# Event-TriggeredH∞ Filtering for Multiagent Systems with Markovian Switching Topologies

Authors: Jiahao Li; Tingting Zhang; Jinfeng Gao; Ping Wu
Journal: Journal of Control Science and Engineering
 (2018)

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## Abstract

This paper is concerned with the problem of event-triggeredH∞ filtering for multiagent systems with Markovian switching topologies and network-induced delay. An event-triggered mechanism is given to ease the information transmission. Consider that the network topology is directed in this paper, which represents the communication links among agents. Due to the existence of network-induced delay, the time-delay approach is adopted, which can effectively deal with filtering error system. By constructing a Lyapunov-Krasovskii functional and employing linear matrix inequality technique, sufficient conditions are established to ensure the filtering error system to achieve asymptotically stable with H∞ performance index. A simulation example is given to illustrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

Recently, multiagent systems (MASs) have received much interest due to their widespread applications in various fields such as formation control [1, 2], sensor network [3], synchronization [4], and flocking [5, 6]. Due to the fact that communication networks consist of lots of agents, their states are usually not fully available in network outputs. Hence, filtering or state estimation problem is to estimate the states of the agent by the available output measurement, which is very important both in theory and in practice. In the past decades, the consensus-based filtering or estimation problem for a special multiagent system, i.e., the multi-sensor networked system, has received much attention. For example, Kalman filters are the filters primarily proposed in [7, 8] for multisensor networked systems, while H∞ filters are given in [9] for sensor networks with multiple missing measurements. In most existing literatures, the study of multiagent systems has been investigated in the ideal situation without considering external interference. However, in practical engineering, the existence of external disturbances cannot be ignored, which may have a great impact on agent dynamics. Furthermore, the research history of multiagent systems on H∞ filtering or state estimation is very short; there are many problems which need to be studied extensively. Thus, it is necessary to study the H∞ filtering problem of multiagent systems with many practical factors.The exchange of information between agents on a communication network is usually completed on the basis of sampled packets. Many articles have presented more and more methods based on rapid development of the theory of sampling data systems. In the early results, time-triggered communication is a common way to transfer every sampled signal to the controller and update control signal periodically, which wastes a lot of network bandwidth resources and increases network load. Compared with the widely used time-triggered sampling scheme [10–13], event-triggered communication schemes [14–16] can avoid the unnecessary transmission and reduce the release times of the sensor and the burden of communication network. Event-triggered filtering/estimation for different systems has received considerable attention in the past few years [17–22]. Paper [17] addresses event-triggered state estimation problem for a class of complex networks with mixed time delays and [20] presents H∞ filter design for a class of neural network systems with quantization. The filtering problem for discrete-time networked control system (NCS) under event-triggered scheme is proposed in [18, 19, 21]. Paper [22] designs the H∞ fuzzy filter for a class of nonlinear networked control system based on event-triggered communication scheme.In the real world, because of the uncertainty of the network, the communication topology may change. Hence, it is significant to design a filtering network with time-varying and switching topology to estimate or monitor a target; see [23–28]. Event-based H∞ filtering for discrete-time and continue-time Markov jump system with network-induced delay is investigated in [23, 24], respectively. Paper [25, 26] addresses asynchronous l2-l∞ filtering with sensor nonlinearity and delay-dependent robust H∞ control and filtering with parameter uncertainties, respectively. Paper [27] investigates the problem of H∞ estimation for a class of discrete-time Markov jump systems with time-varying transition probabilities. The problem of distributed state estimation about vehicle formation with time-varying measurement topology is discussed in [28]. To the best of our knowledge, there is no article that considers H∞ filtering problem for multiagent systems with Markovian switching topologies and network-induced delay based on event-triggered strategy. The theoretical results of such systems will be attractive and have extensive practical applications, which inspired the research presented in this paper.In the view of the above discussion, this paper addresses the event-triggeredH∞ filtering for multiagent systems with Markovian switching topologies and network-induced delay. The main contributions are summarized as follows: (1) different from the general event-triggered sampled-data strategy, this paper takes into account the effect of network-induced delay and Markovian switching topologies. A distributed event-triggered scheme is presented to release the load of the network. (2) By employing the time-delay system method, H∞ filtering performance is derived, the co-design method of the event-triggered condition and the filter design are also given. (3) The Laplacian matrices of switching topologies are not required to be symmetric.The rest of this paper is organized as follows. In Section2, we give some preliminaries and present the problem formulation. The main results for stability analysis and event-triggered H∞ filter design are elaborated in Section 3. In Section 4, a numerical example is provided to illustrate the feasibility and effectiveness of the proposed method. Finally, conclusions are drawn in Section 5.Notations. R n represents the n-dimension Euclidean space and Rn×m denotes the set of n×m real matrices. Let IN represent the N×N identity matrix and 0 denote zero matrix with an appropriate dimension. The superscripts -1 and T stand for the inverse and the transpose of a matrix, respectively. We use the notation ⊗ and ∗ to represent Kronecker product and the elements below the main diagonal of some symmetric matrix, respectively. The real matrix P>0 shows that P is positive definite.


## 2. Problem Formulation

### 2.1. Graph Theory

LetG=(V,ξ,A) denote a directed weighted graph of N order, where the set of nodes is denotes by V={v1,v2,…,vn} and the set of edges is denoted by ξ⊆V×V. An edge in graph G is described by (vj,vi), i, j∈1,2,…,n which represents that the jth agent transmit information to the ith agent. Matrix A=[aij]n×n is called weighted adjacency matrix. The set of neighbors of agent i is denoted as Ni=j∣vj,vi∈ξ. If j∈Ni, aij>0; otherwise, aij=0. The diagonal matrix D=diag{d1,d2,…,dN}∈RN×N is called the degree matrix of graph G with diagonal elements di=Σj∈Niaij. The Laplacian matrix of G is defined as L=D-A=[lij]∈RN×N, where lii=∑j∈Ni‍aij and lij=-aij for i≠j.


### 2.2. System Description

We consider MASs composed ofN interconnected agents in this paper, where each agent is assumed to have the following state-space description: (1)x˙it=Axit+Bwityit=C2Σj∈Niaijrtxit-xjt+Dwitzit=C1Σj∈Niaijrtxit-xjtfor i=1,2,…,N, where xi(t), yi(t), and zi(t) are the state, the measured output, and the signal to be estimated of agent i. wi(t) denotes the external disturbance which belongs to L2[0,∞).The switching communication topologies are described byGr(t)∈{G1,G2,…,Gq}, where r(t) represents the continuous process of Markov chain. Furthermore, it takes values in a given finite set S={1,2,…,q}. We define the transition probability as follows: (2)probrt+Δt=s∣rt=r=πrsΔt+oΔt,r≠s1+πrrΔt+oΔt,r=swherelimΔt→0oΔt/Δt=0,Δt>0, and the transition probability satisfies πrs>0 when r,s∈S and πrr=-∑s=1,s≠rq‍πrs.For MASs (1), we design the following full-order filter to estimate signal zi(t): (3)x˙fit=Afrxfit+Bfryfitzfit=Cfrxfitwhere xfi(t) represents the state of the ith filtering subsystem, zfi(t) is the estimation signal of zi(t), and Afr,Bfr,Cfr are filter parameters to be determined.


### 2.3. Event-Triggered Scheme

In order to reduce the communication network burden, event-triggered scheme is adopted in this paper. The sampler is assumed to be time-driven; the zero-order-hold (ZOH) is event-driven. Then, we construct the following event-triggered strategy.(4)tk+1ih=tkih+minli>1⁡lih∣χiTtkih+lihϕrχitkih+lih≥σryiTtkih+lihϕryitkih+lihwhere li∈N, χi(tkih+lih)≜yi(tkih+lih)-yi(tkih) is the threshold error, tkih represents the latest kth triggering instant of the ith agent, tkih+lih denotes the currently sampled instant, and tk+1ih denotes the next broadcast instant. Then σr>0 represents the threshold and ϕr>0 stands for a symmetric positive definite matrix.Remark 1.
From the event-triggered strategy (4), we can automatically exclude Zeno behavior.Proof.
Zeno behavior in the event-triggered communication framework is defined as the infinite number of communication in a finite time. Due to the fact that each agent's state is periodically sampled and the sampling time sequence is{0,h,2h,…}, we can obtain that the event-triggered time sequence is a subsequence of the sampling time sequence, namely, {t0ih,t1ih,t2ih,…}⊆{0,h,2h,…}, which means that the minimum interevent time mink{tk+1ih-tkih}, ∀i, is lower bounded by the sampling period h. Since the number of agents is finite, the communication events in any finite time cannot be infinite. Therefore, the event-triggered strategy (4) can rule out Zeno behavior automatically.Remark 2.
Different from traditional filtering problem, due to the existence of network-induced delays, taking the property of ZOH into account, we get(5)yfit=yitkihSubstituting (5) into (3), we have(6)x˙fit=Afrxfit+Bfryitkihzfit=Cfrxfit,t∈tkih+τtki,tk+1ih+τtk+1iRemark 3.
Suppose that the network delay is bounded, i.e.,0<τm≤τki≤τ-, where τm and τ- denote the lower and upper bound of τ(t), respectively. It is worth noting that when σr=0, all the sampled signals are transmitted, the event-triggered strategy (4) reduces to a periodic time-triggered scheme.


### 2.4. Time-Delay Modeling

Using the same technique as in [29], we assume that there is a integer q>0, which satisfies tk+1i=tki+q+1. Hence, we can obtain the following equality:(7)tkih+τtki,tk+1ih+τtk+1i=⋃n=0qΩnwith Ωn=[tkih+nh+τ(tki+n),tkih+(n+1)h+τ(tki+n+1)), tk+1i=tki+n+1.For convenience, we define(8)τt=t-tkih-nh,t∈ΩnThen, we can know that (9)0<τm≤τt≤h+τ-=τMTo show the effect of the event-triggered scheme (4) in deriving system stability and stabilization criterion, we define a new variable eki(t), which satisfies the following equality: (10)ekit=yitkih-yitkih+nh,t∈ΩnSubstituting (8), (10) into (6), we have(11)x˙fit=Afrxfit+Bfryit-τt+ekitzfit=CfrxfitFrom (10) and (4), we can obtain (12)ekiTtϕrekit≤σryiTt-τtϕryit-τtwhere t∈[tkih+τ(tki),tk+1ih+τ(tk+1i)).


### 2.5. Event-TriggeredH∞ Filter Problem

By definingξiT(t)=xiT(t)xfiT(t), w^i(t)=wiTtwiTt-τtT, and ei(t)=zi(t)-zfi(t), we can obtain the following filtering error system:(13)ξ˙it=A-ξit+Σj∈NiaijB-Hξit-τt+E-ekit+D-w^iteit=C-1ξit+Σj∈NiaijC-2ξit-ξjtwhere (14)A-=A00Afr,B-=0BfrC2,H=In0,D-=B00BfrD,C-1=0-Cfr,C-2=C10E-=0BfrThen, we define some new augment variables to compact system:(15)ξt=ξ1Ttξ2Tt…ξNTtT,et=e1Tte2Tt…eNTtT,w^t=w^1Ttw^2Tt…w^NTtT,ekt=ek1Ttek2Tt…ekNTtTTherefore, the filtering error system can be rewritten as follows:(16)ξ˙t=IN⊗A-ξt+L⊗B-Hξt-τt+IN⊗E-ekt+IN⊗D-w^tet=IN⊗C-1+L⊗C-2ξtDefinition 4.
The filtering error system (16) with w(t)=0 is asymptotically stable in mean square, if for any initial conditions, such that (17)limt→+∞⁡ξt2=0Definition 5.
Given a positive scalarγ, and for all nonzero w(t)∈L20,∞, the filtering error system (16) is asymptotically stable in mean square with a guaranteed H∞ performance γ if the filtering error e(t) satisfies (18)E∫0∞eTtetdt≤γ2E∫0∞wTtwtdtBefore proceeding further, we introduce the following lemmas that will be helpful for deriving the following results.Lemma 6 (see [30]).
Given scalarτ>0, matrix M>0, and vector function w, 0,τ→Rn satisfy the following inequality: (19)∫0τwsdsTM∫0τwsds≤τ∫0τwTsMwsds.Lemma 7 (see [31]).
Suppose thatf1(t),f2(t),…,fN(t):Rm↦Rn is positive values, which satisfies (20)min⁡∑i=1N1αifit=∑i=1Nfit+max⁡∑i=1N∑j=1,j≠iNgi,jtsubject to (21)gi,j:Rm⟼Rn,gj,it≡gi,jt,fitgi,jgi,jfit≥0


## 2.1. Graph Theory

LetG=(V,ξ,A) denote a directed weighted graph of N order, where the set of nodes is denotes by V={v1,v2,…,vn} and the set of edges is denoted by ξ⊆V×V. An edge in graph G is described by (vj,vi), i, j∈1,2,…,n which represents that the jth agent transmit information to the ith agent. Matrix A=[aij]n×n is called weighted adjacency matrix. The set of neighbors of agent i is denoted as Ni=j∣vj,vi∈ξ. If j∈Ni, aij>0; otherwise, aij=0. The diagonal matrix D=diag{d1,d2,…,dN}∈RN×N is called the degree matrix of graph G with diagonal elements di=Σj∈Niaij. The Laplacian matrix of G is defined as L=D-A=[lij]∈RN×N, where lii=∑j∈Ni‍aij and lij=-aij for i≠j.


## 2.2. System Description

We consider MASs composed ofN interconnected agents in this paper, where each agent is assumed to have the following state-space description: (1)x˙it=Axit+Bwityit=C2Σj∈Niaijrtxit-xjt+Dwitzit=C1Σj∈Niaijrtxit-xjtfor i=1,2,…,N, where xi(t), yi(t), and zi(t) are the state, the measured output, and the signal to be estimated of agent i. wi(t) denotes the external disturbance which belongs to L2[0,∞).The switching communication topologies are described byGr(t)∈{G1,G2,…,Gq}, where r(t) represents the continuous process of Markov chain. Furthermore, it takes values in a given finite set S={1,2,…,q}. We define the transition probability as follows: (2)probrt+Δt=s∣rt=r=πrsΔt+oΔt,r≠s1+πrrΔt+oΔt,r=swherelimΔt→0oΔt/Δt=0,Δt>0, and the transition probability satisfies πrs>0 when r,s∈S and πrr=-∑s=1,s≠rq‍πrs.For MASs (1), we design the following full-order filter to estimate signal zi(t): (3)x˙fit=Afrxfit+Bfryfitzfit=Cfrxfitwhere xfi(t) represents the state of the ith filtering subsystem, zfi(t) is the estimation signal of zi(t), and Afr,Bfr,Cfr are filter parameters to be determined.


## 2.3. Event-Triggered Scheme

In order to reduce the communication network burden, event-triggered scheme is adopted in this paper. The sampler is assumed to be time-driven; the zero-order-hold (ZOH) is event-driven. Then, we construct the following event-triggered strategy.(4)tk+1ih=tkih+minli>1⁡lih∣χiTtkih+lihϕrχitkih+lih≥σryiTtkih+lihϕryitkih+lihwhere li∈N, χi(tkih+lih)≜yi(tkih+lih)-yi(tkih) is the threshold error, tkih represents the latest kth triggering instant of the ith agent, tkih+lih denotes the currently sampled instant, and tk+1ih denotes the next broadcast instant. Then σr>0 represents the threshold and ϕr>0 stands for a symmetric positive definite matrix.Remark 1.
From the event-triggered strategy (4), we can automatically exclude Zeno behavior.Proof.
Zeno behavior in the event-triggered communication framework is defined as the infinite number of communication in a finite time. Due to the fact that each agent's state is periodically sampled and the sampling time sequence is{0,h,2h,…}, we can obtain that the event-triggered time sequence is a subsequence of the sampling time sequence, namely, {t0ih,t1ih,t2ih,…}⊆{0,h,2h,…}, which means that the minimum interevent time mink{tk+1ih-tkih}, ∀i, is lower bounded by the sampling period h. Since the number of agents is finite, the communication events in any finite time cannot be infinite. Therefore, the event-triggered strategy (4) can rule out Zeno behavior automatically.Remark 2.
Different from traditional filtering problem, due to the existence of network-induced delays, taking the property of ZOH into account, we get(5)yfit=yitkihSubstituting (5) into (3), we have(6)x˙fit=Afrxfit+Bfryitkihzfit=Cfrxfit,t∈tkih+τtki,tk+1ih+τtk+1iRemark 3.
Suppose that the network delay is bounded, i.e.,0<τm≤τki≤τ-, where τm and τ- denote the lower and upper bound of τ(t), respectively. It is worth noting that when σr=0, all the sampled signals are transmitted, the event-triggered strategy (4) reduces to a periodic time-triggered scheme.


## 2.4. Time-Delay Modeling

Using the same technique as in [29], we assume that there is a integer q>0, which satisfies tk+1i=tki+q+1. Hence, we can obtain the following equality:(7)tkih+τtki,tk+1ih+τtk+1i=⋃n=0qΩnwith Ωn=[tkih+nh+τ(tki+n),tkih+(n+1)h+τ(tki+n+1)), tk+1i=tki+n+1.For convenience, we define(8)τt=t-tkih-nh,t∈ΩnThen, we can know that (9)0<τm≤τt≤h+τ-=τMTo show the effect of the event-triggered scheme (4) in deriving system stability and stabilization criterion, we define a new variable eki(t), which satisfies the following equality: (10)ekit=yitkih-yitkih+nh,t∈ΩnSubstituting (8), (10) into (6), we have(11)x˙fit=Afrxfit+Bfryit-τt+ekitzfit=CfrxfitFrom (10) and (4), we can obtain (12)ekiTtϕrekit≤σryiTt-τtϕryit-τtwhere t∈[tkih+τ(tki),tk+1ih+τ(tk+1i)).


## 2.5. Event-TriggeredH∞ Filter Problem

By definingξiT(t)=xiT(t)xfiT(t), w^i(t)=wiTtwiTt-τtT, and ei(t)=zi(t)-zfi(t), we can obtain the following filtering error system:(13)ξ˙it=A-ξit+Σj∈NiaijB-Hξit-τt+E-ekit+D-w^iteit=C-1ξit+Σj∈NiaijC-2ξit-ξjtwhere (14)A-=A00Afr,B-=0BfrC2,H=In0,D-=B00BfrD,C-1=0-Cfr,C-2=C10E-=0BfrThen, we define some new augment variables to compact system:(15)ξt=ξ1Ttξ2Tt…ξNTtT,et=e1Tte2Tt…eNTtT,w^t=w^1Ttw^2Tt…w^NTtT,ekt=ek1Ttek2Tt…ekNTtTTherefore, the filtering error system can be rewritten as follows:(16)ξ˙t=IN⊗A-ξt+L⊗B-Hξt-τt+IN⊗E-ekt+IN⊗D-w^tet=IN⊗C-1+L⊗C-2ξtDefinition 4.
The filtering error system (16) with w(t)=0 is asymptotically stable in mean square, if for any initial conditions, such that (17)limt→+∞⁡ξt2=0Definition 5.
Given a positive scalarγ, and for all nonzero w(t)∈L20,∞, the filtering error system (16) is asymptotically stable in mean square with a guaranteed H∞ performance γ if the filtering error e(t) satisfies (18)E∫0∞eTtetdt≤γ2E∫0∞wTtwtdtBefore proceeding further, we introduce the following lemmas that will be helpful for deriving the following results.Lemma 6 (see [30]).
Given scalarτ>0, matrix M>0, and vector function w, 0,τ→Rn satisfy the following inequality: (19)∫0τwsdsTM∫0τwsds≤τ∫0τwTsMwsds.Lemma 7 (see [31]).
Suppose thatf1(t),f2(t),…,fN(t):Rm↦Rn is positive values, which satisfies (20)min⁡∑i=1N1αifit=∑i=1Nfit+max⁡∑i=1N∑j=1,j≠iNgi,jtsubject to (21)gi,j:Rm⟼Rn,gj,it≡gi,jt,fitgi,jgi,jfit≥0


## 3. Main Results

### 3.1. Asymptotical Stability Analysis

Theorem 8.
For given scalarsσr>0, τM>τm>0, the filtering error system (16) without external disturbance can achieve mean square stable under Definition 4 if there exist appropriate dimensional matrices Pr>0, Q1>0, Q2>0, R1>0, R2>0, ϕr>0 and S1 such that the following LMIs hold: (22)R2S1∗R2>0(23)ΠτmΓTH^TR1τM-τmΓTH^TR2∗-R10∗∗-R2<0(24)∑s=1qπrsPs≤Prwhere(25)Π=Ξ11Ξ12Ξ130Ξ15∗Ξ22Ξ23Ξ240∗∗Ξ33Ξ340∗∗∗Ξ440∗∗∗∗-ϕ^rΓ=IN⊗A-L⊗B-00IN⊗E-Ξ11=IN⊗A-TPr+PrIN⊗A-+Pr+H^TQ1H^+H^TQ2H^-H^TR1H^Ξ12=PrL⊗B-Ξ13=H^TR1Ξ15=PrIN⊗E-Ξ22=σrL⊗C2Tϕ^rL⊗C2-2R2+S1+S1TΞ23=-S1T+R2Ξ24=-S1+R2Ξ33=-Q1-R1-R2Ξ34=S1Ξ44=-Q2-R2H^=IN⊗H,H^2=IN⊗H2,ϕ^r=IN⊗ϕr,H2=0InProof.
Consider that there is no external disturbance, i.e.,w(t)=0. We construct the following Lyapunov functional: (26)Vt,ξt,ξ˙t,rt=∑i=15Vit,ξt,ξ˙t,rtwhere(27)V1t,ξt,ξ˙t,rt=ξTtPrξtV2t,ξt,ξ˙t=∫t-τmtξTsH^TQ1H^ξsdsV3t,ξt,ξ˙t=∫t-τMtξTsH^TQ2H^ξsdsV4t,ξt,ξ˙t=τm∫-τm0∫t+θtξ˙TsH^TR1H^ξ˙sdsdθV5t,ξt,ξ˙t=τM-τm∫-τM-τm∫t+θtξ˙TsH^TR2H^ξ˙sdsdθthen its derivative along system (16) with w(t)=0 is (28)V˙1=ξ˙TtPrξt+ξTtPrξ˙t+ξTt∑s=1qπrsPsξtV˙2=ξTtH^TQ1H^ξt-ξTt-τmH^TQ1H^ξt-τmV˙3=ξTtH^TQ2H^ξt-ξTt-τMH^TQ2H^ξt-τMV˙4=τm2ξ˙TtH^TR1H^ξ˙t-τm∫t-τmtξ˙TsH^R1H^ξ˙sdsV˙5=τM-τm2ξ˙TtH^TR2H^ξ˙t-τM-τm∫t-τtt-τmξ˙TsH^R2H^ξ˙sds-τM-τm∫t-τMt-τtξ˙TsH^R2H^ξ˙sdsApplying Lemmas 6 and 7 to deal with the integral items and combining the event-triggered scheme (12), we have (29)V˙t,ξt,ξ˙t,rt≤∑i=15Vit,ξt,ξ˙t,rt-ekTtϕ^rekt+σryTt-τtϕ^ryt-τt=ζTtψζtwhere (30)ζTt=ξTtξTt-τtH^TξTt-τmH^TξTt-τMH^TekTtψ=Π+τm2ΓTH^TR1H^Γ+τM-τm2ΓTH^TR2H^ΓBy employing the Schur complement, we can find easily that (23) ensures ψ<0. According to Definition 4, the filtering error system (16) is asymptotically stable in mean square under the event-triggered scheme (12) if (22), (23), and (24) hold. The proof is completed.


### 3.2.H∞ Performance Analysis

Theorem 9.
For given scalarsσr>0, τM>τm>0, considering the existence of external disturbance, the filtering error system (16) can achieve being asymptotically stable with an H∞ performance index γ under Definition 5 if there exist appropriate dimensional matrices Pr>0, Q1>0, Q2>0, R1>0, R2>0, ϕr>0 and S1 such that the following LMIs hold: (31)R2S1∗R2>0(32)ΘτmΓ1TH^TR1τM-τmΓ1TH^TR2Γ2T∗-R100∗∗-R20∗∗∗-I<0(33)∑s=1qπrsPs≤Prwhere(34)Θ=Ξ11Ξ12Ξ130Ξ15Ξ16∗Ξ22Ξ23Ξ240Ξ26∗∗Ξ33Ξ3400∗∗∗Ξ4400∗∗∗∗-ϕ^r∗∗∗∗∗Ξ66Γ1=IN⊗A-L⊗B-00IN⊗E-IN⊗D-Γ2=IN⊗C-1+L⊗C-200000Ξ16=PrIN⊗D-Ξ26=σrL⊗C2Tϕ^rIN⊗DH^2Ξ66=-γ2I+σrH^2TIN⊗DTϕ^rIN⊗DH^2and the other parameters are defined as in Theorem 8.Proof.
Consider the existence of external disturbance, i.e.,w(t)≠0, and choose the same Lyapunov functional like Theorem 8; we have (35)V˙t,ξt,ξ˙t,rt=V˙t,ξt,ξ˙t,rt+eTtet-γ2w^Ttw^t-eTtet+γ2w^Ttw^t=ηTtΩηt-eTtet+γ2w^Ttw^twhere (36)ηTt=ζTtw^TtΩ=Θ+τm2Γ1TH^TR1H^Γ1+τM-τm2Γ1TH^TR2H^Γ1+Γ2TΓ2By using the Schur complement, we can find easily that (32) guarantees Ω<0. According to Definition 5, the filtering error system (16) is asymptotically stable with H∞ norm bound γ if (31), (32), and (33) are satisfied. The proof is completed.


### 3.3. Filter Design

Theorem 10.
For given scalarsσr>0,τM>τm>0 and H∞ performance index γ, the filtering error system (16) under the event-triggered scheme (12) is asymptotically stable in mean square for w(t)=0 and satisfies the H∞ performance constraint (18) for all nonzero w(t)∈L20,+∞ under the zero initial condition if there exist real matrices Pr>0,Wr>0,ϕr>0 (r∈S), Q1>0,Q2>0,R1>0,R2>0 and matrices S1,A-fr,B-fr,C-fr(r∈S) with appropriate dimension such that the following LMIs hold: (37)R2S1∗R2>0,(38)ϕ11ϕ12ϕ13ϕ14∗-R100∗∗-R20∗∗∗-I<0,(39)∑s=1qπrsU1s-Ws≤U1r-Wrwhere(40)ϕ11=Ξ^11Ξ^12Ξ^130Ξ^15Ξ^16∗Ξ22Ξ23Ξ240Ξ26∗∗Ξ33Ξ3400∗∗∗Ξ4400∗∗∗∗-ϕ^r0∗∗∗∗∗Ξ66ϕ12=τmR1IN⊗AH^0000τmR1IN⊗BH^T,ϕ13=τM-τmR2IN⊗AH^0000τM-τmR2IN⊗BH^T,ϕ14=IN⊗-C-frH^2+L⊗C1H^00000T,Ξ^11=IN⊗U1r+ATU1r+U1rAWr+A-fr+ATWr∗Wr+A-fr+A-frT+H^TQ1+Q2-R1H^,Ξ^12=L⊗B-frC2B-frC2,Ξ^13=H^TR1,Ξ^15=IN⊗B-frB-fr,Ξ^16=IN⊗U1rBB-frDWrBB-frDMoreover, if the above conditions are feasible, the parameter matrices of the filter are given by (41)Afr=Wr-1A-frBfr=Wr-1B-frCfr=C-frProof.
LetPr=IN⊗Ur for conditions (32) in Theorem 8. Let matrix Ur be partitioned as (42)Ur=U1rU2r∗U3rwhere U1r>0, U2r, and U3r are nonsingular matrices, satisfying Wr=U2rU3r-1U2rT. By Schur complement, we can obtain that Ur>0 is equivalent to U1r-Wr>0. Define the following invertible matrix: (43)J1=I00U2rU3r-1Then, pre- and postmultiply (32) by J2 and J2T, respectively, where J2=diag{J1,I,…,I︸8} and define new variables: (44)A-fr=U2rAfrU3r-1U2rTB-fr=U2rBfrC-fr=CfrU3r-1U2rTBy the linear transformation above, we can find easily that (32) is equivalent to (38). Therefore, we can obtain (41). The proof is completed.Remark 11.
Comparing with the previous results [32], lower bounds theorem [31] is adopted in this paper, which not only achieves performance behavior identical to approaches based on the integral inequality lemma, but also decreases the number of decision variables dramatically.


## 3.1. Asymptotical Stability Analysis

Theorem 8.
For given scalarsσr>0, τM>τm>0, the filtering error system (16) without external disturbance can achieve mean square stable under Definition 4 if there exist appropriate dimensional matrices Pr>0, Q1>0, Q2>0, R1>0, R2>0, ϕr>0 and S1 such that the following LMIs hold: (22)R2S1∗R2>0(23)ΠτmΓTH^TR1τM-τmΓTH^TR2∗-R10∗∗-R2<0(24)∑s=1qπrsPs≤Prwhere(25)Π=Ξ11Ξ12Ξ130Ξ15∗Ξ22Ξ23Ξ240∗∗Ξ33Ξ340∗∗∗Ξ440∗∗∗∗-ϕ^rΓ=IN⊗A-L⊗B-00IN⊗E-Ξ11=IN⊗A-TPr+PrIN⊗A-+Pr+H^TQ1H^+H^TQ2H^-H^TR1H^Ξ12=PrL⊗B-Ξ13=H^TR1Ξ15=PrIN⊗E-Ξ22=σrL⊗C2Tϕ^rL⊗C2-2R2+S1+S1TΞ23=-S1T+R2Ξ24=-S1+R2Ξ33=-Q1-R1-R2Ξ34=S1Ξ44=-Q2-R2H^=IN⊗H,H^2=IN⊗H2,ϕ^r=IN⊗ϕr,H2=0InProof.
Consider that there is no external disturbance, i.e.,w(t)=0. We construct the following Lyapunov functional: (26)Vt,ξt,ξ˙t,rt=∑i=15Vit,ξt,ξ˙t,rtwhere(27)V1t,ξt,ξ˙t,rt=ξTtPrξtV2t,ξt,ξ˙t=∫t-τmtξTsH^TQ1H^ξsdsV3t,ξt,ξ˙t=∫t-τMtξTsH^TQ2H^ξsdsV4t,ξt,ξ˙t=τm∫-τm0∫t+θtξ˙TsH^TR1H^ξ˙sdsdθV5t,ξt,ξ˙t=τM-τm∫-τM-τm∫t+θtξ˙TsH^TR2H^ξ˙sdsdθthen its derivative along system (16) with w(t)=0 is (28)V˙1=ξ˙TtPrξt+ξTtPrξ˙t+ξTt∑s=1qπrsPsξtV˙2=ξTtH^TQ1H^ξt-ξTt-τmH^TQ1H^ξt-τmV˙3=ξTtH^TQ2H^ξt-ξTt-τMH^TQ2H^ξt-τMV˙4=τm2ξ˙TtH^TR1H^ξ˙t-τm∫t-τmtξ˙TsH^R1H^ξ˙sdsV˙5=τM-τm2ξ˙TtH^TR2H^ξ˙t-τM-τm∫t-τtt-τmξ˙TsH^R2H^ξ˙sds-τM-τm∫t-τMt-τtξ˙TsH^R2H^ξ˙sdsApplying Lemmas 6 and 7 to deal with the integral items and combining the event-triggered scheme (12), we have (29)V˙t,ξt,ξ˙t,rt≤∑i=15Vit,ξt,ξ˙t,rt-ekTtϕ^rekt+σryTt-τtϕ^ryt-τt=ζTtψζtwhere (30)ζTt=ξTtξTt-τtH^TξTt-τmH^TξTt-τMH^TekTtψ=Π+τm2ΓTH^TR1H^Γ+τM-τm2ΓTH^TR2H^ΓBy employing the Schur complement, we can find easily that (23) ensures ψ<0. According to Definition 4, the filtering error system (16) is asymptotically stable in mean square under the event-triggered scheme (12) if (22), (23), and (24) hold. The proof is completed.


## 3.2.H∞ Performance Analysis

Theorem 9.
For given scalarsσr>0, τM>τm>0, considering the existence of external disturbance, the filtering error system (16) can achieve being asymptotically stable with an H∞ performance index γ under Definition 5 if there exist appropriate dimensional matrices Pr>0, Q1>0, Q2>0, R1>0, R2>0, ϕr>0 and S1 such that the following LMIs hold: (31)R2S1∗R2>0(32)ΘτmΓ1TH^TR1τM-τmΓ1TH^TR2Γ2T∗-R100∗∗-R20∗∗∗-I<0(33)∑s=1qπrsPs≤Prwhere(34)Θ=Ξ11Ξ12Ξ130Ξ15Ξ16∗Ξ22Ξ23Ξ240Ξ26∗∗Ξ33Ξ3400∗∗∗Ξ4400∗∗∗∗-ϕ^r∗∗∗∗∗Ξ66Γ1=IN⊗A-L⊗B-00IN⊗E-IN⊗D-Γ2=IN⊗C-1+L⊗C-200000Ξ16=PrIN⊗D-Ξ26=σrL⊗C2Tϕ^rIN⊗DH^2Ξ66=-γ2I+σrH^2TIN⊗DTϕ^rIN⊗DH^2and the other parameters are defined as in Theorem 8.Proof.
Consider the existence of external disturbance, i.e.,w(t)≠0, and choose the same Lyapunov functional like Theorem 8; we have (35)V˙t,ξt,ξ˙t,rt=V˙t,ξt,ξ˙t,rt+eTtet-γ2w^Ttw^t-eTtet+γ2w^Ttw^t=ηTtΩηt-eTtet+γ2w^Ttw^twhere (36)ηTt=ζTtw^TtΩ=Θ+τm2Γ1TH^TR1H^Γ1+τM-τm2Γ1TH^TR2H^Γ1+Γ2TΓ2By using the Schur complement, we can find easily that (32) guarantees Ω<0. According to Definition 5, the filtering error system (16) is asymptotically stable with H∞ norm bound γ if (31), (32), and (33) are satisfied. The proof is completed.


## 3.3. Filter Design

Theorem 10.
For given scalarsσr>0,τM>τm>0 and H∞ performance index γ, the filtering error system (16) under the event-triggered scheme (12) is asymptotically stable in mean square for w(t)=0 and satisfies the H∞ performance constraint (18) for all nonzero w(t)∈L20,+∞ under the zero initial condition if there exist real matrices Pr>0,Wr>0,ϕr>0 (r∈S), Q1>0,Q2>0,R1>0,R2>0 and matrices S1,A-fr,B-fr,C-fr(r∈S) with appropriate dimension such that the following LMIs hold: (37)R2S1∗R2>0,(38)ϕ11ϕ12ϕ13ϕ14∗-R100∗∗-R20∗∗∗-I<0,(39)∑s=1qπrsU1s-Ws≤U1r-Wrwhere(40)ϕ11=Ξ^11Ξ^12Ξ^130Ξ^15Ξ^16∗Ξ22Ξ23Ξ240Ξ26∗∗Ξ33Ξ3400∗∗∗Ξ4400∗∗∗∗-ϕ^r0∗∗∗∗∗Ξ66ϕ12=τmR1IN⊗AH^0000τmR1IN⊗BH^T,ϕ13=τM-τmR2IN⊗AH^0000τM-τmR2IN⊗BH^T,ϕ14=IN⊗-C-frH^2+L⊗C1H^00000T,Ξ^11=IN⊗U1r+ATU1r+U1rAWr+A-fr+ATWr∗Wr+A-fr+A-frT+H^TQ1+Q2-R1H^,Ξ^12=L⊗B-frC2B-frC2,Ξ^13=H^TR1,Ξ^15=IN⊗B-frB-fr,Ξ^16=IN⊗U1rBB-frDWrBB-frDMoreover, if the above conditions are feasible, the parameter matrices of the filter are given by (41)Afr=Wr-1A-frBfr=Wr-1B-frCfr=C-frProof.
LetPr=IN⊗Ur for conditions (32) in Theorem 8. Let matrix Ur be partitioned as (42)Ur=U1rU2r∗U3rwhere U1r>0, U2r, and U3r are nonsingular matrices, satisfying Wr=U2rU3r-1U2rT. By Schur complement, we can obtain that Ur>0 is equivalent to U1r-Wr>0. Define the following invertible matrix: (43)J1=I00U2rU3r-1Then, pre- and postmultiply (32) by J2 and J2T, respectively, where J2=diag{J1,I,…,I︸8} and define new variables: (44)A-fr=U2rAfrU3r-1U2rTB-fr=U2rBfrC-fr=CfrU3r-1U2rTBy the linear transformation above, we can find easily that (32) is equivalent to (38). Therefore, we can obtain (41). The proof is completed.Remark 11.
Comparing with the previous results [32], lower bounds theorem [31] is adopted in this paper, which not only achieves performance behavior identical to approaches based on the integral inequality lemma, but also decreases the number of decision variables dramatically.


## 4. Simulation

The parameters of MASs (1) are given as follows: (45)A=-110.8-4,B=0.50.1C1=0.10,C2=0.20,D=0.1In addition, the external disturbances of MASs (1) are defined as follows: (46)wit=0.4e-0.2tcos⁡0.6t.All the possible information transmission relationships among agents are shown in Figure 1. Then, the corresponding Laplacian matrices are given as follows, respectively. (47)L1=1-100010-10-110-1001,L2=100-1-11000-1100-101Figure 1
Directed communication topology graphs.Figure2 shows the switching of two modes in a Markov chain. Suppose that the probability transition matrix is defined as (48)Π=-552-2Figure 2
Evolution of Markov chain.Given thath=0.1, σ1=0.1, σ2=0.2, minimum allowable delay τm=0.1 and the maximum allowable delay τM=0.5. Let the initial states x1(0)=[3;-1],x2(0)=[1;-1],x3(0)=[1;-3],x4(0)=[-3;3]. By solving LMIs in Theorem 9, we can obtain parameters of the designed filter (3),(49)Af1=-24.884052.462-9.676120.1822,Bf1=-0.6878-0.2918Cf1=-1.35350.0581Af2=-58.5910137.6004-22.913353.7343,Bf2=0.88930.3558Cf2=-0.1373-0.0623×10-5,and the event-triggered parameters, (50)ϕ1=15.5816,ϕ2=7.9041In Figure 3, it shows the filtering error of agent i(i=1,2,3,4). In Figure 4, it depicts the state curves of zi(t) and estimation signal zfi(t)(i=1,2,3,4). The event-triggered release instants and intervals of agent i(i=1,2,3,4) are shown in Figure 5. Letting the simulation time t=30, we can get that agent 1 triggers 84 times, agent 2 triggers 36 times, agent 3 triggers 40 times, and agent 4 triggers 92 times. If σr=0 (time-triggered), there are 300 times transmitted. We can find clearly that the event-triggered control strategy (12) efficiently saves the network resources.Figure 3
Filtering error response.Figure 4
z i ( t ) and their estimation zfi(t).Figure 5
The release instants and release interval of each agent.


## 5. Conclusion

In this paper, we study the problem ofH∞ filtering for MASs with Markovian switching topologies. Considering the effect of switching topologies and network-induced delay, we adopt a reasonable event-triggered mechanism to reduce the amount of network transmission. By employing Lyapunov stability theory and LMI technique, some sufficient conditions are obtained which can ensure filtering error system to achieve mean square stable with an H∞ norm bound. Finally, a numerical example is provided to show that the method we proposed is feasible and effective.



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*Source: 2572846-2018-08-14.xml*