Extended passive filtering for discrete-time singular Markov jump systems with time-varying delays

Author’s Accepted Manuscript Extended passive filtering for discrete-time singular Markov jump systems with time-varying delays Hao Shen, Lei Su, Ju H. Park

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To appear in: Signal Processing Received date: 2 December 2015 Revised date: 6 February 2016 Accepted date: 13 March 2016 Cite this article as: Hao Shen, Lei Su and Ju H. Park, Extended passive filtering for discrete-time singular Markov jump systems with time-varying delays, Signal Processing, http://dx.doi.org/10.1016/j.sigpro.2016.03.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Extended passive filtering for discrete-time singular Markov jump systems with time-varying delays 1

∗

Hao Shen1 , Lei Su 1 , Ju H. Park2 † School of Electrical and Information Engineering, Anhui University of Technology, Ma’anshan 243002, PR China 2 Department of Electrical Engineering, Yeungnam University, 280 Daehak-Ro, Kyongsan 38541, Republic of Korea

Abstract: In this paper, the extended passive filtering problem is investigated for a class of discretetime singular Markov jump systems (SMJSs) with time-varying delays. Our attention is focused on the design of a general filter which contains the mode-independent part and the mode-dependent part to address the filtering issue. By employing some novel inequalities based on Abel lemma, some sufficient conditions which ensure the considered SMJSs to be stochastically admissible with a mixed H∞ and passive performance γ are established. Based on the conditions, an explicit expression for the desired filter is given. Three numerical examples and a dynamical Leontief model of economic systems are given to demonstrate the reduced conservatism and effectiveness of the presented general filter technique. Keywords: Singular systems; Markov jump systems; mixed H∞ and passive filtering; time-varying delays

1

Introduction

Singular systems have received extensively attention during the past decades due basically to their powerful applications in many practical systems, such as electric circuits, robotic systems, economic systems. In comparison with the regular systems, singular systems are more general and complex owing to the existence of algebraic equation constraints. Although some essential results on regular systems have been extended to singular systems, there is still room for investigation of many difficult problems for singular systems. For instance, when the stability analysis problem is taken into account, one has to consider simultaneously the regularity and impulses free (for continuous-time singular systems) and causality (for discrete-time singular systems) of singular systems. So far, a good many of important and interesting results on singular systems have been reported in the existing literature, see, for instance [3, 35] and the references therein. As a special class of hybrid systems, Markovian jump systems (MJSs) are popular in modeling many practical systems that may encounter random abrupt changes in their structures or parameters [16, 20, 27, 28, 29, 36, 37]. The past years have seen a surge of research interest on MJSs, and a great number of results ∗

This work was supported by the National Natural Science Foundation of China under Grant 61304066,61473171,61503002, the Natural Science Foundation of Anhui Province under Grant 1308085QF119, the Research Project of State Key Laboratory of Mechanical System and Vibration under Grant MSV201509, the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2013R1A1A2A10005201). † Corresponding author: J.H. Park, Email addresses: [email protected](H. Shen); [email protected](L. Su); [email protected](J.H. Park)

1

have been reported in the literature. To mention just a few, in [33] , the filtering problem for discrete-time MJSs was investigated. In [21], the control problem of nonlinear MJSs with general transition probabilities allowed to be known, uncertain and unknown was proposed. When singular Markov jump systems (SMJSs) were considered, the stabilization problem for continuous-time SMJSs with generally uncertain transition rates was studied in [6], and such a problem was further investigated in [13]. Nevertheless, it should be pointed out that the above mentioned papers ignored the time delays which are inherent features of many physical processes. It is therefore significant to investigate SMJSs taking time delays into account. To date, despite many existing methods can be used to obtain delay-dependent results on time-delay systems [14, 38, 39], they have not been appropriately applied for SMJSs with time delays. Accordingly, how to obtain some less conservative results on SMJSs with time delays is a naturally interesting question. On another research front, studies on the H∞ filtering problem have been gaining considerable momentum over the past few years. The main aim is to design a filter such that the mapping from the external input to the filtering error is minimized or is less than a prescribed level according to the H∞ norm. Up to now, many results about the H∞ filtering have been reported, see, for example [2, 4, 7, 8, 15, 26] . Especially, the problems of performance analysis and filter design for continuous- and discrete-time SMJSs were addressed in [25] and [5], respectively. In addition, passivity theory plays a key role in analyzing the stability of a dynamical system [30, 34]. There is no doubt that many scholars devote themselves to investigating the passive filtering problem [1, 17, 18, 23]. It is worth mentioning that, an H∞ filter may be needed by the researchers now, but a passive filter will be more appropriate in the next time in view of the practical working situation. In this case, how to design a filter such that the filtering error system is passive and simultaneously satisfies a prescribed level of H∞ performance is an attractive issue. Although such an issue was investigated in [12] for Markovian jump impulsive networked control systems with norm bounded uncertainties, little attention has been paid for discrete-time SMJSs with time-varying delays so far. In this paper, we endeavor to study the extended passive filtering problem for discrete-time SMJSs with time-varying delays. The main contributions of this paper are three-fold: 1. Compared with the sole mode-dependent/independent filter design method, in this paper we construct a mixed filter, which can switch between the mode-dependent filter and the mode-independent filter by tuning the weighting parameter α. 2. A trade-off index between the H∞ and the passive performance is presented in designing the filter which makes the considered issue more general. 3. The Abel lemma-based finite-sum inequality is adopted to cope with reducing the conservatism of the established delay-dependent conditions. Meanwhile, the burden of computing loads could be alleviated because no slack variables are brought for computation. The rest of this paper is organized as follows. The addressed problem is formulated in Section 2. Our main results are presented in Section 3, where some mixed H∞ and passive performance conditions are established. The method to calculate the parameters of the filter is also given. Section 4 provides three examples to demonstrate the effectiveness and the reduced conservatism of the proposed method. Finally, we conclude the paper in Section 5. Notation : The following notations will be used throughout the paper: Rn and Rm×n denote the ndimensional Euclidean space and the set of all m × n real matrices, respectively. S > 0 means that matrix S is real symmetric and positive definite. The superscript ”T” stands for the transpose. Sym {X} is defined as X + X T . E {·} denotes the expectation operator with respect to some probability measure P. The symbol ”∗” is used to represent a matrix which can be inferred by symmetry. If not explicitly stated, all matrices are assumed to have compatible dimensions for algebraic operations. 2

2

Problem Formulation

Fix a probability space (Ω, F, P) and consider the following discrete-time singular Markovian jump system (SMJS) with time-varying delays: Ex (k + 1) = A (σ (k)) x (k) + Ad (σ (k)) x (k − d (k)) + B (σ (k)) ω (k) , y (k) = C (σ (k)) x (k) + Cd (σ (k)) x (k − d (k)) + D1 (σ (k)) ω (k) , z (k) = L (σ (k)) x (k) + Ld (σ (k)) x (k − d (k)) + D2 (σ (k)) ω (k) , x (k) = φ (k) ,

k = −d2 , − d2 + 1, . . . , 0,

(1)

where x (k) ∈ Rn is the state vector, ω (k) ∈ Rq is the state of exogenous inputs belongs to l2 [0, ∞), y (k) ∈ Rm is the measurement vector, z (k) ∈ Rp is the signal to be estimated. The matrix E ∈ Rn×n is singular and has rank r (r < n). d (k) is assumed to be interval varying and d1 ≤ d (k) ≤ d2 , where d1 , d2 satisfying d1 > 1 and d2 − d1 > 1, denote the lower and the upper bounds of d (k) , respectively. Here, let σ (k) be a discrete-time Markov chain taking values in a finite set N = {1, 2, . . . , N } with transition probability matrix Π  {πin } given by: Pr {σ (k + 1) = n |σ (k) = i } = πin ,

(2)

where 0 ≤ πin ≤ 1, ∀i, n ∈ N , and ΣN n=1 πin = 1, ∀i ∈ N . For notational simplicity, A (σ (k)) will be denoted by Ai for each σ (k) = i ∈ N , and the other symbols are similarly denoted. In this paper, we introduce the following classes of filters described by  xf (k + 1) = Af i xf (k) + Bf i y (k) , (3) Σ1f = zf (k) = Lf i xf (k)  Σ2f =

xf (k + 1) = Af xf (k) + Bf y (k) , zf (k) = Lf xf (k)

(4)

where xf (k) ∈ Rn and zf (k) ∈ Rp are the states of the filter and the output vector, respectively; Af i , Bf i , Lf i , Af , Bf and Lf are the filter gains to be designed. Roughly speaking, the filter (3) is said to be mode-dependent, while the filter (4) is said to be mode-independent. Remark 1 It has come to be widely recognized that the mode-dependent filter (3) is a powerful tool to cope with the state estimation for SMJSs with fully available modes information. However, in practice, the modes information can not sometimes successfully be transmitted to the filter especially in communication network medium. In this case, the normal mode-dependent filter design method is inapplicable, and consequently the mode-independent filter design approach can be a good choice to overcome this gap. Therefore, how to comprehensively apply the respective advantages of mode-dependent/independent filters in light of practical situations is a naturally interesting question. In view of Remark 1, it is reasonable that we will be interested in the following filter according to practical situations, which is described by Σf = (1 − α) Σ1f + αΣ2f ,

α ∈ {0, 1} .

(5)

Remark 2 It should be pointed out that the expression (5) is a filter with mixed design, which contains the mode-independent filter and the mode-dependent filter. By turning the weighting parameter α, the 3

designed filter can switch between the mode-dependent/independent filter. For example, when α ≡ 1, the expression (5) reduces to the mode-independent filter; and when α ≡ 0, the expression in ( 5) denotes the mode-dependent filter. Combining filter (5) with system (1), results in the filtering error system as follows: ⎧ ¯x ¯i ω (k) , ⎪ E ¯ (k + 1) = A¯i x ¯ (k) + A¯di H x ¯ (k − d (k)) + B ⎪ ⎪ ⎪ ⎨ ¯ ix ¯ (k)+ Ldi Hx ¯ (k − d (k)) + D2i ω (k) , z˜ (k) = L ¯f = Σ ⎪ φ (k) ⎪ ⎪ ¯ (k) = φ¯ (k) = , k = −d2 , . . . , −1, 0, ⎪ ⎩ x 0 where





(6)



 A 0 i ¯ = E , A¯i = , αBf Ci + (1 − α) Bf i Ci αAf + (1 − α) Af i     A B i di ¯i = , B , A¯di = αBf Cdi + (1 − α) Bf i Cdi αBf D1i + (1 − α) Bf i D1i  



x (k) ¯i = , H= I 0 , x ¯ (k) = L Li −αLf − (1 − α) Lf i , xf (k) E 0 0 I

z˜ (k) = z (k) − zf (k) . Before proceeding further, the following definitions and lemmas are given to facilitate the subsequent discussion. Definition 1 [10] The system (6) with ω (k) = 0 is said to be  ¯ A¯i is regular and causal for each i ∈ N . (a) regular and causal if the pair E, (b) stochastically stable, if for any initial state (φ (k) , σ (0)) such that the following condition holds :  ∞

x (k)2 |φ, σ0 < ∞. (7) E k=0

(c) stochastically admissible if it is regular, causal and stochastically stable. Definition 2 [22] Given a scalar δ ∈ [0, 1], γ > 0, the system (6) is said to be stochastically admissible with a mixed H∞ and passive performance γ. Then, under zero initial state, the following conditions are satisfied: (1) the system (6) is stochastically admissible according to Definition 1; (2) under zero initial condition, the following condition is satisfied:  ∞

  T T 2 T ≥ 0, (8) −δ˜ z (k) z˜ (k) + 2 (1 − δ) γ z˜ (k) ω (k) + γ ω (k) ω (k) E k=0

for any non-zero ω (k) ∈ l2 [0, ∞). Remark 3 The index (8) covers the H∞ performance index and the passive performance index as special cases, which are stated as follows: a) if we set δ = 1, the index (8) converts to the H∞ performance index. b) if we choose δ = 0, the index (8) switches to the passive performance index. c) if the scalar δ ∈ (0, 1), the index (8) stands for the mixed H∞ and passive performance index. 4

Lemma 1 [38] (An Abel Lemma-based Finite-sum Inequality) For a positive matrix R, integers τ1 and τ2 with τ2 − τ1 > 1, the following inequality holds: ΩR (τ1 , τ2 ) ≥

1 T 3 2 T χ1 Rχ1 + χ Rχ2 ,

1

1 3 2

(9)

where τ 2 −1

ΩR (τ1 , τ2 ) 

η T (j) Rη (j) ,

η (j)  x (j + 1) − x (j) ,

1  τ 2 − τ 1 ,

j=τ1

2  τ2 − τ1 − 1, χ2

3  τ2 − τ1 + 1, χ1  x (τ2 ) − x (τ1 ) , τ 2 −1 2  x (τ2 ) + x (τ1 ) − x (j) . τ2 − τ1 − 1 j=τ1 +1

Lemma 2 [39] For any constant positive matrix W, two positive integers τ2 and τ1 satisfying τ2 ≥ τ1 ≥ 1, the following inequality holds: (τ2 − τ1 + 1)

τ2

⎛ xT (j) W x (j) ≥ ⎝

j=τ1

3

τ2

j=τ1

⎞T

⎛

x (j)⎠ W ⎝

τ2

⎞ x (j)⎠ .

j=τ1

Main Results

In this section, we first give a sufficient condition, which guarantees that the system (6) is stochastically admissible with a mixed H∞ and passive performance γ. Theorem 1 For given scalars d1 , d2 , γ, and δ ∈ [0, 1] , if there exist matrices Pi > 0, Q1 > 0, Q2 > 0, Q3 > 0, R1 > 0, R2 > 0, matrices Si and Gi such that the following matrix inequalities hold for each mode i ∈ N  (10) Ψi  ΨTi = Ψiab < 0, a, b = 1, 2, 3, 4, where Ψi11

¯ T Pi E ¯+  −E

2

 H

m=1 3

  ¯ T A¯i + +Sym Si Θ

Ψi13 Ψi23

 3 (dm − 1) T E Rm E H (dm − 1) Rm − E Rm E − dm + 1 T

H T Qm H + (d2 − d1 ) H T Q3 H,

m=1

1 −1) T E T R E H T E T R E − 3(d2 −1) H T E T R E S Θ ¯ T Adi H , H H T E T R1 E − 3(d 1 2 2 i d1 +1 d2 +1   3 (d1 − 1) T 3 (d2 − 1) T T T T E R1 E, −Q2 − E R2 E − E R2 E, −H Q3 H ,  diag −Q1 − E R1 E − d1 + 1 d2 + 1

¯ T + Si Θ ¯TB ¯i ,  d16+1 H T E T R1 E d26+1 H T E T R2 E − (1 − δ) γ L i   6 6 E T R1 E, E T R2 E, 0 ,  diag d1 + 1 d2 + 1

Ψi12  Ψi22

T



5



−12 1 −12 1 E T R1 E − R1 , E T R2 E − R2 , (d1 + 1) (d1 − 1) d1 − 1 (d2 + 1) (d2 − 1) d2 − 1  −γ 2 I − (1 − δ) γSym {D2i } , √ T

¯ , Ψi14  A¯Ti Xi d1 H T (Ai − E)T R1 d2 H T (Ai − E)T R2 δL i ⎡ ⎤ 0 0 0 0 ⎢ ⎥ i Ψ24  ⎣ 0 0 0 0 ⎦, √ δH T LTdi H T A¯Tdi Xi d1 H T ATdi R1 d2 H T ATdi R2 ⎡ ⎤ 0 0 0 0 ⎢ ⎥ i Ψ34  ⎣ 0 0 0 0 ⎦, √ T T T T ¯ δD B Xi d1 B R1 d2 B R2 Ψi33

 diag

i

i

2i

i

Ψi44  diag {−Xi , −R1 , −R2 , −I} ,

Xi 

N

πin Pn ,

n=1

¯ = 0, then the ¯ is any matrix with appropriate dimensions and full column rank that satisfies E ¯T Θ and Θ system (6) is stochastically admissible with a mixed H∞ and passive performance γ. Proof. In view of Definition 1, system (6) is readily proven to be regular and causal by following the line of the proof of Theorem 1 in [31]. Now, we are ready to demonstrate that the system (6) is stochastically admissible with a mixed H∞ and passive performance γ. Define η (j) = x (j + 1) − x (j) and choose a Lyapunov functional candidate as: V (k) =

4

Vl (k) ,

(11)

l=1

where ¯ T Pi E ¯x ¯T (k) E ¯ (k) , V1 (k) = x V2 (k) =

k−1

T

x (s) Q1 x (s) +

s=k−d1

V3 (k) = d1

−1

k−1

x (s) Q2 x (s) +

s=k−d2 k−1

−1

k−1

s=−d1 +1 j=k+s

xT (s) Q3 x (s) ,

s=k−d(k)

η T (j) E T R1 Eη (j) + d2

s=−d1 j=k+s

V4 (k) =

k−1

T

−1

k−1

η T (j) E T R2 Eη (j) ,

s=−d2 j=k+s

xT (j) R1 x (j) +

−1

k−1

s=−d2 +1 j=k+s

xT (j) R2 x (j) +

−d1

k−1

xT (j) Q3 x (j) .

s=−d2 +1 j=k+s

(12)

6

Then, define ΔV (k) = E {V (¯ x (k + 1) |σ (k) = i ) − V (¯ x (k))} , and we can obtain ¯ T Xi E ¯ T Pi E ¯x ¯x ¯T (k + 1) E ¯ (k + 1) − x ¯T (k) E ¯ (k) , ΔV1 (k) = x

(13)

ΔV2 (k) ≤ x ¯T (k) H T (Q1 + Q2 + Q3 ) H x ¯ (k) − xT (k − d1 ) Q1 x (k − d1 ) − xT (k − d2 ) Q2 x (k − d2 ) T

k−d

1

T

¯ (k − d (k)) + −¯ x (k − d (k)) H Q3 H x

xT (s) Q3 x (s) ,

(14)

s=k−d2 +1

ΔV3 (k) =

d21 η T

T

(k) E R1 Eη (k) +

d22 η T

k−1

T

(k) E R2 Eη (k) − d1

η T (s) E T R1 Eη (s)

s=k−d1

−d2

k−1

η T (s) E T R2 Eη (s) ,

(15)

s=k−d2 T

T

T

k−1

T

¯ (k) H R1 H x ¯ (k) + (d2 − 1) x ¯ (k) H R2 H x ¯ (k) − ΔV4 (k) = (d1 − 1) x

xT (s) R1 x (s)

s=k−d1 +1

−

k−1

T

T

k−d

1

T

x (s) R2 x (s) + (d2 − d1 ) x ¯ (k) H Q3 H x ¯ (k) −

s=k−d2 +1

xT (s) Q3 x (s) .

(16)

s=k−d2 +1

Then, according to Lemma 1 and Lemma 2, we can achieve −d1

k−1

η T (s) E T R1 Eη (s) ≤ − [H x ¯ (k) − x (k − d1 )]T E T R1 E [H x ¯ (k) − x (k − d1 )] −

s=k−d1

⎡ × ⎣H x ¯ (k) + x (k − d1 ) − ⎡ × ⎣H x ¯ (k) + x (k − d1 ) −

−d2

k−1

k−1

2 d1 −1

x (s)⎦ E T R1 E

s=k−d1 +1

⎤ x (s)⎦ ,

⎡ × ⎣H x ¯ (k) + x (k − d2 ) − k−1

xT (s) R1 x (s) ≤ −

s=k−d1 +1 k−1

xT (s) R2 x (s) ≤ −

s=k−d2 +1

1 d1 − 1 1 d2 − 1

k−1

k−1

d2 −1

xT (s) R1

⎤T

3 (d2 − 1) d2 + 1

x (s)⎦ E T R2 E

s=k−d2 +1

2

s=k−d1 +1 k−1

k−1

2 d2 −1

(17)

s=k−d1 +1

⎡ × ⎣H x ¯ (k) + x (k − d2 ) −

−

d1 −1

⎤T

η T (s) E T R2 Eη (s) ≤ − [H x ¯ (k) − x (k − d2 )]T E T R2 E [H x ¯ (k) − x (k − d2 )] −

s=k−d2

−

k−1

2

3 (d1 − 1) d1 + 1

⎤ x (s)⎦ ,

(18)

s=k−d2 +1

k−1

x (s) ,

(19)

x (s) .

(20)

s=k−d1 +1

xT (s) R2

s=k−d2 +1

k−1

s=k−d2 +1

On the other hand, we have that ¯ (k) + Adi H x ¯ (k − d (k)) + Bi ω (k) , Eη (k) = (Ai − E) H x ¯TE ¯x ¯ (k + 1) . 0 = 2¯ xT (k) Si Θ 7

(21) (22)

Defining U (k) = δ˜ z T (k) z˜ (k) − 2 (1 − δ) γ z˜T (k) ω (k) − γ 2 ω T (k) ω (k) , then it follows from (13)-(22) that   ΔV (k)+U (k) ≤ ξ T (k) Ψiab ξ (k)+ξ T (k) ΛT1 Xi Λ1 + d21 ΛT2 R1 Λ2 + d22 ΛT2 R2 Λ2 + δΛT3 Λ3 ξ (k) , a, b = 1, 2, 3, (23) where



¯i , Λ2 = (Ai − E) H 0 0 Adi H 0 0 Bi , Λ1 = A¯i 0 0 A¯di H 0 0 B

¯ i 0 0 Ldi H 0 0 D2i , Λ3 = L T  k−1 k−1 ! ! T T T T T T T ¯ (k − d (k)) x (s) x (s) ω (k) x ¯ (k) x (k − d1 ) x (k − d2 ) x . ξ (k) = s=k−d1 +1

s=k−d2 +1

Using Schur complement with respect to ΛT1 Xi Λ1 , d21 ΛT2 R1 Λ2 , d22 ΛT2 R2 Λ2 , δΛT3 Λ3 in (23), we can obtain (10). If the condition (10) is satisfied, it is easy to achieve that ΔV (k) + U (k) < 0. Therefore, under zero initial condition, we have   T   T

[ΔV (k) + U (k)] − E {V (T )} ≤ E [ΔV (k) + U (k)] < 0, E k=0

k=0

which implies (8) holds for any non-zero ω (k) ∈ l2 [0, ∞) . Then the condition (10) indicates that the following condition holds " # ˆ Ti = Ψ ˆ i < 0, a, b = 1, 2, 3, 4, ˆi  Ψ Ψ ab where



i i i i i 6 6 T T T T T ˆ ˆ ˆ ¯ Ψ12  Ψ12 , Ψ13  d1 +1 H E R1 E d2 +1 H E R2 E Ai Xi , Ψ11  Ψ11 ,

T T T T ˆ i  Ψi , ˆi  Ψ Ψ d1 H (Ai − E) R1 d2 H (Ai − E) R2 , 14 22 22 ⎡ ⎤ 0 0   6 6 ⎥ ˆi  ⎢ ˆ i  diag E T R1 E, E T R2 E, H T A¯Tdi Xi , Ψ Ψ 0 0 ⎣ ⎦, 23 24 d1 + 1 d2 + 1 T T T T d1 H Adi R1 d2 H Adi R2   1 −12 1 −12 ˆ i33  diag E T R1 E − R1 , E T R2 E − R2 , −Xi , Ψ (d1 + 1) (d1 − 1) d1 − 1 (d2 + 1) (d2 − 1) d2 − 1 i i ˆ 44  diag {−R1 , −R2 } , ˆ 34  0, Ψ Ψ

which guarantees ΔV (k) < 0 such that the system (6) with ω (k) = 0 is stochastically stable. Then applying the same method as in [31], the system (6) is stochastically admissible. This completes the proof. Based on Theorem 1, the extended passive filter synthesis problem can be developed in terms of LMIs for SMJS (1). Theorem 2 Considering system (1), given scalars d1 , d2 , γ, α and δ ∈ [0, 1] , the filtering error system (6) is stochastically admissible with a mixed H∞ and passive performance γ, if there exist matrices Pi =   P1i P2i ¯f , B ¯f i , > 0, Qm > 0 (m = 1, 2, 3) , R1 > 0, R2 > 0, matrices S1i, S2i , G1i , G2i , Y, A¯f , A¯f i , B ∗ P3i Lf , Lf i with appropriate dimensions such that the following matrix inequalities hold for each mode i ∈ N  Υi  ΥTi = Υiab < 0, a, b = 1, 2, 3, 4, 5, 8

(24)

where Υi11



Υi12  Υi13  Υi15  Υi22  Υi23  Υi24 

% 2 $ 3

3 (dm − 1) T T Qm (dm − 1) Rm − E Rm E − E Rm E − E T P1i E + dm + 1 m=1 m=1   + (d2 − d1 ) Q3 + Sym S1i ΘT Ai ,

1 −1) T T R E E T R E − 3(d2 −1) E T R E , E T R1 E − 3(d E −E T P2i + ATi ΘS2i 1 2 2 d1 +1 d2 +1



Υi14  − (1 − δ) γLTi + S1i ΘT Bi ϑ1i ϑ2i , S1i ΘT Adi d16+1 E T R1 E d26+1 E T R2 E , √ T

δLi , d1 (Ai − E)T R1 d2 (Ai − E)T R2   3 (d1 − 1) T 3 (d2 − 1) T T T E R1 E, −Q2 − E R2 E − E R2 E , diag −P3i , −Q1 − E R1 E − d1 + 1 d2 + 1   6 6 E T R1 E, E T R2 E , diag S2i ΘT Adi , d1 + 1 d2 + 1 ⎤ ⎡ S2i ΘT Bi + (1 − δ) (1 − α) γLTfi + α (1 − δ) γLTf ϑ3i ϑ4i ⎥ ⎢ 0 0 0 ⎦, ⎣ 0

0

0

⎤ ⎤ ⎡ √ √ 0 0 −α δLTf − (1 − α) δLTfi −γ (1 − δ) LTdi ϑ5i ϑ6i ⎥ ⎥ ⎢ ⎢ Υi34  ⎣  ⎣ 0 0 0 0 0 0 ⎦, ⎦, 0 0 0 0 0 0 ⎤ ⎤ ⎡ ⎡ √ √ T d1 ATdi R1 d2 ATdi R2 d1 BiT R1 d2 BiT R2 δLTdi δD2i ⎥ ⎥ ⎢ ⎢ Υi45  ⎣  ⎣ 0 0 0 0 0 0 ⎦, ⎦, 0 0 0 0 0 0   −12 1 −12 1 T T E R1 E − R1 , E R2 E − R2 ,  diag −Q3 , (d1 + 1) (d1 − 1) d1 − 1 (d2 + 1) (d2 − 1) d2 − 1 ⎤ ⎡ −γ 2 I − (1 − δ) γSym {D2i } ϑ7i ϑ8i ⎥ ⎢ N N ! ! ⎥ ⎢ T −Y + ⎥ ⎢ ∗ π P − Sym {G } −G π P in 1n 1i in 2n 2i  ⎢ ⎥, n=1 n=1 ⎥ ⎢ N ⎦ ! ⎣ ∗ ∗ πin P3n − Sym {Y } ⎡

Υi25

Υi35 Υi33

Υi44

n=1

Υi55  diag {−R1 , −R2 , −I} , with ¯ T + (1 − α) C T B ¯T , ϑ1i  ATi GT1i + αCiT B f i fi ϑ3i  αA¯Tf + (1 − α) A¯Tfi ,

¯ T + (1 − α) C T B ¯T , ϑ2i  ATi GT2i + αCiT B f i fi

ϑ4i  αA¯Tf + (1 − α) A¯Tfi ,

T ¯T T ¯T Bf + (1 − α) Cdi Bf i , ϑ5i  ATdi GT1i + αCdi

T ¯T T ¯T ϑ6i  ATdi GT2i + αCdi Bf + (1 − α) Cdi Bf i ,

T ¯T T ¯T Bf + (1 − α) D1i Bf i , ϑ7i  BiT GT1i + αD1i

T ¯T T ¯T ϑ8i  BiT GT2i + αD1i Bf + (1 − α) D1i Bf i ,

and Θ is any matrix with appropriate dimensions and full column rank that satisfies E T Θ = 0, then system (6) is stochastically admissible with a mixed H∞ and passive performance γ. In this case, the parameters of the desired filter can be chosen by Af

= Y −1 A¯f ,

Af i = Y −1 A¯f i ,

Bf

¯f , = Y −1 B

¯f i . Bf i = Y −1 B 9

(25)

Proof. Denote & Ξi = diag I I I I I I I Gi Xi−T   G1i Y . Gi = G2i Y

I I

I

'

,

Pre- and post-multiplying the left hand side of (10) by Ξi and ΞTi , respectively. Noting that −Gi Xi−1 GTi ≤ Xi − GTi − Gi , we can obtain (24), which completes the proof. In order to show the effectiveness of the presented method, we consider the following SMJS with time-varying delays: Ex (k + 1) = Ai x (k) + Adi x (k − d (k)) , k = −d2 , − d2 + 1, . . . , 0.

x (k) = φ (k) ,

(26)

Corollary 1 For given scalars d1 and d2 , the system (26) is stochastically admissible, if there exist matrices Pi > 0, Q1 > 0, Q2 > 0, Q3 > 0, R1 > 0, R2 > 0, Si , Gi , such that the following matrix inequalities hold for each mode i ∈ N  (27) Φi  ΦTi = Φiab < 0, a, b = 1, 2, 3, 4, where Φi11  −E T Pi E +

2 

(dm − 1) Rm − E T Rm E −

m=1

3 (dm − 1) T E Rm E dm + 1



3   T Qm + (d2 − d1 ) Q3 , +Sym Si Θ Ai +

Φi12



Φi13  Φi22 Φi23 Φi33 Φi34



m=1

E T R1 E

−

3(d1 −1) T d1 +1 E R1 E

6 T d1 +1 E R1 E

6 T d2 +1 E R2 E

E T R2 E ATi GTi

−

3(d2 −1) T d2 +1 E R2 E

Φi14 



Si ΘT Adi



,

T

T



d1 (Ai − E) R1 d2 (Ai − E) R2 ,   3 (d1 − 1) T 3 (d2 − 1) T T T E R1 E, −Q2 − E R2 E − E R2 E, −Q3 ,  diag −Q1 − E R1 E − d1 + 1 d2 + 1 ⎡ ⎤ 0 0   6 6 ⎢ ⎥ E T R1 E, E T R2 E, ATdi GTi ,  diag Φi24  ⎣ 0 0 ⎦, d1 + 1 d2 + 1 d1 ATdi R1 d2 ATdi R2   −12 1 −12 1 T T E R1 E − R1 , E R2 E − R2 , Xi − Sym {Gi } ,  diag (d1 + 1) (d1 − 1) d1 − 1 (d2 + 1) (d2 − 1) d2 − 1 ˆi , ˆi ,  Ψ Φi  Ψ 34

44

,

44

and the other parameters follow the same definitions as those in Theorem 1. The proof of the Corollary 1 is similar to that of Theorem 1 and thus is omitted here.

4

Numerical Examples

In this section, two numerical examples and a dynamical Leontief model of economic systems are given to illustrate the validness of our results. The first example is employed to explain less conservatism of the conditions presented in Corollary 1. The second example is adopted to analyze the mode-dependent and 10

mode-independent filtering error and study the relation among the upper bound of time-varying delay, i.e. d2 , δ and the minimum allowable scalar γ in Theorem 2. Finally, as the third example, the dynamical Leontief model of economic systems is researched to show the practical application in this note. Example 1. Consider the SMJS with time-varying delays (26), the following parameters are borrowed from [5]:     1 −0.5 1.05 1.6 , A2 = , A1 = 1 2.1 −2 1     0.5 −0.1 0.25 −0.2 , Ad2 = , Ad1 = 0.5 −0.1 0.1 −0.1     6 3 0 0 E = , Θ= . 0 0 0 1 To compare our delay-dependent stochastic stability condition with that in [5, 11], we assume that the transition rate π11 = 0.25, π21 = 0.3, and d1 = 3. Table 1 lists the comparison results which show that the condition in Corollary 1 is less conservative than that in [11]. In addition, we achieve fewer variables than that in [5]. It is clear that the results of Corollary 1 are better than those in [5, 11]. Table 1. Comparisons of maximum allowed d2 between different methods for Example 1 Methods

[11]

[5]

Corollary 1

d2

9

10

10

Number of variables

34

45

37

Remark 4 In order to establish delay-dependent conditions, some important and essential methods have been reported in the literature such as Jensen’s inequality, free weighting matrix method, reciprocally convex approach and Wirtinger’s inequality. It is noted that the Abel lemma-based finite-sum inequality is adopted in the proof of Theorem 1 which may bring some conservation. Compared with reciprocally convex approach, the Abel lemma finite-sum inequality could be more computationally efficient, because no slack variables are brought for computation. Example 2. Consider the system (1) with two modes and the following parameters:

11

 A1 =  Ad1 =  B1 = C1 = Cd1 =



 −2.5 0.8 , A2 = , 0 −5.0    −0.5 0.25 −1.6 0.1 , Ad2 = , 0.2 −0.1 0.2 −0.4    0.8 1.0 , B2 = , 0.6 −0.3



C2 = 1.0 −0.6 , −1.0 0.3 ,



Cd2 = 0.5 −0.7 , −0.03 0.3 , −5 0 0.3 −3.9

D12 D11 = 0.8, L1 = −0.8 0.7 Ld1 = −0.9 0.4





= −0.5,



, L2 = 0.5 −0.6 ,



, Ld2 = 0.7 −0.5 ,

D22 = 0.7, D21 = 0.3,     12 0 0 0 E = , Θ= . 0 0 0 1 In this example, the main purpose is to analyze the mode-dependent and the mode-independent filtering errors and study the relation among the upper bound of time-varying delay (d2 ), δ and the minimum allowable scalar γ in Theorem 2. Case 1: First, we set α = 0, as stated in Remark 2, the mode-dependent filter is designed. The other parameters are given as d1 = 2, d2 = 9, γ = 2, δ = 0.5, π11 = 0.2 and π21 = 0.4. Then, by using the Matlab LMI Toolbox to solve the conditions ( 24), the desired filter gains can be obtained as follows:     −0.3514 0.0368 −0.1233 0.0100 , Af 2 = , Af 1 = 0.0359 −0.0041 −0.0187 0.0020     −0.9301 1.9256 , Bf 2 = , Bf 1 = −0.1872 −0.0902



Lf 2 = −0.4638 0.0507 . Lf 1 = −0.4056 0.0446 , For simulation purposes, we assume the initial condition x (0) = xf (0) = exogenous disturbance signal  sin (k) , 0 < k < 15 ω (k) = . 0, otherwise



0.5 0.0625

T

, and the

The simulation of filtering error z˜ (k) , the possible time sequences of the system mode jumps and the possible realizations of time delays are shown in Fig. 1. It is clear from Fig. 1 that the mixed H∞ and passive mode-dependent filter is feasible and effective. Then, we try to investigate the relation among d2 , δ and the minimum allowable scalar γ in the desired mode-dependent filter. We fix d1 = 2, and the other parameters are the same as before except scalars γ, d2 and δ. The relationship among γ, d2 and δ is given in Table 2. From Table 2, it is easy to observe that the scalar γ is increasingly large with the increasement of d2 . On the other hand, the scalar γ has the same trend with the increasement of δ for each fixed d2 . Therefore, it is significant to research the relationship among d2 , δ, and γ to cope with some special requirements in practice. 12

Table 2. Comparisons of minimum allowed γ between different d2 for Example 2 d2

18

19

20

21

δ=0

0.0214

0.0600

0.1093

0.1741

δ = 0.5

0.3813

0.3982

0.4218

0.4540

δ=1

0.9759

0.9897

1.0052

1.0228

Case 2: In order to design a mode-independent filter, we can set α = 1 . The other parameters are given as d1 = 2, d2 = 9, γ = 2, δ = 0.5, π11 = 0.2, π21 = 0.4. Then, by using the Matlab LMI Toolbox to solve the conditions (24), the desired filter gains can be given as follows:     −0.2671 0.2591 0.1427 , Bf = , Af = 0.0065 −0.0479 −0.1512

Lf = −0.3034 0.2861 . The initial condition and the exogenous disturbance signal ω (k) are the same as before (Case 1). The simulation results are plotted in Fig. 2 which imply the mixed H∞ and passive mode-independent filter is feasible and effective. Then, we research the relationship among d2 , δ and the minimum allowable scalar γ in the desired mode-independent filter. The other parameters are the same as before (Case 1) and we also find the relationship among d2 , δ, and γ in mode-independent filter has the same trend as in mode-dependent filter which is given in Table 3. Table 3. Comparisons of minimum allowed γ between different d2 for Example 2 d2

18

19

20

21

δ=0

0.1936

0.2551

0.3221

0.4008

δ = 0.5

0.4670

0.5108

0.5651

0.6311

δ=1

1.0853

1.1221

1.1634

1.2104

Remark 5 It should be pointed out that if the system modes information is successfully transmitted to the filter all the time, the minimum allowable scalar γ in mode-dependent filter is smaller than that in mode-independent filter which can be obviously observed from Table 2 and Table 3. Example 3. Consider the dynamical Leontief model of economic systems given by [9]. x (k) = Ax (k) + B [x (k + 1) − x (k)] + v (k) ,

(28)

where x (k) is the vector of output level, v (k) is the vector of final demands (excluding investment), A is the Leontief input-output matrix, and B is the capital coefficient matrix. B [x (k + 1) − x (k)] is the amount required for capacity expansion to be able to produce x (k + 1) in the next period. In [32], the v (k) is controlled by u (k) such that v (k) = Hu (k) , where u (k) ∈ Rp (1 ≤ p < n) . Then (28) can be rewritten as Bx (k + 1) = (I − A + B) x (k) − Hu (k) , (29) the control law with fault model is given by uF (k) = F (σ (k)) [K1 x (k) + K2 x (k − d (k))] , 13

(30)

where F (σ (k)) = diag {f1 (σ (k)) , f2 (σ (k)) , . . . , fp (σ (k))} , 0 ≤ fq (σ (k)) ≤ 1 (q = 1, 2, . . . , p) . Then, consider a Leontief model described by       1 0 2.04 1 −1 B= , A= , H= . 0 0 0.8 1 3.05 The system (29) can be rewritten as       1 0 −0.04 −1 −1 x (k + 1) = x (k) − u (k) . 0 0 −0.8 0 3.05

(31)





Choose K1 = −0.01 0.6 , K2 = 0.1676 0.1170 , F1 = 0.3, F2 = 0.8, and F3 = 1, according to (30 )-(31), the closed-loop system can be modeled by discrete-time singular Markov jump delayed system (26) with the following parameters     −0.0430 −0.8200 0.0503 0.0351 , Ad1 = , A1 = −0.7909 −0.5490 −0.1534 −0.1071     −0.0480 −0.5200 0.1341 0.0936 , Ad2 = , A2 = −0.7756 −1.4640 −0.4089 −0.2855     −0.0500 −0.4000 0.1676 0.1170 , Ad3 = , A3 = −0.7695 −1.8300 −0.5112 −0.3569   1 0 E = . 0 0  0 As stated in [32], system (31) is regular, causal and stochastically stable on conditions that R = , 1 2 ≤ d (k) ≤ 8. Under this circumstance, we aim to apply the desired mode-independent filter in Theorem 2 to estimate the demands v (k). The other parameters in our paper are given as follows  



−0.1440 , C1 = −0.07 0.21 , Cd1 = −0.3 0.13 , B1 = 1.28



D11 = −1.2, D21 = −0.3, L1 = −0.18 0.17 , Ld1 = −0.2 −0.1 ,  



−0.08 , C2 = 0.07 −0.182 , Cd2 = −0.5 −0.17 , B2 = 0.2



D12 = −1.5, D22 = 0.5, L2 = −0.25 −0.16 , Ld2 = −0.04 −0.05 ,  



−0.08 , C3 = −0.35 −0.42 , Cd3 = −0.2 0.27 , B3 = −0.24



D13 = −0.3, D23 = 0.7, L3 = −0.1 −0.26 , Ld3 = −0.17 −0.1 , ⎡ ⎤ 0.1 0.2 0.7 ⎢ ⎥ α = 1, δ = 0.1, γ = 2, d1 = 2, d2 = 6, Π = ⎣ 0.1 0.1 0.8 ⎦ . 

0.3 0.5 0.2 Then, by using the Matlab LMI Toolbox to solve the conditions (24), the desired filter gains can be given

14

as follows:

 Af

=

Lf

=



0.0510 0.0173 0.0025 0.0007



 ,

−0.0554 −0.0169

Bf =

−0.0335 −0.0073

 ,

.



T For simulation purposes, we assume that the initial condition x (0) = xf (0) = 0.5 −0.7196 , the exogenous disturbance signal ω (k) is the same as before. The simulation results are depicted in Fig. 3, which show that the mode-independent filter is feasible and effective.

5

Conclusions

In this paper, we have studied the extended passive filtering problem for discrete-time singular Markov jump systems (SMJSs) with time-varying delays. In order to reduce the conservatism and improve computational efficiency, some novel inequalities based on Abel lemma have been adopted. In this way, sufficient criteria which ensure that the SMJSs to be stochastically admissible with a mixed H∞ and passive performance γ have been presented. Three examples and a dynamical Leontief model of economic systems have been provided to show the reduced conservatism and effectiveness of the proposed method. It is pointed out that our main results can be extended to dissipative filtering/control for network-based/nonlinear singular Markov jump systems with distributed time-delays based on the fuzzy model-based approach. This will be a direction of future research [19, 24].

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17

1.5

System mode

Filtering error

1

0.5

2 1 0

100

50 Time (k)

0

0

Time−varying delay d(k)

z˜(k) 3

10 8 6 4 2 50 Time (k)

0

100

−0.5

−1

0

10

20

30

40

50 Time (k)

60

70

80

90

Figure 1: Filtering error z˜ (k) for Case 1 in Example 2

18

100

0.8 3 System mode

0.6

Filtering error

0.4

0.2

2 1 0

0

0

50 Time (k)

100

40

50 Time (k)

Time−varying delay d(k)

z˜(k) 10 8 6 4 2 0

50 Time (k)

100

−0.2

−0.4

−0.6

0

10

20

30

60

70

80

90

100

Figure 2: Filtering error z˜ (k) for Case 2 in Example 2

19

1 z˜(k) 0.8

Filtering error

0.6 0.4 0.2

Time−varying delay d(k)

System mode

4 3 2 1 0

0

50 Time (k)

0

6 4 2

100

0

50 Time (k)

60

50 Time (k)

100

−0.2 −0.4 −0.6

0

10

20

30

40

70

80

90

Figure 3: Filtering error z˜ (k) for Example 3

20

100