# Delay-Dependent Stability Analysis of TS Fuzzy Switched Time-Delay Systems

Authors: Nawel Aoun; Marwen Kermani; Anis Sakly
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

This paper proposes a new approach to deal with the problem of stability under arbitrary switching of continuous-time switched time-delay systems represented by TS fuzzy models. The considered class of systems, initially described by delayed differential equations, is first put under a specific state space representation, called arrow form matrix. Then, by constructing a pseudo-overvaluing system, common to all fuzzy submodels and relative to a regular vector norm, we can obtain sufficient asymptotic stability conditions through the application of Borne and Gentina practical stability criterion. The stability criterion, hence obtained, is algebraic, is easy to use, and permits avoiding the problem of existence of a common Lyapunov-Krasovskii functional, considered as a difficult task even for some low-order linear switched systems. Finally, three numerical examples are given to show the effectiveness of the proposed method.


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## Body

## 1. Introduction

Switched systems, seen as an important class of hybrid dynamical systems, are defined as a family of subsystems described by differential or difference equations and a rule that governs the switching between them [1, 2]. They have a strong engineering background in various areas and are often used to suitably model a great number of real-world systems, such as mechanical systems, chemical processes, communication networks, robotic systems, and aircraft and air traffic.On the other hand, since the significant development that has witnessed the field of fuzzy modeling and control and more particularly model-based control, a new theory about fuzzy switched systems has emerged as an answer to more complicated real systems analysis and synthesis requirements such as multiple nonlinear systems, switched nonlinear systems, and second-order nonholonomic systems [3–10].Originally inspired from the concept of sector nonlinearity, the main idea of Takagi-Sugeno fuzzy modeling is to partition the nonlinear system dynamics into several locally linearized models so that the overall nonlinear system could be represented by a sufficiently accurate approximation [11]. Hence, model-based control is considered as a powerful universal approximation tool and a reliable approach to deal with complex and ill-determined systems [12–15].A fuzzy switched system is defined as a switched system which involves fuzzy models among its subsystems.Despite the considerable efforts made for the analysis of nonlinear systems, stability study of switched systems and in particular of fuzzy switched systems is still complex. Indeed, the example of asymptotically stable subsystems which, due to a specific switching sequence, yield to an unstable behavior of the overall system is well known. Besides, the case of unstable subsystems that, via a particular stabilizing switching law, lead to a stable global system also exists.The stability problem of such systems becomes more challenging when time delay is involved [16]. In fact, time delay often occurs in many dynamical systems, namely, in biological systems, chemical systems, metallurgical processing systems, and network systems. Its existence, whether in the state variables, control inputs, or the measurement outputs, is frequently a source of instability and poor performance.The existing results on stability of TS fuzzy switched time-delay systems can be classified into two types: delay-independent criteria, which are applicable to delay of arbitrary size [17–20] and delay-dependent criteria, which include information on the size of the delay [21]. It is generally recognized that delay-dependent results are usually less conservative than delay-independent ones, especially when the size of the delay is small.Stability analysis of switched systems in general has been conducted mainly on the basis of Lyapunov stability theory and concerns two major problems [22–24].The first is related to the asymptotic stability of switched systems under arbitrary switching. An important result in this area states that a sufficient condition for the asymptotic stability of this class of systems is that all the subsystems share a common Lyapunov function [25–31]. Sufficient stability conditions are then derived through the resolution of a set of Linear Matrix Inequalities (LMIs).In the case of switched time-delay systems, a common Lyapunov-Krasovskii functional is searched [32–34]. However, checking the existence of such a functional is a hard task even for some simple cases. Moreover, the method becomes less reliable when the number of subsystems switching between each other is important or when the number of fuzzy rules required to model each subsystem with a good accuracy is high.A second problem concerns the stabilization of TS fuzzy switched systems by restricting the class of admissible switching signals to those in which the interval between any two consecutive switching instants is no smaller than a numberτ called dwell time [35].To overcome limitations due to the existence of Lyapunov-Krasovskii functionals, we propose, in this paper, to study the stability of TS fuzzy switched systems through the convergence of a regular vector norm, associated with a specific characteristic matrix, called arrow form matrix [36–43]. The proposed method is based on the construction of a common overvaluing/comparison system for all the fuzzy submodels and whose stability permits concluding to that of the original system. The obtained results, valuable for the case of arbitrary switching, are expressed in terms of simple algebraic conditions and explicitly involve the time delay.The application of vector norms to switched systems has already been introduced in [44–47]. It has been extended later to switched time-delay systems in [48–53].The remainder of the paper is organized as follows: Section2 presents the problem formulation and some preliminaries. In Section 3, new delay-dependent stability conditions are derived for a class of TS fuzzy switched time-delay systems described by differential equations and having a single constant delay. Section 4 generalizes the main result to the case of TS fuzzy switched systems with multiple delays. Three numerical examples are provided in Section 5 to demonstrate the effectiveness of the proposed method. Finally, some concluding remarks are given in Section 6.Notations. The notations used throughout this paper are fairly standard. Rn denotes the n-dimensional Euclidean space, In is the identity matrix with appropriate dimensions, and · denotes Euclidean vector norm. For any u=ui1≤i≤n,  v=vi1≤i≤n∈Rn, we define the scalar product of the vectors u and v as u,v=∑i=1nuivi. M=mi,j1≤i,j≤n,  MT and M-1 are its transpose and its inverse, respectively. We denote M∗=mi,j∗1≤i,j≤n with mi,j∗=mi,j if i=j and mi,j∗=mi,j if i≠j and M=mi,j,  ∀1≤i,j≤n.


## 2. Problem Formulation and Preliminaries

### 2.1. Problem Formulation

Let us consider the unforced switched nonlinear time-delay systems that are described by a differential equation of the form:(1)x˙t=fσtxt,xt-τxt=Φt,t∈-τ,0,where x(t)∈Rn is the state vector, τ>0 is the time delay, Φ:-τ,0→Rn is a differentiable vector valued initial function, fσ:Rn→Rn are sufficiently regular functions that are parametrized by the index set I=1,2,…,N,  σ(t):Rn→I is a piecewise constant function depending on time, called switching signal and assumed to be available in real time, and N is the number of subsystems.The switching sequence is defined through a switching vectorξ(t)=ξ1(t),…,ξN(t)T whose components ξi(t) are given by(2)ξit=1ifσt=i,i∈I(i.e. subsystemiis active)0otherwise.It is obvious that ∑i=1Nξi(t)=1  ∀t≥0.Therefore, the switched system is composed ofN subsystems expressed as (3)x˙t=fixt,xt-τ,i∈Ixt=Φt,t∈-τ,0.In addition, each subsystem∑fi is described by a set of IF-THEN rules; each rule is related to a region of the state space where the subsystem could be approximated by a local linear model.Thus, thelth fuzzy rule associated with the ith subsystem is given by(4)Ril:IFz1tisMli1and…andzptisMlipTHENx˙t=Alixt+Dlixt-τxt=Φt,t∈-τ,0,where z1(t),…,zp(t) are the premise variables, Ali and Dli (l=1,…,r and i=1,…,N) are matrices of appropriate dimensions, Mlij  (j=1,…,p) are the fuzzy sets, and r and p are the number of fuzzy rules and premise variables, respectively.By using the product inference engine and the center of average defuzzification, theith TS fuzzy subsystem can be inferred as(5)x˙t=∑l=1rhliztAlixt+Dlixt-τ, where z(t)=z1(t),…,zp(t)T and hli(z(t)):Rp→0,1 are the normalized weighting functions expressed by(6)hlizt=wlizt∑l=1rwlizt=∏j=1pMlijzjt∑l=1r∏j=1pMlijzjt.Mlij(zj(t)) is the firing strength of the membership function Mlij. It is assumed that wli(zj(t))≥0 and ∑l=1rwli(zj(t))>0.Hence,hli(z(t))≥0 and ∑l=1rhli(z(t))=1.Finally, the fuzzy switched system (3) can be represented by(7)x˙t=∑i=1N∑l=1rξithliztAlixt+Dlixt-τ. Equation (7) can also be written as(8)x˙t=∑l=1rhlσtztAlσtxt+Dlσtxt-τ.By using the Newton-Leibniz formula,(9)xt-τ=xt-∫t-τtx˙θdθ. Equation (8) can be written as(10)x˙t=∑i=1Nξit∑l=1rhliAli+Dlixt-DliAli∫t-τtxθdθ-Dli2∫t-τtxθ-τdθ. In the sequel, hli(z(t)) will be simplified by hli.


### 2.2. Preliminaries

This subsection recalls some of the definitions and remarks that will be useful throughout this paper.Kotelyanski Lemma (see [54]). The real parts of the eigenvalues of matrixA, with non-negative off-diagonal elements, are less than a real numberμif and only if all those of matrixM;M=μIn-Aare positive, withInthenidentity matrix.Definition 1 (see [43]).
The matrixA(·)=ai,j(·)1≤i,j≤n is called an M-matrix, if the following conditions are met:(i)
aii(·)>0  (i=1,…,n),  aij(·)≤0  (i≠j;  i,j=1,…,n);(ii)
the principal minors ofA(·) are all positive:(11)A·12⋯j12⋯j>0,j=1,…,n;(iii)
for any positive real vectorη=η1,…,ηnT, the algebraic equations A(·)x=η have a positive solution w=w1,…,wnT.Remark 2.
A
(
·
)
=
a
i
,
j
(
·
)
1
≤
i
,
j
≤
n is the opposite of an M-matrix if (-A(·)) is an M-matrix.Remark 3.
A continuous-time system characterized byA(·) is stable if A(·) is the opposite of an M-matrix. In this case, the main minors of A(·) are of alternating signs (the first is negative) and the Kotelyanski lemma permits concluding to the stability of the system characterized by A(·).Definition 4 (see [43, 55]).
IfMc· is a pseudo-overvaluing matrix of the system x˙(t)=Ai(·)x(t) with respect to the vector norm p(x)=x1,…,xnT, the following inequality is satisfied:(12)D+px≤Mc·px, where D+ denotes the right hand derivative operator.
Consequently, the stability of the comparison system,z˙(t)=Mc(·)z(t) with the initial conditions such as z0=p(x0), implies the same property for the initial system.


## 2.1. Problem Formulation

Let us consider the unforced switched nonlinear time-delay systems that are described by a differential equation of the form:(1)x˙t=fσtxt,xt-τxt=Φt,t∈-τ,0,where x(t)∈Rn is the state vector, τ>0 is the time delay, Φ:-τ,0→Rn is a differentiable vector valued initial function, fσ:Rn→Rn are sufficiently regular functions that are parametrized by the index set I=1,2,…,N,  σ(t):Rn→I is a piecewise constant function depending on time, called switching signal and assumed to be available in real time, and N is the number of subsystems.The switching sequence is defined through a switching vectorξ(t)=ξ1(t),…,ξN(t)T whose components ξi(t) are given by(2)ξit=1ifσt=i,i∈I(i.e. subsystemiis active)0otherwise.It is obvious that ∑i=1Nξi(t)=1  ∀t≥0.Therefore, the switched system is composed ofN subsystems expressed as (3)x˙t=fixt,xt-τ,i∈Ixt=Φt,t∈-τ,0.In addition, each subsystem∑fi is described by a set of IF-THEN rules; each rule is related to a region of the state space where the subsystem could be approximated by a local linear model.Thus, thelth fuzzy rule associated with the ith subsystem is given by(4)Ril:IFz1tisMli1and…andzptisMlipTHENx˙t=Alixt+Dlixt-τxt=Φt,t∈-τ,0,where z1(t),…,zp(t) are the premise variables, Ali and Dli (l=1,…,r and i=1,…,N) are matrices of appropriate dimensions, Mlij  (j=1,…,p) are the fuzzy sets, and r and p are the number of fuzzy rules and premise variables, respectively.By using the product inference engine and the center of average defuzzification, theith TS fuzzy subsystem can be inferred as(5)x˙t=∑l=1rhliztAlixt+Dlixt-τ, where z(t)=z1(t),…,zp(t)T and hli(z(t)):Rp→0,1 are the normalized weighting functions expressed by(6)hlizt=wlizt∑l=1rwlizt=∏j=1pMlijzjt∑l=1r∏j=1pMlijzjt.Mlij(zj(t)) is the firing strength of the membership function Mlij. It is assumed that wli(zj(t))≥0 and ∑l=1rwli(zj(t))>0.Hence,hli(z(t))≥0 and ∑l=1rhli(z(t))=1.Finally, the fuzzy switched system (3) can be represented by(7)x˙t=∑i=1N∑l=1rξithliztAlixt+Dlixt-τ. Equation (7) can also be written as(8)x˙t=∑l=1rhlσtztAlσtxt+Dlσtxt-τ.By using the Newton-Leibniz formula,(9)xt-τ=xt-∫t-τtx˙θdθ. Equation (8) can be written as(10)x˙t=∑i=1Nξit∑l=1rhliAli+Dlixt-DliAli∫t-τtxθdθ-Dli2∫t-τtxθ-τdθ. In the sequel, hli(z(t)) will be simplified by hli.


## 2.2. Preliminaries

This subsection recalls some of the definitions and remarks that will be useful throughout this paper.Kotelyanski Lemma (see [54]). The real parts of the eigenvalues of matrixA, with non-negative off-diagonal elements, are less than a real numberμif and only if all those of matrixM;M=μIn-Aare positive, withInthenidentity matrix.Definition 1 (see [43]).
The matrixA(·)=ai,j(·)1≤i,j≤n is called an M-matrix, if the following conditions are met:(i)
aii(·)>0  (i=1,…,n),  aij(·)≤0  (i≠j;  i,j=1,…,n);(ii)
the principal minors ofA(·) are all positive:(11)A·12⋯j12⋯j>0,j=1,…,n;(iii)
for any positive real vectorη=η1,…,ηnT, the algebraic equations A(·)x=η have a positive solution w=w1,…,wnT.Remark 2.
A
(
·
)
=
a
i
,
j
(
·
)
1
≤
i
,
j
≤
n is the opposite of an M-matrix if (-A(·)) is an M-matrix.Remark 3.
A continuous-time system characterized byA(·) is stable if A(·) is the opposite of an M-matrix. In this case, the main minors of A(·) are of alternating signs (the first is negative) and the Kotelyanski lemma permits concluding to the stability of the system characterized by A(·).Definition 4 (see [43, 55]).
IfMc· is a pseudo-overvaluing matrix of the system x˙(t)=Ai(·)x(t) with respect to the vector norm p(x)=x1,…,xnT, the following inequality is satisfied:(12)D+px≤Mc·px, where D+ denotes the right hand derivative operator.
Consequently, the stability of the comparison system,z˙(t)=Mc(·)z(t) with the initial conditions such as z0=p(x0), implies the same property for the initial system.


## 3. Delay-Dependent Stability Conditions for TS Fuzzy Switched Systems with Single Delay

### 3.1. Main Results

In this section, new delay-dependent conditions for global asymptotic stability of system (10) under arbitrary switching are stated.Theorem 5.
System (10) is globally asymptotically stable under an arbitrary switching rule σ(t)=i,  i∈I, if matrix(13)Tm=max1≤i≤N1≤l≤rTli is the opposite of an M-matrix, where(14)Tli=Ali+Dli∗+τDliAli+Dli2.Proof.
Letw∈R+∗n with components (wm>0,  ∀m=1,…,n) and consider the radially unbounded candidate Lyapunov functional for each fuzzy submodel ∑Ali,Dli given by(15)Vt=V1t+V2t+V3t+V4t, where(16)V1t=xt,wV2t=DliAli∫-τ0∫t+θtxsdsdθ,wV3t=Dli2∫-τ0∫t+θtxs-τdsdθ,wV4t=τDli2∫t-τtxsds,w. It is clear that V(t0)<∞ and that a common Lyapunov functional for all the submodels can be given by(17)Vct=max1≤i≤N1≤l≤rVt.The right Dini derivative of V(t) along the trajectory of system (10) gives(18)D+Vt10=∑j=14D+Vjt10D+V1t10=d+xtdt+,w=sgnxtd+xtdt+,wwith(19)sgnxt=sgn⁡x1t⋱sgn⁡xnt.Then,(20)D+V1t10=sgnxtAli+Dlixt-DliAli∫t-τtxθdθ-Dli2∫t-τtxθ-τdθ,w≤Ali+Dli∗xt,w+DliAli∫t-τtxθdθ,w+Dli2∫t-τtxθ-τdθ,wD+V2t10=DliAliτxt-∫t-τtxsds,w=τDliAlixt,w-DliAli∫t-τtxsds,wD+V3t10=Dli2τxt-τ-∫t-τtxs-τds,w=τDli2xt-τ,w-Dli2∫t-τtxs-τds,wD+V4t10=τDli2xt-xt-τ,w=τDli2xt,w-τDli2xt-τ,w.Finally, we obtain(21)D+Vt10≤Ali+Dli∗+τDliAli+Dli2xt,w≤Tmxt,w,where(22)Tm=max1≤i≤N1≤l≤rAli+Dli∗+τDliAli+Dli2.
On the other hand, if we suppose thatTm is the opposite of an M-matrix and according to the M-matrices properties, we can find a vector ρ∈R+∗n  (ρl>0  ∀l=1,…,n) satisfying the relation: TmTw=-ρ,  ∀w∈R+∗n.
Knowing that(23)Tmxt,w=TmTw,xt,we can write(24)D+Vt10≤TmTw,xt=-ρ,xt=-∑l=1nρlxlt<0.This completes the proof of Theorem 5.


### 3.2. Extension of the Results to the Case of TS Fuzzy Switched Systems Described by Delayed Differential Equations with Single Delay

In this section, we consider the class of TS fuzzy switched time-delay systems that are governed by the following differential equation:(25)ynt+∑i=1Nξit∑l=1rhli∑j=0n-1alijyjt+∑j=0n-1dlijyjt-τ=0,where y(t)∈Rn is the state vector and alij and dlij (l=1,…,r;  i=1,…,N; and j=0,…,n-1) are constant coefficients.A change of variable under the formxj+1(t)=y(j)(t),  j=0,…,n-1, allows system (25) to be represented in the state space as follows:(26)x˙t=∑i=1Nξit∑l=1rhliAlixt+Dlixt-τxt=Φt,t∈-τ,0,where matrices Ali and Dli are given by(27)Ali=01⋯0⋮⋱⋱⋮0⋯01-ali0⋯⋯-alin-1,Dli=0⋯⋯0⋮⋮0⋯⋯0-dli0⋯⋯-dlin-1. A change of base of the form z(t)=Px(t) of (26) under the arrow form matrix gives(28)z˙t=∑i=1Nξit∑l=1rhliMlizt+Nlizt-τzt=PΦt,t∈-τ,0,where Mli=P-1AliP,  Nli=P-1DliP, and P is the corresponding passage matrix such that(29)P=11⋯10α1α2⋯αn-10α12α22⋯αn-12⋮⋮⋮⋯⋮0α1n-1α2n-1⋯αn-1n-11.Arrow form matrices Mli,  l=1,…,r and i∈I, are given by(30)Mli=α1β1⋱⋮αn-1βn-1γli1⋯γlin-1γlinwith (31)βj=∏q=1q≠jn-1αj-αq-1∀j=1,…,n-1γlij=-PAliαj∀j=1,…,n-1γlin=-alin-1-∑j=1n-1αj,whereas matrices Nli are given by(32)Nli=0n-1,n-10n-1,1δli1⋯δlin-1δlinwith(33)δlij=-PDliαj∀j=1,…,n-1δlin=-dlin-1PDliλ=∑j=0n-1dlijλj.Note that αj,  j=1,…,n-1, are distinct constant parameters that can be chosen arbitrarily and PAli(λ) is the instantaneous characteristic polynomial of matrix Ali given by(34)PAliλ=λn+∑j=0n-1alijλj.Therefore, the expression of the common comparison matrix in the new base is given through the sum of the following terms:(35)Mli+Nli∗=α1β1⋱⋮αn-1βn-1γli1+δli1⋯γlin-1+δlin-1γlin+δlinNliMli=0n-1,n-10n-1,1ψliα1⋯ψliαn-1δlinγlin+∑j=1n-1δlijβjNli2=0n-1,n-10n-1,1-δli1dlin-1-δlin-1dlin-1dlin-12.The new polynomial ψli(αj) is defined by(36)ψliαj=-PDliαj×αj+dlin-1PAliαj. Notice that NliMli can be simplified. In fact, δlinγlin+∑j=1n-1δlijβj=trace(NliMli). However, trace(NliMli)=trace(DliAli)=alin-1dlin-1-dlin-2. Consequently, we can construct for each individual submodel ∑Ali,Dli the following overvaluing system:(37)Tli=Mli+Nli∗+τNliMli+Nli2, where(38)Tli=α1β1⋱⋮αn-1βn-1tli1⋯tlin-1tlin,tlij=γlij+δlij+τψliαj+-δlijdlin-1tlin=γlin+δlin+τalin-1dlin-1-dlin-2+dlin-12.Finally, the common overvaluing matrix is given by(39)Tm=α1β1⋱⋮αn-1βn-1t1⋯tn-1tnwith(40)tj=max1≤i≤N1≤l≤rtlij∀j=1,…,n. At this level, we can state the following theorem.Theorem 6.
System (26) is globally asymptotically stable under an arbitrary switching rule σ(t)=i,  i∈I, if there exist αj<0,  ∀j=1,…,n-1,  αj≠αq,  ∀j≠q, satisfying the following condition:(41)-tn+∑j=1n-1tjβjαj-1>0.Proof.
It is sufficient to check that the matrix(42)Tm=max1≤i≤N1≤l≤rMli+Nli∗+τNliMli+Nli2 is the opposite of an M-matrix by verifying that all the successive principal minors of Tm, denoted by Δj, are of alternating signs; that is,(43)-1jΔj>0∀j=1,…,n-1.It is clear that, for an arbitrary choice of αj<0, relation (43) is satisfied for j=1,…,n-1.
Condition (43) becomes for j=n(44)-1nΔn=-1ndetTm=-1ntn-∑j=1n-1tjβjαj-1∏j=1n-1αj.By dividing (44) by κ=(-1)n-1∏j=1n-1αj, we obtain(45)-tn+∑j=1n-1tjβjαj-1>0. This achieves the proof of Theorem 6.Remark 7.
The maximum value ofτ ensuring the asymptotic stability of each individual model ∑Ali,Dli (τ<τmax(li)) is computed by(46)τmaxli=-γlin+δlin+∑j=1n-1αj-1-PAliαj+PDliαjβjdlin-12+alin-1dlin-1-dlin-2-∑j=1n-1αj-1ψliαj+PDliαjdlin-1βj.Remark 8.
Expression (46) can be widely simplified. We can give, for instance, two of the possible combinations.
Case 1. One has(47)PDliαj+PAliαjβj<0ψliαjβj>0PDliαjdlin-1βj>0.Then, stability conditions will depend on the sign of alin-1dlin-1-dlin-2 as follows:(48)τmaxli=PAli0+PDli02κdlin-2-alin-1dlin-1+dlin-1PAli0+PDli0ifalin-1dlin-1<dlin-2τmaxli=1dlin-1ifalin-1dlin-1>dlin-2,dlin-1>0.
Case 2. One has (49)PDliαj+PAliαjβj<0ψliαjβj<0PDliαjdlin-1βj<0(50)τmaxli=PAli0+PDli02κalin-1dlin-1-dlin-2+dlin-12-dlin-1PAli0+PDli0ifalin-1dlin-1>dlin-2(51)τmaxli=PAli0+PDli02κdlin-12-dlin-1PAli0+PDli0ifalin-1dlin-1<dlin-2.Proof.
See the Appendix.


## 3.1. Main Results

In this section, new delay-dependent conditions for global asymptotic stability of system (10) under arbitrary switching are stated.Theorem 5.
System (10) is globally asymptotically stable under an arbitrary switching rule σ(t)=i,  i∈I, if matrix(13)Tm=max1≤i≤N1≤l≤rTli is the opposite of an M-matrix, where(14)Tli=Ali+Dli∗+τDliAli+Dli2.Proof.
Letw∈R+∗n with components (wm>0,  ∀m=1,…,n) and consider the radially unbounded candidate Lyapunov functional for each fuzzy submodel ∑Ali,Dli given by(15)Vt=V1t+V2t+V3t+V4t, where(16)V1t=xt,wV2t=DliAli∫-τ0∫t+θtxsdsdθ,wV3t=Dli2∫-τ0∫t+θtxs-τdsdθ,wV4t=τDli2∫t-τtxsds,w. It is clear that V(t0)<∞ and that a common Lyapunov functional for all the submodels can be given by(17)Vct=max1≤i≤N1≤l≤rVt.The right Dini derivative of V(t) along the trajectory of system (10) gives(18)D+Vt10=∑j=14D+Vjt10D+V1t10=d+xtdt+,w=sgnxtd+xtdt+,wwith(19)sgnxt=sgn⁡x1t⋱sgn⁡xnt.Then,(20)D+V1t10=sgnxtAli+Dlixt-DliAli∫t-τtxθdθ-Dli2∫t-τtxθ-τdθ,w≤Ali+Dli∗xt,w+DliAli∫t-τtxθdθ,w+Dli2∫t-τtxθ-τdθ,wD+V2t10=DliAliτxt-∫t-τtxsds,w=τDliAlixt,w-DliAli∫t-τtxsds,wD+V3t10=Dli2τxt-τ-∫t-τtxs-τds,w=τDli2xt-τ,w-Dli2∫t-τtxs-τds,wD+V4t10=τDli2xt-xt-τ,w=τDli2xt,w-τDli2xt-τ,w.Finally, we obtain(21)D+Vt10≤Ali+Dli∗+τDliAli+Dli2xt,w≤Tmxt,w,where(22)Tm=max1≤i≤N1≤l≤rAli+Dli∗+τDliAli+Dli2.
On the other hand, if we suppose thatTm is the opposite of an M-matrix and according to the M-matrices properties, we can find a vector ρ∈R+∗n  (ρl>0  ∀l=1,…,n) satisfying the relation: TmTw=-ρ,  ∀w∈R+∗n.
Knowing that(23)Tmxt,w=TmTw,xt,we can write(24)D+Vt10≤TmTw,xt=-ρ,xt=-∑l=1nρlxlt<0.This completes the proof of Theorem 5.


## 3.2. Extension of the Results to the Case of TS Fuzzy Switched Systems Described by Delayed Differential Equations with Single Delay

In this section, we consider the class of TS fuzzy switched time-delay systems that are governed by the following differential equation:(25)ynt+∑i=1Nξit∑l=1rhli∑j=0n-1alijyjt+∑j=0n-1dlijyjt-τ=0,where y(t)∈Rn is the state vector and alij and dlij (l=1,…,r;  i=1,…,N; and j=0,…,n-1) are constant coefficients.A change of variable under the formxj+1(t)=y(j)(t),  j=0,…,n-1, allows system (25) to be represented in the state space as follows:(26)x˙t=∑i=1Nξit∑l=1rhliAlixt+Dlixt-τxt=Φt,t∈-τ,0,where matrices Ali and Dli are given by(27)Ali=01⋯0⋮⋱⋱⋮0⋯01-ali0⋯⋯-alin-1,Dli=0⋯⋯0⋮⋮0⋯⋯0-dli0⋯⋯-dlin-1. A change of base of the form z(t)=Px(t) of (26) under the arrow form matrix gives(28)z˙t=∑i=1Nξit∑l=1rhliMlizt+Nlizt-τzt=PΦt,t∈-τ,0,where Mli=P-1AliP,  Nli=P-1DliP, and P is the corresponding passage matrix such that(29)P=11⋯10α1α2⋯αn-10α12α22⋯αn-12⋮⋮⋮⋯⋮0α1n-1α2n-1⋯αn-1n-11.Arrow form matrices Mli,  l=1,…,r and i∈I, are given by(30)Mli=α1β1⋱⋮αn-1βn-1γli1⋯γlin-1γlinwith (31)βj=∏q=1q≠jn-1αj-αq-1∀j=1,…,n-1γlij=-PAliαj∀j=1,…,n-1γlin=-alin-1-∑j=1n-1αj,whereas matrices Nli are given by(32)Nli=0n-1,n-10n-1,1δli1⋯δlin-1δlinwith(33)δlij=-PDliαj∀j=1,…,n-1δlin=-dlin-1PDliλ=∑j=0n-1dlijλj.Note that αj,  j=1,…,n-1, are distinct constant parameters that can be chosen arbitrarily and PAli(λ) is the instantaneous characteristic polynomial of matrix Ali given by(34)PAliλ=λn+∑j=0n-1alijλj.Therefore, the expression of the common comparison matrix in the new base is given through the sum of the following terms:(35)Mli+Nli∗=α1β1⋱⋮αn-1βn-1γli1+δli1⋯γlin-1+δlin-1γlin+δlinNliMli=0n-1,n-10n-1,1ψliα1⋯ψliαn-1δlinγlin+∑j=1n-1δlijβjNli2=0n-1,n-10n-1,1-δli1dlin-1-δlin-1dlin-1dlin-12.The new polynomial ψli(αj) is defined by(36)ψliαj=-PDliαj×αj+dlin-1PAliαj. Notice that NliMli can be simplified. In fact, δlinγlin+∑j=1n-1δlijβj=trace(NliMli). However, trace(NliMli)=trace(DliAli)=alin-1dlin-1-dlin-2. Consequently, we can construct for each individual submodel ∑Ali,Dli the following overvaluing system:(37)Tli=Mli+Nli∗+τNliMli+Nli2, where(38)Tli=α1β1⋱⋮αn-1βn-1tli1⋯tlin-1tlin,tlij=γlij+δlij+τψliαj+-δlijdlin-1tlin=γlin+δlin+τalin-1dlin-1-dlin-2+dlin-12.Finally, the common overvaluing matrix is given by(39)Tm=α1β1⋱⋮αn-1βn-1t1⋯tn-1tnwith(40)tj=max1≤i≤N1≤l≤rtlij∀j=1,…,n. At this level, we can state the following theorem.Theorem 6.
System (26) is globally asymptotically stable under an arbitrary switching rule σ(t)=i,  i∈I, if there exist αj<0,  ∀j=1,…,n-1,  αj≠αq,  ∀j≠q, satisfying the following condition:(41)-tn+∑j=1n-1tjβjαj-1>0.Proof.
It is sufficient to check that the matrix(42)Tm=max1≤i≤N1≤l≤rMli+Nli∗+τNliMli+Nli2 is the opposite of an M-matrix by verifying that all the successive principal minors of Tm, denoted by Δj, are of alternating signs; that is,(43)-1jΔj>0∀j=1,…,n-1.It is clear that, for an arbitrary choice of αj<0, relation (43) is satisfied for j=1,…,n-1.
Condition (43) becomes for j=n(44)-1nΔn=-1ndetTm=-1ntn-∑j=1n-1tjβjαj-1∏j=1n-1αj.By dividing (44) by κ=(-1)n-1∏j=1n-1αj, we obtain(45)-tn+∑j=1n-1tjβjαj-1>0. This achieves the proof of Theorem 6.Remark 7.
The maximum value ofτ ensuring the asymptotic stability of each individual model ∑Ali,Dli (τ<τmax(li)) is computed by(46)τmaxli=-γlin+δlin+∑j=1n-1αj-1-PAliαj+PDliαjβjdlin-12+alin-1dlin-1-dlin-2-∑j=1n-1αj-1ψliαj+PDliαjdlin-1βj.Remark 8.
Expression (46) can be widely simplified. We can give, for instance, two of the possible combinations.
Case 1. One has(47)PDliαj+PAliαjβj<0ψliαjβj>0PDliαjdlin-1βj>0.Then, stability conditions will depend on the sign of alin-1dlin-1-dlin-2 as follows:(48)τmaxli=PAli0+PDli02κdlin-2-alin-1dlin-1+dlin-1PAli0+PDli0ifalin-1dlin-1<dlin-2τmaxli=1dlin-1ifalin-1dlin-1>dlin-2,dlin-1>0.
Case 2. One has (49)PDliαj+PAliαjβj<0ψliαjβj<0PDliαjdlin-1βj<0(50)τmaxli=PAli0+PDli02κalin-1dlin-1-dlin-2+dlin-12-dlin-1PAli0+PDli0ifalin-1dlin-1>dlin-2(51)τmaxli=PAli0+PDli02κdlin-12-dlin-1PAli0+PDli0ifalin-1dlin-1<dlin-2.Proof.
See the Appendix.


## 4. Delay-Dependent Stability Conditions for TS Fuzzy Switched Systems with Multiple Delays

Stability criteria in Theorem5 can be generalized to systems with multiple delays.


### 4.1. Main Result

Consider a switched system composed ofN subsystems; each subsystem is a TS fuzzy time-delay system as shown in the following differential equation:(52)x˙t=∑i=1Nξit∑l=1rhliAlixt+∑k=1mDli,kxt-τkxt=Φt,t∈-max1≤k≤mτk,0,where Ali,  Dli,k  (l=1,…,r;  i=1,…,N; and k=1,…,m) are constant matrices of appropriate dimensions, hli are fuzzy weighting factors, previously defined in (6), and τk>0  ∀k=1,…,m.System (52) can also be put in the following form:(53)x˙t=Aσtxt+∑k=1mDσt,kxt-τkxt=Φt,t∈-max1≤k≤mτk,0,where(54)Aσt=∑i=1Nξit∑l=1rhliAli,Dσt,k=∑i=1Nξit∑l=1rhliDli,k.Theorem 9.
System (52) is globally asymptotically stable under arbitrary switching rule σ(t)=i∈I, if matrix(55)Tm,M=max1≤i≤N1≤l≤rAli+∑k=1mDli,k∗+∑k=1mτkDli,kAli+Dli,k2is the opposite of an M-matrix.Proof.
Following the same steps as those given in the proof of Theorem5 and by choosing the common radially unbounded Lyapunov functional as follows,(56)Vc,Mt=max1≤i≤N1≤l≤rVMt=max1≤i≤N1≤l≤r∑p=14Vp,Mt,where(57)V1,Mt=xt,wV2,Mt=∑k=1mDli,kAli∫-τk0∫t+θtxsdsdθ,wV3,Mt=∑k=1mDli,k2∫-τk0∫t+θtxs-τkdsdθ,wV4,Mt=∑k=1mτkDli,k2∫t-τktxsds,w,we deduce that it suffices that Tm,M is the opposite of an M-matrix to conclude to the asymptotic stability of system (52).


### 4.2. Extension of the Results to the Case of TS Fuzzy Switched Systems Described by Differential Equations with Multiple Delays

The same method is applied to determine delay-dependent stability criteria for systems described by the following multiple time-delayed differential equation:(58)ynt+∑i=1Nξit∑l=1rhli∑j=0n-1alijyjt+∑k=1m∑j=0n-1dli,kjyjt-τk=0yjt=Φjt∀t∈-max1≤k≤mτk,0,j=0,…,n-1.The same change of variable xj+1(t)=y(j)(t),  j=0,…,n-1, as in Section 3.2 yields to the new state space representation:(59)x˙t=∑i=1Nξit∑l=1rhliAlixt+∑k=1mDli,kxt-τkxt=Φt,t∈-max1≤k≤mτk,0,where x(t)=x1t,…,xn(t)T is the state vector and for each i∈I,  k∈1,…,m, Ali and Dli,k are given by(60)Ali=01⋯0⋮⋱⋱⋮0⋯01-ali0⋯⋯-alin-1Dli,k=0n-1,n-10n-1,1-dli,k0⋯-dli,kn-1. A change of base of the form z(t)=Px(t) allows system (59) to be represented by the arrow form matrix:(61)z˙t=∑i=1Nξit∑l=1rhliMlizt+∑k=1mNli,kzt-τkzt=PΦt,t∈-max1≤k≤mτk,0,where Mli is given by (30) and Nli,k becomes(62)Nli,k=0n-1,n-10n-1,1δli,k1⋯δli,kn-1δli,knwith(63)δli,kj=-PDli,kαjδli,kn=-dli,kn-1.Define the new polynomials:(64)PDli,kλ=∑j=0n-1dli,kjλj,ψli,kλ=-PDli,kλ×λ+dli,kn-1PAliλ.Finally, we can compute the common overvaluing matrix as follows:(65)Tm,M=α1β1⋱⋮αn-1βn-1t1,M⋯tn-1,Mtn,Mwith(66)tj,M=max1≤i≤N1≤l≤rγlij+∑k=1mδli,kj+∑k=1mτkψli,kαj+-dli,kn-1δli,kj,∀j=1,…,n-1,tn,M=max1≤i≤N1≤l≤rγlin+∑k=1mδli,kn+∑k=1mτkalin-1dli,kn-1-dli,kn-2+dli,kn-12.Hence, we can state Theorem 10.Theorem 10.
System (59) is globally asymptotically stable under an arbitrary switching rule σ(t)=i∈I, if there exist αj<0,  j=1,…,n-1,  αj≠αq,  ∀j≠q such that(67)-tn,M+∑j=1n-1tj,Mβjαj-1>0.Proof.
The same proof is as in Theorem6; just replace matrix Tm by Tm,M.


## 4.1. Main Result

Consider a switched system composed ofN subsystems; each subsystem is a TS fuzzy time-delay system as shown in the following differential equation:(52)x˙t=∑i=1Nξit∑l=1rhliAlixt+∑k=1mDli,kxt-τkxt=Φt,t∈-max1≤k≤mτk,0,where Ali,  Dli,k  (l=1,…,r;  i=1,…,N; and k=1,…,m) are constant matrices of appropriate dimensions, hli are fuzzy weighting factors, previously defined in (6), and τk>0  ∀k=1,…,m.System (52) can also be put in the following form:(53)x˙t=Aσtxt+∑k=1mDσt,kxt-τkxt=Φt,t∈-max1≤k≤mτk,0,where(54)Aσt=∑i=1Nξit∑l=1rhliAli,Dσt,k=∑i=1Nξit∑l=1rhliDli,k.Theorem 9.
System (52) is globally asymptotically stable under arbitrary switching rule σ(t)=i∈I, if matrix(55)Tm,M=max1≤i≤N1≤l≤rAli+∑k=1mDli,k∗+∑k=1mτkDli,kAli+Dli,k2is the opposite of an M-matrix.Proof.
Following the same steps as those given in the proof of Theorem5 and by choosing the common radially unbounded Lyapunov functional as follows,(56)Vc,Mt=max1≤i≤N1≤l≤rVMt=max1≤i≤N1≤l≤r∑p=14Vp,Mt,where(57)V1,Mt=xt,wV2,Mt=∑k=1mDli,kAli∫-τk0∫t+θtxsdsdθ,wV3,Mt=∑k=1mDli,k2∫-τk0∫t+θtxs-τkdsdθ,wV4,Mt=∑k=1mτkDli,k2∫t-τktxsds,w,we deduce that it suffices that Tm,M is the opposite of an M-matrix to conclude to the asymptotic stability of system (52).


## 4.2. Extension of the Results to the Case of TS Fuzzy Switched Systems Described by Differential Equations with Multiple Delays

The same method is applied to determine delay-dependent stability criteria for systems described by the following multiple time-delayed differential equation:(58)ynt+∑i=1Nξit∑l=1rhli∑j=0n-1alijyjt+∑k=1m∑j=0n-1dli,kjyjt-τk=0yjt=Φjt∀t∈-max1≤k≤mτk,0,j=0,…,n-1.The same change of variable xj+1(t)=y(j)(t),  j=0,…,n-1, as in Section 3.2 yields to the new state space representation:(59)x˙t=∑i=1Nξit∑l=1rhliAlixt+∑k=1mDli,kxt-τkxt=Φt,t∈-max1≤k≤mτk,0,where x(t)=x1t,…,xn(t)T is the state vector and for each i∈I,  k∈1,…,m, Ali and Dli,k are given by(60)Ali=01⋯0⋮⋱⋱⋮0⋯01-ali0⋯⋯-alin-1Dli,k=0n-1,n-10n-1,1-dli,k0⋯-dli,kn-1. A change of base of the form z(t)=Px(t) allows system (59) to be represented by the arrow form matrix:(61)z˙t=∑i=1Nξit∑l=1rhliMlizt+∑k=1mNli,kzt-τkzt=PΦt,t∈-max1≤k≤mτk,0,where Mli is given by (30) and Nli,k becomes(62)Nli,k=0n-1,n-10n-1,1δli,k1⋯δli,kn-1δli,knwith(63)δli,kj=-PDli,kαjδli,kn=-dli,kn-1.Define the new polynomials:(64)PDli,kλ=∑j=0n-1dli,kjλj,ψli,kλ=-PDli,kλ×λ+dli,kn-1PAliλ.Finally, we can compute the common overvaluing matrix as follows:(65)Tm,M=α1β1⋱⋮αn-1βn-1t1,M⋯tn-1,Mtn,Mwith(66)tj,M=max1≤i≤N1≤l≤rγlij+∑k=1mδli,kj+∑k=1mτkψli,kαj+-dli,kn-1δli,kj,∀j=1,…,n-1,tn,M=max1≤i≤N1≤l≤rγlin+∑k=1mδli,kn+∑k=1mτkalin-1dli,kn-1-dli,kn-2+dli,kn-12.Hence, we can state Theorem 10.Theorem 10.
System (59) is globally asymptotically stable under an arbitrary switching rule σ(t)=i∈I, if there exist αj<0,  j=1,…,n-1,  αj≠αq,  ∀j≠q such that(67)-tn,M+∑j=1n-1tj,Mβjαj-1>0.Proof.
The same proof is as in Theorem6; just replace matrix Tm by Tm,M.


## 5. Illustrative Examples

Example 1.
Consider a switched system composed of three second-order subsystems∑Ai,Di,  i∈1,2,3; each subsystem is represented by a TS fuzzy time-delay model as follows:Subsystem(1)(68)A11=01-16-15,D11=00-1.5-0.02A21=01-16-13.5,D21=00-1.50.Subsystem(2)(69)A12=01-20-18,A22=01-16-15D12=D22=00-0.50.Subsystem(3)(70)A13=01-14-12,D13=00-1.750A23=01-14.5-10,D23=00-10.A transformation of state matrices under the arrow form Fli=P-1AliP and Nli=P-1DliP (l∈1,2 and i∈1,2,3) with P=10α1 gives(71)Fli=α1γli1γli2,Nli=00δli1δli2.For i∈1,2,3 and l∈1,2, parameters γli and δli are computed as follows:(72)γ111=-PA11α=-α2+15α+16γ112=-15-α;γ211=-PA21α=-α2+13.5α+16γ212=-13.5-α;γ121=-PA12α=-α2+18α+20γ122=-18-α;γ221=-PA22α=-α2+15α+16γ222=-15-α;γ131=-PA13α=-α2+12α+14γ132=-12-α;γ231=-PA23α=-α2+10α+14.5γ232=-10-α;δ111=-PD11α=-0.02α+1.5δ112=-0.02;δ211=-PD21α=-1.5δ212=0.For an arbitrary choice of α=-1, we obtain, for example, for subsystem (1)(73)F11=-11-2-14,N11=00-1.48-0.02;F21=-11-3.5-12.5,N21=00-1.50.For h2=1-h1 and by using the center of average defuzzification, state matrix Fi=h1F1i+h2F2i and delayed-state matrix Ni=h1N1i+h2N2i  (i∈1,2,3) are given, respectively, by(74)F1=-111.5h1-3.5-1.5h1+12.5,N1=000.02h1-1.5-0.02h1,F2=-11-h1+2-3h1+14,N2=00-0.50,F3=-11-5.5-2.5h1-2h1+9,N3=00-0.75h1+10.Denote by Tli,  l∈1,2 and i∈1,2,3, the minimal pseudo-overvaluing matrix of the fuzzy submodel ∑Ali,Dli such that(75)Tli=-11tli1tli2,where(76)T11=-113.48+1.5496τ-14.02+1.2τT21=-115+1.5τ-12.5+1.5τT12=-113.5+0.5τ-17+0.5τT22=-112.5+0.5τ-14+0.5τT13=-114.75+1.75τ-11+1.75τT23=-116.5+τ-9+τ.At this level, we can derive sufficient conditions for the asymptotic stability of each submodel individually as shown in Table 1.
From Table1, we can already see that it is sufficient to have a time delay τ inferior to 1.25s to ensure the asymptotic stability of all the submodels individually. Taking into account this value, we can notice that ∑T23 can be chosen as the common comparison system for the six submodels ∑Ali,Dli (l∈1,2 and i∈1,2,3). Then, the TS fuzzy switched time-delay system is asymptotically stable under an arbitrary switching law if(77)τ<1.25s.The switching signal is plotted on Figure 1. For a time delay equal to τ=1.2s, a final simulation time tf=4s, weighting factors h1=h2=0.5, and an initial state vector x(t)=1-2T  ∀t∈-1.2,0, the evolution of state vectors, the system’s trajectory, and the state’s norm are illustrated on Figures 2, 3, and 4, respectively.
This example shows that the obtained stability conditions are sufficient and very close to be necessary. Moreover, the proposed method makes it possible to avoid searching a common Lyapunov-Krasovskii functional, which is very difficult in this case.Table 1
Maximum allowable time delay for each submodel (τmax(li)).
Submodel(li)
(
11
)
(
21
)
(
12
)
(
22
)
(
13
)
(
23
)
τ
m
a
x
(
l
i
)
(
s
)
3.83
2.5
13.5
11.5
1.78
1.25Figure 1
Switching signal (Example1).Figure 2
Evolution of the state vectors (Example1).Figure 3
System’s trajectory (Example1).Figure 4
State’s norm (Example1).Example 2.
Consider the TS fuzzy switched system with two delaysτ1 and τ2 as follows:(78)x˙t=∑i=12ξit∑l=12hliAlixt+Dli,1xt-τ1+Dli,2xt-τ2xt=Φt,t∈-max1≤k≤2τk,0.
Subsystem(1)(79)A11=01-4-3,D11,1=000-1,D11,2=00-0.50A21=01-3-3,D21,1=000-1,D21,2=00-0.20.
Subsystem(2)(80)A12=01-5-10,D12,1=00-2-1,D12,2=00-0.10A22=01-7-12,D22,1=00-3-1,D22,2=00-0.10.For α=-1, a transformation under the arrow form matrices gives(81)F11=-11-2-2,N11,1=001-1,N11,2=00-0.50F21=-11-1-2,N21,1=001-1,N21,2=00-0.20F12=-114-9,N12,1=00-1-1,N12,2=00-0.10,F22=-114-11,N22,1=00-2-1,N22,2=00-0.10.Denote by Tli,M,  l,i∈1,2, the minimal overvaluing matrix of subsystem of index (li) and relative to the vector norm p(x)=x1,…,xnT such that(82)Tli,M=-11tli,M1tli,M2,where(83)T11,M=-111.5+2τ1+0.5τ2-3+4τ1+0.5τ2T21,M=-110.2+τ1+0.2τ2-3+4τ1+0.2τ2T12,M=-112.9+4τ1+0.1τ2-10+9τ1+0.1τ2T22,M=-111.9+4τ1+0.1τ2-12+10τ1+0.1τ2.Applying Theorem 10, we can draw Table 2 which gives a sufficient stability condition 0<τ2<τ2,max(li) for each submodel.
Finally, we can construct the following common comparison system∑Tm,M such that(84)Tm,M=-112.9+4τ1+0.1τ2-3+4τ1+0.5τ2.Consequently, the TS fuzzy switched time-delay system is asymptotically stable under arbitrary switching if(85)0<τ2<0.1-8τ10.6.In this example, we can notice that maximum bounds found for the delays τ1 and τ2 are small. Indeed, this is justified by the multiplicity of time delay, by the choice of delay matrices Nli,k,  l,i,k∈1,2, and by the fact that arbitrary switching strategy brings conservativeness because of the “strict” condition that requires finding a comparison system which is common to all fuzzy local models.
Nevertheless, the simplicity of vector norms based approach makes the study almost feasible in most cases in the opposite of Lyapunov-based method which proves to be difficult when dealing with the problem of stability under arbitrary switching.
For a switching signal as shown on Figure5, a choice of time delays τ1=0.01s and τ2=0.033s, particular values of weighting factors h1=0.2 and h2=0.8, and an initial state vector x(t)=1-2T  ∀t∈-0.033,0, the evolution of state responses is illustrated on Figure 6 whereas Figures 7 and 8 show the system’s trajectory and the norm’s state, respectively.Table 2
Maximum allowable time delay for each submodel (τ2,max(li)).
Submodel(li)
(
11
)
(
21
)
(
12
)
(
22
)
τ
2
,
m
a
x
(
l
i
)
1.5
-
6
τ
1
7
-
12.5
τ
1
35.5
-
65
τ
1
50.5
-
70
τ
1Figure 5
Switching signal (Example2).Figure 6
Evolution of state vectors (Example2).Figure 7
System’s trajectory (Example2).Figure 8
State’s norm (Example2).Example 3 (see [38, 45]).
Consider a DC motor with separate excitation (Figure9). The system acts under variable mechanical load and is subject to a retarded state feedback control input. Our aim is to synthesize a state feedback control for each local model so that the overall system is asymptotically stable under arbitrary switching despite the time delay.
A PDC controller scheme is employed such that thelth controller fuzzy rule of the ith subsystem is given by(86)Ril:IFz1tisMli1andz2tisMli2THENut=-Klixt,where Kli=kli0kli1,  l,i∈1,2, is the local feedback gain vector.
The TS fuzzy switched system is then represented in the state space by(87)x˙t=∑i=12ξit∑l=12hliAlixt-∑q=12hqiBKqixt-τ,where Ali=010-σli and B=01T.
We can also write(88)x˙t=∑i=12ξit∑l=12hliAlixt-∑q=12hqi∑l=12hliBKqixt-τ.Hence, (89)x˙t=∑i=12ξit∑l=12hliAlixt-BKlixt-τ.Denote Dli=-BKli, and the problem is then reduced to determining admissible values of parameters kli0,kli1,  l,i∈1,2, for a given value of time delay τ.
A transformation under the arrow form matrix withα=-1 yields(90)Fli=-11-1-σli-σli+1Nli=00kli1-kli0-kli1.To tune parameters kli0,kli1, suppose that for kli1>0,  σli<1 we have(91)kli0<σlikli1. Thus, we can verify that(92)-δli1dli1=-kli1-kli0×kli1<0ψli-1=kli0-σlikli1<0. Since all the assumptions of Remark 8 (Case 2) are met, this allows us, according to inequality (50), to write(93)τ<kli02σlikli1-kli0+kli12-kli1kli0. Therefore, each fuzzy model is asymptotically stable individually if(94)kli0>2τkli1σli+kli11+τ2+kli1. The pseudo-overvaluing matrix for each submodel is constructed as follows:(95)Tli=-11tli1tli2with(96)tli1=kli1-kli0+σli-1+τψli-1+kli1kli1-kli0tli2=1-σli-kli1+τσlikli1-kli0+kli12.For instance, if the time delay τ is equal to 0.5s, σ11=0.86, σ21=0.97, σ12=0.76, and σ22=0.89, and local feedback gains are chosen as K11=0.70.9, K21=1.151.2, K12=0.921.22, and K22=0.81, we obtain(97)T11=-110.187-0.318T21=-110.057-0.443T12=-110.233-0.2352T22=-110.2350-0.345. Finally, the common comparison matrix that verifies the properties of the opposite of an M-matrix is given by(98)Tm=-110.2350-0.2352.This example has shown that the proposed approach is appropriate for the synthesis of a stabilizing state feedback retarded control for an initially unstable nonlinear system described by a set of linear TS fuzzy local linear models.Figure 9
DC motor with separate excitation.


## 6. Conclusion

In this paper, new delay-dependent stability conditions by using the vector norms approach have been presented. In fact, most of the research works that have been carried out on stability analysis of TS fuzzy switched time-delay systems under arbitrary switching are based on the research of a common Lyapunov-Krasovskii functional for all the fuzzy models, which is considered as a hard task. The idea of the proposed method consists in putting the switched system under a special form of state space representation using arrow form matrices and then finding a common pseudo-overvaluing system for all the constituent submodels. The stability analysis of this comparison system, based on the aggregation techniques and theM-matrices properties, permits concluding to that of the original TS fuzzy switched system.Vector norms approach, applied to TS fuzzy switched time-delay systems, whether with single or multiple delay, is suitable for the case of arbitrary switching and provides delay-dependent algebraic criteria. The applicability of the obtained conditions is shown through three numerical examples. It would be interesting to generalize the study to the case of systems with control input.



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