Event-triggered mixed H∞ and passive filtering for discrete-time networked singular Markovian jump systems

Applied Mathematics and Computation 368 (2020) 124803

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Event-triggered mixed H∞ and passive filtering for discrete-time networked singular Markovian jump systems Qiyi Xu a,b, Yijun Zhang a,∗, Wenhai Qi b, Shunyuan Xiao a a b

School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China School of Engineering, Qufu Normal University, Rizhao 276800, China

a r t i c l e

i n f o

Article history: Received 2 April 2019 Revised 7 August 2019 Accepted 30 September 2019

Keywords: Event-triggered H∞ filtering Passive filtering Singular Markovian jump systems

a b s t r a c t The mixed H∞ and passive filtering problem for a class of discrete-time networked singular Markovian jump systems (SMJSs) is investigated in this paper. To reduce transmission of signals in the limited bandwidth networks, a novel event-triggered scheme (ETS) is proposed. A networked singular Markovian jump filtering error system (FES) with time-delay is established by considering the network communication characteristic and the ETS. By utilizing Lyapunov–Kravoskii stability theory, some new criteria are derived in the form of linear matrix inequalities (LMIs) such that the FES is stochastically admissible (SA) and has a prescribed level of mixed H∞ and passive performance. The co-design method is given, which can realize the synchronous design for the filter and ETS parameters. The proposed method can deal with the mode-dependent or mode-independent filter design problems for SMJSs including H∞ , passive and mixed filter (i.e., mixed H∞ and passive filter) in a unified form. Finally, three numerical examples are provided to illustrate the effectiveness of the proposed method. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Markovian jump systems (MJSs) have gained continuously considerable attention over the past decades, which can accurately describe a lot of actual industrial process control systems, such as power systems, manufacturing systems, chemical systems, aerospace systems, and economic systems, in which structures or parameters or sub-systems random abrupt changes may occur [1]. The research on MJSs is in the ascendant, numerous scholars devoted themselves to this field and achieved many important and interesting results. In [2], the stability and stabilization problems of the MJSs with time-delay have been investigated. Taking the MJSs with partially unknown transition rates into account, the fault detection and control problems have been studied in [3]. The filtering problem of continuous- and discrete-time MJSs has been respectively considered in [4] and [5]. In [6], the authors studied the asynchronous passive control problem for MJSs. The sliding mode control methods for MJSs was considered in [7]. Recently, the singular Markovian jump systems (SMJSs), as the typical MJSs, have attracted the extensive interest of the researchers due to the fact that the singular MJSs model is more accurate than regular ones in describing the dynamics of MJSs which contain the algebraic constraints of states. During the recent decades, scholars have studied SMJSs from various angles, including, but not limited to, analysis and synthesis [8,9], filtering problem [10], H∞ control [11], sliding mode control [12]. ∗

Corresponding author. E-mail addresses: [email protected] (Q. Xu), [email protected] (Y. Zhang), [email protected] (W. Qi), [email protected] (S. Xiao).

https://doi.org/10.1016/j.amc.2019.124803 0096-3003/© 2019 Elsevier Inc. All rights reserved.

2

Q. Xu, Y. Zhang and W. Qi et al. / Applied Mathematics and Computation 368 (2020) 124803

On the other hand, recently, the state estimation problem has been attracted tremendous attention in various applicational fields due to the system states cannot always be obtained directly, since the limitations of detection devices or large costs in practice [13–16]. As one of the celebrated approaches, the Kalman filtering has been fully investigated for systems, in which it is assumed that the system dynamics is accurate known and the system interference is white noise [17]. In a lot of cases, however, the hypothesises are not always satisfied. The H∞ filtering approach is a way to deal with such issues [18]. Recently, for various dynamical systems, the H∞ filtering problem is getting more and more attention. Reference [19] investigated the H∞ filtering problem for a class of stochastic neural networks with multiple delays. In [20] and [21], authors studied the nonfragile H∞ filtering problem for fuzzy system with or without uncertainties, respectively. Considering SMJSs with delays and general unknown transition probabilities, the delay-dependent H∞ filter design method was proposed in [10]. Authors of [22] dealt with the quantization-based H∞ filtering problem for a kind of switched LPV systems with sojourn probabilities and constrained communication. Besides, the passivity theory plays an important role in the study of stability analysis for the linear or nonlinear dynamical systems [6]. The problem of passive filter design has been wildly studied such as references [23,24], in which the designed filter guaranteed that the FES was passive. Combining the H∞ filtering method and passive filtering method whether there exists a way, by which one can construct the passive filter, H∞ filter and mixed filter in a unified approach. Such an issue is of importance and interesting question. In such case, a designed mixed filter should guarantee that the corresponding FES is admissible and satisfy a new performance index, that is the mixed H∞ and passive performance index. Recently, many scholars have begun to pay attention to such an issue for MJSs. Considering the norm bounded uncertainties and random packet dropouts, the mixed H∞ and passive filter design problem of Markovian jump impulsive networked control systems was studied in [25], in which the mode-dependent conditions were established to guarantee the FES to be stochastically stable and achieve a prescribed performance index. In [26], the reliable mixed passive and H∞ filtering problem for uncertain semi-Markov jump delayed systems subject to sensor failures was investigated. The finite-time mixed H∞ and passive performance analysis and filter design problems for a class of uncertain nonlinear discrete-time MJSs, which were described by Takagi–Sugeno fuzzy model with nonhomogeneous jump processes, were considered in [27]. In networked communication systems, it is of great importance to improve the utilization efficiency of communication resources, especially in networked control systems (NCSs) wherein both state signals and control signals are transmitted via communication networks [28]. Since the network bandwidth is limited, it is a challenging topic to transmit signals efficiently and guarantee system performance in a limited bandwidth network. The practice has proved that the event-triggered control strategy is an effective method, which can reduce signal transmission while ensuring system performance. The system and control researchers have developed a good many of event-triggered control methodologies in recent decades, such as absolute event-triggered control methodology[29], relative event-triggered methodology [30,31], discrete event-triggered communication methodology [32], distributed event-triggered methodology [33], etc. The issues related to event-triggered strategies have become a research focus on the control and communication fields. A large number of scholars have devoted themselves to the research of event-based control and filtering problems for different systems. Examples include linear systems [31], nonlinear systems [34], Takagi–Sugeno (T-S) fuzzy systems [35], multi-agent systems [36]. For MJSs, the problem of H∞ control for the continuous networked MJSs based on ETS was considered in [37], in which MJSs with delays have been modeled to describe the ETS and the networked behavior. In [38], the event-triggered fuzzy sliding mode control problem of networked control systems regulated by semi-Markov process has been well studied. An expectation based event-triggered condition is proposed and the stochastic stability issue is well studied. However, it is noted that there are few results about the mixed H∞ and passive filtering design for the discrete-time networked SMJSs with ETS. Due to the complex dynamics of the networked SMJSs, when designing the mixed H∞ and passive filter for the SMJSs, the stability, regularity, and causality of the corresponding FES should be considered simultaneously. Moreover, the co-design of the filter and ETS for the SMJSs is also a challenging problem. This paper investigates the filter design problem for a class of the SMJSs by using the ETS method. A novel approach to filter design will be proposed, by which three types of filters for SMJSs are designed in a unified framework. By utilizing the Lyapunov theory, new sufficient conditions are derived, which guarantee the stochastically admissibility and the preset mixed H∞ and passive performance of the constructed filtering error system. The main contributions of this paper are summarised as the follows: (1) Different from the design methods for the sole mode-dependent or mode-independent filter, this paper construct a unified framework to design filters by introducing a switch parameter, which can cover the above two kinds of filters. (2) To improve the utilization efficiency of the networked communication resources and save the limited communication bandwidth, an ETS is proposed for SMJSs, which is a mode-dependent scheme, i.e., the parameters of the ETS is relied on the mode of the SMJSs. (3) To obtain the filter and ETS parameters, the co-design method is given, by which the solutions of the ETS parameters are transferred to the feasible solution problem of the LMIs, and the explicit expressions of the filter parameters are presented. Also, by the co-design method, the mode-dependent or mode-independent eventtriggered H∞ filter design, passive filter design and mixed filter (i.e., mixed H∞ and passive filter) design for SMJSs can be dealt with in a unified framework. Throughout this paper, Rn stands for the n-dimensional Euclidean space, Rm×n denotes the set of all m × n real matrices. N and N+ respectively represent the set of all non-negative integers and the set of all positive integers. The transpose of vector or matrix is denoted by those with superscript ”T”. The symbol ”∗ ” denotes the symmetric terms in a symmetric matrix. P > 0(P ≥ 0) means that P is positive definite (positive semi-definite). The diag {} stands for a block-diagonal matrix. E {·} denotes the expectation operator.  ·  stands for the Euclidean norm for a vector. Rank{E} denotes the rank of the matrix E.

Q. Xu, Y. Zhang and W. Qi et al. / Applied Mathematics and Computation 368 (2020) 124803

3

Fig. 1. The block diagram of the event-triggered filter for SMJSs.

2. Problem formulation In the fixed probability space (, F, P), consider the following discrete-time SMJSs:

⎧ ⎨Ex(k + 1 ) = A(g(k ))x(k ) + B(g(k ))w(k ) y(k ) = C (g(k ))x(k ) + D(g(k ))w(k ) ⎩ z(k ) = L(g(k ))x(k )

(1)

where x(k ) ∈ Rn , y(k ) ∈ Rm and z(k ) ∈ R p are, respectively, the state vector, the original measured output vector and the signal to be estimated. w(k ) ∈ l2 [0, +∞ ) is disturbance input. E ∈ Rn×n is a singular matrix and it is supposed that Rank{E } = r < n. g(k) represents the systems mode, which is a discrete-time homogeneous Markovian chain taking values in a finite set M = {1, 2, . . . , N} with transition probability matrix . {π ij } is given by

Prob{g(k + 1 ) = j|g(k ) = i} = πi j ,

 where 0 ≤ πi j ≤ 1, ∀i, j ∈ M and Nj=1 πi j = 1, ∀i ∈ M. For a fixed system mode g(k), A(g(k)), B(g(k)), C(g(k)), D(g(k)), L(g(k)) are known real constant system matrices with appropriate dimensions. For ease of expression, simplify notation, when g(k ) = i ∈ M, Ai denotes A(g(k)), and the other symbols are similar denoted. In this paper, to reduce the signal transmission in the networks, an ETS is employed for the SMJSs. The framework of the filter for SMJSs based on ETS is shown in Fig. 1. The event-detector, constructed between sensors and the filter, is used to select which sampling signals will be transmitted to the filter via the communication networks. Let y(k) and y(k ) be the current measured signal and the latest transmitted signal, respectively, where k stands for the instant when the signal is released to networks, and  means the event th triggered, obviously 0 = k0 < k1 < k2 < · · · . The ith sensor measured signal will be released to networks and then be sent to the filter only when the current signal y(k) satisfies the following inequality.

[y(k ) − y(k )]T (g(k ))[y(k ) − y(k )] ≥ δ (g(k ))yT (k )(g(k ))y(k ).

(2)

Remark 1. It is noted from Fig. 1 that the ETS is on the sensor side. The modes of the SMJSs are available to ETS. In this paper, the ETS (2) is mode-dependent, where δ (g(k)) ∈ [0, 1] and (g(k)) > 0 are mode-dependent scalars and modedependent matrices, respectively. For the different modes of SMJSs, the event-triggered conditions are different. The transmission delay is the inevitable phenomenon during signals transmit through networks. Let dk is the transmission delay, dk ∈ [0, dM ], where dk and dM are the positive integers. By taking transmission delay into account, the released signal y(k ) reaches the filter at instant k + dk . The input signal to filter is holden as y(k ) by the zero-order-holder (ZOH) during k ∈ [k + dk , k+1 + dk+1 ). Consequently, we construct the following full-order filter described by

⎧ ⎨x f (k + 1 ) = (α A f i + α¯ A f )x f (k ) + (α B f i + α¯ B f )y(k ) z f (k ) = (αC f i + α¯ C f )x f (k ) + (α D f i + α¯ D f )y(k ) ⎩ k ∈ [k + dk , k+1 + dk+1 )

(3)

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Q. Xu, Y. Zhang and W. Qi et al. / Applied Mathematics and Computation 368 (2020) 124803

where x f (k ) ∈ Rn and z f (k ) ∈ R p are, respectively, the filter state vector and the estimation signal; Afi , Bfi , Cfi , Dfi , Af , Bf , Cf and Df are the filter parameters to be designed later. α ∈ {0, 1}, and α¯ = 1 − α . Remark 2. By utilizing α and α¯ , the filter (3) covers two types filter. When α = 1, the designed filter is a mode-dependent filter described as follows.



x f (k + 1 ) = A f i x f (k ) + B f i y(k ) z f (k ) = C f i x f (k ) + D f i y(k )

(4)

when α = 0, the designed filter become the following mode-independent filter:



x f (k + 1 ) = A f x f (k ) + B f y(k ) z f (k ) = C f x f (k ) + D f y(k )

(5)

Remark 3. In dealing with the state estimation problem for MJSs, the mode-dependent filter design method has been proved to be a productive method under the premise that the modes are fully available [39]. In engineering practice, however, such a condition may sometimes not be satisfied. The mode-dependent approach is invalid any more for the filter design of MJSs. As an alternative way, the mode-independent method is proposed to deal with these issues. Thus, in the filter design problem, how to exploit the advantages of the above two approaches in practice become a naturally exciting topic. Consequently, it is reasonable to design the filter as (3). The released states y(k )( ∈ N, k0 = 0 ) will be obtained by the filter at instants k + dk . Obviously, k + dk < k+1 + dk+1 for  = 1, 2, . . . , ∞. In what follows, By utilizing the delay system method [31] to model the SMJSs with event-triggered scheme, and then we deal with the filtering design problem based on ETS. When k + dM + 1 ≥ k+1 + dk+1 , define d (k ) = k − k , k + dk ≤ k < k+1 + dk+1 . We can have dk ≤ d (k ) ≤ k+1 − k + dk+1 ≤ 1 + dM . When k + dM + 1 < k+1 + dk+1 , consider the intervals [k + dk , k + dM ] and [k + dM + l, k + dM + l + 1 ), where l ∈ N+ . For dk ≤ dM , there exists an integer q ∈ N+ such that k + q + dM < k+1 + dk+1 − 1 ≤ k + q + dM + 1. y(k ) and y(k + l ) satisfy the ETS (2) for l = 1, 2, · · · , q. Define d(k) as follows.

d (k ) =

⎧ ⎨k − k ,

k ∈ S1

k − k − l, ⎩ k − k − q,

k ∈ S2l k ∈ S3

(6)

where S1 = [k + dk , k + dM ), S2 = [k + dM + l, k + dM + l + 1 ), l = 1, 2, . . . , q − 1 and S3 = [k + dM + q, k+1 + dk+1 ). From the above, if k + dM + 1 ≥ k+1 + dk+1 , for k ∈ [k + d , k+1 + d+1 ), we define  (k ) = 0. Otherwise, we define

⎧ ⎨0,  (k ) = y(k ) − y(k + l ), ⎩ y(k ) − y(k + q ),

k ∈ S1 k ∈ S2l k ∈ S3

(7)

From (7) and (2), the following inequality holds.

 (k )T i  (k ) < δi y(k − d (k ))T i y(k − d (k )) k ∈ [k + dk , k+1 + dk+1 )

(8)

In this paper, d(k) is supposed to be bounded and satisfies 0 < d1 ≤ d (k ) ≤ dM + 1  d2 , d1 , d2 ∈ N+ . Combining (6) and (7), rewrite the filter (3) as



x f (k + 1 ) = (α A f i + α¯ A f )x f (k ) + (α B f i + α¯ B f )( (k ) + y(k − d (k ))) z f (k ) = (αC f i + α¯ C f )x f (k ) + (α D f i + α¯ D f )( (k ) + y(k − d (k )))

(9)

where α , α¯ ∈ {0, 1}, α + α¯ = 1, k ∈ [k + dk , k+1 + dk+1 ).



T

Let ξ T (k ) = xT (k )

xTf (k ) . Define the filter error as e(k ) = z(k ) − z f (k ). The FES is described by the follows.



˜ (k ) Eˆ ξ (k + 1 ) = Aˆ i ξ (k ) + Bˆdi H ξ (k − d (k )) + Bˆei  (k ) + Bˆwi w ˜ (k ) e(k ) = Cˆi ξ (k ) + Dˆ di H ξ (k − d (k )) + Dˆ ei  (k ) + Dˆ wi w

where Eˆ = Bˆwi =





Bi 0



E 0 0 I

, Aˆ i =



Ai



0

∗ α A f i + α¯ A f

0

α B f i Di + α¯ B f Di



, Cˆi =



, Bˆdi =



0

, Bˆei =



0



α B f iCi + α¯ B f Ci α B f i + α¯ B f

T, D ˆ ei = −α D f i − α¯ D f , − α¯ CT

,

LT i

−αCTf i

Dˆ di = −α D f iCi − α¯ D f Ci , Dˆ wi = 0



(10)

f





−α D f i Di − α¯ D f Di , H = I



˜ (k ) = 0 ,w



w (k )



w(k−d (k ))

.

Q. Xu, Y. Zhang and W. Qi et al. / Applied Mathematics and Computation 368 (2020) 124803

5

Definition 1 [40]. ˆ Aˆ i ) is regular and causal for each system mode ˜ (k ) = 0 is called regular and causal if the pair (E, (1) The FES (10) with w i ∈ M; ˜ (k ) = 0 is said to be stochastically stable if the following condition holds: (2) The FES (10) with w



E

∞ 



ξ (k ) |ξ (0 ), g(0 ) < ∞. 2

(11)

k=0

˜ (k ) = 0 is said to be SA if it is regular, causal and stochastically stable. (3) The FES (10) with w Definition 2 [25]. The FES (10) is said to be SA with the mixed H∞ and passive performance index γ , if the following conditions are satisfied: ˜ (k ) = 0 is stochastically stable; (1) the FES (10) with w ˜ (k ) ∈ l2 [0, ∞ ) (2) under zero initial condition, for any nonzero w



∞ 

β eT (k )e(k ) − 2(1 − β )γ eT (k )w˜ (k ) − γ 2 w˜ T (k )w˜ (k )

E



≤0

(12)

k=0

where γ > 0, β ∈ [0, 1]. Remark 4. The mixed H∞ and passive performance definition was first proposed in [25]. In the condition (12), the H∞ performance index, passive performance index, and the mixed performance are included. When β = 1, equation (12) reflects the H∞ norm constraint; on the contrary, when β = 0, condition (12) becomes the passivity performance condition; and when β ∈ (0, 1), the expression in (12) states a mixed performance. Based on the above analysis, the problem of the event-based mixed H∞ and passive filter for discrete-time SMJSs can be summarized as design the filter (9) and ETS (8), which can guarantee that the FES (10) is SA and has the mixed performance index γ represented in (12). Before presenting the main results, the following lemmas are needed. Lemma 1 [41]. For the given integers g1 and g2 with g2 − g1 > 1, and a constant matrix R ∈ Rn×n with R = RT > 0, the following inequality holds. G (g1 , g2 ) ≥

1

κ1

S1T RS1 +

3κ2 T S RS κ1 κ3 2 2

(13)

where G (g1 , g2 ) 

g2 −1



ηT (s )Rη (s ), η (s ) = x(k + 1 ) − x(k );

s=g1

S1  x(g1 ) − x(g2 ), S2  x(g1 ) + x(g2 ) −

g2 −1  2 x ( s ); g2 − g1 − 1 s=g1 +1

κ1  g2 − g1 , κ2  g2 − g1 − 1, κ3  g2 − g1 + 1. Lemma 2 [42]. For given integers ρ 1 and ρ 2 with ρ2 − ρ1 > 1, and any positive definite matrix M ∈ Rn×n , the following inequality holds.





ρ2 −1 x(k ) T M

k=ρ1





ρ2 −1 x(k )

≤ ( ρ2 − ρ1 )

k=ρ1



ρ2 −1 xT (k )Mx(k ).

(14)

k=ρ1

3. Main results In this section, we will give the mixed H∞ and passive performance analysis for SMJSs (10), and then present co-design method of filter and ETS for the discrete-time SMJSs. We first show the mixed performance analysis for SMJSs (10) and give the following theorem. Theorem 1. For given scalars d1 , d2 , α , β , δ i , γ , the FES (10) is SA with a prescribed mixed H∞ and passive performance index γ , if there exist positive definite matrices Pi , Q1 , Q2 , Q3 , R1 , R2 , i , and matrices Si and Gi such that for each mode i ∈ M

⎡ i 11 ∗ ⎢ i = ⎣ ∗ ∗

i12 i22 ∗ ∗

i13 i23 i33 ∗

⎤ i14 i 24 ⎥ <0 i34 ⎦ i44

(15)

6

Q. Xu, Y. Zhang and W. Qi et al. / Applied Mathematics and Computation 368 (2020) 124803

where 2 

i11 = −EˆT Pi Eˆ + 

i 12

=

i13 = i22 = i23 = i33 = i14 = 

i 24



H T (dm − 1 )Rm + E T Rm E −

m=1



3 ( dm − 1 ) T ˆ T Aˆ i + Aˆ T ST + (d2 − d1 + 1 )H T Q3 H, E Rm E + Qm H + Si  i i dm + 1



4 − 2d1 T T 4 − 2d1 T T ˆ T Bˆdi , H E R1 E H E R2 E Si  d1 + 1 d1 + 1

6 6 ˆ T Bˆei , H T E T R1 E H T E T R2 E Si  d1 + 1 d2 + 1 2 − 4d1 T 2 − 4d2 T diag{−Q1 + E R1 E, −Q2 + E R2 E, −Q3 + δiCiT iCi }, d1 d2 6 6 H T E T R1 E, H T E T R2 E, 0}, diag{ d1 + 1 d2 + 1 1 12 1 12 R1 − R2 − diag{− E T R1 E, − E T R2 E, −i }, d1 − 1 (d1 − 1 )(d1 + 1 ) d2 − 1 (d2 − 1 )(d2 + 1 ) ¯ CˆT Aˆ T Xi d1 H T (Ai − E )T R1 d2 H T (Ai − E )T R2 ], [−βγ i



=

 i34 =

i

0 0 ¯ Dˆ T + δiC T i [0, Di ] −βγ i di 0 0 ¯ Dˆ T −βγ ei

0 0 BˆTei Xi

0 0 0



0 0 BˆTdi Xi

0 0 0

⎡

ϑi 0 ⎢ 0 , i44 = ⎣ 0 0 0 0



0 0 , 0 BˆTwi Xi −Xi 0 0

0 0 −R1 0

⎤

0 0 ⎥ ⎦, 0 −R2

¯ Dˆ T H + δi [0, Di ]T i [0, Di ], ϑi = −γ 2 I − βγ wi Xi 

N

m=1

ˆ is a full column rank matrix satisfying  ˆ T Eˆ = 0. πim Pm , β¯ = 1 − β and 

˜ (k ) = 0 is regular and causal. Next, Proof. By utilizing the similar method as in [43], it is easy to check the FES (10) with w we prove FES (10) is stochastically stable. Construct the following Lyapunov function:

V (x(k ), g(k ))  V1 (k ) + V2 (k ) + V3 (k ) + V4 (k )

(16)

with

V1 (k ) = V2 (k ) =

ξ T (k )EˆT Pi Eˆξ (k ) k−1 

ξ T (s )H T Q1 H ξ (s ) +

s=k−d1

V3 (k ) = d1

k−1 

−1 

k−1 

k−1 

ξ T (s )H T Q3 H ξ (s )

s=k−d (k )

s=k−d2

k−1 −1  

ηT ( j )E T R1 E η ( j ) + d2

s=−d1 j=k+s

V4 (k ) =

ξ T (s )H T Q2 H ξ (s ) + k−1 −1  

ηT ( j )E T R2 E η ( j )

s=−d2 j=k+s

ξ T ( j )H T R1 H ξ ( j ) +

s=−d1 +1 j=k+s

−1 

k−1 

−d1 

ξ T ( j )H T R2 H ξ ( j ) +

s=−d2 +1 j=k+s

k−1 

ξ T ( j )H T Q3 H ξ ( j )

s=−d2 +1 j=k+s

where η (k ) = x(k + 1 ) − x(k ). Define E {V (k )}  E {V (ξ (k + 1 ), g(k + 1 )|ξ (k ), g(k ))} − V (ξ (k ), g(k )). Along the trajectory of (10), for g(k + 1 ) = j, g(k ) = i (i, j ∈ M ), we obtain

E {V1 (k )} =

ξ T (k + 1 )EˆT

N 

 πi j Pj Eˆξ (k + 1 ) − ξ T (k )EˆT Pi Eˆξ (k )

j=1

=

E {V2 (k )} ≤

ξ

T

(k + 1 )Eˆ T Xi Eˆξ (k + 1 ) − ξ T (k )EˆT Pi Eˆξ (k )

(17)

ξ T (k )H T (Q1 + Q2 + Q3 )H ξ (k ) − ξ T (k − d1 )H T Q1 H ξ (k − d1 ) − ξ T (k − d2 )H T × Q2 H ξ (k − d2 ) − ξ T (k − d (k ))H T Q3 H ξ (k − d (k )) +

k−d 1 s=k−d2 +1

xT (s )Q3 x(s )

(18)

Q. Xu, Y. Zhang and W. Qi et al. / Applied Mathematics and Computation 368 (2020) 124803

k−1 

E {V3 (k )} = d12 ηT (k )E T R1 E η (k ) + d22 ηT (k )E T R2 E η (k ) − d1

ηT (s )E T R1 E η (s ) − d2

s=k−d1

E {V4 (k )} =

k−1 

7

ηT (s )E T R2 E η (s )

(19)

s=k−d2

ξ T (k )H T ((d1 − 1 )R1 + (d2 − 1 )R2 + (d2 − d1 )Q3 )H ξ (k ) −

k−1 

k−1 

xT (s )R1 x(s ) −

s=k−d1 +1

xT (s )R2 x(s ) −

s=k−d2 +1

k−d 1

xT (s )Q3 x(s )

(20)

s=k−d2 +1

According to Lemma 1, we get

−d1

k−1 

ηT (s )E T R1 E η (s )

s=k−d1



3 ( d1 − 1 ) 2 ≤ −(H ξ (k ) −H ξ (k − d1 )) E R1 E (H ξ (k ) −H ξ (k − d1 )) − H ξ (k ) + H ξ (k − d1 ) − d1 + 1 d1 − 1 T T



2 × E R1 E H ξ (k ) + H ξ (k − d1 ) − d1 − 1



k−1 

T

x (s )

− d2

s=k−d1 +1

k−1  s=k−d2



3 ( d2 − 1 ) 2 ≤ −(H ξ (k ) −H ξ (k − d2 )) E R2 E (H ξ (k ) −H ξ (k − d2 )) − H ξ (k ) + H ξ (k − d2 ) − d2 + 1 d2 − 1



2 × E R2 E H ξ (k ) + H ξ (k − d2 ) − d2 − 1



k−1 

x (s )

s=k−d1 +1

ηT (s )E T R2 E η (s ) T

k−1 

T T

T

T

k−1 

x (s )

s=k−d2 +1

x (s )

(21)

s=k−d2 +1

Applying Lemma 2, we obtain k−1 

−

xT (s )R1 x(s )

s=k−d1 +1



1 ≤− d1 − 1

 ≤−

1 d2 − 1

k−1 

T  x (s )

s=k−d1 +1 k−1 

R1

T  x (s )

R2

s=k−d2 +1

k−1 

 x (s )

s=k−d1 +1 k−1 

−



k−1 

xT (s )R2 x(s )

s=k−d2 +1

x (s )

(22)

s=k−d2 +1

From η ( j ) = x( j + 1 ) − x( j ) and (10), we can easily obtain the following equation.

E η ( k ) = ( Ai − E ) H ξ ( k )

(23)

ˆ T Eˆ = 0, there exists matrix Si , i = 1, . . . , N such that Since 

ˆ T Eˆ ξ (k + 1 ) = 0 2 ξ T ( k ) Si 

(24)



Combining (17)–(24) and the event-triggered scheme (8), and letting ζ (k ) = ξ T (k ), ξ T (k − d1 )H T , ξ T (k − d2 )H T , ξ T (k −    d (k ))H T , k−1 xT (s ), k−1 xT (s ), T (k ) T , we have s=k−d +1 s=k−d +1 1

2

˜ i + )ζ (k ), E {V (k )} ≤ ζ T (k )(

⎡

where



⎤

i11 i12 i13 ˜ i = ⎣ ∗  i i ⎦,  22 23 ∗ ∗ i33

( A i − E )H

 = 1T Xi 1 + d12 2T R1 2 + d22 2T R2 2 ,



(25)

 1 = Aˆ i

0

0

Bˆi H

0

0



Bˆei ,

2 =

0 0 0 0 0 . ⎡˜i T ⎤  1 Xi d1 2T R1 d2 2T R2 0 0 ⎢ ∗ −X i ⎥ Let  = ⎣ ⎦. From (15), we easily have  < 0. By utilizing the Schur complement, we can obtain ∗ ∗ −R1 0 ∗ ∗ ∗ −R2 ˜ i +  < 0, which guarantees E {(k )} < 0. Thus, the system (10) with w(k ) = 0 is stochastically stable. 

8

Q. Xu, Y. Zhang and W. Qi et al. / Applied Mathematics and Computation 368 (2020) 124803

In what follows, we discuss the mixed H∞ and passive performance of the system (10). Let

˜ (k ) − γ 2 w ˜ T ( k )w ˜ (k ) J (k ) = β zˆT (k )zˆ(k ) − 2(1 − β )γ zˆT (k )w and

ψ T (k )

=

˜ T (k )], [ ζ T ( k ), w

(26)

then

E {(k ) + J (k )} ≤ ψ (k )(i )ψ (k ), T

(27)

Form condition (15), we can obtain

E {(k ) + J (k )} < 0

(28)

Consequently, under zero initial condition, we have



E

N  

(k ) + J (k )





− E V (N )





≤E

k=0

N  

J (k )



<0

(29)

k=0

which implies (12) holds. According to 2, we can conclude from the above that FES (10) is SA and has the mixed performance index γ .  Remark 5. Theorem 1 can deal with the three types performances analyses of SMJSs, namely H∞ performance analysis, passive performance analysis and the mixed H∞ & passive performance analysis. In (15), if β = 1, the condition given in Theorem 1 can guarantee that FES is SA and has the prescribed H∞ performance, the same problem was investigated in [43]; if β = 0, the problem turns to the passive performance analysis problem [23]; when β ∈ (0, 1), the problem becomes the mixed H∞ and passive performance analysis [24–26]. The criterion obtained in this paper is general. Remark 6. The mixed H∞ and passive filter design problem for the continuous-time SMJSs [25] and discrete-time MJSs [24], were respectively studied. In this paper, the network communication problem and the ETS are taken into account in the filter design for the networked SMJSs. The using of the ETS improves the efficiency of signals transmission and saves the network bandwidth. By using the Lyapunov theory and LMIs technology, the co-design method of the ETS and the mixed filter design is given in this paper. Next, we deal with the co-design problem of filter and ETS for SMJSs. To this end, the following theorem is presented based on Theorem 1. Theorem 2. Given scalars γ > 0, d1 , d2 ∈ N+ , α ∈ {0, 1}, β ∈ [0, 1],under ETS  (8), the FES (10) is admissible with mixed perforP1i P2i > 0, Q1 > 0, Q2 > 0, Q3 > 0, R1 > 0, R2 > 0, i > 0, mance index γ represented in (12), if there exist real matrices Pi = ∗ P3i and S1 , S2 , G1i , G2i , Y, A¯ f , A¯ f i , B¯ f , B¯ f i , C¯ f , C¯ f i , D¯ f , D¯ f i (i ∈ M) such that

⎡ i 11 ∗ ⎢ i = ⎣ ∗

i 12 i 22

i 13 i 23 i 33

∗ ∗

∗

∗

⎤ i 14 i 24 ⎥ i ⎦<0 34 i 44

(30)

where i 11 = −E T P1i E +



T

=

i 13 =

−E P2i +



T ((dm − 1 )Rm + E T Rm E + Qm ) + (d2 − d1 + 1 )Q3 + S1i T Ai + ATi S1i ,

m=1

 i 12

2 

ATi



6 E T R2 E d2 + 1

6 E T R1 E d1 + 1

¯ LT 14 = [−βγ i

0

4 − 2d1 T E R1 E d1 + 1

T S2i

ATi GT1i , ATi GT2i

i 22 = diag{−P3i , −Q1 + ⎡ i 23

0 ⎢6/d1 + 1E T R1 E =⎣ 0 0



i 34 =

0 0 ¯ D βγ

0 0 0

0 0 ρ1 B







4 − 2d2 T E R2 E, 0 , d2 + 1

0 ,

β LTi d1 (Ai − E )T R1 d2 (Ai − E )T R2 ],

2 − 4d1 T 2 − 4d2 T E R1 E, −Q2 + E R2 E, −Q3 + δiCiT iCi }, d1 d2 0 0 6/d2 + 1E T R2 E 0 0 0 ρ2 B

0 0 − βD

⎤

0 0⎥ , 0⎦ 0 0 0 0

 i 24 = 241 

0 0 , 0

i 44 =

 i 242 , 33 = i33 ,  441 ∗ ∗

442 443 ∗

444 0

445

 ,

Q. Xu, Y. Zhang and W. Qi et al. / Applied Mathematics and Computation 368 (2020) 124803

⎡ ¯ βγ C ⎢ 0 with 241 = ⎣ 0

⎡

⎤

0

ρ1 A

⎢ ⎥ ⎦, 242 = ⎢ ⎣ρ

0 T 1Ci B 0

0 0

¯ CT D βγ i −γ 2 I

9

⎤



ρ2 A

− βC

0

0 ρ2CTi B 0

0

0

0 − β CTi D 

0

0

0⎥

⎥,

0⎦

δCTi i Di 0 0   BTi GT1i BTi GT2i β¯ DTi D 441 = , 442 = , ρ1 DTi B ρ2 DTi B ∗ −γ 2 I + δi DTi i Di     0 0 0 X − G1i − GT1i Xi2 − Y − GT2i  443 = i1 ,  = , 445 = −diag{I, R1 , R2 }, A = α A¯ Tf i + (1 − α )A¯ Tf , 444 ∗ Xi3 − Y − Y T 0 0 − β DTi D  B = α B¯ Tf i + (1 − α )B¯ Tf , C = αC¯ Tf i + (1 − α )C¯ Tf , D = α D¯ Tf i + (1 − α )D¯ Tf , Xi j = N n=1 πin Pjn , j = 1, 2, 3, and  is a full column rank matrix satisfying T E = 0. Furthermore, the parameters of the filter are designed by 

A f i = Y −1 A¯ f i , B f i = Y −1 B¯ f i , C f i = C¯ f i , D f i = D¯ f i , A f = Y −1 A¯ f , B f = Y −1 B¯ f , C f = C¯ f , D f = D¯ f .



Proof: Define the matrix  = diag{I, I, I, I, I, I, I, I, Gi Xi−T , I, I, I} with Gi =

G1i G2i

 ρ1Y . Pre- and post-multiplying the both ρ2Y

sides of (15) by  and its transpose, respectively. By utilizing −Gi Xi−1 GTi ≤ Xi − Gi − GTi , we can obtain (30).  Remark 7. Theorem 2 presents a new criterion, which can dealt with the mode-dependent or mode-independent filter design problems for SMJSs, including H∞ , passive and mixed filter (i.e., mixed H∞ and passive filter), in a unified form. When α = 0 and β = 1, Theorem 2 reduces to be the method of mode-independent H∞ filter design for SMJSs, and such issue was investigated in [43]. Based on Theorem 2, we present the following corollary, which is a new method to deal with the mode-independent H∞ filtering problem for SMJSs, Corollary 1. Let scalars γ > 0, d1 > 0 and d2 > 0 to be the ETS (8), the SMJSs (10) is admissible and has the H∞  given, under  performance index γ , if there exist real matrices Pi =

P 1i

∗

P 2i P 3i

> 0, Q1 > 0, Q2 > 0, Q3 > 0, R1 > 0, R2 > 0, i > 0, and S1 , S2 ,

G1i , G2i , Y, A¯ f , B¯ f , C¯ f , D¯ f for each mode i ∈ M such that

⎡ˆi 11 ⎢ ∗ i ˆ  =⎣

ˆi  12 i ˆ 22

∗ ∗

where

 ˆ 14 = 0  

ˆi  34

0 = 0 0



ˆi  44

∗ ∗

ˆi ⎤  14 ˆi ⎥  24 <0 ˆi ⎦  34 ˆi  44

ˆi  13 i ˆ 23 i ˆ 33 ∗

0

ATi GT1i

ATi GT2i

LTi

0 0 0

0 0 ρ1 B¯Tf

0 0 ρ2 B¯Tf

0 0 −D¯ Tf

ˆ 441  = ∗ ∗

442 443 ∗

0 0

445

(31)

d1 (Ai − E )T R1

 ˆ 441 = ,

0 0 0





d2 (Ai − E )T R2 ,

⎡

0  0 ⎢0 i ⎢ ˆ = 0 , 24 ⎣0 0 0

−γ 2 I ∗

0 0 0

ρ1 A¯ Tf 0

ρ1CiT B¯Tf δCiT i Di 0 

ρ2 A¯ Tf 0

ρ2CiT B¯Tf 0

−C¯ Tf 0 0 −C T D¯ T i

f

0 0 0 0

⎤

0 0⎥ ⎥, 0⎦ 0

0 , −γ 2 I + δi DTi i Di

and the other matrices are follow the same definitions as those in 2. The parameters of the desired filter (5) are obtained as follows.

A f = Y −1 A¯ f , B f = Y −1 B¯ f , C f = C¯ f , D f = D¯ f Remark 8. Let α = 0 and β = 1, the rest prove process is similar to the proof of Theorem 2 and is omitted here. Remark 9. Corollary 1 provides the mode-independent H∞ filter design method for SMJSs (1), by which the modeindependent H∞ filter and ETS’ parameters can be obtained synchronously, i.e., the co-design for ETS and filter is realized. Remark 10. The mode-independent H∞ filter design problem was also investigated in [43], where they use the delay partitioning technique to obtain the conditions. Different with [43], in this paper, we constructed the different Lyapunov function and utilized the Abel lemma-based finite-sum inequality approach [41] to obtain Corollary 1. In dealing with the design problem of the mode-independent H∞ filter, the Corollary 1 needs fewer variables than [43] and has less conservative, which will be illustrated in the following simulation examples.

10

Q. Xu, Y. Zhang and W. Qi et al. / Applied Mathematics and Computation 368 (2020) 124803 Table 1 The minimum allowed γ and the number of variables under different methods. Methods

[43] 0.4430 27

γ min

Number of variables

Corollary 1 0.3199 24

4. Numerical examples

Example 1. We consider SMJSs (1) with the following parameters, which are borrowed from [43].



E =

1 0

 T 2 5

C1 =







3 0.8 , A1 = 0 1.7

 T

1 , C2 = 1





2.4 0.70 , A2 = 3.3 1.85











0.3

0.7

0.8

0.2



 T

−3 5 , D1 = 0.9, D2 = −1, L1 = , L2 = −9 15

The transition probability matrix is  =



2.10 0.1 0 , B1 = , B2 = , 3.75 1 −1 ,

 . Let the lower bound and upper bound of the time-varying delay are 6

and 9, respectively, that is d1 = 6, d2 = 9. Applying Theorem 2 with I = In , n = 2, m = 3 in [43], we can obtain the full order and mode-independent filter as follows.

 ⎧ ⎨x (k + 1 ) = 0.3165 f −0.0064  ⎩ z f (k ) = −1.6662







−0.0062 0.0016 x (k ) + y (k ) −0.0005 f −0.0011

(32)



−0.0592 x f (k )

And by using Table 1 in this paper, we also design the filter as

 ⎧ ⎨x (k + 1 ) = 0.4188 f 1.5218  ⎩ z f (k ) = 0.6157







−0.0244 0.0265 x (k ) + y (k ) −0.0836 f −0.0032

(33)



−0.0307 x f (k ) + −0.0646y(k )

From above, we can see that both Theorem 2 in [43] and Corollary 1 in this paper can deal with the mode-independent H∞ filter design problem for SMJSs. Table 1 lists the minimum allowable performance indicators γ under different methods. From Table 1, under the condition in Corollary 1, the minimum allowed γ is smaller than that in [43], which implies that there is less conservative in the condition given in Corollary 1 than those in [43]. On the other hand, Table 1 also lists the number of the decision variables to solve LMSs, which shows that Corollary 1 needs fewer variables than [43]. Example 2. Consider the discrete-time SMJSs with the following parameters:



−5 A1 = 0.3









0.8 −0.8 −1 , B1 = , C1 = −3.9 0.6 0.8



−0.8 D1 =0.06, D2 = 0.02, L1 = 0.7

T

T





2.5 , A2 = 0.9



0.5 , L2 = −0.6

T







T

1.7 1 1 , B2 = , C2 = −5 −0.3 −0.6



7 ,E = 0





0 , w (k ) = 0

,

1 , k2 +1.0

0 ≤ k ≤ 20 sin(k ), 20 < k ≤ 30 0, else

The lower bound and upper bound of the time delay d(k) are set as d1 =2 and d2 =  9, respectively. The performance index 0.2 0.8 γ = 1.5. Let δ1 = 0.4, δ2 = 0.3. The transition probability matrix is  = . The Markovian system mode g(k) and 0.4 0.6 the exogenous disturbance signal w(k) are presented in Fig. 2. Let β = 0.5, the mixed filter for SMJSs is designed. We first analyze the mode-dependent filter design, from Remark 2, we set α = 1. By utilizing the Matlab LMI Toolbox to solve (30), we obtain the parameters of the designed mixed H∞ and passive filter as follows:



Af1

−0.1759 = −0.1082



C f 1 = −0.1627





−0.2391 0.3360 , Af2 = −0.1344 −0.0744





−0.1569 , C f 2 = 0.0852

1 =14.8770, 2 = 18.4862.











0.5201 0.5188 0.2600 , Bf1 = , Bf2 = , −0.1151 0.0181 0.3213



0.1312 , D f 1 = −0.0357, D f 2 = −0.0721,

Q. Xu, Y. Zhang and W. Qi et al. / Applied Mathematics and Computation 368 (2020) 124803

11

Fig. 2. The disturbance signal and system mode.

Fig. 3. State trajectories under mixed passive and H∞ filter.

Letting α = 0, the parameters of the corresponding mode-independent mixed H∞ and passive filter are listed as follows.



0.1112 Af = −0.0940







 0.2069 0.3581 , Bf = , C f = 0.2224 −0.1326 0.0861



0.3364 , D f = −0.0467,

1 =14.2831, 2 = 17.5413.  Let the initial states of system and the initial states of filter as x(0 ) =

0.5



0.0038

 and x f (0 ) =

0 0

, respectively. Simulation

results of mixed passive and H∞ filter design are shown in Figs. 3 and 4. Fig. 3a and b, respectively, depict the state trajectories under mode-independent case and mode-independent case. Fig. 4a contains filter errors in both mode-dependent and mode-independent cases. Filter error under mode-independent case is shown in Fig. 4a, and those of mode-independent case is present in Fig. 4b. It can be concluded from Figs. 3 and 4 that the mixed H∞ and passive filter design method for SMJSs is feasible and effective. Fig. 5 depicts the event-triggering instants and release intervals of measurement outputs. From Fig. 5, it can be seen that for the 100 sampled measurement output signals, only 55 signals are released to the communication network and be transmitted to the filter, which implies that the bandwidth of communication network is saved. And the energy consumption caused by signal transmission is reduced.

12

Q. Xu, Y. Zhang and W. Qi et al. / Applied Mathematics and Computation 368 (2020) 124803

Fig. 4. Filter errors under mixed passive and H∞ filter.

Fig. 5. Event-triggered release instant and release interval. Table 2 Minimum γ for various values of β .

β

0

0.3

0.5

0.7

1

α=0 α=1

0.0410 0.0461

0.0295 0.0337

0.0330 0.0305

0.0284 0.0293

0.0412 0.0409

On the other hand, we investigate the relationship between β and the minimum allowable performance index γ under different α . Table 2 lists the results. It is clear that, from Table 2, the minimum of scalar γ under β ∈ (0, 1) is smaller than the values under β = 0 (the passive filter design case) or β = 1 (H∞ filter design case). It is worth pointing out that, from Table 2, the system performance γmin under β ∈ (0, 1) is smaller than the values under β = 0 or β = 1, which implies that the mixed passive and H∞ performance case is better than the other solo case such as H∞ performance case and passivity performance case. In other words, compared to the other two cases, the mixed H∞ and passive filter design method more effective and practical in tackling the filtering problem for SMJSs.

Q. Xu, Y. Zhang and W. Qi et al. / Applied Mathematics and Computation 368 (2020) 124803

13

Fig. 6. Diagram of DC-motor.

Fig. 7. Simulation results of Example 3.

Example 3. Consider the DC-motor model as shown in Fig. 6. In the practical systems, the DC-motor model can be described by generalized markovian jump system because the running parameters or load of the motor may change suddenly [44]. Let i(t), ω(t) and u(t) represent the current, speed and voltage of DC-motor, respectively. According to electronic circuit and mechanical theory, we can get



ω˙ (t ) = − bJqq ω (t ) + KJqt i(t ) u(t ) = Kω ω (t ) + R(t ),

(34)

14

Q. Xu, Y. Zhang and W. Qi et al. / Applied Mathematics and Computation 368 (2020) 124803 Table 3 The parameters of DC-motor system.

where Jq = Jm + rewritten as

Jcq n2

Jm

Jc1

Jc2

bc1

bc2

R

bm

Kt

Kω

n

0.5 Kg m

50 Kg m

150 Kg m

100

240

1

1

3 N m/A

1 V s/rad

10

bcq

, bq = b m +

n2



, q ∈ M = {1, 2}. Let x(t ) = x1 (t )

Ex(t ) = Gq x(t ) + Hq u(t )



where E =





1

0

0

0



, Gq =

b

− Jqq

Kt Jq

Kω

R



T

x2 (t ) , x1 (t ) = ω (t ), x2 (t ) = i(t ), system (35) can be

(35)







1 , u(t ) = Kq x(t ). The parameters of DC-motor system are given in Table 3.

, Hq = 0





Let K1 = −0.1519 −0.1123 , K2 = −0.1278 −0.3200 . Putting the parameters into the system and discretizing the system we can have the discrete-time SMJSs described as (1), where



0.8000 A1 = 0.8481







0.3000 0.8300 , A2 = 0.8877 0.8722



0.1500 1 ,E = 0.6800 0





The other parameters are supposed as follows: B1 = −0.12 1





−0.6 , L1 = −0.18









d2 = 7. The transition probability matrix is  =



Af =

0.1642 0.0139





T



1.28 , B2 = −0.28

T



0.21 , C1 = −1



0.8 , C2 =

0.15 , D1 = 0.03, D2 = 0.05, α = 0, β = 0.5, δ1 = 0.3, δ2 = 0.3, d1 = 2,

0.17 , L2 = −0.25

independent filter as follows.



0 . 0





0.2 0.4

0.8 . Applying Corollary 1, we get the parameters of the mode0.6



T

0.0304 −0.0352 −0.6011 , Bf = , Cf = 0.0012 −0.0172 −0.1262

, D f = −0.0146.

Fig. 7 presents the simulation results. Simulation results show that the event-triggered mixed H∞ and passive filter design method is effective. 5. Conclusions In this paper, we have concerned the event-triggered mixed H∞ and passive filtering problem for discrete-time SMJSs. To save the limited networked bandwidth, a novel ETS is proposed. By introducing a switchable parameter, a unified filter design framework is constructed, which can cover the mode-dependent filter and the mode-independent filter. A networkbased SMJSs under ETS is established by taking the network-induced delays into account. Sufficient criteria in the form of LMIs are derived, which guarantee the FES to be SA and have a certain mixed performance index. The co-design method is also given, which can tackle the synchronous design problem for filter and ETS. Finally, three numerical examples are conducted to show the effectiveness of the proposed method. In addition, the semi-Markovian jump systems (semi-MJSs) have much broader applications than the conventional MJS due to their relaxed conditions on the probability distributions. The semi-MJSs has received attention in the control field in recent years, for example, [45,46] investigated the sliding mode control problem of semi-MJSs. Extension of our obtained results to the event-triggered mixed filtering problem for SMJSs with various forms of uncertainties and for the semi-MJSs, will be our future research topic. On the other hand, the package dropout of the networked SMJSs is not considered in this paper, which is another challenging research topic. Acknowledgment This work is supported by National Natural Science Foundation of China under Grant No. 61973166, and by the Fundamental Research Funds for the Central Universities under Grant No. 30919011409. References [1] Z. Wang, Y. Liu, X. Liu, Exponential stabilization of a class of stochastic system with Markovian jump parameters and mode-dependent mixed time-delays, IEEE Trans. Autom. Control 55 (7) (2010) 1656–1662. [2] Z. Yan, Y. Song, P.J. H., Finite-time stability and stabilization for stochastic Markov jump systems with mode-dependent time delays, ISA Trans. 68 (2017) 141–149. [3] X. Liu, D. Zhai, D.-K. He, X.-H. Chang, Simultaneous fault detection and control for continuous-time Markovian jump systems with partially unknown transition probabilities, Appl. Math. Comput. 337 (2018) 469–486. [4] M. Shen, J.H. Park, H∞ filtering of Markov jump linear systems with general transition probabilities and output quantization, ISA Trans. 63 (2016) 204–210. [5] Q. Zhong, J. Cheng, Y. Zhao, J. Ma, B. Huang, Finite-time H∞ filtering for a class of discrete-time Markovian jump systems with switching transition probabilities subject to average dwell time switching, Appl. Math. Comput. 225 (2013) 278–294.

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