Circuits Syst Signal Process DOI 10.1007/s00034-016-0288-5

State estimation for Markovian jump systems with an event-triggered communication scheme Yushun Tan1 · Dongsheng Du2 · Qiong Qi1

Received: 29 December 2014 / Revised: 21 February 2016 / Accepted: 23 February 2016 © Springer Science+Business Media New York 2016

Abstract This paper investigates the state estimation problem of Markovian jump systems with time-varying delay. The purpose is to design a state estimator to estimate system states through available output measurements. A novel event-triggered scheme is proposed, which is used to determine whether the sampled state information should be transmitted. The main idea is that the proposed event-triggered scheme provides a supervision of the system state in discrete instants and the newly sampled sensor measurements violating specified triggering condition will be transmitted to the estimator. Firstly, a state estimation model is constructed which describes the transmission delay, Markov parameters and an event-triggered mechanism in a unified framework. Secondly, based on the model, the criteria for the exponential mean square stability are proposed by using Lyapunov functional method and convexity property of the matrix inequality. Under the obtained conditions and the event-triggered scheme, the desired state estimator of Markovian jump systems with time delay is established by solving some linear matrix inequalities. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.

B

Yushun Tan [email protected] Dongsheng Du [email protected] Qiong Qi [email protected]

1

Department of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210046, Jiangsu, People’s Republic of China

2

College of Automatica, Huaiyin Institute of Technology, Huaian 223003, Jiangsu, People’s Republic of China

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Keywords Time delay · Event-triggered scheme · State estimation · Markovian jump systems

1 Introduction Markovian jump systems (MJSs) have been an active field of research during the past few decades. Due to the fact that many dynamical systems subject to random abrupt variations, sudden changes of environment, modification of the operating point of the systems and changes in the interconnections of subsystems can be modeled by MJSs. The system parameters usually jump in a finite mode set, in which the transitions among different modes are governed by a Markov chain. As an important factor, the transition probabilities in the jumping process determine the system behavior. In society life, many applications of MJSs can be showed, such as power systems, failure-prone manufacturing systems, communication systems, biochemical systems with diverse changes of environmental conditions and economy system. So far, MJSs have been widely investigated and a lot of outstanding results have been made such as the controllability and observability, stability and stabilization, robust control, optimal control, H∞ control, l2 − l∞ control and sliding-mode control, H∞ filtering and state estimation. More details on this topic can be found in the literature [3,5,18,27] and the references therein. In recent years, state estimation has been widely studied, and many researches devoted themselves into this interesting issue. Up to now, many related research results on state estimation for different systems can be found in the literature [4,14–16,20,25, 30]. Meanwhile, note that many important results about state estimation for MJSs have been reported in the literature [1,6,13,17,28,31]. In [1] the authors considered state estimation for Markovian jumping delayed continuous-time recurrent neural networks, where only matrix parameters were mode-dependent. In [6] the authors solve the optimal state estimates for a class of jump Markov linear systems using an efficient simulation-based algorithms called particle filters. In [13] the authors discussed state estimation and sliding-mode control problems for phase-type semi-Markovian jump systems. In [17] the authors studied state estimation problem for a class of discretetime neural networks with Markovian jumping parameters and mode-dependent mixed time delay. The authors in [28] concerned the problem of state estimation for Markov jump systems with time delay and missing measurement. The authors in [31] studied sliding-mode control and state estimation problems for continuous-time Markovian jump singular systems with unmeasured states. With the wide application of network, networked control systems (NCSs) have arisen in many fields. There are a lot of data to be collected in NCSs. When the mathematical model is unknown or hard to obtain, data-based method is becoming an efficient alternative way due to their simplicities and excellent ability to handle large amounts of data for some modern industrial issues under complex conditions [35– 38]. On the contrary, if the mathematical model is known or easy to obtain, periodic triggered control method is widely used due to easy implementation and analysis. However, this kind of triggering method sends many unnecessary sampling signals through the network, which will cause high utilization of the communication band-

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width. Recently, event-triggered scheme provided an effective way of determining when the sampling action is carried out. Compared with periodic sampling method, the event-triggered scheme has some advantages as following: (1) It could reduce the burden of the communication, since this communication scheme only samples when necessary; (2) the computation cost of the sensor is reduced. So far, the application on event-triggered scheme could be found in many literature. In [24], how to reduce communication requirements on networked control systems was addressed, where the maximum time allowed to elapse was obtained to guarantee the stability of the system. However, their approach leads to an inherently periodic transmission. In [10], the authors studied the control design problem of event-triggered networked systems with both state and control input quantization. The authors in [21] investigate the reliable control design for networked control system under event-triggered scheme. In [19], the authors studied the problem of event-based fault detection for networked systems with communication delay and nonlinear perturbation, and design a favorable eventbased fault detection. The authors in [40] studied the event-triggered H∞ controller design for networked control systems. In most recent works concerning the study of event-triggered estimation, the representative works can be found in such literature [8,11,12,23,26,29,32]. Besides, time delay in networked systems is a common phenomenon. A great number of research results about this topic can be found in the literature [7] and the references therein. In [7], the authors present a new delay system approach to networkbased control, and new results on stability and H∞ performance condition developed are exploited to investigate the problem of network-based H∞ control. However, to the best of authors knowledge, there are no papers to deal with the state estimation problem for event-triggered MJSs with time-varying delay, which still remains as a challenging problem. Motivated by the idea of above papers, the event-triggered state estimation for MJSs with time delayed is investigated through available output measurements in this paper. By using Lyapunov functional method and convexity property of the matrix inequality, the criteria for the exponential mean square stability and the criteria for co-designing both the desired state estimator of systems and the trigger parameters are derived in the form of linear matrix inequalities. The main contributions of this paper are highlighted as follows:

1. The novel event-triggered scheme is proposed so that the number of the transmitted state signals from sensor to estimator could be reduced to certain extent. Compared with time-triggered periodic communication scheme, the proposed event-triggered scheme has some advantages since the sampled sensor measurements will be transmitted to the estimator only in the case of violating specified triggering condition. 2. In order to reduce conservative of the results, a new Lyapunov function including the lower and upper delay bound  t of interval time-varying delay established. Based ¯ can depart into two parts on above analysis, the item t−τ( t) x¯ T (s)Q 2 (θt )x(s)ds to deal with. On the other hand, the convexity of the matrix functions is used to avoid the conservative caused by enlarging τ (t) to τ M in the deriving results. 3. Criteria are obtained in the form of linear matrix inequalities which can be readily solved by using the LMI toolbox in MATLAB. Furthermore, the solvability of

Circuits Syst Signal Process

derived conditions is connected with both trigger parameters and the size of the delay. The rest of this paper is organized as follows: Sect. 2 presents the problem of statement and preliminaries. By using Lyapunov functional method, a sufficient condition for the exponential mean square stability and criteria for state estimator designed in terms of LMIs are proposed in Sect. 3. A numerical example illustrates that the proposed event-triggered scheme is superior to some existing ones in Sect. 4, and we conclude this paper in Sect. 5. Notation: The superscript “T ” represents matrix transposition, Rn and Rn×m denote the n-dimensional Eculidean space and the set of n × m real matrices;  ·  represents the Euclidean vector norm or the induced matrix 2-norm as appropriate; the notation (Ω, F, Pr ) represents the probability space with the sample space Ω, σ -algebra F of subsets of the sample space and probability measure Pr ; I is the identity matrix of appropriate dimension. E{x} represents the expectation of x when x is a stochastic   A ∗ variable. denote a symmetric matrix, where * denotes the entries implied by BC symmetry, for a matrix B and two symmetric matrices A and C. The notation X > 0 (respectively, X ≥ 0), for X ∈ Rn×n , means that the matrix X is real symmetric positive definite (respectively, positive semi-definite).

2 Problem statement and preliminaries Fix a probability space (Ω, F, P ) and consider the following class of uncertain linear stochastic systems with Markovian jump parameters and time-varying delays ⎧ x(t) ˙ = A(θt )x(t) + Ad (θt )x(t − τ (t)) + Aω (θt )ω(t), ⎪ ⎪ ⎨ y(t) = C(θt )x(t), z(t) = L(θt )x(t) + L d (θt )x(t − τ (t)) + L ω (θt )ω(t), ⎪ ⎪ ⎩ x(t) = φ(t), ∀t ∈ [−τ M , 0].

(1)

where x(t) ∈ Rn is the state vector, y(t) ∈ Rr is the measurement vector, z(t) ∈ R p is the signal to be estimated, ω(t) ∈ L 2 [0, ∞) is the exogenous disturbance signal, φ(t) is a vector-valued initial continuous function defined on the interval [−τ M , 0], {θt } is a continuous-time Markovian process which has right continuous trajectories and takes values in a finite set S = {1, 2, . . . , N } with stationary transition probabilities: Pr {θt+h = j|θt = i} =

πi j h + o(h), 1 + πii h + o(h),

i = j, i = j.

(2)

where h > 0, limh→0 o(h) h = 0, and πi j ≥ 0, for j  = i is the transition rate from mode i at time t to the mode j at time t + h and πii = −

N

j=1, j=i

πi j .

(3)

Circuits Syst Signal Process

In the system (1), the time delay τ (t) is a time-varying continuous function satisfying the following assumption. 0 ≤ τm ≤ τ (t) ≤ τ M < ∞, τ˙ (t) ≤ μ, ∀t > 0,

(4)

where τ M is the upper bound and τm is the lower bound of the communication delay, and μ is the upper bound of change rare of communication delay. Similar to [19], we introduce an event generator between the sensor and the state estimator. The sensor measurements are sampled regularly by the sampler of the smart sensor with period h, which will be given in sequel. Whether or not the newly sampled sensor measurements will be sent out to the estimator is determined by the following judgement algorithm: [x((k + j)h) − x(kh)]T W [x((k + j)h) − x(kh)] ≤ ρx((k + j)h)W x((k + j)h),

(5)

where W is a symmetric positive definite matrix, j = 1, 2, . . . , and ρ ∈ [0, 1). Only when the current sampled sensor measurements x((k + j)h) and the latest transmitted sensor measurements x(kh) variate the specified threshold (5), the current sampled sensor measurements x((k + j)h) can be transmitted by the event generator and sent into the state estimator. Remark 1 The sensor measurement is sampled at time kh by sampler with a given period h. The next sensor measurement is at time (k + 1)h. Suppose that the release times are t0 h, t1 h, t2 h, . . ., it is easily seen that si h = ti+1 h − ti h denotes the release period of event generator in (5). Remark 2 There is no transmission delay when sampled sensor measurements x((k + j)h) are transmitted by the event generator and sent into the state estimator. Therefore, the state x(t0 h), x(t1 h), x(t2 h), . . ., will arrive at the state estimator side still at the instants t0 h, t1 h, t2 h, . . ., respectively. Notice that the set of the release instants, i.e., {t0 , t1 , t2 , . . .} is a subset of {0, 1, 2, . . .}. The amount of {t0 , t1 , t2 , . . .} depends on not only the variation of the state but also the value of ρ. When ρ = 0, {t0 , t1 , t2 , . . .} = {0, 1, 2, . . .}, it reduces to the case with periodic release times. For technical convenience, similar to [19], consider the following two cases about the interval [tk h, tk+1 ), where k is a positive integer and h is a sampling period. Case 1: If tk+1 h = tk h + h, define a function d(t) as d(t) = t − tk h, t ∈ [tk h, tk+1 h).

(6)

It can easily be obtained that 0 ≤ d(t) < h. Case 2: If tk+1 h > tk h + h, there exists a positive integer m, such that tk+1 h = tk h + mh.

(7)

Circuits Syst Signal Process

It can be easily shown that

[tk h, tk+1 h) =

m−1 

Ii ,

(8)

i=0

where

I0 = [tk h, tk h + h), Ii = [tk h + i h, tk h + i h + h).

(9)

We define d(t) =

t ∈ I0 , t − tk h, t − tk h − i h, t ∈ Ii , i = 1, 2, . . . , m − 1.

(10)

It easy to see that 0 ≤ d(t) < h. Moreover, x(tk h) and tk h+i h with i = 1, 2, . . . , m−1 satisfy (5). In Case 1, for t ∈ [tk h, tk+1 h), define ek (t) = 0. In Case 2, define ek (t) =

0, t ∈ I0 , x(tk h + i h) − x(tk h), t ∈ Ii , i = 1, 2, . . . , m − 1.

(11)

From the definition of ek (t) and the triggering algorithm (5), it can be easily seen that for t ∈ [tk h, tk+1 h) ekT (t)W ek (t) ≤ ρx T (t − d(t))W x(t − d(t)).

(12)

In this paper, the measurement output is the sampled before it enters the estimator. Based on the zero-order hold and the sampling technique, the actual output in system (1) can be described as y(t) = y(tk h) = C(θt )x(tk h), t ∈ [tk h, tk+1 h].

(13)

Based on the measurement y(t), we consider the following state estimator for system (1): ⎧ ˙ˆ = A(θt )x(t) ˆ + Ad (θt )x(t ˆ − τ (t)) + G(θt )( yˆ( t) − y(t)), ⎨ x(t) ˆ yˆ (t) = C(θt )x(t), ⎩ ˆ + L d (θt )x(t ˆ − τ (t)). zˆ (t) = L(θt )x(t)

(14)

where G(θt ) is the feedback gain matrix to be designed, and x(t) ˆ is estimator state vector, yˆ (t) is estimator measurement and zˆ (t) is estimator signal vector, respectively.

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The set S contains the various operation modes of system (1), and for each possible value of θt = i, i ∈ S, the matrices connected with “i-th mode” will be denoted by Ai := A(θt = i), Adi := Ad (θt = i), Aωi := Aω (θt = i), Ci := C(θt = i), Cdi := Cd (θt = i), Cωi := Cω (θt = i), L i := L(θt = i), L di := L d (θt = i), L ωi := L ω (θt = i), where Ai , Adi , Aωi , Ci , Cdi , Cωi , L i , L di , L ωi are constant matrices for any i ∈ S. In this paper we assume that the jumping process {θt } is accessible, i.e., the operation mode of system (1) is known for every t ≥ 0. By setting the estimation error e(t) = x(t) ˆ − x(t) and z˜ (t) = zˆ (t) − z(t). Then the following error dynamics of the state estimation system will be showed from (1) and (14), and it follows that

e(t) ˙ = (Ai + G i Ci )e(t) + Adi e(t − τ (t)) + Gi − Aωi ω(t), z˜ (t) = L i e(t) + L di e(t − τ (t)) − L ωi ω(t).

(15)

where Gi = G i Ci x(t) − G i Ci x(t − d(t)) + G i Ci ek (t). T

Denoting x(t) ¯ = x T (t) e T (t) , we can get the following augmented system from (1) and (15)

˙¯ = A¯ i x(t) x(t) ¯ + B¯ i x(t ¯ − d(t)) + A¯ di x(t ¯ − τ (t)) + G¯ i ek (t) + A¯ ωi ω(t), (16) ¯ ¯ z˜ (t) = L i x(t) ¯ + L di x(t ¯ − d(t)) − L ωi ω(t).

where      0 Ai 0 0 Adi 0 ¯ ¯ , Adi = , Bi = , G i C i Ai + G i C i 0 Adi −G i Ci 0      

0 Aωi , A¯ ωi = , L¯ i = 0 L i , L¯ di = 0 L di . G¯ i = −Aωi G i Ci

A¯ i =



The state estimation problem which is addressed in this paper is to design a state estimator of form (16) such that • The estimation error systems (16) are exponentially stable when ω(t) = 0; • For all nonzero ω(t) ∈ L 2 [0, ∞) and a prescribed γ > 0, the H∞ performance ˜z (t)2 < γ ω2 is sure under the condition e(t) = 0, ∀t ∈ [−τ M , −τm ]. Before giving the main results, the following definitions and lemmas are needed in the proof of our main results. Definition 1 [22] The system (6) is considered to be exponentially stable in the mean square sense (EMSS), if there exist constants λ > 0, α > 0, such that t > 0   E x(t)2 ≤ αe−λt

sup

−τ M
  φ(s)2 .

(17)

Circuits Syst Signal Process

Definition 2 [22] For a given function V : C Fb 0 ([−τ M , 0], R n ) × S → R, its infinitesimal operator L is defined as LV (xt ) = lim

Δ→0+

 1 E(V (xt+Δ |xt ) − V (xt )) . Δ

(18)

Lemma 1 [9] For any constant matrix R ∈ R, R = R T > 0, scalar 0 ≤ τ (t) ≤ τ M and vector function x(t) ˙ : [−τ M , 0] → Rn and constant τ M > 0 such that the following integration is well defined, and then the following inequality holds: 



t

− τM

x˙ (s)R x(s)ds ˙ ≤ T

t−τ M

 −τ M

x(t) x(t − τ M )

T 

−R R

⎡

t t−τ M

⎤T ⎡ x(t) −R x˙ T (s)R x(s)ds ˙ ≤ ⎣ x(t−τ (t)) ⎦ ⎣ −R x(t−τ M ) 0

R −R

R −2R R



 x(t) , (19) x(t − τ M )

⎤⎡ ⎤ 0 x(t) R ⎦ ⎣ x(t−τ (t)) ⎦. −R x(t−τ M ) (20)

Lemma 2 [39] Suppose Ξ1 , Ξ2 and Ω are constant matrices of appropriate dimensions, 0 ≤ τm ≤ τ (t) ≤ τ M , then (τ M − τm )Ξ1 + Ω < 0, (τ M − τm )Ξ2 + Ω < 0.

(21) (22)

3 Main results In this section, we will invest the estimation problem for MJSs (16) with time delay based on event-triggered control. LMI conditions are established to ensure the estimation error to be exponentially mean square stable. Then, according to the analysis results, the methods to design the estimator gain matrices G i are derived in terms of the solution to certain matrix inequalities. We first present some sufficient conditions for state estimation error dynamics (16). Theorem 1 For some given constants 0 ≤ τm ≤ τ M , γ , μ and matrix G 1 , G 2 , the systems (16) are EMSS with a prescribed H∞ performance γ under the event trigger scheme (12), if there exist Pi > 0, Q 0 > 0, Q 1 > 0, Q 2i > 0, Q 3 > 0, R0 > 0, R1 > 0, R2 > 0, Z 1 > 0, Z 2 > 0, W > 0, Mik and Nik (i ∈ S, k = 1, 2, . . . , 6) with appropriate dimensions so that the following matrix inequalities hold

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⎡

Ψ11 + Υ + Υ T ⎢ Ψ21 ⎢ ⎢ Ψ31 Ψ =⎢ ⎢ Ψ 41 ⎢ ⎣ Ψ51 Ψ61 (s) N

∗ −R0 0 0 0 0

∗ ∗ −R1 0 0 0

∗ ∗ ∗ −R2 0 0

∗ ∗ ∗ ∗ −I 0

⎤ ∗ ∗ ⎥ ⎥ ∗ ⎥ ⎥ , s = 1, 2, (23) ∗ ⎥ ⎥ ∗ ⎦ −R1

πi j Q 2 j ≤ Z k , k = 1, 2,

(24)

j=1

where ⎡

Ψ11 =

Ψ21 = Ψ31 = Ψ41 = Ψ51 = Ψ61 (1) =

⎤ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎢ A¯ T Pi (μ − 1)Q 2i ∗ ∗ ∗ ∗ ∗ ∗ ⎥ di ⎢ ⎥ ⎢ 0 −R0 + Q 0 ∗ ∗ ∗ ∗ ∗ ⎥ R0 ⎢ ⎥ ⎢ ∗ ∗ ∗ ⎥ 0 0 −Q 1 ∗ ⎢ T 0 ⎥, ⎢ B¯ Pi + R2 ⎥ 0 0 0 ∗ ∗ ∗ ⎢ i ⎥ ⎢ ∗ ⎥ 0 0 0 0 R2 −R0 − Q 3 ∗ ⎢ ⎥ ⎣ A¯ T Pi 0 0 0 0 0 −γ 2 I ∗ ⎦ ωi C¯ iT Pi 0 0 0 0 0 0 −W

 τm R0 A¯ i τm R0 A¯ di 0 0 τm R0 B¯ i 0 τm R0 A¯  i τm R0 C¯ i , 

√ √ √ √ √ δ R1 A¯ i δ R1 A¯ di 0 0 δ R1 B¯ i 0 δ R1 A¯  i δ R1 C¯ i ,

 h R2 A¯ i h R2 A¯ di 0 0 h R2 B¯ i 0 h R0 A¯  i h R2 C¯ i ,

 L¯ i L¯ di 0 0 0 0 L¯ ωi 0 , √ T √ δ Mi , Ψ61 (2) = δ NiT , δ = τ M − τm , Γ

Γ = Pi A¯ i + A¯ iT Pi + Q 0 + Q 1 + Q 2i − R0 + τm Z 1 + δ Z 2 +  Ω=

W 0



0 ,Υ = 0 0

N

πi j P j ,

j=1

−Mi + Ni

Mi

Ni

0

0

0

 0 ,

and A¯ i , A¯ di , A¯ ωi , B¯ i , C¯ i are all as defined in (16). T

 Proof Introduce a vector ζ (t) = X1 X2 , where X1 = x¯ T (t) x¯ T (t − τ (t)) ,

T  X2 = x¯ (t − τm ) x¯ T (t − τ M ) x¯ T (t − d(t)) x¯ T (t − d M ) ω T (t) ekT (t) . And let

A1 = A¯ i

A2 = L¯ i

A¯ di L¯ di

0

0

B¯ i

0

0

0

0 0

A¯ ωi − L¯ ωi

 C¯ i ,  C¯ i .

Then the system (16) can be rewritten as

˙¯ = A1 ζ (t), x(t) z˜ (t) = A2 ζ (t).

(25)

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Let xt (s) = x(t + s), (−τ (t) ≤ s ≤ 0). Then, the same as [2], {(xt , θt ), t ≥ 0} is a Markov process. Choosing the following Lyapunov functional candidate as

V (x(t), ¯ θt ) =

4

Vi (x(t), ¯ θt ),

(26)

i=1

where ¯ θt ) = x¯ T (t)P(θt )x(t), ¯ V1 (x(t),  t  V2 (x(t), ¯ θt ) = x¯ T (s)Q 0 x(s)ds ¯ + +

t−τm  t



x¯ T (s)Q 2 (θt )x(s)ds ¯ +

 V4 (x(t), ¯ θt ) =

t

x¯ T (s)Q 3 x(s)ds, ¯

t−h



t

¯ θt ) = τm V3 (x(t),



x¯ T (s)Q 1 x(s)ds ¯

t−τ M

t−τ( t)  t

+h

t

˙ + x¯˙ T (v)R0 x(v)dvds



t−τ M

t−τm s t  t s t



t−τm

t

¯ x¯˙ T (v)R1 x(v)dvds

s

¯ x¯˙ T (v)R2 x(v)dvds,

τ −h t



t−τm

 x¯ T (v)Z 1 x(v)dvds ¯ +

t−τm



t−τ M

s

t

x¯ T (v)Z 2 x(v)dvds. ¯

s

Let L be the weak infinite generator of the random process {xt , θt }. Then, for each θt = i (i ∈ S), we have L[V (x(t), ¯ θ )] ⎛t ≤ x¯ T (t) ⎝2Pi A¯ i +

N

⎞ ¯ πi j P j + Q 0 + Q 1 + Q 2i + Q 3 + τm Z 1 + δ Z 2 ⎠ x(t)

j=1

+ 2 x¯ T (t)Pi A¯ di x(t ¯ − τ (t)) + 2 x¯ T (t)Pi B¯ i x(t ¯ − d(t)) + 2 x¯ T (t)Pi C¯ i ek (t)  t  t−τm  t N

x(s) ¯ πi j Q 2 j x(s)ds− ¯ x¯ T (s)Z 1 x(s)ds− ¯ x¯ T (s)Z 2 x(s)ds ¯ + t−τ (t)

j=1

+ 2 x¯ T (t)Pi Aωi ω(t) − τm



t−τm t

˙¯ x˙¯ T (s)R0 x(s)ds −h

t−τm



t−τ M t

˙¯ ˙¯ x(s)R 2 x(s)ds

t−h

¯ − τm ) − x(t ¯ − τ M )Q 1 x(t ¯ − τM ) − x(t ¯ − τm )Q 0 x(t ˙¯ ˙¯ − x(t ¯ − τ (t))Q 2i x(t ¯ − τ (t))(1 − μ) − x(t ¯ − h)Q 3 x(t ¯ − h) + τm2 x(t)R 0 x(t)  t−τm ˙¯ ˙¯ + h 2 x(t)R ˙¯ ˙¯ − ˙¯ ˙¯ x(s)R (27) + (τ M − τm )x(t)R 1 x(t) 2 x(t) 1 x(s)ds. t−τ M

Circuits Syst Signal Process

Note that ⎛



t

x¯ T (s) ⎝

t−τ( t)



⎞ πi j Q 2 j ⎠ x(s)ds ¯ =

j=1

⎛

t

+

N

x¯ T (s) ⎝

t−τm

N

⎛



t−τm

x¯ T (s) ⎝

t−τ( t)

⎞

N

⎞ πi j Q 2 j ⎠ x(s)ds ¯

j=1

¯ πi j Q 2 j ⎠ x(s)ds.

(28)

j=1

From (24) and (28), we have 

⎛ ⎞  N

x¯ T (s) ⎝ πi j Q 2 j ⎠ x(s)ds ¯ −

t t−τ( t)



−

t

x¯ T (s)Z 1 x(s)ds ¯

t−τm

j=1 t−τm

x¯ T (s)Z 2 x(s)ds ¯ ⎛ ⎞  N

x¯ T (s) ⎝ πi j Q 2 j ⎠ x(s)ds ¯ +

t−τ M

 =

t−τm

t−τ( t)



−

t

j=1

⎛ ⎞ N

x¯ T (s) ⎝ πi j Q 2 j ⎠ x(s)ds ¯

t

t−τm



j=1

t−τm

x¯ T (s)Z 1 x(s)ds ¯ − x¯ T (s)Z 2 x(s)ds ¯ t−τ M ⎞ ⎛ N

¯ x¯ T (s) ⎝ πi j Q 2 j − Z 1 ⎠ x(s)ds

t−τm

 =

t

t−τm



j=1

t−τm

+

t−τ( t)

 ≤

t

+

⎛

x¯ T (s) ⎝

t−τm



⎛ ⎞  N

x¯ (s) ⎝ πi j Q 2 j ⎠ x(s)ds ¯ − T

t−τm t−τ M

j=1 N

x¯ T (s) ⎝

x¯ T (s)Z 2 x(s)ds ¯

t−τ M

⎞

πi j Q 2 j − Z 1 ⎠ x(s)ds ¯

j=1

⎛

t−τm

N

⎞ πi j Q 2 j − Z 2 ⎠ x(s)ds ¯ < 0.

(29)

j=1

Remark 3 In (29), the derivation process uses the following skills. First, the item   t−τm T N x(s)ds ¯ is decomposed into two parts. Then one of them x ¯ (s) π Q j=1 i j 2 j t−τ( t)    t−τm T N ¯ is amplified through changing τ (t) to τ M for the j=1 πi j Q 2 j x(s)ds t−τ( t) x¯ (s) nonnegativeness of the integrand. Finally, the results are get based on the (24).

Circuits Syst Signal Process

It follows from Lemma 1 that T    R0 −R0 x(t) ¯ x(t) ¯ , −τm (30) R0 −R0 x(t ¯ − τm ) x(t ¯ − τm ) t−τm ⎤ ⎤⎡ ⎤T ⎡ ⎡  t R2 0 −R2 x(t) ¯ x(t) ¯ ˙¯ ˙¯ −2R2 R2 ⎦ ⎣ x(t ¯ − d(t)) ⎦ . ¯ − d(t)) ⎦ ⎣ R2 x(s)R ≤ ⎣ x(t −h 2 x(s)ds t−h x(t ¯ − h) 0 R2 −R2 x(t ¯ − h) 

t

˙¯ x˙¯ T (s)R0 x(s)ds ≤



(31) Combining (27), (29)–(31) and introducing slack matrices Mi , Ni (i ∈ S), we obtain L[V (x(t), ¯ θt )] − γ 2 ω T (t)ω(t) + z˜ T (t)˜z (t) ⎛ ⎞ N

≤ x¯ T (t) ⎝2Pi A¯ i + πi j P j + Q 0 + Q 1 + Q 2i + Q 3 + τm Z 1 + δ Z 2 ⎠ x(t) ¯ j=1

+ 2 x¯ T (t)Pi A¯ di x(t ¯ − τ (t)) + 2 x¯ T (t)Pi B¯ i x(t ¯ − d(t)) + 2 x¯ T (t)Pi Aωi ω(t) ¯ − τm )Q 0 x(t ¯ − τm ) − x(t ¯ − τ M )Q 1 x(t ¯ − τM ) + 2 x¯ T (t)Pi C¯ i ek (t) − x(t ¯ − τ (t))(1 − u) − x(t ¯ − h)Q 3 x(t ¯ − h) − x(t ¯ − τ (t))Q 2i x(t 2˙ 2 ˙¯ + δ x(t)R ˙¯ ˙¯ + h x(t)R ˙¯ ˙¯ + ξ T (t)A2T A2 ξ(t) + τm x(t)R ¯ 0 x(t) 1 x(t) 2 x(t) ⎡ ⎤T ⎡ ⎤⎡ ⎤ x(t) ¯ −R2 R2 x(t) ¯ 0 ¯ − d(t)) ⎦ ⎣ R2 −2R2 R2 ⎦ ⎣ x(t ¯ − d(t)) ⎦ + ⎣ x(t x(t ¯ − h) x(t ¯ − h) 0 R2 −R2   T    t−τm −R0 R0 x(t) ¯ x(t) ¯ ˙¯ ˙¯ + − x(s)R 1 x(s)ds R0 −R0 x(t ¯ − τm ) x(t ¯ − τm ) t−τ M    t−τm T ˙ ¯ − τm ) − x(t + 2ζ (t)Mi x(t ¯ − τ (t)) − x(s)ds ¯ − γ 2 ω T (t)ω(t) t−τ (t)   t−τ (t) T ˙¯ + 2ζ (t)Ni x(t ¯ − τ (t)) − x(t ¯ − τM ) − x(s)ds . (32) t−τ M

where

T MiT = Mi1

T NiT = Ni1

T Mi2 T Ni2

T Mi3 T Ni3

T Mi4 T Ni4

T Mi5 T Ni5

Mi6 Ni6

0 0

 0 ,  0 .

Note that  −2ζ (t)Mi T

 ≤

t−τm

t−τ (t)

t−τm

t−τ (t)

˙¯ x(s)ds

˙¯ x¯˙ T (s)R1 x(s)ds + (τ (t) − τm )ζ T (t)Mi R1−1 MiT ζ (t),

(33)

Circuits Syst Signal Process

 −2ζ T (t)Ni  ≤

t−τ (t)

˙¯ x(s)ds

t−τ M t−τ (t)

t−τ M

˙¯ x˙¯ T (s)R1 x(s)ds + (τ M − τ (t))ζ T (t)Ni R1−1 NiT ζ (t).

(34)

Then, (12) can be rewritten as ˙¯ ˙¯ − ekT (t)W ekT (t) > 0, ρ x(t)Ω x(t)

(35)

where 

 W 0 Ω= . 0 0 Combining (32)–(35), we can obtain L[V (x(t), ¯ θt )] − γ 2 ω T (t)ω(t) + z˜ T (t)˜z (t) ⎛ ⎞ N

≤ x¯ T (t) ⎝2Pi A¯ i + πi j P j + Q 0 + Q 1 + Q 2i + Q 3 + τm Z 1 + δ Z 2 ⎠ x(t) ¯ j=1

+ 2 x¯ (t)Pi A¯ di x(t ¯ − τ (t)) + 2 x¯ T (t)Pi B¯ i x(t ¯ − d(t)) + 2 x¯ T (t)Pi Aωi ω(t) ¯ − τm )Q 0 x(t ¯ − τm ) − x(t ¯ − τ M )Q 1 x(t ¯ − τM ) + 2 x¯ T (t)Pi C¯ i ek (t) − x(t T

− x(t ¯ − τ (t))Q 2i x(t ¯ − τ (t))(1 − u) − x(t ¯ − h)Q 3 x(t ¯ − h) 2˙ 2 ˙¯ + δ x(t)R ˙¯ ˙¯ + h x(t)R ˙¯ ˙¯ + τm x(t)R ¯ 0 x(t) 1 x(t) 2 x(t) 2 T T T ˙¯ − ekT (t)W ekT (t) ¯˙ x(t) − γ ω (t)ω(t) + ξ (t)A2 A2 ξ(t) + ρ x(t)Ω ⎡ ⎤T ⎡ ⎤⎡ ⎤ x(t) ¯ −R2 R2 x(t) ¯ 0 ¯ − d(t)) ⎦ ⎣ R2 −2R2 R2 ⎦ ⎣ x(t ¯ − d(t)) ⎦ + ⎣ x(t x(t ¯ − h) x(t ¯ − h) 0 R2 −R2   T   −R0 R0 x(t) ¯ x(t) ¯ + R0 −R0 x(t ¯ − τm ) x(t ¯ − τm ) + 2ζ T (t)Mi [x(t ¯ − τ (t))] + 2ζ T (t)Ni [x(t ¯ − τm ) − x(t ¯ − τ (t)) − x(t ¯ − τ M )] −1 T −1 T T + (τ (t) − τm )ζ (t)Mi R1 Mi ζ (t) + (τ M − τ (t))ζ (t)Ni R1 Ni ζ (t). (36) By using Lemma 2 and Schur complement, it is easy to see that (23) with s = 1, 2 are sufficient conditions to guarantee L[V (x(t), ¯ θt )] − γ 2 ω T (t)ω(t) + z˜ T (t)˜z (t) < 0.

(37)

Then, the following inequality can be concluded E{LV (x(t), ¯ i, t)} < −λmin (Ψ )E{ζ T (t)ζ (t)}.

(38)

Circuits Syst Signal Process

Define a new function as ¯ i, t). W (x(t), ¯ i, t) = et V (x(t),

(39)

Its infinitesimal operator L is given by W(x(t), ¯ i, t) = et V (x(t), ¯ i, t) + et LV (x(t), ¯ i, t).

(40)

By the generalized I t oˆ formula [22] , we can obtain from (40) that E{W (x¯t , i, t)} − E{W (x¯0 , i)}  t  t es E{V (x¯s , i)}ds + es E{LV (x¯s , i)}ds. = 0

(41)

0

Then, similar to the method of [41], we can see that there exist a positive number α such that for t > 0 E{V (x¯t , i, t)} ≤ α

sup

−τ M ≤s≤0

  φ(s)2 e−t .

(42)

¯ it can be shown from (42) that for t ≥ 0 Since V (x¯t , i, t) ≥ {λmin (Pi )}x¯ T (t)x(t), ¯ ≤ α¯ −t E{x¯ T (t)x(t)}

sup

−τ M ≤s≤0

  φ(s)2 .

where α¯ = α/(λmin Pi ). Recalling definition 1, the proof can be completed.

(43)



Remark 4 A delay-dependent stochastic stability condition for MJSs is provided in Theorem 1.  a Lyapunov function is constructed, and the term In above proof, t N T (s) ¯ in (29) is separated into two parts. It is easy to x ¯ π Q j=1 i j 2 j x(s)ds t−τ( t) see that this method has less conservative than the ones in the literature [33,34]. Theorem 1 established some analysis results. In the following, the problem of state estimator design based on event-triggered is to be considered and the following results can be readily obtained from Theorem 1. Theorem 2 For given constants γ , μ and 0 ≤ τm ≤ τ M , the augmented systems (16) are EMSS with a prescribed H∞ performance γ under the event trigger scheme (12) if there exist Pi = diag{Pi1 , Pi2 } > 0, Q 0 = diag{Q0 , Q0 } > 0, Q 1 = diag{Q1 , Q1 } > 0, Q 2i = diag{Q2i , Q2i } > 0, R0 = diag{R0 , R0 } > 0, R1 = diag{R1 , R1 } > 0, Z 1 = diag{Z1 , Z1 } > 0, Z 2 = diag{Z2 , Z2 } > 0, W > 0, G¯ i , Mik = diag{Mik , Mik } and Nik = diag{Nik , Nik }(i ∈ S, k = 1, 2, . . . , 6) with appropriate dimensions so that the following linear matrix inequalities hold for a given ε > 0.

Circuits Syst Signal Process

⎡ ˆ Ψ11 + Υ + Υ T ⎢ Ψˆ 21 ⎢ ⎢ Ψˆ 31 Ψˆ = ⎢ ⎢ Ψˆ 41 ⎢ ⎣ Ψ51 Ψ61 (s) N

∗ Δ0 0 0 0 0

∗ ∗ Δ1 0 0 0

∗ ∗ ∗ Δ2 0 0

∗ ∗ ∗ ∗ −I 0

⎤ ∗ ∗ ⎥ ⎥ ∗ ⎥ ⎥ < 0, s = 1, 2, ∗ ⎥ ⎥ ∗ ⎦ −R1

πi j Q 2 j ≤ Z k , k = 1, 2,

(44)

(45)

j=1

where ⎡

Γˆ

∗ ∗ ⎢ ∗ ∗ ⎢ ⎢ ∗ ∗ ⎢ ⎢ ∗ ∗ 0 Ψˆ 11 = ⎢ ⎢ B¯ T Pi + R2 ∗ ∗ ⎢ i ⎢ 0 −R0 − Q 3 ∗ ⎢ ⎣ A¯ T Pi 0 −γ 2 I ωi C¯ iT Pi 0 0

 ¯ ¯ ¯ ¯ ¯ ˆ Ψ21 = τm Pi Ai τm Pi Adi 0 0 τm Pi Bi 0 τm Pi A i τm Pi Ci , √ √ √ √

√  Ψˆ 31 = δ Pi A¯ i δ Pi A¯ di 0 0 δ Pi B¯ i 0 δ Pi A¯  i δ Pi C¯ i ,

 Ψˆ 41 = h Pi A¯ i h Pi A¯ di 0 0 h Pi B¯ i 0 h R0 A¯  i h Pi C¯ i ,       0 0 Ci T G¯ iT Γˆ ¯ i = Pi1 Ai ¯i = A C Γˆ = ¯ 1 , P , P , i i ¯ ¯ ¯ G i Ci G i Ci Pi1 Ai + G i Ci G i Ci Γˆ2       0 0 0 Pi1 Adi Pi1 Aωi , Pi B¯ i = , , Pi A¯  i = Pi A¯ di = 0 Pi2 Adi Pi2 Aωi −G¯ i Ci 0 T P A¯ di i R0

∗ −(1 − μ)Q 2i 0 0 0 0 0 0

∗ ∗ −R0 + Q 0 0 0 0 0 0

Γˆ1 = Pi1 Ai + AiT Pi1 + Π + τm Z1 + δZ2 +

N

∗ ∗ ∗ −Q 1 0 0 0 0

∗ ∗ ∗ ∗ ∗ R2 0 0

∗ ∗ ∗ ∗

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ∗ ⎥ ⎥ ∗ ⎦ −W

πi j P j1 ,

j=1

Γˆ2 = Pi2 Ai + AiT Pi2 + G¯ i Ci + Ci T G¯ iT + Π + τm Z1 + δZ2 +

N

πi j P j2 ,

j=1

Π = Q0 + Q1 + Q 2i + Q3 − R0 − R2 , Δk = ε2 Rk − 2ε Pi , (k = 0, 1, 2),

and W , Ψ51 ,Ψ61 ,Υ and δ are all as defined in Theorem 1. Moreover, the desired estimator gain matrix is given in the form of (14) as following : −1 ¯ Gi . G i = Pi2

(46)

Proof Define R = diag{I, Pi R0 −1 , Pi R1 −1 , Pi R2 −1 , I, I }. Performing a congruence transformation of R to (23). It can be derived that the matrix inequality (23) holds if and only if

Circuits Syst Signal Process

⎡

Ψ11 + Υ + Υ T ⎢ Ψˇ 21 ⎢ ⎢ ˇ 31 Ψ Ψˇ = ⎢ ⎢ ˇ 41 Ψ ⎢ ⎣ Ψ51 Ψ61 (s)

∗ −Pi R0 −1 Pi 0 0 0 0

∗ ∗ −Pi R1 −1 Pi 0 0 0

∗ ∗ ∗ −Pi R2 −1 Pi 0 0

∗ ∗ ∗ ∗ −I 0

∗ ∗ ∗ ∗ ∗ −R1

⎤ ∗ ∗⎥ ⎥ ∗⎥ ⎥ <0, ∗⎥ ⎥ ∗⎦

(47) where 

Ψˇ 21 = τm Pi A¯ i τm Pi A¯ di 0 0 τm Pi B¯ i 0 τm Pi A¯  i τm Pi C¯ i , 

√ √ √ √ √ Ψˇ 31 = δ Pi A¯ i δ Pi A¯ di 0 0 δ Pi B¯ i 0 δ Pi A¯  i δ Pi C¯ i , 

Ψˇ 41 = h Pi A¯ i h Pi A¯ di 0 0 h Pi B¯ i 0 h R0 A¯  i h Pi C¯ i , and    0 0 Pi1 Adi ¯ , Pi Adi = , Pi1 Ai + Pi2 G i Ci 0 Pi2 Adi       0 0 Pi1 Aωi 0 , Pi C¯ i = . Pi B¯ i = , Pi A¯  i = Pi2 Aωi −Pi2 G i Ci 0 Pi2 G i Ci

Pi A¯ i =



Pi1 Ai Pi2 G i Ci

Due to (Rk − ε−1 Pi )Rk−1 (Rk − ε−1 Pi ) ≥ 0, i ∈ S, k = 0, 1, 2,

(48)

− Pi Rk−1 Pi ≤ −2ε Pi + ε2 Rk , i ∈ S, k = 0, 1, 2.

(49)

we can have

Defining G¯ i = Pi2 G i in (47). Then substituting −Pi Rk−1 Pi with −2ε Pi +ε2 Rk (k = 0, 1, 2) into (47), and we obtain (44), so if (44) holds, we have (23) holds, and from −1 ¯ Gi . above proof, we know that the desired state estimator gain matrix is G i = Pi2 This completes the proof.

Remark 5 Theorem 2 illustrates that for given ρ and ε, the feedback gain G i can be obtained by solving a set of LMIs in (44) and (45); meanwhile, using Theorem 2, for the preselected W and the feedback gain G i , we can get event-triggered parameter ρ. Therefore, Theorem 2 provides a method to co-design the feedback gain and eventtriggered parameter. Remark 6 In this paper, event-triggered scheme has been taken into account in MJSs. The main purpose is to build an mechanism which provides a useful way to determine when sampling action is carried out. In another words, the samples are transmitted while a specified trigger condition prevails, so event-triggered scheme has some advantages to reduce the burden of sensor communication and the computation cost of

Circuits Syst Signal Process

estimator. Under the event-triggered mechanism, Theorem 2 provides a state estimator design method of the MJSs with time delay. In the next section, a numerical example is provided to show the usefulness of the proposed design procedure for the desired state estimation.

4 Numerical example In this section, well-studied example is considered to illustrate the effectiveness of above approaches proposed and also to explain the proposed method on state estimator design. Consider linear MJSs in the form (1) with two modes. For mode 1 and mode 2, the dynamics of system with following parameters are described as [18]: ⎡ A1 = C1 = L1 = A2 = C2 = L2 =

⎤ −3 1 0 ⎣ 0.3 −2.5 1 ⎦, Ad1 −0.1 0.3 −3.8



0.8 0.3 0 , Cd1 = 0.2



0.5 −0.1 1 , L d1 = 0 ⎡ ⎤ −2.5 0.5 −0.1 ⎣ 0.1 −3.5 0.3 ⎦, Ad2 −0.1 1 −2



0.5 0.2 0.3 , Cd2 = 0



0 1 0.6 , L d2 = 0 0

⎡

⎡ ⎤ ⎤ 0.1 0.6 1 −1 −0.8 ⎦, Aω1 = ⎣ 0 ⎦ , 1 −2.5 1  −0.6 , Cω1 = 0.2,

−0.2 = ⎣ 0.5 0 −0.3

 0 , L ω1 = 0, ⎡ ⎡ ⎤ ⎤ 0 −0.3 0.6 −0.6 0.5 0 ⎦, Aω2 = ⎣ 0.5 ⎦ , = ⎣ 0.1 −0.6 1 −0.8 0  −0.6 0.2 , Cω2 = 0.5,  0 , L ω2 = 0. 0

T Suppose the initial conditions are given by x(0) = 0.8 0.2 −0.9 , x(0) ˆ = 

T 0.4 0.6 0 0.2 0 and the transition probability matrix π = . Setting h = 0.3 0.7 0.1 and ρ = 0.1, by applying Theorem 2, we get the maximum allowable delay τ M = 0.976 for τm = 0.05, μ = 0.2 , ε = 10 and γ = 1.2. More detailed calculation results for different values of ρ are given in Table 1, which describes the upper bound of τ M varying along the ρ. From Table 1, it can be shown that the larger ρ, the smaller τ M . For given h = 0.1, τm = 0.05, τ M = 0.68, t = 30, ε = 10, Table 2 gives the relation of the trigger parameter, trigger times and the percentage of data transmissions; it shows that the larger ρ , the smaller trigger times, the smaller percentage of data transmission, which are reasonable results. Table 1 h = 0.1, τm = 0.05, t = 30, ε j = 10( j = 0, 1, 2)

ρ

0

0.01

0.1

0.3

The upper bound of τ M

1.131

1.047

0.976

0.965

Circuits Syst Signal Process Table 2 h = 0.1, τm = 0.05, τ M = 0.68, t = 30, ε j = 10 ( j = 0, 1, 2)

ρ

0

0.01

0.1

0.3

Trigger times

300

286

112

78

Data transmission

100 %

95.3 %

37.3 %

22.7

2.2

2

Mode

1.8

1.6

1.4

1.2

1

0.8

0

5

10

15

20

25

30

20

25

30

Time(s) Fig. 1 Operation modes 0.8 0.6

State response

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

0

5

10

15

Time(s) Fig. 2 State response for system (1)

Circuits Syst Signal Process

0.4

0.2

Estimated signals error

0

−0.2 ρ=0 ρ=0.1

−0.4

−0.6

−0.8

0

5

10

15

20

25

30

20

25

30

Time(s)

Event−based release instants and release interval

Fig. 3 Estimated signals error z˜ (t) 0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

0

5

10

15

Time(s) Fig. 4 The release instants and release interval for given ρ = 0.01

To illustrate the performance of the designed state estimator, choose the disturbance function as follows ⎧ ⎨ −0.625, 5 < t < 10, 0.495, 15 < t < 20, ω(t) = ⎩ 0, others. By using LMI toolbox in LMIs (44) and (45), the desired estimator and the triggered matrix for h = 0.1, ρ = 0.1, τm = 0.05, τ M = 0.68, t = 30, ε = 10 can be obtained as follows:

Event−based release instants and release interval

Circuits Syst Signal Process

1.5

1

0.5

0

0

5

10

15

20

25

30

20

25

30

Time(s)

Event−based release instants and release interval

Fig. 5 The release instants and release interval for given ρ = 0.1 1.5

1

0.5

0

0

5

10

15

Time(s) Fig. 6 The release instants and release interval for given ρ = 0.3

⎡

⎡ ⎡ ⎤ ⎤ −0.0977 0.1423 1.6155 G 1 = ⎣ −0.1002 ⎦ , G 2 = ⎣ −0.0273 ⎦, W = ⎣ −0.3108 −0.0764 −0.1116 −0.3554

−0.3108 0.8123 −0.2094

⎤ −0.3554 −0.2094 ⎦ . 0.9700

With above state estimators obtained, the simulation results are shown in Figs. 1, 2, 3, 4, 5 and 6. Figures 1, 2 show the operation modes of the MJSs and state response for system (1), and Fig. 3 shows the estimated signal error z˜ (t) = z(t) − zˆ (t) under ρ = 0 and ρ = 0.1, respectively. It is obvious to see that the simulation results are almost

Circuits Syst Signal Process

the same from Figs. 3, but the percentage of data transmission under event-triggered scheme used much small number than time-triggering scheme. Figures 4, 5, 6 show the event-triggered release instants and intervals for ρ = 0.01, ρ = 0.1, ρ = 0.3, respectively, when other parameters are taken the same as h = 0.1, τm = 0.05, τ M = 0.68, t = 30, ε = 10. These results illustrate that event-triggered scheme is more advantageous than the time-triggered scheme in improving the resource utilization. This means that the event-triggered scheme could not only reduce the burden of the communication but also preserve the desired properties of the ideal continuous state feedback system, such as stability and convergence compared with previous similar topic in the literature [13,18].

5 Conclusions In order to reduce the communication burden, a novel event-triggered scheme is proposed to determine whether the sampled sensor measurements will be transmitted to the estimator. Under this event-triggered scheme, this paper considers state estimation problem for MJSs with time-varying delay. A delay model has been used to describe the prosperities of the event trigger and effects of the transmission delay on the MJSs. Based on Lyapunov functional method, an sufficient condition for the EMSS and criteria for the existence of the desired state estimator derived are proposed, which are expressed in the form of linear matrix inequalities. At last, a numerical simulation example has shown that our event-triggered scheme can lead to a larger average release period than those by some existing methods. Acknowledgments The authors would like to acknowledge the Natural Science Foundation of China (Nos. 71571092, 61403185), the Natural Science Foundation of Jiangsu Province (No. BK20140457), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 15KJB120002).

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