ =m.

ARTICLE IN PRESS B. Zhang et al. / Signal Processing 89 (2009) 605–614

Now, under zero initial conditions and for all nonzero uðtÞ 2 L2 ½0; 1Þ, we define V˜ ðxðtÞÞ ¼ xðtÞT PðaÞxðtÞ and ˜ aÞxðtÞ þ Bð ˜ aÞuðtÞ. Then, it can be verified that j˜ ðtÞ ¼ Að

3T

xðtÞ 6 7 6 ˜ ðtÞ 7 ¼ 4j 5 uðtÞ 2 ˜ F1 ðaÞ 6 6 4 

F2 ðaÞ

˜ aÞ þ F˜ 2 Q ðaÞBð

RðaÞ  RðaÞT

˜ aÞ RðaÞBð



gI þ F˜ 3



32

3 xðtÞ 7 76 7 76 j ˜ 54 ðtÞ 5, uðtÞ (35)

˜ 1 ðaÞ is given in (22). It follows from (34) that where F 2 ˜ F1 ðaÞ 6  4 

F2 ðaÞ RðaÞ  RðaÞT 

3 ˜ aÞ þ F˜ 2 Q ðaÞBð 7 ˜ aÞ RðaÞBð 5o0. gI þ F˜ 3





Z

t

uðsÞT uðsÞ ds.

(36)

0

On the other hand, applying the Schur complement ˜ aÞogPðaÞ. Hence ˜ aÞT Cð equivalence to (33) leads to Cð

EfeðtÞT eðtÞgpEfgxðtÞT PðaÞxðtÞg ¼ gEfV˜ ðxðtÞÞg.

(37)

From (36) and (37), we have T

EfeðtÞ eðtÞgpg

2

Z

t

T

2

uðsÞ uðsÞ dspg

0

Z

1

T

uðtÞ uðtÞ dt.

P 2 ðaÞ



P 3 ðaÞ

# pI,



P3 ðaÞ

T C^ f





gI

T

3 7 7 740, 5

(39)

C11 ðaÞ C12 ðaÞ C13 ðaÞ C14 ðaÞ C15 ðaÞ

3 7

C22 ðaÞ C23 ðaÞ C24 ðaÞ C25 ðaÞ 7 7



7

C33 ðaÞ C34 ðaÞ C35 ðaÞ 7 7o0,

















7

(40)

C44 ðaÞ C45 ðaÞ 7 5 C55 ðaÞ



where Cij ðaÞ, 1pipjp5, are given in Theorem 1 with DðaÞ replaced by D. In this case, the parameters of a desired robust L2 2L1 deconvolution filter in ðSf Þ are given by 3 " # " #2 ^ ^ A f Bf W 1 0 4 Af Bf 5 ¼ . (41) ^ C f Df C^ f D 0 I

Theorem 4. Consider the uncertain stochastic system ðSÞ. Under Assumption 1 and for prescribed scalars s40, g40, d1 and d2 with d2 a0, the robust L2 2L1 deconvolution filtering problem is solvable, if there exist a constant scalar 40 and constant real matrices S40, W, A^ f , B^ f , C^ f , P 1;k , P2;k , P 3;k , Q 1;k , Q 2;k , R1;k , R2;k , k 2 KðrÞ, such that the following LMIs hold for k 2 Kðr þ 1Þ: X i2½1;2;...;s ki 40

"

2

2

Theorem 3. Consider the uncertain stochastic system ðSÞ. Under Assumption 1 and for prescribed scalars s40, g40, d1 and d2 with d2 a0, the robust L2 2L1 deconvolution filtering problem is solvable, if there exist parameter-dependent matrices P1 ðaÞ ¼ P 1 ðaÞT , P 2 ðaÞ, P3 ðaÞ ¼ P 3 ðaÞT , Q 1 ðaÞ, Q 2 ðaÞ, R1 ðaÞ, R2 ðaÞ, constant real matrices S40, W, A^ f , B^ f , C^ f , and constant scalars m40 and 40, such that the following conditions hold: P 1 ðaÞ

^ CðaÞT D

ki 40

0

The following two theorems provide sufficient conditions for the solvability of the robust L2 2L1 deconvolution filtering problem for system ðSÞ. Their proofs can be established based on Lemma 2 and using similar techniques to those in the previous subsection, and thus are omitted here.

"

P2 ðaÞ

X 6 6 6 6 i2½1;2;...;s4

This implies the condition in (12) and thus completes the proof. &

mIp

6 6 6 6 6 6 6 6 4

P 1 ðaÞ

Moreover, a corresponding decay coefficient is given by r ¼ =m.

This, together with (35), implies that LV˜ ðxðtÞÞpguðtÞT uðtÞ, which further yields

E V˜ ðxðtÞÞ pg

6 6 6 4 2

LV˜ ðxðtÞÞ  guðtÞT uðtÞ 2

2

611

(38)

6 6 X 6 6 6 6 i2½1;2;...;s6 6 ki 40 4

P1;ðkIi Þ

P 2;ðkIi Þ



P 3;ðkIi Þ

P 1;ðkIi Þ

P 2;ðkIi Þ



P 3;ðkIi Þ





# p

X

ck;i I,

(42)

i2½1;2;...;s ki 40

^ ck;i C Ti D

T

3

7 7 T 7 ^ ck;i C f 740, 5 ck;i gI

(43) 3

Gk;11

Gk;12

Gk;13

Gk;14

Gk;15



Gk;22

Gk;23

Gk;24

Gk;25 7





Gk;33

Gk;34







Gk;44









7 7 7 Gk;35 7 7o0, 7 Gk;45 7 5

(44)

Gk;55

where ck;i and Gk;jl , 1pjplp5, are given in Theorem 2 with Di replaced by D. In this case, the parameters of a desired robust L2 2L1 deconvolution filter in ðSf Þ are given by (41). Moreover, a corresponding decay coefficient is given by (28). Remark 4. By using the HPPDM approach together with combinatoric mathematics, LMI-based conditions are obtained in Theorems 2 and 4 for the design of robust H1 and L2 2L1 deconvolution filters for polytopic uncertain stochastic systems. We note that, if the LMIs of (26)–(27) or (42)–(44) are feasible for a given degree r^ , then the LMIs corresponding to any degree r4r^ are also feasible [14,15]. Therefore, the LMI-based conditions in Theorems 2 and 4 will be less conservative as the degree r increases. This is also shown numerically in the next section.

ARTICLE IN PRESS 612

B. Zhang et al. / Signal Processing 89 (2009) 605–614

Remark 5. The conditions of Theorems 2 and 4 are dependent on the decay rate s. This enables one to design robust H1 and L2 2L1 deconvolution filters such that the resulting deconvolution error system converges with a specified decay rate, which may be practically important. Moreover, the left-hand side of (27) or (44) is a monotonic increasing matrix function with respect to s. Thus, when the scalars g40, d1 and d2 are given, a maximum allowed value of the decay rate s can be obtained by using the convex optimization algorithms. Remark 6. It is worth noting that the conditions of Theorems 2 and 4 are LMIs not only over the matrix variables but also over the scalar g. Hence, when the scalars s40, d1 and d2 are given, we can obtain the minimum allowed values of the H1 and L2 2L1 performance level by solving the corresponding LMIs. It is also worth mentioning that the HPPDM approach has been applied to study the robust H2 and H1 filtering problems for discrete-time systems with polytopic uncertainties in [17,18,42], respectively, where some valuable results have been reported. We note that, by using the approaches developed in [17,18,42] together with the ideas given in this paper, one can further investigate the robust H1 and L2 2L1 deconvolution filtering problems for discrete-time stochastic systems with polytopic uncertainties. 4. Numerical examples In this section, we provide two examples to demonstrate the effectiveness of the proposed design methods. Example 1 is for the robust H1 deconvolution filtering problem, while Example 2 is for the robust L2 2L1 deconvolution filtering problem. Example 1. Consider the following stochastic system: (" # " # ) 0:5a 0:1 3 þ 0:5a dxðtÞ ¼ xðtÞ þ uðtÞ dt 0:9a 3 4 (" # " # ) 0:5 0 0:1 (45) þ xðtÞ þ uðtÞ doðtÞ, 0 0:5 0:1 yðtÞ ¼ ½0:8 0:8ð1 þ aÞxðtÞ þ ð0:45  0:5aÞuðtÞ,

(46)

where a is a bounded constant uncertain parameter satisfying jajp1. This system can be rewritten as ðSÞ with " # " # 0:1 2:5 0:5 A1 ¼ ; B1 ¼ ; C 1 ¼ ½0:8 0, 0:9 3 4 D1 ¼ 0:5, " 0:1 A2 ¼ 3

3:5

D2 ¼ 0:4, " 0:25 F1 ¼ 0

0

#

" B2 ¼

;

4

0:25

#

0:5 0:9

" ;

F2 ¼

0:05 0:05

# ;

obtained when the degree r of the homogeneous polynomial matrices is higher. Now, by Theorem 2, we shall design a robust H1 deconvolution filter for system (45)–(46) and give an exponential estimate for the filtering error system. To this end, we choose s ¼ 1:5, g ¼ 0:6, d1 ¼ 0:5, d2 ¼ 0:5 and r ¼ 2. Then, by Theorem 2, the parameters of a desired H1 deconvolution filter are obtained as follows: " # " # 0:2378 4:5615 0:9918 ; Bf ¼ , Af ¼ 5:0745 6:0789 1:7961 C f ¼ ½0:7227 0:3734;

Df ¼ 1:3559.

Moreover, by Theorem 2 we also obtain a decay coefficient as r ¼ 19:3243; that is, an exponential estimate for the resulting deconvolution error system in the form of ðSe Þ is

EfjxðtÞj2 gp19:3243e1:5t jx0 j2 . Example 2. Consider system ðSÞ with " # " # 1:2 6 0 ; B1 ¼ ; C 1 ¼ ½0 2, A1 ¼ 8 1:2 1 D1 ¼ 0:2, " 0 A2 ¼ 4 D2 ¼ 0:2, " 0:01 F1 ¼ 0

3:5

#

" B2 ¼

;

5 0 0:01

1

#

0 "

;

F2 ¼

# ;

0:01 0:01

C 2 ¼ ½3 0, # ;

F 3 ¼ 0:02.

By applying Theorem 4 with d1 ¼ 0:5 and d2 ¼ 0:5, Table 3 gives the minimum allowed values of the L2 2L1 performance g for different s and r, and Table 4 gives the maximum allowed values of the decay rates s for different g and r. It is seen from these tables that better results can be obtained when the degree r of the homogeneous polynomial matrices is higher. In addition, we choose s ¼ 0:9, g ¼ 7, d1 ¼ 0:5, d2 ¼ 0:5 and r ¼ 2. Then, by Theorem 4, the parameters of a desired

Table 1 Minimum allowed values of H1 performance g for Example 1 ðd1 ¼ d2 ¼ 0:5Þ. Cases

s¼0

s ¼ 0:5

s ¼ 1:0

s ¼ 2:0

s ¼ 3:0

r¼1 r¼2 r¼3

0.4594 0.4456 0.4436

0.4967 0.4816 0.4793

0.5435 0.5267 0.5241

0.6838 0.6635 0.6615

0.9658 0.9474 0.9473

C 2 ¼ ½0:8 1:6,

# ;

F 3 ¼ 0:02.

By using Theorem 2 with d1 ¼ 0:5 and d2 ¼ 0:5, Table 1 gives the minimum allowed values of the H1 performance g for different s and r, and Table 2 gives the maximum allowed values of the decay rates s for different g and r. It is seen from these tables that better results can be

Table 2 Maximum allowed values of decay rates s for Example 1 ðd1 ¼ d2 ¼ 0:5Þ. Cases

g ¼ 0:45

g ¼ 0:5

g ¼ 0:6

g ¼ 0:7

g ¼ 0:8

r¼1 r¼2 r¼3

Infeasible 0.0679 0.0978

0.5394 0.7191 0.7463

1.4743 1.6089 1.6267

2.0839 2.1850 2.1922

2.5044 2.5838 2.5841

ARTICLE IN PRESS B. Zhang et al. / Signal Processing 89 (2009) 605–614

Table 3 Minimum allowed values of L2  L1 performance g for Example 2 ðd1 ¼ d2 ¼ 0:5Þ. Cases

s¼0

s ¼ 0:5

s ¼ 1:0

s ¼ 1:5

s ¼ 2:0

r¼1 r¼2 r¼3

5.7705 5.7029 5.7029

6.3635 6.2068 6.2068

7.3525 6.9344 6.9344

9.1595 8.0103 7.9839

13.4721 9.9132 9.7842

Table 4 Maximum allowed values of decay rates s for Example 2 ðd1 ¼ d2 ¼ 0:5Þ. Cases

g ¼ 6:0

g ¼ 7:0

g ¼ 8:0

g ¼ 9:0

g ¼ 10

r¼1 r¼2 r¼3

0.2205 0.3162 0.3162

0.8530 1.0372 1.0372

1.2177 1.4961 1.5062

1.4672 1.8081 1.8278

1.6483 2.0140 2.0359

L2 2L1 deconvolution filter are obtained as follows: " # " # 6:3549 22:5071 9:1675 ; Bf ¼ , Af ¼ 6:1933 5:6630 1:8490 C f ¼ ½2:5780 6:6519;

Df ¼ 5.

Moreover, by Theorem 4 we also obtain a decay coefficient as r ¼ 18:2779; that is, an exponential estimate for the resulting deconvolution error system in the form of ðS0e Þ is

EfjxðtÞj2 gp18:2779e0:9t jx0 j2 . 5. Conclusions In this paper, we have investigated the robust H1 and L2 2L1 deconvolution filtering problems for stochastic systems with time-invariant polytopic uncertainties. Based on the HPPDM approach and using the combination mathematics techniques, decay-rate-dependent conditions for the solvability of the two problems have been presented in terms of LMIs. By using the solutions of these LMIs, the desired robust deconvolution filters have been designed and the exponential estimates of the deconvolution error system have been given. The effectiveness of the proposed design methods has been demonstrated by two examples provided. References [1] J.C. Geromel, M.C. de Oliveira, H2 and H1 robust filtering for convex bounded uncertain systems, IEEE Trans. Automat. Control 46 (1) (2001) 100–107. [2] J.C. Geromel, Optimal linear filtering under parameter uncertainty, IEEE Trans. Signal Process. 47 (1) (1999) 168–175. [3] S.H. Jin, J.B. Park, Robust H1 filtering for polytopic uncertain systems via convex optimisation, IEE Proc. Control Theory Appl. 148 (1) (2001) 55–59. [4] S. Zhou, B. Zhang, W.X. Zheng, Gain-scheduled H1 filtering of parameter-varying systems, Internat. J. Robust Nonlinear Control 16 (8) (2006) 397–411. [5] N.T. Hoang, H.D. Tuan, P. Apkarian, S. Hosoe, Gain-scheduled filtering for time-varying discrete systems, IEEE Trans. Signal Process. 52 (9) (2004) 2464–2476. [6] J.C. Geromel, M.C. de Oliveira, J. Bernussou, Robust filtering of discrete-time linear systems with parameter dependent Lyapunov functions, SIAM J. Control Optim. 41 (3) (2002) 700–711.

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