# Simple Algebraic Criteria for Asymptotic Stability of a Class of TS Fuzzy Time-Delay Switched Systems

Authors: Nawel Aoun; Marwen Kermani; Anis Sakly
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper investigates the asymptotic stability of a class of TS fuzzy switched systems when an arbitrary switching strategy is adopted. The proposed method, applied to neutral and retarded-type systems, is based on the vector-norms approach. The idea consists in defining a common comparison system to all the fuzzy models. If this comparison system can be described by a state matrix that fulfills the properties of the opposite of an M-matrix, then we can conclude on the asymptotic stability of the initial system via simple algebraic delay-independent conditions.


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## Body

## 1. Introduction

Switched systems are defined as an important class of hybrid dynamical systems that consists of several continuous-time or discrete-time subsystems. Each subsystem is activated during a certain time interval according to a specific signal that governs the switching between all the subsystems [1, 2]. In the literature, this signal is sometimes referred to as switching rule or sequence and may be function of time, which is the case in this work, or of the state variables or any other external input.Switched systems are widely used in many industrial fields as they can properly model many real-world systems which exhibit switching features such as mechanical systems, chemical processes, communication networks, switching power converters, robotic systems, air traffic and aircraft, etc.Besides and during the past decades, fuzzy theory about modeling and control and more particularly about Takagi-Sugeno (TS) fuzzy models has been recognized as a powerful approximation tool when dealing with complex and ill-determined systems [3–6].This model-based approach consists in modeling the dynamic of a complex system as a convex sum of the dynamics of several subsystems where each subsystem results from the linearization of the initial nonlinear system around a point of the state space.Obviously, we live in an era where technology witnesses a rapid evolution. Technological advances lead inevitably to an increasing complexity of systems and thus, new tools and concepts need to be developed in order to solve efficiently the problem of analysis/synthesis of such systems. Multiple nonlinear systems, switched nonlinear systems, and nonholonomic systems for instance are part of these complicated real-world systems whose study requirements have given rise to the new concept of TS fuzzy switched systems [7–9]. By definition, a switched fuzzy system is a switched system whose all subsystems are fuzzy subsystems.On the other hand, time delay is an inherent feature of many physical processes and it is well known that this phenomenon may cause the deterioration of the system’s performances and can eventually lead to its instability [10]. Time-delay, whether constant or variable, single or multiple, is unavoidable in many engineering applications including rolling mills, pneumatic and hydraulic systems, aircraft and robotic systems, etc.Time-delay systems can be classified into two types. The first type is called neutral-type systems in which the time-delay argument appears in the derivative of the state variables. They arise mainly in heat exchangers, distributed networks containing lossless transmission lines, etc. The second type, probably the most known, is referred to as retarded-type systems.The problem of stability and stabilization of switched systems in general (this is also the case for TS fuzzy switched systems) is quite difficult. The problem lies in the fact that the asymptotic stability of each individual system does not necessarily imply that of the whole system. Indeed, the nature of the switching signal, i.e., the instants of switching between the different constituent subsystems, plays an important role in the stability of the overall system [11].In this context, three strategies are generally adopted. The first one is based on searching sufficient conditions guaranteeing the asymptotic stability of such systems under an arbitrary switching law [12, 13]. This kind of switching, which is the subject of this paper, is also called random or unconstrained switching. An important result in this area states that it suffices to find a common Lyapunov function for all the subsystems to conclude to the asymptotic stability of the overall system. In the case of time-delay switched systems, a Lyapunov-Krasovskii functional is generally searched yielding delay-dependent or delay-independent stability criteria [14].The second strategy aims at stabilizing the set of modes or subsystems by restricting the class of admissible switching sequences to those in which the interval between any two consecutive switching instants is no smaller than a numberτ called dwell time [15].The last strategy consists in constructing a stabilizing switching law in order to stabilize a set of initially unstable subsystems.Undoubtedly, Lyapunov theory has been at the origin of most of the results on stability of TS fuzzy switched time-delay systems under arbitrary switching strategy. The reported results are thus expressed in terms of Linear Matrix Inequalities (LMIs) [16]. In particular, in the case of TS fuzzy neutral or retarded time-delay switched systems, finding a common Lyapunov-Krasovskii functional for all the fuzzy models is a hard task. In fact, the difficulty arises from the computational complexity over LMIs or even from the problem of inexistence of such a functional.In this paper, we propose to study the stability of TS fuzzy switched time-delay systems under arbitrary switching via the vector-norms approach as an alternative to the Lyapunov method [17–19]. The idea consists in constructing a common pseudo-overvaluing/comparison system that is described by the arrow form matrix. This matrix description permits exploiting the properties of the M-matrices and consequently the application of the Kotelyanski lemma [20]. In the case of nonlinear systems, we recall that this special form is suitable for the application of the Borne and Gentina practical stability criterion since it isolates the nonlinear elements in only one row or column [21].Stability of this common comparison system permits to conclude to that of the original system by means of simple algebraic and practical conditions. The vector-norms approach, applied to switched time-delay systems, has already been introduced in [22–27]. For instance, in [26], the authors have derived stability conditions for nonlinear retarded-type switched time-delay systems under arbitrary switching. Moreover, delay-dependent stability criteria are established in [27] for switched retarded-type systems modeled by Takagi-Sugeno fuzzy models. However, to the best of the authors’ knowledge, the same approach has not been applied yet to switched systems with neutral-type time delay.The remainder of this paper is organized as follows. Section2 gives the description as well as the main results related to the neutral type of the considered class of systems. In Section 3, the retarded type is investigated followed by an extension of the results to the case of systems with multiple delays. Section 4 illustrates the results through two examples. Finally, some concluding remarks are given in Section 5.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. u,v=∑i=1nuivi is defined as the scalar product of vectors u=ui1≤i≤n∈Rn and v=vi1≤i≤n∈Rn. In addition, matrices are assumed to be compatible for algebraic operations, if their dimensions are not explicitly stated. MT and M-1 stand for the transpose and the inverse of matrix M=(mi,j)1≤i,j≤n, 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. Neutral-Type TS Fuzzy Switched Time-Delay Systems Described by Functional Differential Equations

### 2.1. Problem Statement

In this section, we are interested in the autonomous switched systems of neutral type that are described by the following functional differential equation:(1)ynt+∑i=1Nξit∑j=0n-1ai,j.yjt+∑j=0ndi,j.yjt-τ=0,t>0yθ=ϕθ,θ∈-τ,0where y(t)∈Rn, ai,j(.) and di,j(.) are nonlinear coefficients for each i∈S=1,..,N and j=0,..,n, ai,n(.)=1, τ>0 is a constant time delay, ϕ(t)=ϕ1(t),..,ϕn-1(t)T is a continuously differentiable vector-valued initial function, N is the number of subsystems switching among each other, and ξi(t),i∈S is an exogenous function indicating the active subsystem of index i at instant t and defined by(2)ξit=1if  subsystemiis  active0otherwiseA change of variable of the form xj+1(t)=y(j)(t),j=0,..,n-1 allows system (1) to be represented in the state space by(3)x˙jt=xj+1tx˙nt=-∑i=1Nξit∑j=0n-1ai,j.xj+1t+∑j=0n-1dij.xj+1t-τ+di,n.x˙nt-τ=0Therefore, the neutral-type time-delay switched system is described by the following state space representation:(4)x˙t=∑i=1NξitAi.xt+Di.xt-τ+D~i.x˙t-τxθ=ϕθ,θ∈-τ,0or equivalently by(5)x˙t=Aσt.xt+Dσt.xt-τ+D~σt.x˙t-τxθ=ϕθ,θ∈-τ,0where x(t)∈Rn is the state vector, Aσ(t)(.), Dσ(t)(.), and D~σ(t)(.) are matrices of appropriate dimensions, and σ(t):R+→S is the switching signal assumed to be available in real time.MatricesAi(.), Di(.), and D~i(.), i∈S are given by(6)Ai.=010⋯0⋮⋱⋱⋱⋮⋮⋱⋱00⋯⋯01-ai,0.⋯⋯⋯-ai,n-1.,Di.=0……0⋮⋮0……0-di,0.……-di,n-1.and(7)D~i.=0n-1,n-10n-1,10…0-di,n.In addition, each subsystem is described by a set of IF-THEN fuzzy rules where each rule is related to a region of the state space in which the subsystem could be approximated by a local linear model.Thus, thelth rule associated with theith subsystem is given by(8)Rli:IFz1tisMli,1and…andzptisMli,pTHENx˙t=Alixt+Dlixt-τ+D~lix˙t-τxθ=ϕθ,θ∈-τ,0where Mli,j,l∈R=1,..,r,i∈S,j=1,..,p are the fuzzy sets, z1(t),..,zp(t) are the premise variables, Ali, Dli, and D~li are matrices of appropriate dimensions, and r and p are the number of fuzzy rules and premise variables, respectively.For each fuzzy ruleRli, we attribute a weighting factor ωli(z(t)) which depends on the degree of membership of zj(t) to the fuzzy sets Mli,j (denoted by Mli,j(zj(t))) as well as on the choice of the conjunctive operator “and”. The latter, usually considered as the product operator, is computed as follows:(9)ωlizt=∏j=1pMli,jzjtThe final output of the switched system is then inferred as(10)x˙t=∑i=1N∑l=1rξithliztAlixt+Dlixt-τ+D~lix˙t-τwhere(11)hlizt=ωlizt∑l=1rωliztMli,j(zj(t)) is the firing strength of membership function Mli,j. It is assumed that ωli(zj(t))≥0 and ∑l=1rωli(zj(t))>0. hli(z(t)) are thus the normalized weighting functions satisfying the properties of convex sums such that hli(z(t))≥0 and ∑l=1rhli(z(t))=1.MatricesAli, Dli, and D~li are given by(12)Ali=010⋯0⋮⋱⋱⋱⋮⋮⋱⋱00⋯⋯01-ai,0⋯⋯⋯-ai,n-1,Dli=0……0⋮⋮0……0-dli,0……-dli,n-1and(13)D~li=0n-1,n-10n-1,10…0-dli,nRelatively to the regular vector norm p(x)=x1,..,xi,..,xnT, system (10) admits the following comparison system:(14)z˙t=Alim∗zt+Dli,mmzt-τ+D~li,mmz˙t-τ


### 2.2. Preliminaries

In this subsection, we recall some of the definitions and remarks that will be useful throughout the paper.Kotelyanski Lemma (see [20]). The real parts of the eigenvalues of matrix A(.), with nonnegative off-diagonal elements, are less than a number μ if and only if those of matrix M(.); M(.)=μIn-A(.) are positive, with In the n×n identity matrix.Definition 1 (see [21]).
The matrixA(.)=ai,j(.)1≤i,j≤n is called an M-matrix, if the following conditions are met:(i)
ai,i(.)>0,∀i=1,..,nandai,j(.)≤0∀i≠j,i,j=1,..,n(ii)
The principal minors ofA(.) are all positive(15)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 sign-alternate (the first is negative) and the Kotelyanski lemma permits to conclude to the stability of the system characterized by A(.).Definition 4.
A dynamic systemA is said to be a comparison system to another dynamic system B according to a specific property P (for instance, the stability of the origin), if the verification of the property P for system A implies the same property for system B.Definition 5 (see [19, 21]).
IfMc(.) is a pseudo-overvaluing matrix of the system x˙(t)=A(.)x(t) with respect to the vector norm p(x)=x1(t),..,xi(t),..,xn(t)T, the following inequality is satisfied:(16)D+px≤Mc.pxwhere D+ denotes the right hand derivative operator.Consequently, the stability of the comparison systemz˙(t)=Mc(.)z(t) with initial conditions z0=p(x0) implies the same property for the initial system.


### 2.3. Stability Analysis

In the rest of the paper, for any set of matricesMli=(mj,k(li))1≤j,k≤n∈Rn×n,l∈R and i∈S, we denote by Mlim- the maximum value of Mli such that(17)Mlim=maxl∈R,i∈S⁡Mli=maxl∈R,i∈S⁡m1,1li⋯maxl∈R,i∈S⁡m1,nli⋮⋮maxl∈R,i∈S⁡mn,1li⋯maxl∈R,i∈S⁡mn,nliand by Mlim∗ the following matrix:(18)Mlim∗=maxl∈R,i∈S⁡Mli∗=maxl∈R,i∈S⁡m1,1li⋯maxl∈R,i∈S⁡m1,nli⋮⋮maxl∈R,i∈S⁡mn,1li⋯maxl∈R,i∈S⁡mn,nliTheorem 6.
System (10) is asymptotically stable under an arbitrary switching law σ(t)=i, i∈S if the matrix Tm is the opposite of an M-matrix, with(19)Tm=Alim∗+11-dli,nmDlim+D~limAlimand dli,n∈-1,1,∀l∈R and i∈S.Proof.
Letw∈R+∗n be a strictly positive vector (wm>0,∀m=1,..,n) and choose a common radially unbounded candidate Lyapunov functional for all the fuzzy model as follows:(20)Vxt=V1xt+V2xt+V3xtwith(21)V1xt=xt,w(22)V2xt=11-dli,nmDlim∫t-τtxθdθ,wand(23)V3xt=11-dli,nmD~lim∫t-τtx˙θdθ,wIt is clear that V(x(t0))<∞.
The right derivative ofV(x(t)) along the trajectory of system (10) is expressed as follows:(24)D+Vxt10=∑j=13D+Vjxt10where(25)D+V1xt10=d+xtdt+,w=sgn⁡xtd+xtdt+,wwith(26)sgn⁡xt=sgn⁡x1t⋱sgn⁡xntThen,(27)D+V1xt10=sgn⁡xt∑i=1N∑l=1rξithliztAlixt+Dlixt-τ+D~lix˙t-τ,w≤Alim∗xt,w+Dlimxt-τ,w+D~limx˙t-τ,wAlso, we have(28)D+V2xt10=11-dli,nmDlimxt,w-11-dli,nmDlimxt-τ,wand(29)D+V3xt10=11-dli,nmD~limx˙t,w-11-dli,nmD~limx˙t-τ,w≤11-dli,nmD~limAlimxt,w+11-dli,nmD~limDlimxt-τ,w+11-dli,nmD~lim2x˙t-τ,w-11-dli,nmD~limx˙t-τ,wNotice that D~limDlim=dli,nmDlim and D~lim2=dli,nmD~lim. This permits the simplification of (24) through the following expressions:(30)D~limx˙t-τ,w+dli,nm1-dli,nmD~limx˙t-τ,w-11-dli,nmD~limx˙t-τ,w=0Similarly,(31)Dlixt-τ,w-11-dli,nDlixt-τ,w+dli,n1-dli,nDlixt-τ,w=0Finally, we obtain(32)D+Vxt10≤Alim∗+11-dli,nmDlim+D~limAlimxt,w=Tmxt,wwhere Tli, l∈R, and i∈S is the minimal pseudo-overvaluing matrix for subsystem li given by(33)Tli=Ali∗+11-dli,nDli+D~liAliand Tm is a common pseudo-overvaluing matrix already defined in (19).
On the other hand, assume thatTm is the opposite of an M-matrix and according to the M-matrices properties, we can find a vector ρ∈R+∗n, (ρs>0,∀s=1,..,n) such that TmTw=-ρ,∀w∈R+∗n. Therefore,(34)D+Vxt10≤TmTw,xt=-ρ,xt=-∑s=1nρsxsk<0This completes the proof of Theorem 6.Now, to apply Theorem6, a system’s description by a special matrix form, called the arrow form matrix, can be considered useful. Indeed, a change of base of the form v(t)=Px(t) of system (10) leads to the new state space representation:(35)v˙t=∑i=1N∑l=1rξithliztFlivt+Glivt-τ+G~liv˙t-τvθ=Pxθ,θ∈-τ,0where matrices Fli, Gli, and G~li,l∈R and i∈S are in the arrow form and determined as follows:(36)Fli=P-1AliP=α1β1⋱⋮αn-1βn-1fli,1⋯fli,n-1fli,n,(37)Gli=P-1DliP=0n-1,n-10n-1,1gli,1…gli,n-1gli,n,and(38)G~li=P-1D~liP=0n-1,n-10n-1,1g~li,1…g~li,n-1g~li,nP is the corresponding passage matrix given by(39)P=11⋯10α1α2⋯αn-10α12α22⋯αn-12⋮⋮⋮⋯⋮0α1n-1α2n-1⋯αn-1n-11The elements of matrices Fli,l∈R and i∈S are computed by(40)βj=∏q=1q≠jn-1αj-αq-1∀j=1,..,n-1fli,j=-PAliαj∀j=1,..,n-1fli,n=-ali,n-1-∑j=1n-1αjwhereas the elements of matrices Gli and G~li, l∈R and i∈S are defined, respectively, by(41)gli,j=-PDliαj∀j=1,..,n-1gli,n=-dli,n-1and(42)g~li,j=-PD~liαj=-dli,nαj∀j=1,..,n-1g~li,n=-dli,nNote that αj,j=1,..,n-1 are distinct constant parameters that are chosen arbitrarily and that PAli(λ), PDli(λ), and PD~li(λ) denote the respective instantaneous characteristic polynomials of matrices Ali, Dli, and D~li such that(43)PAliλ=λn+∑j=0n-1ali,jλjand(44)PDliλ=∑j=0n-1dli,jλjPD~liλ=dli,nλThe computation of G~limFlim gives(45)G~limFlim=0n-1,n-10n-1,1g~li,1mα1+-dli,nmfli,1m…g~li,n-1mαn-1+-dli,nmfli,n-1m∑j=1n-1g~li,jmβj+-dli,nmfli,nmNotice that G~limFlim can be simplified using the fact that traceG~limFlim=traceD~limAlim. Hence, ∑j=1n-1g~li,jmβj+-dli,nmfli,nm=dli,nmali,n-1m. Therefore, for l∈R and i∈S, it is possible to define for all the fuzzy models the following comparison system:(46)y˙t=Tmytsuch that the arrow from pseudo-overvaluing matrices Tm is given by(47)Tm=α1β1⋱⋮αn-1βn-1tm,1…tm,n-1tm,nwith(48)tm,j=fli,jm+11-dli,nmgli,jm+g~li,jmαj+-dli,nmfli,j,j=1,..,n-1tm,n=fli,nm+11-dli,nm-dli,n-1m+dli,nmali,n-1mAt this level, we can state the following theorem.Theorem 7.
System (10) is asymptotically stable under an arbitrary switching law σ(t)=i,i∈S, if there exist αj<0,j=1,..,n-1, αj≠αq,∀j≠q such that the following inequality is satisfied:(49)-tm,n+∑j=1n-1tm,jβjαj-1>0Proof.
The common pseudo-overvaluing matrix has the form:(50)Tm=α1β1⋱⋮αn-1βn-1tm,1…tm,n-1tm,nSince αj<0,∀j=1,..,n-1, it suffices to check that(51)-1ndet⁡Tm>0to conclude to the asymptotic stability of system (10).
Knowing that the determinant of the arrow form matrixTm is computed as follows:(52)det⁡Tm=tm,n-∑j=1n-1tm,jβjαj-1∏k=1n-1αkand by denoting κ=-1n-1∏k=1n-1αk=∏k=1n-1(-αk)>0, we can write(53)-tm,n+∑j=1n-1tm,jβjαj-1×κ>0Hence, we deduce relation (49).
This achieves the proof of Theorem7.Remark 8.
To study the asymptotic stability of each individual fuzzy model separately, it is obvious that it suffices to verify for eachl∈R and i∈S that(54)-tli,n+∑j=1n-1tli,jβjαj-1>0where(55)tli,j=fli,j+11-dli,ngli,j+g~li,jαj+-dli,nfli,j,j=1,..,n-1tli,n=fli,n+11-dli,n-dli,n-1+dli,nali,n-1Remark 9.
Relation (54) can be widely simplified if some further conditions are met. We give, for instance, two situations in which dli,n-1<0.Situation 10.
If the following inequalities hold:(56)PAliαjβj<0(57)PDliαjβj<0(58)PD~liαjαjβj>0and(59)0<dli,n<1condition (54) is reduced to(60)PAli0+PDli0-2κdli,nali,n-1>0Situation 11.
If in addition to (56) and (57), we have(61)PD~liαjαjβj<0and(62)-1<dli,n<0then, the asymptotic stability condition becomes(63)PAli0+PDli0+2κdli,nali,n-1>0Proof.
See the Appendix.Remark 12.
Relations (60) and (63) are considered useful when the common comparison system is identical to one of the fuzzy models.


## 2.1. Problem Statement

In this section, we are interested in the autonomous switched systems of neutral type that are described by the following functional differential equation:(1)ynt+∑i=1Nξit∑j=0n-1ai,j.yjt+∑j=0ndi,j.yjt-τ=0,t>0yθ=ϕθ,θ∈-τ,0where y(t)∈Rn, ai,j(.) and di,j(.) are nonlinear coefficients for each i∈S=1,..,N and j=0,..,n, ai,n(.)=1, τ>0 is a constant time delay, ϕ(t)=ϕ1(t),..,ϕn-1(t)T is a continuously differentiable vector-valued initial function, N is the number of subsystems switching among each other, and ξi(t),i∈S is an exogenous function indicating the active subsystem of index i at instant t and defined by(2)ξit=1if  subsystemiis  active0otherwiseA change of variable of the form xj+1(t)=y(j)(t),j=0,..,n-1 allows system (1) to be represented in the state space by(3)x˙jt=xj+1tx˙nt=-∑i=1Nξit∑j=0n-1ai,j.xj+1t+∑j=0n-1dij.xj+1t-τ+di,n.x˙nt-τ=0Therefore, the neutral-type time-delay switched system is described by the following state space representation:(4)x˙t=∑i=1NξitAi.xt+Di.xt-τ+D~i.x˙t-τxθ=ϕθ,θ∈-τ,0or equivalently by(5)x˙t=Aσt.xt+Dσt.xt-τ+D~σt.x˙t-τxθ=ϕθ,θ∈-τ,0where x(t)∈Rn is the state vector, Aσ(t)(.), Dσ(t)(.), and D~σ(t)(.) are matrices of appropriate dimensions, and σ(t):R+→S is the switching signal assumed to be available in real time.MatricesAi(.), Di(.), and D~i(.), i∈S are given by(6)Ai.=010⋯0⋮⋱⋱⋱⋮⋮⋱⋱00⋯⋯01-ai,0.⋯⋯⋯-ai,n-1.,Di.=0……0⋮⋮0……0-di,0.……-di,n-1.and(7)D~i.=0n-1,n-10n-1,10…0-di,n.In addition, each subsystem is described by a set of IF-THEN fuzzy rules where each rule is related to a region of the state space in which the subsystem could be approximated by a local linear model.Thus, thelth rule associated with theith subsystem is given by(8)Rli:IFz1tisMli,1and…andzptisMli,pTHENx˙t=Alixt+Dlixt-τ+D~lix˙t-τxθ=ϕθ,θ∈-τ,0where Mli,j,l∈R=1,..,r,i∈S,j=1,..,p are the fuzzy sets, z1(t),..,zp(t) are the premise variables, Ali, Dli, and D~li are matrices of appropriate dimensions, and r and p are the number of fuzzy rules and premise variables, respectively.For each fuzzy ruleRli, we attribute a weighting factor ωli(z(t)) which depends on the degree of membership of zj(t) to the fuzzy sets Mli,j (denoted by Mli,j(zj(t))) as well as on the choice of the conjunctive operator “and”. The latter, usually considered as the product operator, is computed as follows:(9)ωlizt=∏j=1pMli,jzjtThe final output of the switched system is then inferred as(10)x˙t=∑i=1N∑l=1rξithliztAlixt+Dlixt-τ+D~lix˙t-τwhere(11)hlizt=ωlizt∑l=1rωliztMli,j(zj(t)) is the firing strength of membership function Mli,j. It is assumed that ωli(zj(t))≥0 and ∑l=1rωli(zj(t))>0. hli(z(t)) are thus the normalized weighting functions satisfying the properties of convex sums such that hli(z(t))≥0 and ∑l=1rhli(z(t))=1.MatricesAli, Dli, and D~li are given by(12)Ali=010⋯0⋮⋱⋱⋱⋮⋮⋱⋱00⋯⋯01-ai,0⋯⋯⋯-ai,n-1,Dli=0……0⋮⋮0……0-dli,0……-dli,n-1and(13)D~li=0n-1,n-10n-1,10…0-dli,nRelatively to the regular vector norm p(x)=x1,..,xi,..,xnT, system (10) admits the following comparison system:(14)z˙t=Alim∗zt+Dli,mmzt-τ+D~li,mmz˙t-τ


## 2.2. Preliminaries

In this subsection, we recall some of the definitions and remarks that will be useful throughout the paper.Kotelyanski Lemma (see [20]). The real parts of the eigenvalues of matrix A(.), with nonnegative off-diagonal elements, are less than a number μ if and only if those of matrix M(.); M(.)=μIn-A(.) are positive, with In the n×n identity matrix.Definition 1 (see [21]).
The matrixA(.)=ai,j(.)1≤i,j≤n is called an M-matrix, if the following conditions are met:(i)
ai,i(.)>0,∀i=1,..,nandai,j(.)≤0∀i≠j,i,j=1,..,n(ii)
The principal minors ofA(.) are all positive(15)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 sign-alternate (the first is negative) and the Kotelyanski lemma permits to conclude to the stability of the system characterized by A(.).Definition 4.
A dynamic systemA is said to be a comparison system to another dynamic system B according to a specific property P (for instance, the stability of the origin), if the verification of the property P for system A implies the same property for system B.Definition 5 (see [19, 21]).
IfMc(.) is a pseudo-overvaluing matrix of the system x˙(t)=A(.)x(t) with respect to the vector norm p(x)=x1(t),..,xi(t),..,xn(t)T, the following inequality is satisfied:(16)D+px≤Mc.pxwhere D+ denotes the right hand derivative operator.Consequently, the stability of the comparison systemz˙(t)=Mc(.)z(t) with initial conditions z0=p(x0) implies the same property for the initial system.


## 2.3. Stability Analysis

In the rest of the paper, for any set of matricesMli=(mj,k(li))1≤j,k≤n∈Rn×n,l∈R and i∈S, we denote by Mlim- the maximum value of Mli such that(17)Mlim=maxl∈R,i∈S⁡Mli=maxl∈R,i∈S⁡m1,1li⋯maxl∈R,i∈S⁡m1,nli⋮⋮maxl∈R,i∈S⁡mn,1li⋯maxl∈R,i∈S⁡mn,nliand by Mlim∗ the following matrix:(18)Mlim∗=maxl∈R,i∈S⁡Mli∗=maxl∈R,i∈S⁡m1,1li⋯maxl∈R,i∈S⁡m1,nli⋮⋮maxl∈R,i∈S⁡mn,1li⋯maxl∈R,i∈S⁡mn,nliTheorem 6.
System (10) is asymptotically stable under an arbitrary switching law σ(t)=i, i∈S if the matrix Tm is the opposite of an M-matrix, with(19)Tm=Alim∗+11-dli,nmDlim+D~limAlimand dli,n∈-1,1,∀l∈R and i∈S.Proof.
Letw∈R+∗n be a strictly positive vector (wm>0,∀m=1,..,n) and choose a common radially unbounded candidate Lyapunov functional for all the fuzzy model as follows:(20)Vxt=V1xt+V2xt+V3xtwith(21)V1xt=xt,w(22)V2xt=11-dli,nmDlim∫t-τtxθdθ,wand(23)V3xt=11-dli,nmD~lim∫t-τtx˙θdθ,wIt is clear that V(x(t0))<∞.
The right derivative ofV(x(t)) along the trajectory of system (10) is expressed as follows:(24)D+Vxt10=∑j=13D+Vjxt10where(25)D+V1xt10=d+xtdt+,w=sgn⁡xtd+xtdt+,wwith(26)sgn⁡xt=sgn⁡x1t⋱sgn⁡xntThen,(27)D+V1xt10=sgn⁡xt∑i=1N∑l=1rξithliztAlixt+Dlixt-τ+D~lix˙t-τ,w≤Alim∗xt,w+Dlimxt-τ,w+D~limx˙t-τ,wAlso, we have(28)D+V2xt10=11-dli,nmDlimxt,w-11-dli,nmDlimxt-τ,wand(29)D+V3xt10=11-dli,nmD~limx˙t,w-11-dli,nmD~limx˙t-τ,w≤11-dli,nmD~limAlimxt,w+11-dli,nmD~limDlimxt-τ,w+11-dli,nmD~lim2x˙t-τ,w-11-dli,nmD~limx˙t-τ,wNotice that D~limDlim=dli,nmDlim and D~lim2=dli,nmD~lim. This permits the simplification of (24) through the following expressions:(30)D~limx˙t-τ,w+dli,nm1-dli,nmD~limx˙t-τ,w-11-dli,nmD~limx˙t-τ,w=0Similarly,(31)Dlixt-τ,w-11-dli,nDlixt-τ,w+dli,n1-dli,nDlixt-τ,w=0Finally, we obtain(32)D+Vxt10≤Alim∗+11-dli,nmDlim+D~limAlimxt,w=Tmxt,wwhere Tli, l∈R, and i∈S is the minimal pseudo-overvaluing matrix for subsystem li given by(33)Tli=Ali∗+11-dli,nDli+D~liAliand Tm is a common pseudo-overvaluing matrix already defined in (19).
On the other hand, assume thatTm is the opposite of an M-matrix and according to the M-matrices properties, we can find a vector ρ∈R+∗n, (ρs>0,∀s=1,..,n) such that TmTw=-ρ,∀w∈R+∗n. Therefore,(34)D+Vxt10≤TmTw,xt=-ρ,xt=-∑s=1nρsxsk<0This completes the proof of Theorem 6.Now, to apply Theorem6, a system’s description by a special matrix form, called the arrow form matrix, can be considered useful. Indeed, a change of base of the form v(t)=Px(t) of system (10) leads to the new state space representation:(35)v˙t=∑i=1N∑l=1rξithliztFlivt+Glivt-τ+G~liv˙t-τvθ=Pxθ,θ∈-τ,0where matrices Fli, Gli, and G~li,l∈R and i∈S are in the arrow form and determined as follows:(36)Fli=P-1AliP=α1β1⋱⋮αn-1βn-1fli,1⋯fli,n-1fli,n,(37)Gli=P-1DliP=0n-1,n-10n-1,1gli,1…gli,n-1gli,n,and(38)G~li=P-1D~liP=0n-1,n-10n-1,1g~li,1…g~li,n-1g~li,nP is the corresponding passage matrix given by(39)P=11⋯10α1α2⋯αn-10α12α22⋯αn-12⋮⋮⋮⋯⋮0α1n-1α2n-1⋯αn-1n-11The elements of matrices Fli,l∈R and i∈S are computed by(40)βj=∏q=1q≠jn-1αj-αq-1∀j=1,..,n-1fli,j=-PAliαj∀j=1,..,n-1fli,n=-ali,n-1-∑j=1n-1αjwhereas the elements of matrices Gli and G~li, l∈R and i∈S are defined, respectively, by(41)gli,j=-PDliαj∀j=1,..,n-1gli,n=-dli,n-1and(42)g~li,j=-PD~liαj=-dli,nαj∀j=1,..,n-1g~li,n=-dli,nNote that αj,j=1,..,n-1 are distinct constant parameters that are chosen arbitrarily and that PAli(λ), PDli(λ), and PD~li(λ) denote the respective instantaneous characteristic polynomials of matrices Ali, Dli, and D~li such that(43)PAliλ=λn+∑j=0n-1ali,jλjand(44)PDliλ=∑j=0n-1dli,jλjPD~liλ=dli,nλThe computation of G~limFlim gives(45)G~limFlim=0n-1,n-10n-1,1g~li,1mα1+-dli,nmfli,1m…g~li,n-1mαn-1+-dli,nmfli,n-1m∑j=1n-1g~li,jmβj+-dli,nmfli,nmNotice that G~limFlim can be simplified using the fact that traceG~limFlim=traceD~limAlim. Hence, ∑j=1n-1g~li,jmβj+-dli,nmfli,nm=dli,nmali,n-1m. Therefore, for l∈R and i∈S, it is possible to define for all the fuzzy models the following comparison system:(46)y˙t=Tmytsuch that the arrow from pseudo-overvaluing matrices Tm is given by(47)Tm=α1β1⋱⋮αn-1βn-1tm,1…tm,n-1tm,nwith(48)tm,j=fli,jm+11-dli,nmgli,jm+g~li,jmαj+-dli,nmfli,j,j=1,..,n-1tm,n=fli,nm+11-dli,nm-dli,n-1m+dli,nmali,n-1mAt this level, we can state the following theorem.Theorem 7.
System (10) is asymptotically stable under an arbitrary switching law σ(t)=i,i∈S, if there exist αj<0,j=1,..,n-1, αj≠αq,∀j≠q such that the following inequality is satisfied:(49)-tm,n+∑j=1n-1tm,jβjαj-1>0Proof.
The common pseudo-overvaluing matrix has the form:(50)Tm=α1β1⋱⋮αn-1βn-1tm,1…tm,n-1tm,nSince αj<0,∀j=1,..,n-1, it suffices to check that(51)-1ndet⁡Tm>0to conclude to the asymptotic stability of system (10).
Knowing that the determinant of the arrow form matrixTm is computed as follows:(52)det⁡Tm=tm,n-∑j=1n-1tm,jβjαj-1∏k=1n-1αkand by denoting κ=-1n-1∏k=1n-1αk=∏k=1n-1(-αk)>0, we can write(53)-tm,n+∑j=1n-1tm,jβjαj-1×κ>0Hence, we deduce relation (49).
This achieves the proof of Theorem7.Remark 8.
To study the asymptotic stability of each individual fuzzy model separately, it is obvious that it suffices to verify for eachl∈R and i∈S that(54)-tli,n+∑j=1n-1tli,jβjαj-1>0where(55)tli,j=fli,j+11-dli,ngli,j+g~li,jαj+-dli,nfli,j,j=1,..,n-1tli,n=fli,n+11-dli,n-dli,n-1+dli,nali,n-1Remark 9.
Relation (54) can be widely simplified if some further conditions are met. We give, for instance, two situations in which dli,n-1<0.Situation 10.
If the following inequalities hold:(56)PAliαjβj<0(57)PDliαjβj<0(58)PD~liαjαjβj>0and(59)0<dli,n<1condition (54) is reduced to(60)PAli0+PDli0-2κdli,nali,n-1>0Situation 11.
If in addition to (56) and (57), we have(61)PD~liαjαjβj<0and(62)-1<dli,n<0then, the asymptotic stability condition becomes(63)PAli0+PDli0+2κdli,nali,n-1>0Proof.
See the Appendix.Remark 12.
Relations (60) and (63) are considered useful when the common comparison system is identical to one of the fuzzy models.


## 3. Retarded-Type TS Fuzzy Switched Systems

### 3.1. Stability Analysis

In the special casedli,n=0,l∈R,i∈S, it is obvious that system (10) becomes(64)x˙t=∑i=1N∑l=1rξithliztAlixt+Dlixt-τTheorem 13.
System (64) is asymptotically stable under an arbitrary switching rule σ(t)=i,i∈S, if the matrix Tm,r is the opposite of an M-matrix, with(65)Tm,r=Alim∗+DlimProof.
It suffices to choose the following radially unbounded Lyapunov functional:(66)Vxt=xt,w+Dlim∫t-τtxθdθ,wIn this case, all the results of Section 2 are extended to retarded-type TS fuzzy switched systems by replacing dli,n by 0.


### 3.2. Generalization of the Result to the Case of Retarded-Type Systems with Multiple Delay

Consider the class of TS fuzzy switched time-delay systems that are represented by(67)x˙t=∑i=1N∑l=1rξithliztAlixt+∑k=1MDli,kxt-τkxθ=ϕθ,θ∈-max1≤k≤M⁡τk,0where Ali, Dli,k, l∈R, i∈S and k∈1,..,M are matrices of appropriate dimensions, ϕ(t) is a continuous vector-valued function specifying the initial state of the system, τk>0 denotes the multiple delay, and ξi(t) and hli(z(t)) are already defined in (2) and (11), respectively.Delay-independent sufficient stability conditions generalized to systems with multiple delays are given in the following theorem.Theorem 14.
System (67) is asymptotically stable under an arbitrary switching rule σ(t)=i, i∈S, if matrix Tm,rM is the opposite of an M-matrix, with(68)Tm,rM=Alim∗+∑k=1MDli,kmProof.
In this case, choosing a candidate radially unbounded Lyapunov functional of the form(69)Vxt=xt,w+∑k=1MDli,km∫t-τktxθdθ,wand following the same steps as in the proof of Theorem 6 yields the common comparison system given by y˙(t)=Tm,rMy(t).


### 3.3. Stabilization of TS Fuzzy Switched Systems via Delayed PDC Control

Consider the TS fuzzy switched time-delay system given as above:(70)x˙t=∑i=1N∑l=1rξithliztAlixt+Bliut-τwhere matrices Ali and Bli are in the companion form as in (12) and the input matrix Bli=B=0,..,1T is common for all the fuzzy models.Thelth fuzzy rule of the PDC controller stabilizing the fuzzy model of index li is given by(71)Rli:IFz1tisMli,1and..andzptisMli,pTHENut=-Klixtwhere Kli=kli,0,..,kli,n-1,l∈R and i∈S is the local feedback gain vector. We recall that zj(t) and Mli,j,j=1,..,p are, respectively, the premise variables and the fuzzy sets.This leads to(72)x˙t=∑i=1N∑l=1rξithliAlixt-∑q=1rhqiBKqixt-τwhich can be written in the case B=0,..,1T as follows:(73)x˙t=∑i=1N∑l=1rξithliAlixt-BKlixt-τDenoting D~li=-BKli=0n-1,n-10n-1,1-kli,0⋯-kli,n-1,  l∈R and i∈S, the problem is then reduced to determining admissible values of parameters kli,0,..,kli,n-1 ensuring the asymptotic stability of system (73).


## 3.1. Stability Analysis

In the special casedli,n=0,l∈R,i∈S, it is obvious that system (10) becomes(64)x˙t=∑i=1N∑l=1rξithliztAlixt+Dlixt-τTheorem 13.
System (64) is asymptotically stable under an arbitrary switching rule σ(t)=i,i∈S, if the matrix Tm,r is the opposite of an M-matrix, with(65)Tm,r=Alim∗+DlimProof.
It suffices to choose the following radially unbounded Lyapunov functional:(66)Vxt=xt,w+Dlim∫t-τtxθdθ,wIn this case, all the results of Section 2 are extended to retarded-type TS fuzzy switched systems by replacing dli,n by 0.


## 3.2. Generalization of the Result to the Case of Retarded-Type Systems with Multiple Delay

Consider the class of TS fuzzy switched time-delay systems that are represented by(67)x˙t=∑i=1N∑l=1rξithliztAlixt+∑k=1MDli,kxt-τkxθ=ϕθ,θ∈-max1≤k≤M⁡τk,0where Ali, Dli,k, l∈R, i∈S and k∈1,..,M are matrices of appropriate dimensions, ϕ(t) is a continuous vector-valued function specifying the initial state of the system, τk>0 denotes the multiple delay, and ξi(t) and hli(z(t)) are already defined in (2) and (11), respectively.Delay-independent sufficient stability conditions generalized to systems with multiple delays are given in the following theorem.Theorem 14.
System (67) is asymptotically stable under an arbitrary switching rule σ(t)=i, i∈S, if matrix Tm,rM is the opposite of an M-matrix, with(68)Tm,rM=Alim∗+∑k=1MDli,kmProof.
In this case, choosing a candidate radially unbounded Lyapunov functional of the form(69)Vxt=xt,w+∑k=1MDli,km∫t-τktxθdθ,wand following the same steps as in the proof of Theorem 6 yields the common comparison system given by y˙(t)=Tm,rMy(t).


## 3.3. Stabilization of TS Fuzzy Switched Systems via Delayed PDC Control

Consider the TS fuzzy switched time-delay system given as above:(70)x˙t=∑i=1N∑l=1rξithliztAlixt+Bliut-τwhere matrices Ali and Bli are in the companion form as in (12) and the input matrix Bli=B=0,..,1T is common for all the fuzzy models.Thelth fuzzy rule of the PDC controller stabilizing the fuzzy model of index li is given by(71)Rli:IFz1tisMli,1and..andzptisMli,pTHENut=-Klixtwhere Kli=kli,0,..,kli,n-1,l∈R and i∈S is the local feedback gain vector. We recall that zj(t) and Mli,j,j=1,..,p are, respectively, the premise variables and the fuzzy sets.This leads to(72)x˙t=∑i=1N∑l=1rξithliAlixt-∑q=1rhqiBKqixt-τwhich can be written in the case B=0,..,1T as follows:(73)x˙t=∑i=1N∑l=1rξithliAlixt-BKlixt-τDenoting D~li=-BKli=0n-1,n-10n-1,1-kli,0⋯-kli,n-1,  l∈R and i∈S, the problem is then reduced to determining admissible values of parameters kli,0,..,kli,n-1 ensuring the asymptotic stability of system (73).


## 4. Numerical Examples

Example 15.
Consider a switched system composed of two subsystems where each one is represented by two fuzzy models as follows:(74)x˙t=∑i=12∑l=12ξithliAlixt+Dlixt-τ+D~lix˙t-τxθ=ϕθ,θ∈-τ,0where(75)A11,D11,D~11=01-15-20,001.50.1,0000.17A21,D21,D~21=01-10.5-15.75,00-0.9-0.7,0000.08and(76)A12,D12,D~12=01-9-15.5,00-0.061.2,0000.15A22,D22,D~22=01-10-16,0021,0000.15A transformation of matrices Ali, Dli, and D~li under the arrow form Fli=P-1AliP, Gli=P-1DliP and G~li=P-1D~liP (l,i∈1,2) with P=10-11 gives(77)F11,G11,G~11=-114-19,001.40.1,00-0.170.17F21,G21,G~21=-114.25-14.75,00-0.2-0.7,00-0.080.08and(78)F12,G12,G~12=-115.5-14.5,00-1.261.2,00-0.150.15F22,G22,G~22=-115-15,0011,00-0.150.15Consequently, the common pseudo-overvaluing matrix Tm has the form below:(79)Tm=-115.5-14.5+11-0.17001.41.2+000.170.17×005.519and it is calculated as follows:(80)Tm=-118.518-8.957Since Tm is the opposite of an M-matrix, we can conclude that the overall TS fuzzy switched system is asymptotically stable under an arbitrary switching signal and independently of the time delay.For the simulation, we choose for instance the fuzzy weighting factorsh11=0.2, h21=0.8, h12=0.4, and h22=0.6, a time-delay τ equal to 1.2s and an initial state vector x0=1-2T, ∀t∈-1.2,0. The switching signal is plotted on Figure 1, whereas the evolution of the state variables and the phase space are illustrated, respectively, on Figures 2 and 3.Figure 1
Switching signal.Figure 2
Evolution of the state variables.Figure 3
Phase space.The simulation results show that, for a random arbitrary sequence, the TS fuzzy switched system is asymptotically stable regardless of the time delay.Example 16.
Consider the following TS fuzzy switched system of retarded type:(81)x˙t=∑i=12∑l=12ξithliAlixt-BKlixt-τwhere(82)A11,-BK11=01-10-13.5,00-k11,0-k11,1A21,-BK21=01-5.5-12,00-k21,0-k21,1and(83)A12,-BK12=01-4-9,00-k12,0-k12,1A22,-BK22=01-3-11.5,00-k22,0-k22,1For the simplest choice α=-1, a transformation under the arrow form matrix Fli=P-1AliP and Gli=P-1-BKliP yields(84)F11,G11=-112.5-12.5,00k11,1-k11,0-k11,1F21,G21=-115.5-11,00k21,1-k21,0-k21,1and(85)F12,G12=-116-10,00k12,1-k12,0-k12,1F22,G22=-117-10.5,00k22,1-k22,0-k22,1Next, we can derive sufficient delay-independent criteria ensuring the asymptotic stability of each fuzzy model ∑Ali,-BKli by computing the determinant of matrices Tli,r such that(86)det⁡Tli,r=det⁡α1-α2+ali,1α+ali,0+-kli,1α+kli,0-ali,1-α+-kli,1Under the assumption 0<kli,0<kli,1, the condition det⁡Tli,r>0 implies(87)0<kli,1<ali,0+kli,020<kli,0<ali,0In general, when the following assumptions are verified: (i)
PAli(αj)βj<0,∀j=1,..,n-1,(ii)
P-BKli(αj)βj<0,∀j=1,..,n-1,(iii)
kli,n-1>0,
the stability condition is reduced toPAli(0)-2κkli,n-1+P-BKli(0)>0.For instance, for the admissible gain vectorsK11=11.4, K21=1.92.4, K12=1.21.7, and K22=0.81.1, one has(88)Tm=-117.5-7.6Then, we can deduce that the TS fuzzy switched system is asymptotically stable independently of both the switching law and the time delay as the obtained common pseudo-overvaluing is the opposite of an M-matrix.For the simulation, we can consider, for example, the following inputs:h11=0.3, h21=0.7, h12=0.1 and h22=0.9, τ=1.5s and x0=12T, ∀t∈-1.5,0.The switching sequence, the evolution of the state variables, and the phase plane are plotted on Figures4, 5, and 6, respectively.Figure 4
Switching sequence.Figure 5
System’s response.Figure 6
Phase plane.The simulation results show that the values found for the admissible gains of the retarded state feedback control are sufficient to ensure the asymptotic stability of the whole switched system independently of the switching sequence and the time delay.


## 5. Conclusion

In this paper, the stability of a class of time-delay switched systems represented by TS fuzzy models has been investigated. The study has included the neutral type as well as the retarded type.The vector-norms approach, applied to this class of systems and consisting of defining a common comparison system for all the fuzzy models, is well appropriate for the case of arbitrary switching. Making use of the Kotelyanski lemma and the properties of the M-matrices, the comparison approach has also proved to be practical and easy to use and yields to algebraic stability criteria.Compared with the existing results, the main contribution of this work is that it permits to avoid the search of a common Lyapunov-Krasovskii functional which is a difficult matter. Two numerical examples have been provided to illustrate the obtained results.



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