# Event-Based NonfragileH∞ Filter Design for Networked Control Systems with Interval Time-Varying Delay

Authors: Zhongda Lu; Guangtao Ran; Guoliang Zhang; Fengxia Xu
Journal: Journal of Control Science and Engineering
 (2018)

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## Abstract

This paper first investigates the event-triggered nonfragileH∞ filter design for a class of nonlinear NCSs with interval time-varying delay. An event-triggered scheme is addressed to determine sampled data to be transmitted so that network communication resource can be saved significantly. The nonfragile filter design is assumed to include multiplicative gain variations according to the filter’s implement. Under the event-triggered scheme, the filtering error system is modeled as a system with interval time-varying delay. By constructing a new Lyapunov-Krasovskii functional and employing Wirtinger inequality, a sufficient condition is derived, which guarantees that the filtering error system is asymptotically stable with the prescribed H∞ performance. The nonfragile filter parameters are obtained by solving a set of linear matrix inequalities. Two numerical examples are given to show the usefulness and the effectiveness of the proposed method.


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## Body

## 1. Introduction

During the past several years, Takagi-Sugeno (T-S) fuzzy model approach has got considerable attention due to its great merits on modeling for nonlinear networked control systems (NCSs) [1]. Many efforts have been proposed based on T-S fuzzy model for nonlinear systems [2–5]. On one hand, some state variables cannot be directly measured in practical system; therefore, the filtering problem attracted the attention of many researchers to estimate the unmeasured states. In comparison with traditional Kalman filtering, the H∞ filter does not require statistical assumptions on the exogenous signals. Thus, the H∞ filtering theory has got considerable development. The phenomenon of H∞ filter design with time-varying delays, sensor faults, and packet dropouts was studied for nonlinear systems [6]. The authors in [7] proposed new results on a delay-derivative-dependent fuzzy H∞ filter.It should be mentioned that all the above works are based on accuracy assumptions that filter can be implemented exactly. However, inaccuracies or uncertainties do occur in filter implementation, which will reduce the performance of systems and lead the filter system to be fragile. Thus, a nonfragile filter must be designed to handle uncertainties and maintain the performance of systems. Up to now, a few results on the nonfragile filter have been proposed. Recently, the nonfragileH∞ filtering problem for a class of discrete-time Takagi-Sugeno fuzzy systems with both randomly occurring gain variations and channel fading was investigated in [8]. Based on vertex theory and probabilistic algorithm, deterministic and randomised filtering algorithms are proposed in [9]. Aiming at the fuzzy stochastic systems, the authors in [10] proposed a nonfragile robust H∞ filter design and a desired H∞ performance level. It should be pointed out that the nonfragile H∞ filtering problem has not been fully studied, which is one of motivations of this paper.On the other hand, one topic that has been increasingly important is how to mitigate the network bandwidth while guaranteeing the stability and other desired performance of systems [11–15]. In traditional time-triggered control systems, the sample data will be transmitted into controller via network communication whatever the data are desired or not, which leads to network resource waste and low efficiency. Therefore, an event-triggered scheme was proposed to reduce data transmission and improve the effective utilities of network resource. Recently, a novel discrete model for networked control systems was introduced in [16], which contained trigger parameters, dynamic quantization, and fault detection. In [17], event-triggering load frequency control was employed in multiarea power systems. It should be noted that although the works on event-triggered NCSs are rich, the limitations still remain and there still exist some problems to be handled [18–20]. To the best of our knowledge, there are no works that investigate how to use the event-triggered scheme in nonfragile H∞ filtering systems, which is another motivation of this paper.According to the above discussion, we first consider the problem of the event-triggered nonfragileH∞ filter design for a class of nonlinear networked control systems. Our main contributions of this paper are summarized as follows:( 1 ) An event-triggered communication scheme is proposed to save network resource significantly. By considering the event-triggered scheme, filter’s multiplicative gain variations, and interval time-varying delays, a novel filtering error system is established.( 2 ) Different from some existing works, the Wirtinger inequality is used to tackle the integral items of the derivative of Lyapunov-Krasovskii; a more relaxed H∞ performance stability criterion is derived.The rest of this paper is organized as follows. The problem formulation is given in Section2; under the event-triggered scheme, the filtering error system is modeled as a system with interval time-varying delay. Stability analysis for filtering error system is presented in Section 3. The nonfragile H∞ filter design method is first addressed in Section 4. Numerical examples are provided in Section 5 to demonstrate the effectiveness of the proposed method.Notations. Throughout this paper, Rn denote the n-dimensional Euclidean space and Rn×m is the set of n×m real matrices. Superscript (•)T stands for the matrix transposition, I represents the identity matrix, and diag{⋯} denotes the block-diagonal matrix. The notation P>0 means that the matrix P is a real symmetric positive define matrix. In symmetric block matrices, “∗” is used as ellipsis for terms induced by symmetry.


## 2. Problem Formulation

### 2.1. T-S Fuzzy System

Consider the following nonlinear system, which can be described by T-S fuzzy model withi plant rules:Plant Rule i (1) IF ς 1 t is M i 1 , ς 2 t is M i 2 , … , and ς r t is M i p , THEN (2) x ˙ t = A i x t + B i ω t z t = C i x t y t = D i x t ,where Mij are the fuzzy sets, i=1,2,…,r, and j=1,2,…,p is the number of IF-THEN rules. ς1t,ς2t⋯ςr(t) are the premise variables. x(t)∈Rn and y(t)∈Rm are the system state vector and the measured output, respectively. ω(t)∈Rq is the external disturbance and ω(t)∈l20,∞. z(t)∈Rp is the signal vector to be estimated; Ai, Bi, Ci, and Di are known parameter matrices with appropriate dimensions.By using center-average defuzzifier, product inference, and a singleton fuzzifier, the T-S fuzzy system (2) can be rewritten as follows: (3)x˙t=∑i=1rμiςtAixt+Biωtzt=∑i=1rμiςtCixtyt=∑i=1rμiςtDixtwhere ς=ς1,…,ςrT; the fuzzy basis functions are given by (4)μiςt=hiςt∑i=1rhiςt,hiςt=∏j=1pMijςjt.Mij(ςj(t)) represents the grade of membership for ςjt in Mij; 0≤μi(ς(t))≤1,(i=1,2,…,r),∑i=1rμi(ς(t))>0.Therefore, we have(5)∑i=1rμiςt=1,μiςt≥0i=1,2,…,r.


### 2.2. Event-Triggered Scheme

In this section, we consider the communication networks with limited bandwidth; an event generator under the event-triggered scheme is employed. Inspired by the method in [12], the event-triggered scheme is adopted as follows:(6)tk+1h=tkh+minj≥1Δtkh+jhfor Δtkh+jh=ektkh+jhTψektkh+jh≥λytkhTψytkh, where ek(tkh+jh)=y(tkh+jh)-y(tkh) is the threshold error among the current sampling data and the last transmitted data. ψ and λ are the triggering parameters.Remark 1.
It should be mentioned that the current sampling data will be transmitted only when the condition proposed in (6) is satisfied. Therefore, in comparison with periodic transmission communication scheme, the event-triggered mechanism can reduce the transmission rate and utilize the limited bandwidth effectively.


### 2.3. Nonfragile Fuzzy Filter

In some previous results of filter design, the implicit assumption is made that there are no multiplicative gain uncertainties. However, in fact, there inevitably exist filter parameter uncertainties in filter implementation. Therefore, in this section, we consider a full-order nonfragile fuzzy filter with gain variations as follows:Plant Rule j (7) I F   ς 1 t k h  i s  W i 1 , ς 2 t k h   i s   W i 2 , … , a n d  ς r t k h  i s  W i p , T H E N (8) x ˙ f t = A f j + Δ A f j t x f t + B f j + Δ B f j t y f t z f t = C f j + Δ C f j t x f t ,where xf(t)∈Rn is the filter state vector, yf(t)∈Rn is the real input of the filter, zf(t) is the estimated signal vector of z(t), and Afj, Bfj, and Cfj are the filter parameters to be designed.Similarly, we represent the nonfragile fuzzy filter as(9)x˙ft=∑j=1rμjςtkhAfj+ΔAfjtxft+Bfj+ΔBfjtyftzft=∑j=1rμjςtkhCfj+ΔCfjtxft,where ΔAfj, ΔBfj, and ΔCfj are the multiplicative gain uncertainties, which can be defined as(10)ΔAfjt=AfjM1jKAtN1jΔBfjt=BfjM2jKBtN2jΔCfjt=CfjM3jKCtN3j,where Mij and Nij(i=1,2,3) are constant matrices with appropriate dimension. KA(t),KB(t),KC(t) are uncertain bounded matrices:(11)KATtKAt≤I,KBTtKBt≤I,KCTtKCt≤I.Next, consider the effect of the logic ZOH; the last transmission data instant is maintained with the holding interval as follows:(12)yz=ytkh,t∈tkh+τtk,tk+1h+τtk+1In order to facilitate analysis, the holding interval can be represented by the following subsets:(13)tkh+τtk,tk+1h+τtk+1=⋃l=0mΩl,where (14)Ωi=tkh+lh+τtk+i,tkh+lh+h+τtk+i+1,l=0,1,…,m.Then, we define the function of the interval time-varying delays as follows:(15)τt=1-Δt0tt∈Ω01-Δt1tt∈Ω1⋮⋮1-Δtmtt∈Ωm,where Δtm=αt=tkh+lh, 0≤α<1. Therefore, we can easily derive(16)0<τ1≤τt≤τ2,τ˙t≤1-α=d≤1,where τ1 and τ2 are lower bound and upper bound of the delays, respectively. d is the bound of the delay variation.To summarize, the inputs of the filter are described as(17)yft=yzt=yt-τt-ekt-τt


### 2.4. Fuzzy Filtering Error System

For simplicity, we letμi represent μi(ς(t)) and let μj represent μj(ς(tkh)), and(18)ξt=colxt,xft,eft=zt-zft.By combining (3), (9), and (17), we can obtain the following filtering error system:(19)ξ˙t=∑i=1r∑j=1rA-ijtξt+B-ij1tξt-τt+B-ij2tekt-τt+B-wijtωteft=∑i=1r∑j=1rC-ijtξt,where(20)A-ijt=A~ij+ΔA^ijt,B-ij1t=B~ij1+ΔB^ij1t,B-ij2t=B~ij2+ΔB^ij2t,C-ijt=C~ij+ΔC^ijt,A~ij=∑i=1r∑j=1rμiμjAi00Afj,ΔA^ijt=∑i=1r∑j=1rμiμj000ΔAfjt,B~ij1=∑i=1r∑j=1rμiμj000BfjDi,ΔB^ij1=∑i=1r∑j=1rμiμj000ΔBfjtDi,B~ij2=∑i=1r∑j=1rμiμj0-Bfj,ΔB^ij2=∑i=1r∑j=1rμiμj0-ΔBfjt,C~ij=∑i=1r∑j=1rμiμjCi-Cfj,ΔC^ij=∑i=1r∑j=1rμiμj0-ΔCfjt,B~wij=∑i=1r∑j=1rμiμjBi0.Before the end of this section, the following definitions and lemmas are needed for the fuzzy filtering error system.Definition 2.
The fuzzy filtering error system is asymptotically stable with anH∞ performance, if the following holds:
( 1 ) The filtering error system (19) is asymptotically stable when ω(t)=0.
( 2 ) The filtering error system (19) has a prescribed H∞ performance γ; under zero initial condition,(21)∫0Let2dt≤γ2∫0Lωt2dtis satisfied for any nonzero ω(t)∈L20,∞.Lemma 3 (see [21]).
LetX, Y, and G be real matrices with appropriate dimensions and GiT(t)Gi(t)≤I. Then, for any constant ε>0, one has (22)XGtY+YTGtXT≤εXXT+ε-1YTYLemma 4 (see [22]).
For any vectorsx,y∈Rn and any scalar ε>0, matrices D, F, and E are real matrices of appropriate dimensions with FT(t)F(t)≤I; then the following inequalities hold:(23)2xDFtEy≤εxTDDTx+ε-1yTEETyLemma 5 (see [23]).
LetV, H, E, Q, and F be real matrices of appropriate dimensions such that Q>0 and FTF≤I. Then, for any scalar ε>0 such that Q-εDDT>0, one has(24)V+HFETQV+HFE≤VTQ-1-εHHTV+ε-1ETELemma 6 (see [24]).
LetP>0 and let Ai and Al be any real matrices of appropriate dimensions. Then(25)AiTPAl+AlTPAi≤AiTPAi+AlTPAlLemma 7 (Seuret and Gouaisbaut [25]).
For a given symmetric and positive matrixR>0 of appropriate dimensions and different signal x in [a,b]→Rn, the following inequality holds:(26)-∫abξ˙TsRξ˙sds≤-1b-aξbξavTa1Ra2Ra3R∗a1Ra3R∗∗a4Rξbξav,where(27)v=1b-a∫abξsdsa1=π24+1,a2=π24-1,a3=-π22,a4=π2.


## 2.1. T-S Fuzzy System

Consider the following nonlinear system, which can be described by T-S fuzzy model withi plant rules:Plant Rule i (1) IF ς 1 t is M i 1 , ς 2 t is M i 2 , … , and ς r t is M i p , THEN (2) x ˙ t = A i x t + B i ω t z t = C i x t y t = D i x t ,where Mij are the fuzzy sets, i=1,2,…,r, and j=1,2,…,p is the number of IF-THEN rules. ς1t,ς2t⋯ςr(t) are the premise variables. x(t)∈Rn and y(t)∈Rm are the system state vector and the measured output, respectively. ω(t)∈Rq is the external disturbance and ω(t)∈l20,∞. z(t)∈Rp is the signal vector to be estimated; Ai, Bi, Ci, and Di are known parameter matrices with appropriate dimensions.By using center-average defuzzifier, product inference, and a singleton fuzzifier, the T-S fuzzy system (2) can be rewritten as follows: (3)x˙t=∑i=1rμiςtAixt+Biωtzt=∑i=1rμiςtCixtyt=∑i=1rμiςtDixtwhere ς=ς1,…,ςrT; the fuzzy basis functions are given by (4)μiςt=hiςt∑i=1rhiςt,hiςt=∏j=1pMijςjt.Mij(ςj(t)) represents the grade of membership for ςjt in Mij; 0≤μi(ς(t))≤1,(i=1,2,…,r),∑i=1rμi(ς(t))>0.Therefore, we have(5)∑i=1rμiςt=1,μiςt≥0i=1,2,…,r.


## 2.2. Event-Triggered Scheme

In this section, we consider the communication networks with limited bandwidth; an event generator under the event-triggered scheme is employed. Inspired by the method in [12], the event-triggered scheme is adopted as follows:(6)tk+1h=tkh+minj≥1Δtkh+jhfor Δtkh+jh=ektkh+jhTψektkh+jh≥λytkhTψytkh, where ek(tkh+jh)=y(tkh+jh)-y(tkh) is the threshold error among the current sampling data and the last transmitted data. ψ and λ are the triggering parameters.Remark 1.
It should be mentioned that the current sampling data will be transmitted only when the condition proposed in (6) is satisfied. Therefore, in comparison with periodic transmission communication scheme, the event-triggered mechanism can reduce the transmission rate and utilize the limited bandwidth effectively.


## 2.3. Nonfragile Fuzzy Filter

In some previous results of filter design, the implicit assumption is made that there are no multiplicative gain uncertainties. However, in fact, there inevitably exist filter parameter uncertainties in filter implementation. Therefore, in this section, we consider a full-order nonfragile fuzzy filter with gain variations as follows:Plant Rule j (7) I F   ς 1 t k h  i s  W i 1 , ς 2 t k h   i s   W i 2 , … , a n d  ς r t k h  i s  W i p , T H E N (8) x ˙ f t = A f j + Δ A f j t x f t + B f j + Δ B f j t y f t z f t = C f j + Δ C f j t x f t ,where xf(t)∈Rn is the filter state vector, yf(t)∈Rn is the real input of the filter, zf(t) is the estimated signal vector of z(t), and Afj, Bfj, and Cfj are the filter parameters to be designed.Similarly, we represent the nonfragile fuzzy filter as(9)x˙ft=∑j=1rμjςtkhAfj+ΔAfjtxft+Bfj+ΔBfjtyftzft=∑j=1rμjςtkhCfj+ΔCfjtxft,where ΔAfj, ΔBfj, and ΔCfj are the multiplicative gain uncertainties, which can be defined as(10)ΔAfjt=AfjM1jKAtN1jΔBfjt=BfjM2jKBtN2jΔCfjt=CfjM3jKCtN3j,where Mij and Nij(i=1,2,3) are constant matrices with appropriate dimension. KA(t),KB(t),KC(t) are uncertain bounded matrices:(11)KATtKAt≤I,KBTtKBt≤I,KCTtKCt≤I.Next, consider the effect of the logic ZOH; the last transmission data instant is maintained with the holding interval as follows:(12)yz=ytkh,t∈tkh+τtk,tk+1h+τtk+1In order to facilitate analysis, the holding interval can be represented by the following subsets:(13)tkh+τtk,tk+1h+τtk+1=⋃l=0mΩl,where (14)Ωi=tkh+lh+τtk+i,tkh+lh+h+τtk+i+1,l=0,1,…,m.Then, we define the function of the interval time-varying delays as follows:(15)τt=1-Δt0tt∈Ω01-Δt1tt∈Ω1⋮⋮1-Δtmtt∈Ωm,where Δtm=αt=tkh+lh, 0≤α<1. Therefore, we can easily derive(16)0<τ1≤τt≤τ2,τ˙t≤1-α=d≤1,where τ1 and τ2 are lower bound and upper bound of the delays, respectively. d is the bound of the delay variation.To summarize, the inputs of the filter are described as(17)yft=yzt=yt-τt-ekt-τt


## 2.4. Fuzzy Filtering Error System

For simplicity, we letμi represent μi(ς(t)) and let μj represent μj(ς(tkh)), and(18)ξt=colxt,xft,eft=zt-zft.By combining (3), (9), and (17), we can obtain the following filtering error system:(19)ξ˙t=∑i=1r∑j=1rA-ijtξt+B-ij1tξt-τt+B-ij2tekt-τt+B-wijtωteft=∑i=1r∑j=1rC-ijtξt,where(20)A-ijt=A~ij+ΔA^ijt,B-ij1t=B~ij1+ΔB^ij1t,B-ij2t=B~ij2+ΔB^ij2t,C-ijt=C~ij+ΔC^ijt,A~ij=∑i=1r∑j=1rμiμjAi00Afj,ΔA^ijt=∑i=1r∑j=1rμiμj000ΔAfjt,B~ij1=∑i=1r∑j=1rμiμj000BfjDi,ΔB^ij1=∑i=1r∑j=1rμiμj000ΔBfjtDi,B~ij2=∑i=1r∑j=1rμiμj0-Bfj,ΔB^ij2=∑i=1r∑j=1rμiμj0-ΔBfjt,C~ij=∑i=1r∑j=1rμiμjCi-Cfj,ΔC^ij=∑i=1r∑j=1rμiμj0-ΔCfjt,B~wij=∑i=1r∑j=1rμiμjBi0.Before the end of this section, the following definitions and lemmas are needed for the fuzzy filtering error system.Definition 2.
The fuzzy filtering error system is asymptotically stable with anH∞ performance, if the following holds:
( 1 ) The filtering error system (19) is asymptotically stable when ω(t)=0.
( 2 ) The filtering error system (19) has a prescribed H∞ performance γ; under zero initial condition,(21)∫0Let2dt≤γ2∫0Lωt2dtis satisfied for any nonzero ω(t)∈L20,∞.Lemma 3 (see [21]).
LetX, Y, and G be real matrices with appropriate dimensions and GiT(t)Gi(t)≤I. Then, for any constant ε>0, one has (22)XGtY+YTGtXT≤εXXT+ε-1YTYLemma 4 (see [22]).
For any vectorsx,y∈Rn and any scalar ε>0, matrices D, F, and E are real matrices of appropriate dimensions with FT(t)F(t)≤I; then the following inequalities hold:(23)2xDFtEy≤εxTDDTx+ε-1yTEETyLemma 5 (see [23]).
LetV, H, E, Q, and F be real matrices of appropriate dimensions such that Q>0 and FTF≤I. Then, for any scalar ε>0 such that Q-εDDT>0, one has(24)V+HFETQV+HFE≤VTQ-1-εHHTV+ε-1ETELemma 6 (see [24]).
LetP>0 and let Ai and Al be any real matrices of appropriate dimensions. Then(25)AiTPAl+AlTPAi≤AiTPAi+AlTPAlLemma 7 (Seuret and Gouaisbaut [25]).
For a given symmetric and positive matrixR>0 of appropriate dimensions and different signal x in [a,b]→Rn, the following inequality holds:(26)-∫abξ˙TsRξ˙sds≤-1b-aξbξavTa1Ra2Ra3R∗a1Ra3R∗∗a4Rξbξav,where(27)v=1b-a∫abξsdsa1=π24+1,a2=π24-1,a3=-π22,a4=π2.


## 3. Stability Analysis

In this section, we first present theH∞ performance analysis for the filtering error system (19) under event-triggered scheme. The following results are established to guarantee that the filtering error system (19) is asymptotically stable.Theorem 8.
For given positive parametersτ1, τ2, d≤1, γ>0, εi>0,(i=1,2,…,10), and 0<λ<1, the filtering error system in (19) is asymptotically stable with H∞ performance γ under the event-triggered scheme (16), if there exist matrices Qi>0, (i=1,2,3), ψ>0, Ri>0(i=1,2), ai>0(i=1,2,3,4), and P>0 and Afj, Bfj, and Cfj with appropriate dimensions such that the following matrix inequalities hold:(28)IM~3∗ε10-1>0(29)R1-1M~1∗ε4-1>0,R2-1M~1∗ε7-1>0(30)R1-1M~21∗ε5-1>0,R2-1M~21∗ε8-1>0(31)R1-1M~22∗ε6-1>0,R2-1M~22∗ε9-1>0(32)Π=Λ11Λ12∗Λ22<0
for(33)Λ11=Π11Π12Π130Π15∗Π22Π230Π25∗∗Π33Π340∗∗∗Π440∗∗∗∗Π55,Λ12=00Π18Π19Π26000Π36Π37000Π47000000,Λ22=Π66000∗Π7700∗∗Π880∗∗∗Π99,
where(34)Π11=PA~ij+A~ijTP+Q1+Q2+Q3-a1R1+ε1PM~1M~1TP+ε1-1N~1TN~1+ε2PM~21M~21TP+ε3PM~22M~22TP+ε10-1N~3TN~3+α12A~ijTR1-1-ε4M~1M~1T-1A~ij+α12ε4-1N~1TN~1+α22A~ijTR2-1-ε7M~1M~1T-1A~ij+α22ε7-1N~1TN~1+C~ijTI-ε10M~3M~3T-1C~ij,Π12=-a2R1,Π13=PB~ij1,Π15=-a3R1,Π18=PB~ij2,Π19=PB~wij,Π22=-Q2-a1R1-a1R2,Π23=-a2R2,Π25=-a3R1,Π26=-a3R2,Π33=-1-dQ1+ε2-1N~21TN~21+λDiTψDi-2a1R2+α12ε5-1N~21TN~21+α22ε8-1N~21TN~21+α12Bij1TR1-1-ε5M~21M~21T-1Bij1+α22Bij1TR2-1-ε8M~21M~21T-1Bij1,Π34=-a2R2,Π36=-a3R2,Π37=-a3R2,Π44=-Q3-a1R2,Π47=-a3R2,Π55=-a4R1,Π66=-a4R2,Π77=-a4R2,Π88=ε3-1N~22TN~22+λ-1ψ+α12ε6-1N~22TN~22+α22ε9-1N~22TN~22+α12Bij2TR1-1-ε6M~22M~22T-1Bij2+α22Bij2TR2-1-ε9M~22M~22T-1Bij2,Π99=α12B~wijTR1B~wij+α22B~wijTR2B~wij-γ2In,M~1=0AfjM1j,M~21=0BfjM2j,M~22=0-BfjM2j,M~3=CfjM3j,N~1=0N1j,N~21=0N2jDi,N~22=0N2j,N~3=0N3j,a1=π24+1,a2=π24-1,a3=-π22,a4=π2,α1=2τ1,α2=2τ2-τ1.Proof.
We construct a candidate of Lyapunov-Krasovskii function as follows:(35)Vt,ξt=∑j=13Vjt,ξt,
where(36)V1t,ξt=ξTtPξt,V2t,ξt=∫t-τttξTsQ1ξsds+∫t-τ1tξTsQ2ξsds+∫t-τ2tξTsQ3ξsds,V3t,ξt=τ1∫-τ10∫t+θtξ˙TsR1ξ˙sdsdθ+τ2-τ1∫-τ2-τ1∫t+θtξ˙TsR2ξ˙sdsdθ.
Taking the time derivation ofV(t) for t, we have(37)V˙t,ξt=V˙1t,ξt+V˙2t,ξt+V˙3t,ξt,
where(38)V˙1t,ξt=2ξTtPA-ijtξt+2ξTtPB-ij1tξt-τt+2ξTtPB-ij2tekt-τt+2ξTtPB-wijtωt=2ξTtPA~ijξt+2ξTtPΔA^ijtξt+2ξTtPB~ij1ξt-τt+2ξTtPΔB^ij1tξt-τt+2ξTtPB~ij2ekt-τt+2ξTtPΔB^ij2tekt-τt+2ξTtPB-wijtωt(39)V˙2t,ξt=ξTtQ1+Q2+Q3ξt-1-τ˙tξTt-τtQ1ξt-τt-ξTt-τ1Q2ξt-τ1-ξTt-τ2Q3ξt-τ2≤ξTtQ1+Q2+Q3ξt-1-dξTt-τtQ1ξt-τt-ξTt-τ1Q2ξt-τ1-ξTt-τ2Q3ξt-τ2(40)V˙3t,ξt=τ12ξ˙TtR1ξ˙t+τ2-τ12ξ˙TtR2ξ˙t-τ1∫t-τ1tξ˙TsR1ξ˙sds-τ2-τ1∫t-τ2t-τ1ξ˙TsR2ξ˙sdsBy Lemma 3, for ∀ε1>0, we can obtain(41)2ξTtPΔA^ijtξt=2ξTtPM~1KAtN~1ξt≤ξTtε1PM~1M~1TP+ε1-1PN~1TN~1PξtBy Lemma 4, if there exist ∀ε2>0 and ∀ε3>0, we have(42)2ξTtPΔB^ij1tξt-τt=2ξTtPM~21KBtN~21ξt-τt≤ξTtε2PM~21M~21TPξt+ξTt-τtε2-1PN~21TN~21Pξt-τt(43)2ξTtPΔB^ij2tekt-τt=2ξTtPM~22KBtN~22ekt-τt≤ξTtε3PM~22M~22TPξt+ekTt-τtε3-1PN~22TN~22Pekt-τtFrom (41) to (43), we derive(44)2ξTtPΔA^ijtξt+2ξTtPΔB^ij1tξt-τt+2ξTtPΔB^ij2tekt-τt≤ξTtε1PM~1M~1TP+ε2PM~21M~21TP+ε1-1PN~1TN~1P+ε3PM~22M~22TPξt+ekTt-τtε3-1PN~22TN~22Pekt-τt+ξTt-τtε2-1PN~21TN~21Pξt-τtBy using Lemmas 5 and 6, if there exists ∀εi>0(i=4,5,6,7,8), we derive(45)τ12ξ˙TtR1ξ˙t=τ12A-ijtξt+B-ij1tξt-τt+B-ij2tekt-τt+B-wijtwtTR1A-ijtξt+B-ij1tξt-τt+B-ij2tekt-τt+B-wijtwt≤4τ12ξTtA~ijTR1-1-ε4M~1M~1T-1A~ij+ε4-1N~1TN~1ξt+4τ12ξTt-τtB~ij1TR1-1-ε5M~21M~21T-1B~ij1+ε5-1N~21TN~21ξt-τt+4τ12ekTt-τtB~ij2TR1-1-ε6M~22M~22T-1B~ij2+ε6-1N~22TN~22ekt-τt+4τ12ωTtB-wijTtR1B-wijtωt(46)τ2-τ12ξ˙TtR2ξ˙t=τ2-τ12A-ijtξt+B-ij1tξt-τt+B-ij2tekt-τt+B-wijtwtTR1A-ijtξt+B-ij1tξt-τt+B-ij2tekt-τt+B-wijtwt≤4τ2-τ12ξTtA~ijTR2-1-ε7M~1M~1T-1A~ij+ε7-1N~1TN~1ξt+4τ2-τ12ξTt-τtB~ij1TR2-1-ε8M~21M~21T-1B~ij1+ε8-1N~21TN~21ξt-τt+4τ2-τ12ekTt-τtB~ij2TR2-1-ε9M~22M~22T-1B~ij2+ε9-1N~22TN~22ekt-τt+4τ2-τ12ωTtB-wijTtR2B-wijtωtBy applying Lemma 5, if there exists ∀ε10>0, we can obtain(47)efTteft=ξTtC-ijTtC-ijξt=ξTtC~ij+ΔC^ijtTC~ij+ΔC^ijtξt≤ξTtC~ijTI-ε10M~3M~3T-1C~ijξTt+ε10-1ξTtN~3TN~3ξt
Furthermore, we use Lemma7 to deal with the integral items in (40); it follows that(48)-τ1∫t-τ1tξ˙TsR1ξ˙sds≤ξtξt-τ1υ1Ta1R1a2R1a3R1∗a1R1a3R1∗∗a4R1ξtξt-τ1υ1(49)-τ2-τ1∫t-τ2t-τ1ξ˙TsR2ξ˙sds=-τ2-τ1∫t-τtt-τ1ξ˙TsR2ξ˙sds-τ2-τ1∫t-τ2t-τtξ˙TsR2ξ˙sds≤-ξt-τ1ξt-τtυ2Ta1R2a2R2a3R2∗a1R2a3R2∗∗a4R2ξt-τ1ξt-τtυ2-ξt-τtξt-τ2υ3Ta1R2a2R2a3R2∗a1R2a3R2∗∗a4R2ξt-τtξt-τ2υ3,where(50)υ1=1τ1∫t-τ1tξsds,υ2=1τt-τ1∫t-τtt-τ1ξsds,υ3=1τ2-τt∫t-τ2t-τtξsds.By combining (44) to (49), we obtain(51)V˙t,ξt+efTteft-γ2ωTtωt≤η⌢TtΠη⌢t
where(52)η⌢t=colξt,ξt-τ1,ξt-τt,ξt-τ2,υ1,υ2,υ3,ekt-τt,ωt.Then, from (28) to (32), under the zero initial condition, we integrate the right and left sides of (51) from 0 to L for all L>0; we can easily derive(53)∫0LefTteftdt≤γ2∫0LωTtωtdtThis completes the proof.


## 4. FuzzyH∞ Filter Design

It is worth mentioning that the condition in (32) cannot be directly used for filter design. Therefore, in this section, we provide a sufficient condition for the fuzzy H∞ filter design, and a suitable filter parameter matrix is obtained.Theorem 9.
For given positive parametersτ1, τ2, d≤1, γ>0, δ>0, εi>0,(i=1,2,…,10), and 0<λ<1, if there exist symmetric positive definite matrices P>0, Q1>0, Q2>0, Q3>0, ψ>0, R1>0, and R2>0 and A~fj, B~fj, and C~fj with appropriate dimensions, the following hold:(54)IM~3∗ε10-1>0(55)R1-1M~1∗ε4-1>0,R2-1M~1∗ε7-1>0(56)R1-1M~21∗ε5-1>0,R2-1M~21∗ε8-1>0(57)R1-1M~22∗ε6-1>0,R2-1M~22∗ε9-1>0(58)Π~=Λ~11Λ~12∗Λ~22<0for(59)Λ~11=Π~11Π~12Π~130Π~15∗Π~22Π~230Π~25∗∗Π~33Π340∗∗∗Π440∗∗∗∗Π55,Λ~12=00Π~18Π~19Π~26000Π36Π37000Π47000000,Λ~22=Π66000∗Π7700∗∗Π~880∗∗∗Π~99,where(60)Π~11=φ~11φ~12N~1Tφ~14φ~15N~3Tφ~17φ~18C~ijT∗-ε1-10000000∗∗φ~33000000∗∗∗-ε2-100000∗∗∗∗-ε3-10000∗∗∗∗∗-ε10000∗∗∗∗∗∗φ~7700∗∗∗∗∗∗∗φ~880∗∗∗∗∗∗∗∗φ~99,φ~11=P11AiA~fjP12TAiA~fj+P11AiA~fjP12TAiA~fjT+Q1+Q2+Q3-a1R1,φ~12=A~fjM1jA~fjM1j,φ~14=B~fjM2jB~fjM2j,φ~15=-φ~14,φ~17=α1P11AiA~fjP12TAiA~fjT,φ~18=α2P11AiA~fjP12TAiA~fjT,φ~33=-ε1-1+α12ε4-1+α22ε7-1-1,φ~77=-2δP+δ2R1φ~12∗-ε4-1,φ~88=-2δP+δ2R2φ~12∗-ε7-1,φ~99=IM~3∗-ε10-1,Π~12=-a2R1,Π~13=0B~fjDi0B~fjDi,Π~15=-a3R1,Π~18=-B~fj-B~fj,Π~19=P11BiP12TBi,Π~22=-Q2-a1R1-a1R2,Π~23=-a2R2,Π~25=-a3R1,Π~26=-a3R2,Π~33=φ-11N~21Tφ-130φ-150∗φ-220000∗∗φ-33φ-3400∗∗∗-ε5-100∗∗∗∗φ-55φ-56∗∗∗∗∗-ε8-1,φ-11=-1-dQ1+λDiTψDi-2a1R2,φ-13=00α1DiTB~fjTα1DiTB~fjT,φ-15=00α2DiTB~fjTα2DiTB~fjT,φ-22=-ε2-1+α12ε5-1+α22ε8-1-1,φ-33=-2δP+δ2R1,φ-34=φ-56=B~fjM2jB~fjM2j,φ-55=-2δP+δ2R2,Π~88=λ-1ψN~22Tκ~130κ~150∗κ~220000∗∗κ~33κ~3400∗∗∗-ε6-100∗∗∗∗κ~55κ~56∗∗∗∗∗-ε9-1,κ~13=-α1BfjT-α1BfjT,κ~15=-α2BfjT-α2BfjT,κ~22=-ε3-1+α12ε6-1+α22ε9-1-1,κ~33=-2δP+δ2R1,κ~34=κ~56=-B~fjM2j-B~fjM2j,κ~55=-2δP+δ2R2,Π~99=-γ2Iκ-12κ-13∗κ-220∗∗κ-33,κ-12=α1BiTP11TBiTP12,κ-13=α2BiTP11TBiTP12,κ-22=-2δP+δ2R1,κ-33=-2δP+δ2R2.Moreover, if inequality is feasible, the filter parameters can be obtained by(61)Afj=P12-1A~fj,Bfj=P12-1B~fj,Cfj=C~fj.Proof.
By applying Schur complement lemma, the matrix inequality conditions (32) in Theorem 8 can be rewritten as the following matrix inequality:(62)Ξ~=Γ11Γ12∗Γ22<0for(63)Γ11=Θ11-a2R1PB~ij10-a3R1∗Θ22-a2R20-a3R1∗∗Θ33-a2R20∗∗∗-Q3-a1R20∗∗∗∗-a4R1,Γ12=00PB~ij2PB~wij-a3R2000-a3R2-a3R2000-a4R2000000,Γ22=-a4R2000∗-a4R200∗∗Θ880∗∗∗Θ99,where(64)Θ11=ϕ11PM~1N~1TPM~21-PM~21N~3Tϕ17ϕ18C~ijT∗-ε1-10000000∗∗ϕ33000000∗∗∗-ε2-100000∗∗∗∗-ε3-10000∗∗∗∗∗-ε10000∗∗∗∗∗∗ϕ7700∗∗∗∗∗∗∗ϕ880∗∗∗∗∗∗∗∗ϕ99,ϕ11=PA~ij+A~ijTP+Q1+Q2+Q3-a1R1,ϕ17=α1A~ijTP,ϕ18=α2A~ijTP,ϕ33=-ε1-1+α12ε4-1+α22ε7-1-1,ϕ77=-PR1-1PPM~1∗-ε4-1,ϕ88=-PR2-1PPM~1∗-ε4-1,ϕ99=IM~3∗-ε10-1,Θ22=-Q2-a1R1-a1R2,Θ33=ϕ-11N~21Tα1B~ij1TP0α2B~ij1TP0∗ϕ-220000∗∗-PR1-1PPM~2100∗∗∗-ε5-100∗∗∗∗-PR2-1PPM~21∗∗∗∗∗-ε8-1,ϕ-11=-1-dQ1+λDiTψDi-2a1R2,ϕ-22=-ε2-1+α12ε5-1+α22ε8-1-1,Θ88=λ-1ψN~22Tα1B~ij2TP0α2B~ij2TP0∗κ~220000∗∗-PR1-1PPM~2200∗∗∗-ε6-100∗∗∗∗-PR2-1PPM~22∗∗∗∗∗-ε9-1,Θ99=-γ2Iα1B~wijTPα2B~wijTP∗-PR1-1P0∗∗-PR2-1P.Then, we define the following matrices:(65)P=P11P12∗P12,A~fj=P12Afj,B~fj=P12Bfj,C~fj=Cfj.We have(66)PA~ij=P11AiA~fjP12TAiA~fj,PB~ij1=0B~fjDi0B~fjDi,PB~ij2=-B~fj-B~fj,PB~wij=P11BiP12TBi,PM~1=A~fjM1A~fjM1,PM~21=B~fjM2B~fjM2,PM~22=-B~fjM2-B~fjM2.Moreover, we can derive -PR-1P≤-2δP+δ2R by the inequality δR-PR-1δR-P≥0 for any scalar δ>0. Then, by applying Schur complement lemma and combining (32) and (62)–(66), we can easily derive inequality (58). This completes the proof.


## 5. Simulation Results

In this section, we provide two examples to demonstrate the merit of the nonfragileH∞ filter based event-triggered scheme.Example 1.
Consider the following fuzzy systems:(67)x˙t=∑i=12uiAixt+Biωtzt=∑i=12uiCixtyt=∑i=12uiDixt,where(68)A1=-2.10.11-2,A2=-1.90-0.2-1.1,B1=1-0.2,B1=0.30.1,C1=1-0.5,C2=-0.20.3,D1=10,D2=0.50.6.And the parameter uncertainties are given as(69)M11=M12=0.050.04,M21=M22=0.01,M31=M32=0.020.01,N11=N12=0.020.03,N21=N22=0.05,N31=N32=0.030.02,Kgt=sin⁡t,g=A,B,C.
The membership functionui(t), i=1,2, and the external disturbance signal ω(t) are given as(70)u1t=sin⁡t2,u2t=1-u1t,ωt=11+3t2.
Consider the interval time-varying delay withτ1=0.03 and τ2=0.2. And choose λ=0.2, d=0.2, δ=10, and sample period h=20ms. By Theorem 9, we derive the desired nonfragile parameter as follows:(71)Af1=-17.43668.52687.7146-11.2453,Af2=-16.98356.27157.0539-9.1581,Bf1=-0.4852-0.3343,Bf2=0.2106-0.1013,Cf1=-2.47150.2475,Cf2=-1.89031.1024.Meanwhile, we obtain the event-triggered parameter ψ=23.6026 and the minimum H∞ performance level is γ=0.99.It is assumed that the initial condition isx0=1-1T. The responses of the original state and the filter state are shown in Figures 1 and 2. The system output signal is depicted in Figure 3. As depicted in Figure 4, we can clearly see that the response of the filtering error is stable after about three seconds, which demonstrates the effective of the proposed approach. Note that, in Figure 5, the transmission rate is mitigated from 100% to 7%, which demonstrates that the event-triggered scheme is effective in reducing data transmission and saving the network resource.Figure 1
State trajectories ofx1.Figure 2
State trajectories ofx2.Figure 3
The trajectories ofy(t).Figure 4
The trajectories of estimate error.Figure 5
Release instant.Example 2.
To further demonstrate the effectiveness of the proposed method, in this example, we consider a truck-trailer system; the practical example is borrowed from [26]. The system parameters are given as follows:(72)A1=-1.63710.5741-0.0157-0.5091000.5091-40,A2=-1.60860.5284-0.0150-0.5091000.0016-0.01270,B1=10.50,B2=10.50,C1=010,C2=010,D1=4.9-20.04,D2=4.9-20.04.
Then, we suppose that the parameter uncertainties are given by(73)M11=M12=0.10.10.21,M21=M22=0.1,M31=M32=-0.200.2,N11=N12=0.20.30.1,N21=N22=0.5,N31=N32=-0.30.20.1,Klt=sint,l=A,B,C. We assume that the lower bound and upper bound of the interval time-varying delay are τ1=0.01 and τ2=0.25, respectively. The derivative of the time delay is d=0.3. Furthermore, we suppose that λ=0.2 and δ=0.1. By solving the LMIs in Theorem 9, we obtain minimum performance level γ=1.1241, which is less than previously studied γ=7.0948 in [26]. Meanwhile, the event-triggered argument ψ=2.0069 is obtained, and the corresponding filter parameters can be derived as follows:(74)Af1=-8.14498.2655-0.6046-3.66463.5043-0.2331-2.38692.7130-1.5149,Af2=-7.6655-8.2644-0.3039-3.20743.1396-0.1352-0.68030.6193-0.9489,Bf1=-0.0942-0.0503-0.1788,Bf2=-0.0742-0.0360-0.2396,Cf1=-0.98620.8778,Cf2=-2.68574.0130.
Figures6, 7, and 8 show the original state and the filter state; it is clear that the filter state is asymptotically stable. The curve of the system output signal is depicted in Figure 9.
The filtering error response is depicted in Figure10, which shows that the designed nonfragile H∞ filter is feasible.
Finally, we choose the sample periodh=10ms; under the event-triggered scheme, it can be seen that Figure 11 shows that only 24 times in all 900 sampling instants are transmitted, which illustrates that the network bandwidth is greatly mitigated.Figure 6
State trajectories ofx1.Figure 7
State trajectories ofx2.Figure 8
State trajectories ofx3.Figure 9
The trajectories ofy(t).Figure 10
The trajectories of estimate error.Figure 11
Release instant.


## 6. Conclusions

In this paper, the problem of the event-triggered fuzzy nonfragileH∞ filter design has been investigated for nonlinear NCSs with interval time-varying delay. The resultant filtering error system has been modeled as a system with an interval time-varying delay. By constructing a new Lyapunov-Krasovskii functional and employing Wirtinger inequality, we have obtained a sufficient condition for designing the fuzzy nonfragile H∞ filter such that the fuzzy filtering error system is asymptotically stable with a prescribed H∞ performance level. Two numerical examples have been given to demonstrate the effectiveness of the proposed method. Future research includes nonfragile fuzzy fault filter design for nonlinear control systems considering time-varying delays. In addition, type-2 nonfragile fuzzy filter design for NCSs with packet dropout also can be further considered for the future investigation.



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*Source: 4541586-2018-06-12.xml*