# Event-Triggered Dissipative Filter Design for Semi-Markovian Jump Systems with Time-Varying Delays

Authors: Huiying Chen; Zuxin Li; Weifeng Xia
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper deals with the event-triggered dissipative filtering problem for semi-Markovian jump systems with time-varying delays. The purpose is to design a filter which guarantees that the filtering error system is not only stochastically stable but also satisfies dissipativity. First, based on Lyapunov–Krasovskii theory and matrix integral inequality, a sufficient condition is established for the existence of such a filter. Then, the codesign method of event-triggered matrices and the desired filter parameters is proposed. Finally, two numerical examples are given to illustrate the advantages and validity of the scheme developed in this paper.


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## Body

## 1. Introduction

Over the past decades, Markovian jump systems (MJSs) have attracted significant attention by the research community mainly due to the fact that they have a powerful framework to model practical systems, such as economy [1], robotics [2], mechanics [3], and solar thermal receivers [4]. However, among the aforementioned works, the jump time of a Markov chain is a time-homogeneous exponential random variable. This result in the MJSs has many limitations in applications. To overcome this obstacle, a more wider class of systems with general sojourn time distributions, named semi-Markovian jump systems (SMJSs), have been studied in recent years. To mention a few, the problem of stability and stabilization for semi-Markovian jump systems was addressed in [5–7]. The controller design problem for semi-Markovian jump systems was investigated in [8–11]. In [12, 13], the filtering problem for semi-Markovian jump systems was considered.It is known that time-delay phenomenon is commonly observed in real-world applications and usually deteriorates the stability of a control system [14–22]. As a consequence, increasing attention has been devoted to studying of SMJSs with time delays. For instance, the problem of finite-time H∞ synchronization for SMJSs with time-varying delays was reported in [23, 24]. The resilient estimation problem for nonlinear descriptor SMJSs was probed in [25], where the T-S fuzzy model and time-varying delays are taken into account. In the case, when the actuator or sensor failures appear in the SMJSs, the reliable controller or filter was designed for delayed SMJSs in [26, 27], respectively. By applying a piecewise analysis method, a stochastic stability condition was established in [28], where the time-varying delays are dependent on system modes. However, the stability condition in [28] may be conservative because Jensen’s inequality was used to bound the integral term, and thus there still exists some room to improve. This is the first motivation of this paper.On the other hand, communication resource saving is also a significant issue for networked systems because of the bandwidth resource becomes more and more limited as the complexity of the network increase [29]. It has been shown that the event-triggered scheme is an effective way to solve this issue. Compared with a time-triggered scheme, not all of sampled data packets in the event-triggered scheme are transmitted through a communication network. Only when an event condition is satisfied, the sampled data packets are transmitted [30]. As a result, the unnecessary communication resources are saved in the event-triggered scheme. In recent years, the event-triggered scheme has been utilized in the controller and filter design problem for various dynamic systems [31–36]. As for SMJSs, the event-triggered leader-following consensus for a multiagent system with semi-Markovian jumping parameters was studied in [37]. In the case of discrete-time systems, an H∞ filter was designed in [38] for a class of semi-Markovian jumping neural networks, where the quantized measurements was used in order to save the network resource, although the dissipative filtering problem for SMJSs was discussed in [39], where time-varying delays were not be considered. Therefore, the event-triggered filtering problem for SMJSs with time-varying delays should be further investigated. This is also a motivation to this paper.Based on the above mentioned discussions, this paper considers the dissipative filter design problem for SMJSs with time-varying delays by using the event-triggered scheme. The main contributions of this paper can be highlighted as follows: (1) the dissipative filter design problem for SMJSs in the presence of time-varying delays is investigated by applying the event-triggered scheme. (2) Based on Lyapunov–Krasovskii theory and matrix integral inequality, a new stochastic stability criterion for SMJSs is established. (3) The method developed in this paper will be helpful for some further development involving this topic, such as nonfragile control and state estimation.The rest part of the paper is organized as follows. InSection 2, we describe the problem formulation. Section 3 presents the event-triggered filter design scheme for the underlying system. Numerical examples and simulation results which demonstrate the effectiveness and advantages of the proposed scheme are provided in Section 4. The conclusion of this paper is made in Section 5.Notations. Throughout this paper, a real symmetric matrix X>0X≥0 means that X is positive definite (respectively, positive semidefinite) matrix; XT represents transpose of the matrix X; ℝn and ℝn×m stand for n−dimensional Euclidean space and set of all n×m real matrices; In denotes the n−dimensional identity matrix; 0n and 0n×m denote the n−dimensional zero matrix and n×m−dimensional zero matrix; symX denotes X+XT; diag⋯ stands for a block-diagonal matrix. The notation ∗ is used as an ellipsis for terms that are induced by symmetry; L20,∞ is the space of the square-integrable vector function over 0,∞; ⋅ denotes the Euclidean norm for vectors. Matrix dimensions, if not explicitly stated are assumed to be compatible for algebraic operations.


## 2. Problem Statement and Preliminaries

Given a probability spaceΩ,F,P, we consider the following semi-Markovian jump systems with time-varying delays:(1)x˙t=Aλtxt+Adλtxt−τt+Bλtωt,yt=Cλtxt+Cdλtxt−τt+Dλtωt,zt=Lλtxt,xt=φt,t∈−τ2,…,0,where xt∈ℝn is the system state vector; yt∈ℝp is the measurement output; zt∈ℝq is the control output; and ωt∈ℝr is the disturbance input, which belongs to L20,∞; φt is a initial condition. The process λt is a time-homogeneous semi-Markov stochastic process, which takes value in a finite state space S=1,2,…,s. The evolution of λt is governed by the following probability transition:(2)Prλt+h=j∣λt=i=ϕijhh+oh,i≠j,1+ϕiihh+oh,i=j,where h>0, limh⟶00h/h=0, and ϕijh is the transition rate from mode i at time t to mode j at time t+h when i≠j and(3)ϕiih=−∑j=1sϕijh.Remark 1.
It is worth noting that when the transition rateϕijh of SMJSs is independent of h, that is, ϕijh=ϕij, then the SMJSs reduce to the common MJSs, which have been investigated extensively in [1, 2, 18, 40, 41].
In system (1), τt denotes the time-varying delay, which satisfies(4)τ1≤τt≤τ2,0≤τ˙t≤μ<1.
To simplify the notation, for eachλt=i∈S, the matrices Aλt, Adλt, Bλt, Cλt, Cdλt, Dλt, and Lλt will be denoted by Ai, Adi, Bi, Ci, Cdi, Di, and Li, respectively.
In this paper, the sampler is time-driven with a sampling periodg>0. In the traditional method, the sampled data should be sent to the filter in every sampling period, and this would lead to need of more band resource. To solve this issue, we introduce an event-triggered processor between the sampler and filter to determine whether or not the current sampled data should be transmitted to the filter. It is assumed that the latest released output is tk, then, the next released instant tk+1 is determined by the following condition [37, 38]:(5)tk+1=inft>tkyt−ytkTϒiyt−ytk≥ρiytTϒiyt,where ρi∈0,1 and ϒi>0 are event-triggered parameters.
Fort∈tk,tk+1, we define the following function:(6)δt=yt−ytk.
Then, by the event-triggered condition (5), we obtain(7)ρiytTϒiyt−δtTϒiδt>0.
This paper aims to design a filter as follows:(8)x˙ft=Afλtxft+Bfλtytk,zft=Cfλtxft,t∈tk,tk+1,where xft∈ℝn is the filter state; zft∈ℝq is the filter output; and Afλt, Bfλt, and Cfλt are the filter parameters to be designed. For each λt=i∈S, the matrices Afλt, Bfλt, and Cfλt will be denoted by Afi, Bfi, and Cfi.
Letξt=xtxftT and et=zt−zft. Then, the filtering error system resulting from (1) and (8) is given by(9)ξ˙t=A˜iξt+A˜dixt−τt+B˜δitδt+B˜ωiωt,et=L˜iξt,where(10)A˜i=Ai0BfiCiAfi,A˜di=AdiBfiCdi,B˜δi=0−Bfi,B˜ωi=BiBfiDi,L˜i=Li−Cfi.
Throughout this paper, the following definitions will be adopted.Definition 1 (see [45]).
System (1) with ωt=0 is said to be stochastically stable if the following inequality holds for any initial mode λ0 and initial condition φ:(11)limt⟶∞ℰ∫0txtTxtdtx0,λ0<∞.Definition 2 (see [9, 43]).
For given matricesZ,Y,X with Z≤0 and X is symmetric, the filtering error system (9) is said to be dissipative, if there exist θ>0, such that for any T>0 and zero initial condition, the following inequality holds:(12)ℰ∫0TJtdt≥θℰ∫0TωtTωtdt,where Jt=etTZet+2etTYωt+ωtTXωt.
Noting thatZ≤0, we can deduce that there always exists matrix Z˜≥0, such that Z=−Z˜TZ˜.
The main purpose of this paper is to design a filter in the form of (8), such that the following conditions hold:(1)
Whenωt=0, the filtering error system (9) is stochastically stable(2)
Whenωt≠0 and ωt∈L20,∞, under zero initial condition, the filtering error system (9) is dissipative in the sense of Definition 2
To realize this purpose, we need to introduce the following lemmas.Lemma 1 (see [44]).
For any matrixZ>0, scalars a,b with b>a, the following inequality holds:(13)b−a∫abx˙αTZx˙αdα≥ϖ1ϖ2TZ003Zϖ1ϖ2,where ϖ1=xb−xaandϖ2=xb+xa−2/b−a∫abxαdα.The following lemma is a direct result of [45].Lemma 2 (see [46]).
For a given matrixZ>0, scalar ε∈0,1, then there exist matrices Y1 and Y2, such that the following inequality holds:(14)1εZ0011−εZ≥symY1In0n+Y20nIn−εY1Z−1Y1T−1−εY2Z−1Y2T.Remark 2.
It is well recognized that the reciprocally convex combination inequality in [47] is a powerful tool to handle the time-varying delays when analysing the stability of a system with time-varying delays. It has been shown that the inequality in Lemma 2 is a more accurate inequality than the reciprocally convex combination inequality; thus, the stability criteria derived by Lemma 2 is expected to be less conservative.


## 3. Main Results

In this section, we carry out the analysis of stochastic stability and dissipativity for the resulting filtering error system (9). Then, on the basis of these results, the desired filter design scheme for system (1) is established.Theorem 1.
For given scalarsτ1,τ2,μ,0<ρi<1 and matrices Z,Y,X with Z≤0 and X is symmetric, the filtering error system (9) is stochastically stable with dissipative, if there exist matrices Pi>0, Q1i>0, Q2i>0, Q3i>0, S>0, Ri>0, U>0, Zi>0, V>0, and ϒi>0, any matrices Y1 and Y2, such that the following inequalities hold, for each i∈S:(15)∑j=1sϕijhQ1j+Q3j−S<0,(16)∑j=1sϕijhQ2j+Q3j−S<0,(17)∑j=1sϕijhQ3j−S<0,(18)τ1∑j=1sϕijhRj−U<0,(19)τ12∑j=1sϕijhZj−V<0,(20)Φiτ1ΨTY2∗−Z˜i<0,(21)Φiτ2ΨTY1∗−Z˜i<0,where(22)Φiτt=W2iTPiW1τt+W1τtTPiW2i+W1τtT∑j=1sϕijhPjW1τt+e1TQ1i+Q2i+Q3i+τ2Se1−e3TQ1ie3−1−μe4TQ2ie4−e5TQ3ie5+WsiTτ12Ri+12τ12U+τ122Zi+12τ22−τ12VWsi−ΨTsymY1I2n02n+Y202nI2nΨ−W3TR˜iW3+ρiCie1+Cdie4+Die10TϒiCie1+Cdie4+Die10−e9Tϒie9−W0TL˜iTZL˜iW0−2W0TL˜iTYe10−e10TXe10,Wsi=Ai0n,2nAdi0n,5nBi,W˜si=A˜i02n,nA˜di02n,4nB˜δiB˜ωi,W0=cole1,e2,τ12=τ2−τ1,W1τt=colW0,τ1e6,τt−τ1e7−τ2−τte8,W2i=colW˜si,e1−e3,e3−e5,W3=cole1−e3,e1+e3−2e6,W4=cole3−e4,e3+e4−2e7,W5=cole4−e5,e4+e5−2e8,ej=0n,j−1nIn0n,8−jn+p+r,j=1,…,8,e9=0p,8nIp0p,r,e10=0r,8n+pIr,Ψ=colW4,W5,Z˜i=diagZi,3Zi,R˜i=diagRi,3Ri.Proof.
We define the following augmented vector:(23)ζt=colξt,∫t−τ1txαdα,∫t−τ2t−τ1xαdα.
Then, we choose a Lyapunov–Krasovskii functional candidate for system (9) as follows:(24)Vξt,λt,t=∑j=14Vjt,where ξt=ξt+σ, σ∈−τ2,0, and(25)V1t=ζtTPiζt,V2t=∫t−τ1txαTQ1ixαdα+∫t−τtt−τ1xαTQ2ixαdα+∫t−τ2txαTQ3ixαdα+∫−τ20∫t+βtxαTSxαdα,V3t=τ1∫−τ10∫t+βtx˙αTRix˙αdαdβ+∫−τ10∫θ0∫t+βtx˙αTUx˙αdαdβdθ,V4t=τ12∫−τ2−τ1∫t+βtx˙αTZix˙αdαdβ+∫−τ2−τ1∫θ0∫t+βtx˙αTVx˙αdαdβdθ.
Letℒ be the weak infinitesimal generator of the random process ξt,λt and ηt=colxtxft,xt−τ1,xt−τt,xt−τ2,1/τ1∫t−τ1txαdα,1/τt−τ1∫t−τtt−τ1xαdα,1/τ2−τt∫t−τ2t−τtxαdα,δt,ωt. Then, for each i∈S, following the similar lines as in [13, 27], we obtain(26)ℒV1t=ζ˙tTPiζt+ζtTPiζ˙t+ζtT∑j=1sϕijhPjζt=ηtTW2iTPiW1τt+W1τtTPiW2i+W1τtT∑j=1sϕijhPjW1τtηt.
By means of (15)–(17), one has(27)ℒV2t≤xtTQ1i+Q2i+Q3i+τ2Sxt−xt−τ1TQ1ixt−τ1−1−μxt−τtTQ2ixt−τt−xt−τ2TQ3ixt−τ2+∫t−τ1txαT∑j=1sϕijhQ1j+∑j=1sϕijhQ3j−Sxαdα+∫t−τtt−τ1xαT∑j=1sϕijhQ2j+∑j=1sϕijhQ3j−Sxαdα+∫t−τ2t−τtxαT∑j=1sϕijhQ3j−Sxαdα≤ηtTe1TQ1i+Q2i+Q3i+τ2Se1−e3TQ1ie3−1−μe4TQ2ie4−e5TQ3ie5ηt.
In terms of (18), we deduce that(28)ℒV3t=τ12x˙tTRix˙t−τ1∫t−τ1tx˙αTRix˙αdα+τ1∫−τ10∫t+βtx˙αT∑j=1sϕijhRjx˙αdαdβ+12τ12x˙tTUx˙t−∫−τ10∫t+βtx˙αTUx˙αdαdβ≤x˙tTτ12Ri+12τ12Ux˙t−τ1∫t−τ1tx˙αTRix˙αdα.
According to Lemma1, one has(29)−τ1∫t−τ1tx˙αTRix˙αdα≤−xt−xt−τ1xt+xt−τ1−2τ1∫t−τ1txαdαTRi003Ri×xt−xt−τ1xt+xt−τ1−2τ1∫t−τ1txαdα=−ηtTe1−e3e1+e3−2e6TR˜ie1−e3e1+e3−2e6ηt=−ηtTW3TR˜iW3ηt.
Then, it follows from (28) and (29) that(30)ℒV3t≤ηtTWsiTτ12Ri+12τ12UWsi−W3TR˜iW3ηt.
Now,ℒV4t is obtained as(31)ℒV4t=τ122x˙tTZix˙t−τ12∫t−τ2t−τ1x˙αTZix˙αdα+τ12∫−τ2−τ1∫t+βtx˙αT∑j=1sϕijhZjx˙αdαdβ+12τ22−τ12x˙tTVx˙t−∫−τ2−τ1∫t+βtx˙αTVx˙αdαdβ.
Splitting the following integral into two terms, we obtain(32)−τ12∫t−τ2t−τ1x˙αTZix˙αdα=−τ12∫t−τtt−τ1x˙αTZix˙αdα−τ12∫t−τ2t−τtx˙αTZix˙αdα.
By Lemma1, it can be verified that(33)−τ12∫t−τtt−τ1x˙αTZix˙αdα≤−τ12τt−τ1xt−τ1−xt−τtxt−τ1+xt−τt−2τt−τ1∫t−τtt−τ1xαdαTZi003Zi×xt−τ1−xt−τtxt−τ1+xt−τt−2τt−τ1∫t−τtt−τ1xαdα=−τ12τt−τ1ηtTe3−e4e3+e4−2e7TZ˜ie3−e4e3+e4−2e7ηt=−τ12τt−τ1ηtTW4TZ˜iW4ηt,(34)−τ12∫t−τ2t−τtx˙αTZix˙αdα≤−τ12τ2−τtxt−τt−xt−τ2xt−τt+xt−τ2−2τ2−τt∫t−τ2t−τtxαdαTZi003Zi×xt−τt−xt−τ2xt−τt+xt−τ2−2τ2−τt∫t−τ2t−τtxαdα=−τ12τ2−τtηtTe4−e5e4+e5−2e8TZ˜ie4−e5e4+e4−2e8ηt=−τ12τ2−τtηtTW5TZ˜iW5ηt.
In light of (33)-(34) and Lemma 2, there exist matrices Y1 and Y2, such that(35)−τ12∫t−τ2t−τ1x˙αTZix˙αdα≤−τ12τt−τ1ηtTW4TZ˜iW4ηt−τ12τ2−τtηtTW5TZ˜iW5ηt≤−ηtTΨTsymY1I2n02n+Y202nI2nΨηt+τt−τ1τ12ηtTΨTY1Z˜i−1Y1TΨηt+τ2−τtτ12ηtTΨTY2Z˜i−1Y2TΨηt.
Then, it follows from (31)–(35) together with (19) that(36)ℒV4t≤x˙tTτ122Zi+12τ22−τ12Vx˙t−τ12∫t−τ2t−τ1x˙αTZix˙αdα≤ηtTWsiTτ122Zi+12τ22−τ12VWsiηt−ηtTΨTsymY1I2n02n+Y202nI2nΨηt+τt−τ1τ12ηtTΨTY1Z˜i−1Y1TΨηt+τ2−τtτ12ηtTΨTY2Z˜i−1Y2TΨηt.
Then, by (24)–(27), (30), and (36), we arrive at(37)ℒVt−Jt+ρiytTϒiyt−δtTϒiδt≤ηtTΦ˜iτtηt,where(38)Φ˜iτt=Φiτt+τt−τ1τ12ΨTY1Z˜i−1Y1TΨ+τ2−τtτ12ΨTY2Z˜i−1Y2TΨ,and Φiτt is defined in Theorem 1.
By convex combination theory, it can be deduced thatΦ˜iτt<0 holds if Φ˜iτ1<0 and Φ˜iτ2<0. These two inequalities can be obtained by applying Schur complement to (20) and (21), respectively.
Therefore, inequalities (7) and (37) lead to(39)ℒVt−Jt<0.
Using the well-known Dykin’s formula to (39), we obtain that, for any T≥0,(40)ℰ∫0TJtdt>ℰVξt,λt,t−Vξ0,λ0,0>0.
LetT⟶∞. It follows from (40) that there exists θ>0, such that inequality (12) holds. Thus, we conclude that the filtering error system (9) is dissipative in the sense of Definition 2.
Next, we will prove that the filtering error system (9) is stochastically stable. Let ωt=0, then (39) leads to(41)ℒVt<etTZet.
Noticing thatZ≤0, we obtain that there exists a sufficient small ς>0, such that ℒVt<−ςxt2. Then, along a similar line as in the proof of [41], we derive that the filtering error system (9) is stochastically stable. The proof is complete here.Remark 3.
Theorem1 provides a sufficient condition such that the filtering error system (9) is stochastically stable and dissipative. It is noted that this condition is dependent not only on system modes and time delays but also on event-triggered parameters.
In order to compare with previous results, we consider the case when system (1) with ωt=0:(42)x˙t=Aixt+Adixt−τt.
The stochastic stability condition of system (42) can be derived from Theorem 1 immediately. We give the result in the following corollary.Corollary 1.
For given scalarsτ1, τ2, and μ, system (42) is stochastically stable if there exist matrices Pi>0, Q1i>0, Q2i>0, Q3i>0, S>0, Ri>0, U>0, Zi>0, V>0, and any matrices Y1 and Y2, such that inequalities (15)–(19) and the following inequalities hold, for each i∈S:(43)Ξiτ1Ψ˜TY2∗−Z˜i<0,Ξiτ2Ψ˜TY1∗−Z˜i<0,where(44)Ξiτt=E2iTPiE1τt+E1τtTPiE2i+E1τtT∑j=1sϕijhPjE1τt+e˜1TQ1i+Q2i+Q3i+τ2Se˜1−e˜2TQ1ie˜2−1−μe˜3TQ2ie˜3−e˜4TQ3ie˜4+Aie˜1+Adie˜3Tτ12Ri+12τ12U+τ122Zi+12τ22−τ12VAie˜1+Adie˜3−E3TR˜iE3−Ψ˜TsymY1I2n02n+Y202nI2nΨ˜,e˜i=0n,i−1nIn0n,7−in,i=1,2,…,7,E1τt=cole˜1,τ1e˜5,τt−τ1e˜6+τ2−τte˜7,E2i=colAie˜1+Adie˜3,e˜1−e˜2,e˜2−e˜4,E3=cole˜1−e˜2,e˜1+e˜2−2e˜5,E4=cole˜2−e˜3,e˜2+e˜3−2e˜6,E5=cole˜3−e˜4,e˜3+e˜4−2e˜7,Ψ˜=colE4,E5.Remark 4.
Corollary1 provides a new stochastic stability criterion for SMJSs (42). It is important to note that the condition in Corollary 1 is less conservative than some of the existing ones in the literature, which will be shown by a numerical example in Section 4.
Based on Theorem1, we give the filter parameter design scheme in the following theorem:Theorem 2.
For given scalarsτ1, τ2, and μ, 0<ρi<1, the filtering error system (9) is stochastically stable with dissipativity, if there exist matrices Q1i>0, Q2i>0, Q3i>0, S>0, Ri>0, U>0, Zi>0, V>0, Mi>0, Xi>0, P2i>0, P3i>0, and ϒi>0, any matrices Y1, Y2, A¯fi, B¯fi, and C¯fi, such that inequalities (15)–(19) and the following inequalities hold, for each i∈S:(45)MiXiXiXi>0,(46)Φ^iτ1ΨTY2W0TL˜iTZ˜T∗−Z˜i0∗∗−I<0,(47)Φ^iτ2ΨTY1W0TL˜iTZ˜T∗−Z˜i0∗∗−I<0,where(48)Φ^iτt=W^2iTW1τt+W1τtTW^2i+W1τtT∑j=1sϕijhPjW1τt+e1TQ1i+Q2i+Q3i+τ2Se1−e3TQ1ie3−1−μe4TQ2ie4−e5TQ3ie5+WsiTτ12Ri+12τ12U+τ122Zi+12τ22−τ12VWsi−ΨTsymY1I2n02n+Y202nI2nΨ−W3TR˜iW3+ρiCie1+Cdie4+Die10TϒiCie1+Cdie4+Die10−e9Tϒie9−2W0TL˜iTYe10−e10TXe10,W^2i=colW^si,P2ie1−e3,P3ie3−e5,W^si=colA^i,02n,n,A^di,02n,4n,B^δi,B^ωi,A^i=MiAi+B¯fiCiA¯fiXiAi+B¯fiCiA¯fi,A^di=MiAdi+B¯fiCdiXiAdi+B¯fiCdi,B^δi=−B¯fi−B¯fi,B^ωi=MiBi+B¯fiDiXiBi+B¯fiDi.
In this case, a desired filter is obtained with the parameters given by(49)Afi=Xi−1A¯fi,Bfi=Xi−1B¯fi,Cfi=C¯fi.Proof.
DefineP1i=MiXiXiXi and Pi=diagP1i,P2i,P3i. It can be seen from (45) that Pi>0.
LetA¯fi=XiAfi, B¯fi=XiBfi, and C¯fi=Cfi. Then, substituting those matrices and Pi into (46)-(47), we obtain (20) and (21). Therefore, according to Theorem 1, we easily draw the conclusion that the filtering error system (9) is stochastically stable and dissipative; thus, the proof is completed.Remark 5.
It should be pointed out that the results given in Theorem1 are nonconvex inequalities, so it should be further discussed. Motivated by [13, 27], the transition rate ϕijh is usually partly measurable in practice while it satisfies ϕ¯ij≤ϕijh≤ϕ¯ij. Then, the ϕijh can be rewritten as the following:(50)ϕijh=∑α=1Nψαϕij,α,∑α=1Nψα=1,ψα≥0,ϕij,α=ϕ¯ij+α−1ϕ¯ij−ϕ¯ijN−1,i≠j,j∈S,ϕ¯ij−α−1ϕ¯ij−ϕ¯ijN−1,i=j,j∈S.
Now, we give the conditions for the solvability of the present dissipative filter design problem.Theorem 3.
For given scalarsτ1, τ2, μ, and ρi, the filtering error system (9) is stochastically stable with dissipativity, if there exist matrices Q1i>0, Q2i>0, Q3i>0, S>0, Ri>0, U>0, Zi>0, V>0, Mi>0, Xi>0, P2i>0, P3i>0, and ϒi>0, any matrices Y1, Y2, A¯fi, B¯fi, and C¯fi, such that inequalities (15)–(19) and (45)–(47) hold, for each i∈S, and ϕijh=ϕij,α, α=1,2,…,N. Then, the desired filter parameters can be shown as in (49).Remark 6.
It is noted that the conditions obtained in Theorem3 are presented in terms of linear matrix inequalities, which can be checked easily by the LMI toolbox of MATLAB.


## 4. Numerical Examples

Example 1.
Consider a semi-Markovian jumping system in (42) with the following parameters [28]:(51)A1=−200−0.9,A2=−10.50−1,Ad1=−10−1−1,Ad2=−10−0.1−1.
To compare the stability criterion in Corollary1 with those in [28, 41, 48, 49], we assume that τ1=1, μ=0.5, ϕ11=−0.1, and ϕ22=−0.8. Table 1 lists the maximum allowed τ2 calculated by the method in [28, 41, 48, 49] and Corollary 1. It can be seen from Table 1 that the result in Corollary 1 is less conservative than those in [28, 41, 48, 49] for this example. It should be noted that the reduced conservatism in Corollary 1 benefits from using the inequalities in Lemma 1 and Lemma 2.Table 1
Maximum allowableτ2 using different methods.
Methods
τ
2
Theorem1 of [41]
1.471
Theorem1 of [48]
1.660
Theorem1 of [49] (m=5)
1.753
Theorem1 of [28] (l=4)
1.807
Corollary1
1.982Example 2.
Consider the semi-Markovian jumping system in (1) with the following parameters:(52)A1=−3.48880.8057−0.6451−3.2684,A2=−2.48980.28951.3396−0.0211,Ad1=−0.8620−1.2919−0.6841−2.0729,Ad2=−2.83060.4978−0.8436−1.0115,B1=0.04030.6771,B2=0.56890.2556,C1=−0.3775−0.2959,C2=−0.4751−0.2340,Cd1=0.2532−0.1684,Cd2=−0.3010−0.3564,L1=−1.3324−1.5532,L2=−1.4562−1.2322,D1=−1.1184,D2=−0.3148.
In this example, the parameters are given byτ1=0.1, τ2=0.5, ϕ11h∈−2.2,−1.8, ϕ22h∈−3.4,−2.6, ρ1=0.1, ρ2=0.2, Z=−4, Y=−1, and X=2. Then, by solving the LMIs in Theorem 3, we obtain the dissipative filter parameters as follows:(53)Af1=−6.86970.8567−3.1332−6.4577,Bf1=−0.42290.4993,Cf1=1.15570.8781T,Af2=−5.8309−1.47481.3407−1.9390,Bf2=3.48550.6502,Cf1=0.60110.9456T,ϒ1=4.8762,ϒ2=13.9442.
In the simulation, we choose the initial conditionx0=−11T and xf0=00T, the external disturbance is(54)ωt=1,2≤t≤3,−1,6≤t≤7,0,otherwise.
Then, the simulation results are provided in Figures1–4. The possible jump mode λt is illustrated in Figure 1. Figure 2 shows the event-triggered release instants and release intervals. Figure 3 plotted the filter state xft. The filter error signal et is presented in Figure 4. These simulation results clearly demonstrate that the filter design scheme developed in this paper is effective.Figure 1
Random jump modesλt.Figure 2
Release instants and release interval.Figure 3
Filter statesxft.Figure 4
Estimated signalzt and filter output signal zft.


## 5. Conclusion

The issue of the dissipative filter for SMJSs with time-varying delays has been investigated in this paper. It is should be noted that the event-triggered scheme is employed in order to save the bandwidth resource. The presented scheme provides a sufficient condition to guarantee the resulting filtering error system is stochastically stable and satisfies dissipative. What should be emphasized is that the stability criterion achieved in this paper is less conservative than some existing results. Two numerical examples are given to illustrate the effectiveness and advantages of the proposed filter design scheme. In addition, it is worth mentioning that the scheme developed in this paper can be extended to other dynamic systems, such as neural networks and sampled-data systems with time-varying delays, which will be our future research.



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