# Delay-DependentH∞ Control for T-S Fuzzy Systems with Local Nonlinear Models: An LMI Approach

Authors: Zhile Xia
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper studies the stabilization design scheme withH∞ performance for a large class of nonlinear discrete-time systems. The system under study is modeled by Takagi-Sugeno (T-S) model with local nonlinearity and state delay. First, the model is changed into an equivalent fuzzy switching model. And then, according to projection theorem and piecewise Lyapunov function (PLF), two new H∞ control methods are proposed for fuzzy switched systems, which consider the time delay information of the system. Finally, the relationship among all fuzzy subsystems is considered. Because the results are only expressed by a series of linear matrix inequalities (LMIs), the controller can be directly designed by the linear matrix inequalities toolbox of MATLAB.


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## Body

## 1. Introduction

As we all know, the T-S fuzzy method is a kind of common and very effective tool for approaching the discrete-time nonlinear complex system [1]. For instance, in [2, 3], it was shown that coupled chaotic systems are a special class of complex systems, which can be processed by the T-S method [4, 5]. In addition, nonlinear neutral differential equations have numerous applications in engineering and natural sciences [6]. By using the T-S method, Pu et al. [7] studied BP neural network and RBF neural network. Bharathi et al. [8] investigated numerical solutions for sophistication single neutrality differential equations with time delay. As shown in [9], asymptotic suboptimality property of the decentralized methods for the linear-quadratic games is proposed. A game-control method based on the fuzzy linearity quadratic adjuster was presented by Ji et al. [10] for emergency collision avoidance. Therefore, it is very important to study the asymptotic stability and controller design of the T-S fuzzy model [11–13]. However, many papers, such as [11–13], depend on a single common positive-definite symmetric matrix P, which needs to satisfy many LMIs. In reality, such a matrix may not exist, especially for systems with high nonlinearity [14]. Therefore, it is conservative to use the common Lyapunov function method to consider the controller design of the T-S model, and its application scope is limited. In order to increase the feasible region of matrix inequalities, a piecewise Lyapunov function (PLF) is proposed in [15, 16], which studied the filter problem for the T-S model with delay. In references [17–19], the fuzzy Lyapunov function (FLF) method was used to study the controller design approaches for the T-S model.Recently, the T-S fuzzy system with a local nonlinear model (FSwLNM) has received considerable research (see, for instance, [20–30] and references therein). As shown in [20, 21], T-S FSwLNM requires less fuzzy rules, which can reduce the computational complexity. It can also decrease modeling error compared with conventional T-S fuzzy systems. For a class of T-S FSwLNM, Yang and Wang [22] investigated the problem of fault detection, Klug et al. [23] proposed a convex way to study an output feedback controller, and Chang and Hsu [24] investigated the sliding mode control problem with multiple performance indexes for stochastic nonlinear systems. Nguyen et al. [25] proposed a new method to design a limited controller, which has different fuzzy rules from the system. Zhai et al. [26] observed the problems of network-based fault detection and isolation observer design. Huang and Yang [27] investigated the problem of fault estimation. In [28, 29], the fuzzy polynomial system was studied, and the subsequent part of fuzzy rules was represented by polynomial function. Compared with the general T-S fuzzy system, the fuzzy polynomial system can express the complex nonlinear system more accurately with fewer fuzzy rules.Based on the above considerations, we will study the delay-dependent controller design method withH∞ performance index for discrete-time T-S FSwLNM. Adopting PLF and the fuzzy switching model [30], two new design methods of H∞ controller are derived. These two methods consider the time-delay information of the system, so they are less conservative. By using projection theorem and introducing relaxation matrix variables, there is no product term of Lyapunov matrices and system dynamic matrices in LMIs constraints. Because the derived condition only contains LMIs, the controller gain matrix can be directly designed by the LMIs toolbox of MATLAB.Notations. This part briefly describes the symbols used in this paper. Symbol ℝn stands for Euclidean space with n dimension. Symbol ℝn×m means the set of real n×m dimensional matrices. Matrices I and O with appropriate dimensions represent unit matrices and zero matrices. Matrix P is strictly greater than 0, which shows that P is a positive definite symmetric matrix (PDSM). The symbol diagA1,A2,…,An denotes block diagonal matrix. The symbol symS indicates S+ST. Elements of symmetric position of symmetric matrix are represented by ∗.


## 2. Preliminaries and Problem Formulation

### 2.1. Preliminaries

Before giving the main conclusions of this paper, we first present some very important lemmas, which are very important in the process of proving the important conclusions of this paper.Lemma 1.
(see [31]). Given an m×m-dimensional symmetric real matrix Z and three matrices Z0, X, and Y of proper dimensions, the two sets of inequalities shown as follows are equivalent:(1)Z+symXTZ0Y<0,X⊥TZX⊥<0,Y⊥TZY⊥<0,where two matrices X⊥ and Y⊥, whose columns are full rank, satisfy the equalities XX⊥=0 and YY⊥=0, respectively.Lemma 2.
(see [32]). Let ϱ be a given positive integer, xh∈ℝn be a vector, and ℳ∈ℝn×n be a semi-PDSM; we have(2)−ϱ∑h=1ϱxhTℳxh≤−∑h=1ϱxhTℳ∑h=1ϱxh.Lemma 3.
(see [31]). Given the proper dimension matrices ℳ, ℋ, and Q, the following two sets of inequalities are equivalent:(3)1Qℳ∗ℋ<0,2ℋ<0,Q−ℳℋ−1ℳT<0.In this case, we say that the matrix productQ−ℳℋ−1ℳT is Schur complement of matrix ℋ.


### 2.2. Problem Formulation

In order to improve the approximation effect of the fuzzy system and reduce the number of rules, we consider T-S FSwLNM as follows:Fuzzy rulem: if ϑ1t is υm1t, …, ϑpt is υmpt, then(4)xt+1=Amxt+Aτ0mxt−τ0+Gmϕt+Gτ0mϕt−τ0+B1mωt+B2mut,zt=Cmxt+Cτ0mxt−τ0+Gzmϕt+Gzτ0mϕt−τ0+D1mωt+D2mut,xt=ςt,−τ0≤t<0,where m∈1,2,…,r0. r0, υmnt, and ϑnt are the rule number, fuzzy sets, and premise variables (usually the state or output of the system), respectively. xt∈ℝn,zt∈ℝm, ςt, and ut∈ℝl represent the system state variables, the system output variables to be estimated, the initial conditions, and the controller to be designed, respectively. Positive integer τ0 represents the constant delay. Disturbance ωt∈l20,∞. ϕt∈ℝn satisfies(5)ϕt2≤θxt2,θ>0.Similar to [33], we define open subspace Ωll=1,…,k in state space. The symbol Ωlc represents the closed subspace and satisfies(6)Ωmc∩Ωnc=∂Ωiυ0,m≠n,m,n=1,…,κ,i∈1,…,r0,where ∂Ωmυ0=ϑhmϑ=1,0≤hmϑ+δ<1,δ≪1,δ≠0. υ0 is the face indexes set of the polyhedron ∂Ωm=∪∂Ωmυ0. hmϑ=ωmϑt/∑m=1rωmϑt, ωmϑt=∏n=1p0μmnϑnt, and ϑt=ϑ1t,ϑ2t,…,ϑpt.Then, we will rewrite system (4) as an equivalent switched fuzzy model according to the idea of [30], as follows:Global rulen: if xt∈Ωn, then there are the following local rules:ifϑ1t is υnq1t, …, ϑpt is υnqpt, then(7)xt+1=Anqxt+Aτ0nqxt−τ0+Gnqϕt+Gτ0nqϕt−τ0+B1nqωt+B2nqut,zt=Cnqxt+Cτ0nqxt−τ0+Gznqϕt+Gzτ0nqϕt−τ0+D1nqωt+D2nqut,xt=ςt,−τ0≤t<0,q=1,…,λn,n=1,…,κ,where κ is the number of subspaces divided. Ωn represents the nth subspace. λn is the rule number in the nth subspace.By using the fuzzy method to deal with system (7), we can achieve(8)xt+1=∑q=1λnhnqAnqxt+Aτ0nqxt−τ0+Gnqϕt+Gτ0nqϕt−τ0+B1nqωt+B2nqut,zt=∑q=1λnhnqCnqxt+Cτ0nqxt−τ0+Gznqϕt+Gzτ0nqϕt−τ0+D1nqωt+D2nqut,xt=ςt,−τ0≤t<0,xt∈Ωn,where hnq=hnqϑt=∏l=1p0υnqlϑt/∑q=1λn∏l=1p0υnqlϑt.In each subspaceΩn, we design independent controllers, that isGlobal rulen: if xt∈Ωn, then there are the following controllers:ifϑ1t is υnl1t, …, ϑpt is υnlpt, then(9)ut=Fanlxt+Faτ0nlxt−τ0+Fbnlϕt+Fbτ0nlϕt−τ0,l=1,2,…,λn.Using fuzzy reasoning technology, fuzzy controller (9) can be written in the following compact form:(10)ut=∑l=1λnhnlFanlxt+Faτ0nlxt−τ0+Fbnlϕt+Fbτ0nlϕt−τ0.Combining (8) with (10), we have(11)xt+1=A˜nqlxt+A˜τ0nqlxt−τ0+B˜1nqωt+G˜nqlϕt+G˜τ0nqlϕt−τ0,zt=C˜nqlxt+C˜τ0nqlxt−τ0+D˜1nqωt+G˜znqlϕt+G˜zτ0nqlϕt−τ0,(12)A˜nqlA˜τ0nqlB˜1nqG˜nqlG˜τ0nqlC˜nqlC˜τ0nqlD˜1nqG˜znqlG˜zτ0nql=∑q=1λn∑l=1λnhnqhnlA^nqlA^τ0nqlB1nqG^nqlG^τ0nqlC^nqlC^τ0nqlD1nqG^znqlG^zτ0nql,with(13)A^nql=Anq+B2nqKanl,A^τ0nql=Aτ0nq+B2nqKaτ0nl,G^nql=Gnq+B2nqKbnl,G^τ0nql=Gτ0nk+B2nqKbτ0nl,C^nql=Cnq+D2nqKanl,C^τ0nql=Cτ0nq+D2nqKaτ0nl,G^znql=Gznq+D2nqKbnl,G^zτ0nql=Gzτ0nq+D2nqKbτ0nl.wherext∈Ωj.Given the upper boundγ of H∞ index, where γ>0, the purpose is to design independent controllers (10) for system (8) in each subspace Ωj, so that the following two conditions are met:(1)
When the disturbanceωt=0, the equilibrium state xt=0 of system (11) is asymptotically stable.(2)
When the initial conditionςt=0, the following norm inequalities hold:(14)zt2<γωt2,∀ωt≠0.


## 2.1. Preliminaries

Before giving the main conclusions of this paper, we first present some very important lemmas, which are very important in the process of proving the important conclusions of this paper.Lemma 1.
(see [31]). Given an m×m-dimensional symmetric real matrix Z and three matrices Z0, X, and Y of proper dimensions, the two sets of inequalities shown as follows are equivalent:(1)Z+symXTZ0Y<0,X⊥TZX⊥<0,Y⊥TZY⊥<0,where two matrices X⊥ and Y⊥, whose columns are full rank, satisfy the equalities XX⊥=0 and YY⊥=0, respectively.Lemma 2.
(see [32]). Let ϱ be a given positive integer, xh∈ℝn be a vector, and ℳ∈ℝn×n be a semi-PDSM; we have(2)−ϱ∑h=1ϱxhTℳxh≤−∑h=1ϱxhTℳ∑h=1ϱxh.Lemma 3.
(see [31]). Given the proper dimension matrices ℳ, ℋ, and Q, the following two sets of inequalities are equivalent:(3)1Qℳ∗ℋ<0,2ℋ<0,Q−ℳℋ−1ℳT<0.In this case, we say that the matrix productQ−ℳℋ−1ℳT is Schur complement of matrix ℋ.


## 2.2. Problem Formulation

In order to improve the approximation effect of the fuzzy system and reduce the number of rules, we consider T-S FSwLNM as follows:Fuzzy rulem: if ϑ1t is υm1t, …, ϑpt is υmpt, then(4)xt+1=Amxt+Aτ0mxt−τ0+Gmϕt+Gτ0mϕt−τ0+B1mωt+B2mut,zt=Cmxt+Cτ0mxt−τ0+Gzmϕt+Gzτ0mϕt−τ0+D1mωt+D2mut,xt=ςt,−τ0≤t<0,where m∈1,2,…,r0. r0, υmnt, and ϑnt are the rule number, fuzzy sets, and premise variables (usually the state or output of the system), respectively. xt∈ℝn,zt∈ℝm, ςt, and ut∈ℝl represent the system state variables, the system output variables to be estimated, the initial conditions, and the controller to be designed, respectively. Positive integer τ0 represents the constant delay. Disturbance ωt∈l20,∞. ϕt∈ℝn satisfies(5)ϕt2≤θxt2,θ>0.Similar to [33], we define open subspace Ωll=1,…,k in state space. The symbol Ωlc represents the closed subspace and satisfies(6)Ωmc∩Ωnc=∂Ωiυ0,m≠n,m,n=1,…,κ,i∈1,…,r0,where ∂Ωmυ0=ϑhmϑ=1,0≤hmϑ+δ<1,δ≪1,δ≠0. υ0 is the face indexes set of the polyhedron ∂Ωm=∪∂Ωmυ0. hmϑ=ωmϑt/∑m=1rωmϑt, ωmϑt=∏n=1p0μmnϑnt, and ϑt=ϑ1t,ϑ2t,…,ϑpt.Then, we will rewrite system (4) as an equivalent switched fuzzy model according to the idea of [30], as follows:Global rulen: if xt∈Ωn, then there are the following local rules:ifϑ1t is υnq1t, …, ϑpt is υnqpt, then(7)xt+1=Anqxt+Aτ0nqxt−τ0+Gnqϕt+Gτ0nqϕt−τ0+B1nqωt+B2nqut,zt=Cnqxt+Cτ0nqxt−τ0+Gznqϕt+Gzτ0nqϕt−τ0+D1nqωt+D2nqut,xt=ςt,−τ0≤t<0,q=1,…,λn,n=1,…,κ,where κ is the number of subspaces divided. Ωn represents the nth subspace. λn is the rule number in the nth subspace.By using the fuzzy method to deal with system (7), we can achieve(8)xt+1=∑q=1λnhnqAnqxt+Aτ0nqxt−τ0+Gnqϕt+Gτ0nqϕt−τ0+B1nqωt+B2nqut,zt=∑q=1λnhnqCnqxt+Cτ0nqxt−τ0+Gznqϕt+Gzτ0nqϕt−τ0+D1nqωt+D2nqut,xt=ςt,−τ0≤t<0,xt∈Ωn,where hnq=hnqϑt=∏l=1p0υnqlϑt/∑q=1λn∏l=1p0υnqlϑt.In each subspaceΩn, we design independent controllers, that isGlobal rulen: if xt∈Ωn, then there are the following controllers:ifϑ1t is υnl1t, …, ϑpt is υnlpt, then(9)ut=Fanlxt+Faτ0nlxt−τ0+Fbnlϕt+Fbτ0nlϕt−τ0,l=1,2,…,λn.Using fuzzy reasoning technology, fuzzy controller (9) can be written in the following compact form:(10)ut=∑l=1λnhnlFanlxt+Faτ0nlxt−τ0+Fbnlϕt+Fbτ0nlϕt−τ0.Combining (8) with (10), we have(11)xt+1=A˜nqlxt+A˜τ0nqlxt−τ0+B˜1nqωt+G˜nqlϕt+G˜τ0nqlϕt−τ0,zt=C˜nqlxt+C˜τ0nqlxt−τ0+D˜1nqωt+G˜znqlϕt+G˜zτ0nqlϕt−τ0,(12)A˜nqlA˜τ0nqlB˜1nqG˜nqlG˜τ0nqlC˜nqlC˜τ0nqlD˜1nqG˜znqlG˜zτ0nql=∑q=1λn∑l=1λnhnqhnlA^nqlA^τ0nqlB1nqG^nqlG^τ0nqlC^nqlC^τ0nqlD1nqG^znqlG^zτ0nql,with(13)A^nql=Anq+B2nqKanl,A^τ0nql=Aτ0nq+B2nqKaτ0nl,G^nql=Gnq+B2nqKbnl,G^τ0nql=Gτ0nk+B2nqKbτ0nl,C^nql=Cnq+D2nqKanl,C^τ0nql=Cτ0nq+D2nqKaτ0nl,G^znql=Gznq+D2nqKbnl,G^zτ0nql=Gzτ0nq+D2nqKbτ0nl.wherext∈Ωj.Given the upper boundγ of H∞ index, where γ>0, the purpose is to design independent controllers (10) for system (8) in each subspace Ωj, so that the following two conditions are met:(1)
When the disturbanceωt=0, the equilibrium state xt=0 of system (11) is asymptotically stable.(2)
When the initial conditionςt=0, the following norm inequalities hold:(14)zt2<γωt2,∀ωt≠0.


## 3. Main Results

The setΩ=m,nxt∈Ωm,xt+1∈Ωn indicates that the system state jumps from subspace Ωm to subspace Ωn. Of course, the system state may stay in a certain subspace all the time. Next, we can prove the result in Theorem 1.Theorem 1.
Given a positive real numberγ, the H∞ control problems of controlled system (11) are solved, if there are symmetric positive definite matrices PnW1n∗W2n, Q11Q12∗Q22, R, the nonsingular matrix Fn and matrices Nnss=1,…,7, Kanl, Kaτ0nl, Kbnl, and Kbτ0ql, n=1,2,…,κ, l=1,2,…,λn, such that(15)Πmnqq<0,q=1,2,…,λn,n,m∈Ω,(16)Πmnql+Πmnlq<0,0<q<l≤λn,n,m∈Ω,where(17)Πmnkl=Π11Π12Π13Π14Π15Π16Π17Nn70∗Φ1Φ2Φ3−KnNn4−Nn5−Nn6Φ4Π1T∗∗Φ5Φ6000−Nn2TΠ2T∗∗∗Φ7000−Nn3T0∗∗∗∗−γ2I00−Nn4TD1nqT∗∗∗∗∗−I0−Nj5TΠ3T∗∗∗∗∗∗−I−Nn6Π4T∗∗∗∗∗∗∗Φ80∗∗∗∗∗∗∗∗−I,Π11=Pm−symKn,Π12=W1m+Nn1+AnqKn+B2qKanl,Π13=Nn2−W1m+Aτ0nqKn+B2nqKaτ0nl,Π14=Nn3+W1m,Π15=KnNn4+KnB1nq,Π16=Nn5+GnqKn+B2nqKbnl,Π17=Nn6+Gτ0nqKn+B2nqKbτ0nl,Π1T=CnqKn+D2nqKanl,Π2T=Cτ0nqKn+D2nqKaτ0nl,Π3T=GznqKn+D2nqKbnl,Π4T=Gzτ0nqKn+D2nqKbτ0nl,Φ1=θI+R−symNn1+W2m+τ02Q11−Q22−Pn,Φ2=θI+Q22−Nn2−W2m,Φ3=W2m−Nn3−Q12T−W1n,Φ4=τ2Q12−Nn1T−Nn7,Φ5=W2m−R−Q22,Φ6=Q12T−W2m,Φ7=W2m−Q11−W2n,Φ8=τ02Q22−symNn7.
Moreover, the controllers are given by(18)Fanl=KanlKn−1,Faτ0nl=Kaτ0nlKn−1,Fbnl=KbnlKn−1,Fbτ0nl=Kbτ0nlKn−1.Proof.
LetUn=Kn−1 and U=diagUn,Un,Un,Un,I,Un,Un,Un,I. On left side of inequalities (15) and (16), postmultiplying UT and premultiplying U, respectively, one has(19)Ωmnqq<0,q=1,2,…,λn,n,m∈Ω,(20)Ωmnql+Ωmnlq<0,0<q<l≤λn,n,m∈Ω,where(21)Ωmnql=Ω11Ω12Ω13Ω14Ω15Ω16Ω17N¯n70∗Φ¯1Φ¯2Φ¯3−Nn4−N¯n5−N¯n6Φ¯4C^nqlT∗∗Φ¯5Φ¯6000−Nn2TC^τnqlT∗∗∗Φ¯7000−N¯n3T0∗∗∗∗−γ2I00−N¯n4TD1nqT∗∗∗∗∗−I¯0−N¯n5TG^znqlT∗∗∗∗∗∗−I¯−N¯n6G^τ0znqlT∗∗∗∗∗∗∗Φ¯80∗∗∗∗∗∗∗∗−Iand X¯ represents UnTXUn. Ω11=P¯m−symUn,Ω12=W¯1m+N¯n1+UnTA^nql, Ω13=N¯n2−W¯1m+UnTA^τnql,Ω14=N¯n3+W¯1m, Ω15=Nn4+B1nq,Ω16=N¯n5+UnTG^nkl, and Ω17=N¯n6+UnTG^τ0nql.
Based on (19) and (20), we have(22)Ω=∑k=1λn∑l=1λnhnkhnlΩmnkl=∑q=1λnhnk2Ωmnqq+∑q<lλnhnqhnlΩmnql+Ωmnlq=Ω0+symΩ1TΩ2<0,where Ω1=Uj,0,0,0,0,0,0,0,0,,Ω2=−I,A˜nql,A˜τ0nql,0,B˜1nq,G˜nql,G˜τ0nql,0,0, and(23)Ω0=P¯mW¯1m+N¯n1N¯n2−W¯1mN¯n3+W¯1mNn4N¯n5N¯n6N¯n70∗Φ¯1Φ¯2Φ¯3−Nn4−N¯n5−N¯n6Φ¯4C˜nqlT∗∗Φ¯5Φ¯6000−Nn2TC˜τnqlT∗∗∗Φ¯7000−N¯n3T0∗∗∗∗−γ2I00−N¯n4TD˜1nqT∗∗∗∗∗−I¯0−N¯n5TG˜znqlT∗∗∗∗∗∗−I¯−N¯n6G˜τ0znqlT∗∗∗∗∗∗∗Φ¯80∗∗∗∗∗∗∗∗−I.
LetΩ2⊥=A˜nql,A˜τ0nql,0,B˜1nq,G˜nql,G˜τ0nql,0,0diagI,…,I︸8blocks, which satisfies Ω2Ω2⊥=0. Moreover, Ω2T,Ω2⊥ is a column full-rank matrix. Then, based on Lemma 1 and inequality (22), we obtain(24)Ω2⊥TΩ0Ω2⊥=Δ11Δ12Δ13Δ14Δ15Δ16Δ17C˜nqlT∗Δ22Δ23Δ24Δ25Δ26Δ27C˜τnqlT∗∗Δ33Δ34Δ35Δ36Δ370∗∗∗Δ44Δ45Δ46Δ47D˜1jqT∗∗∗∗Δ55Δ56Δ57G˜znqlT∗∗∗∗∗Δ66Δ67G˜zτ0nqlT∗∗∗∗∗∗Δ770∗∗∗∗∗∗∗−I<0,where(25)Δ11=A˜nqlTP¯iA˜nql+HeW¯1m+N¯n1TA˜nql+Φ¯1,Δ12=A˜nqlTP¯mA˜τ0nql+A˜nqlTN¯n2−W¯1m+W¯1m+N¯n1TA˜τ0nql+Φ¯2,Δ13=A˜nqlTN¯n3+W¯1m+Φ¯3,Δ14=A˜nqlTP¯mB˜1nq+W¯1m+N¯n1TB˜1nq+A˜nqlTN¯n4−N¯n4,Δ15=A˜nqlTP¯mG˜nql+N¯n1+W¯1nTG˜nql+A˜nqlTN¯n5,Δ16=A˜nqlTP¯mG˜τ0nql+N¯n1+W¯1mTG˜τ0nql+A˜nqlTN¯n6−N¯n6,Δ17=A˜nqlTN¯n7+Φ¯4,Δ22=A˜τ0nqlTP¯mA˜τ0nql+HeN¯n2−W¯1mTA˜τ0nql+Φ¯5,Δ23=A˜τnqlTN¯n3+W¯1m+Φ¯6,Δ24=A˜τ0nqlTP¯mB˜1nq+N¯n2−W¯1mTB˜1nq+A˜τ0nqlTN¯n4,Δ25=A˜τ0nqlTP¯mG˜nql+N¯n2−W¯1mTG˜nql+A˜τ0nklTN¯n5,Δ26=A˜τ0nqlTP¯mG˜τ0nql+N¯n2−W¯1mTG˜τ0nql+A˜τ0nqlTN¯n6,Δ27=A˜τ0nqlTN¯n7−N¯n2T,Δ33=Φ¯7,Δ34=W¯1m+N¯n3TB˜1nq,Δ35=W¯1m+N¯n3TG˜nql,Δ36=W¯1m+N¯n3TG˜τ0nql,Δ37=−N¯n3T,Δ44=B˜1nqTP¯mB˜1nq+HeN¯n4TB˜1nq−γ2I,Δ45=B˜1nqTP¯mG˜nql+N¯n4TG˜nql+B˜1nqN¯n5,Δ46=B˜1nqTP¯mG˜τ0nql+N¯n4TG˜τ0nql+B˜1nqN¯n6,Δ47=B˜1nqTN¯n7−N¯n4T,Δ55=G˜nqlTP¯mG˜nql+HeN¯n5TG˜nql−I¯,Δ57=G˜nqlTN¯n7−N¯n5T,Δ56=G¯nqlTPmG¯τ0nql+Nn5TG¯τ0nql+G¯nqlTNn6,Δ77=Φ¯8Δ66=G˜τ0nqlTP¯mG˜τ0nql+HeN¯n6TG˜τ0nql−I¯,Δ67=G˜τ0nqlTN¯n7−N¯n6T.
Applying the Schur complement described by Lemma3 to the above inequality, we can obtain(26)Ξmnql=Δ11Δ12Δ13Δ14Δ15Δ16Ψ17∗Δ22Δ23Δ24Δ25Δ26Δ27∗∗Δ33Δ34Δ35Δ36Δ37∗∗∗Δ44Δ45Δ46Δ47∗∗∗∗Δ55Δ56Δ57∗∗∗∗∗Δ66Δ67∗∗∗∗∗∗Δ77+C˜nqlTC˜τ0nqlT0D˜1nqTG˜znqlTG˜zτ0nqlT0C˜nqlTC˜τ0nqlT0D˜1nqTG˜znqlTG˜zτ0nqlT0T<0.
Based on (26), we will prove that Theorem 1 is correct.
Construct a discrete-time PLF as follows:(27)Vt=V1t+V2t+V3t,V1t=∑l=t−τ0t−1xlTR¯xv,V2t=τ0∑θ=−τ0−1∑l=t+θt−1xvπlTQ¯11Q¯12∗Q¯22xvπl,V3t=xv∑v=t−τ0t−1xvTP¯nW¯1n∗W¯2nxt∑v=t−τ0t−1xv,where xt∈Ωn,n=1,2,…,κ, and πt=xt+1−xt.
LetΔVt=Vt+1−Vt, we can get(28)ΔV1t=xtTR¯xt−xt−τ0TR¯xt−τ0,(29)ΔV2t=τ02xtπtTQ¯11Q¯12∗Q¯22xtπt−τ0∑v=t−τ0t−1xvπvTQ¯11Q¯12∗Q¯22xvπv.
Using Lemma2, we can obtain(30)−τ0∑v=t−τ0t−1xvπvTQ¯11Q¯12∗Q¯22xvπv≤−∑v=t−τ0t−1xv∑v=t−τ0t−1πvTQ¯11Q¯12∗Q¯22∑l=t−τ0t−1xv∑v=t−τ0t−1πv=∑v=t−τ0t−1xvxt−xt−τ0T−Q¯11−Q¯12∗−Q¯22∑v=t−τ0t−1xvxt−xt−τ0=xtxt−τ0∑v=t−τ0t−1xvT−Q¯22Q¯22−Q¯12T∗−Q¯22Q¯12T∗∗−Q¯11xtxt−τ0∑v=t−τ0t−1xv,(31)ΔV3t=xt+1∑v=t+1−τ0txvTP¯mW¯1m∗W¯2mxt+1∑v=t+1−τ0txv−xt∑v=t−τ0t−1xvTP¯nW¯1n∗W¯2nxt∑v=t−τ0t−1xv=ζtTΘ1P¯mW¯1m∗W¯2mΘ1T−Θ2P¯nW¯1n∗W¯2nΘ2Tζt,(32)Θ1=A˜nqlA˜τ0nqlB˜1nqG˜nqlG˜τ0nql00I−II0000T,Θ2=I00000000I0000T,ζtT=xtTxt−τ0T∑v=t−τ0t−1xlTωtTϕtTϕt−τ0TπtT
where
Observing system (11) and inequality (5), one has(33)N=2ξtTN¯n1N¯n2N¯n3UnTNn4N¯n5N¯n6N¯n7T×A˜nql−Ixt+A˜τ0nqlxt−τ0+B˜1nqωt+G˜nqlϕt+G˜τ0nqlϕt−τ0−πt=0,(34)θxtTUnTUnxt−ϕtTUnTUnϕt≥0,(35)θxt−τ0TUnTUnxt−τ0−ϕt−τ0TUnTUnϕt−τ0≥0.
Then, from (28)–(35) and considering system (11), one can obtain(36)ΔVt+ztTzt−γ2ωtTωt=ΔV1t+ΔV2t+ΔV3t+N+ztTzt−γ2ωtTωt≤ΔV1t+ΔV2t+ΔV3t+N+ztTzt−γ2ωtTωt+θxtTUnTUnxt−ϕtTUnTUnϕt+θxt−τ0TUnTUnxt−τ0−ϕt−τ0TUnTUnϕt−τ0≤ξtTΞmnqlξt.
Based on (26), we have(37)ΔVt+ztTzt−γ2ωtTωt<0.
Therefore, when the disturbance is assumed to be zero, the following can be obtained from (28)–(35):(38)ΔVω=0=ΔV1+ΔV2+ΔV3+Nωt=0≤xtxt−τ0∑v=t−τ0t−1xvπtTΔ11Δ12Δ13Δ17∗Δ22Δ23Δ27∗∗Δ33Δ37∗∗∗Δ77xtxt−τ0∑v=t−τ0t−1xvπt.
Using Schur complement, inequality (26) means that ΔVω=0<0. Therefore, when ωt=0, we can easily obtain the asymptotically stability of system (11). Let(39)J=∑l=0∞zl22−γ2ωl22.
Under the condition that the initial conditions are zero and inequality (37) holds, we can obtain J≤∑l=0∞−ΔVl=−V∞+V0=−V∞<0. That is, z2<γω2.
In order to increase the feasible range of the results obtained in Theorem1, the relationship between the subsystems in each subspace is considered. In addition, the introduced relaxation matrix Mnqlq<l<λn may not be symmetric matrices. Consequently, we can give and prove the following improved results.Theorem 2.
Designing fuzzy controller (10) for system (8), the controlled system (11) is asymptotically stable and has H∞property upper bound γγ>0, if there exist matrices PnW1n∗W2n>0, Q11Q12∗Q22>0, R>0, the nonsingular matrix Fn, matrices Nnιι=1,2,…,7, Kanl, Kaτ0nl, Kbnl, Kbτ0nl, and Mnlq=MnqlT, such that the following are feasible:(40)Πmnqq<Mnqq,q=1,…,λn,n,m∈Ω,(41)Πmnql+Πmnlq<Mnql+MnqlT,0<k<l≤λn,n,m∈Ω,(42)Mn=Mn11Mn12⋯Mn1λnMn21Mn22⋯Mn2λn⋮⋮⋱⋮Mnλn1Mnλn2⋯Nnλnλn<0,n=1,2,…,κ.The feedback matrices of the controller can be designed as(43)Fanl=KanlKn−1,Faτ0nl=Kaτ0nlKn−1,Fbnl=KbnlKn−1,Fbτ0nl=Kbτ0nlKn−1.Proof.
If inequalities (40)–(42) hold, then(44)Πmnkl<∑q=1λnhnq2Mnqq+∑q<lλnhnqhnlMnql+MnqlT=HnTMnHn<0,
whereHn=hn1Ihn2I⋯hnλnIT.
If inequality (44) has a feasible solution, we can obtain(45)ΔVt+ztTzt−ωtTωt<0.
The next proof process is roughly the same as that of Theorem2, so it is omitted here. This completes the proof.


## 4. Conclusion

By using the fuzzy switched system, PLF, and state movement from one subspace to another, two newH∞ control schemes with time delay information are derived. The advantages of the proposed method include that the fuzzy posterior contains nonlinear functions, the Lyapunov function is piecewise, and the condition is expressed by linear matrix inequality. In addition, Theorem 2 considers the relationship between fuzzy subsystems. The disadvantage of the proposed method is the introduction of relaxation variables, which increases the computational complexity. However, in order to expand the feasible region of the result and obtain more feasible solutions at the same time, we sometimes need to introduce the relaxation variables. If the conditions have a feasible solution, the controller feedback matrices can be calculated according to the feasible solutions of a set of LMIs. Since fractional-order systems have more dynamic behaviors than integral-order systems [3], we will consider extending the results of this paper to fractional-order T-S fuzzy systems.



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*Source: 2301085-2020-05-11.xml*