# On Robust Stability Analysis of Uncertain Discrete-Time Switched Nonlinear Systems with Time Varying Delays

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

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## Abstract

This paper provides new sufficient conditions on robust asymptotic stability for a class of uncertain discrete-time switched nonlinear systems with time varying delays. The main focus will be dedicated to development of new algebraic criteria to break with classical criteria in terms of linear matrix inequalities (LMIs). Firstly, by contracting a new common Lyapunov-Krasovskii functional as well as resorting to the M-matrix proprieties, a novel robust stability criterion under arbitrary switching signals is derived. Secondly, the obtained result is extended for a class of switched nonlinear systems modeled by a set of differences equations by applying the aggregation techniques, the norm vector notion, and the Borne-Gentina criterion. Furthermore, a generalization for switched nonlinear systems with multiple delays is proposed. The main contribution of this work is that the obtained stability conditions are algebraic and simple. In addition, they provide a solution of the most difficult problem in switched systems, which is stability under arbitrary switching, and enable avoiding searching a common Lyapunov function considered as a very difficult task even for some low-order linear switched systems. Finally, two examples are given, with numerical simulations, to show the merit and effectiveness of the proposed approach.


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## Body

## 1. Introduction

As a special class of hybrid dynamical systems, switched systems [1] are interestingly used amongst a variety of engineering domains particularly chemical processes, automotive engine control and aircraft control, power systems, power electronics, traffic control, network communications, and many other fields [1–3].From a theoretical point of view, stability represents one of the most significant problems for switched systems. Indeed, it has attracted a growing attention in literature [1, 4–26]. Therefore, stability of switched systems is mainly divided into two aspects: one is how to contract switching laws under which switched systems are stable. In this context, dwell time [4] and average dwell time [5] switching signals methods still play a king role. The second aspect is how to make switched systems stable under arbitrary switching. In this framework, the individual stability of all the subsystems is mandatory. Moreover, the existence of a common Lyapunov function [6, 7] for all subsystems is the unique sufficient condition to ensure stability under arbitrary switching. Unfortunately, getting such a function is a very hard task even for discrete-time switched linear systems. Therefore, this problem becomes more complicated when switched nonlinear systems are involved, and relatively available results in this context are limited [6, 21]. It is worth noting that, in a real frame, switching laws can be unknown, even imposed under a random way. For this, stability under arbitrary switching which will be considered in this paper remains undoubtedly the most interesting issue.On the other hand, time delays especially time varying delays are frequently imposed in diverse real-world engineering systems, which would lead to performance deterioration and, in some cases, it may lead to system malfunction and instability. Consequently, wide efforts have been devoted to address the challenge of switched systems with time varying delays [10, 11, 13, 16, 17, 19, 22–26]. In addition, from the practical standpoint, it is important to tackle uncertain switched systems [18, 19, 23, 25, 26]. Thus, in this investigation, uncertain switched nonlinear systems with time varying delays with polytopic uncertainties type are considered to give a strong practical aspect for this work.Up to now, there are few results concerning stability analysis of uncertain switched nonlinear systems with time varying delays under arbitrary switching [23]. Thus, almost all existing works deal with the linear case or the linearization of the original nonlinear systems by using the TS fuzzy models [19, 20]. It should be noted that in [20] sufficient conditions are derived to ensure the robust stability for discrete-time randomly switched fuzzy systems with known sojourn probabilities, presented in the terms of linear matrix inequalities (LMIs). However, to the best of our knowledge, randomly uncertain switched nonlinear systems with time varying delays have not been largely considered yet.This paper seeks new algebraic practical stability criteria. Firstly, a novel robust asymptotic stability criterion for a class of discrete-time switched nonlinear systems with time varying delays and subject to polytopic uncertainties is established via constructing a new common Lyapunov functional [9], according to the vector norm notion [9–13, 27–31] andM - matrix properties [32]. Secondly, the derived results are extended for a class of switched systems given by a set of difference equations. In fact, new stability conditions are obtained by transforming the considered systems representation under the arrow form matrix [29] and employing the discrete-time Borne and Gentina practical stability criterion [30, 31]. Finally, these proposed results are generalized for a class of switched systems with multiple time varying delays.In contrast with some existing results on underlined filed, the contributions of this paper are twofold. In fact, the obtained results guarantee asymptotic stability of these considered systems under arbitrary switching and may overcome the conservatism of searching a common Lyapunov function and the LMIs constraints. In addition, these stability criteria are expressed in terms of simple algebraic conditions, explicitly, and simple.The rest of this paper is organized as follows. Section2 presents the problem statement and some necessary preliminaries. The main results are proposed in Section 3. New delay-dependent sufficient robust stability conditions for a class of uncertain switched nonlinear systems with time varying delays described by a set of difference equations are given in Section 4. Section 5 generalizes the obtained result for switched systems with multiple delays. Finally, two case studies are presented in Section 6 to show the effectiveness of the provided results. Conclusions are given in Section 7.Notation. The notation used here is fairly standard except where otherwise stated. For a matrix A, we denote the transpose by AT. Let R denote the field of real numbers and Rn denote an n dimensional linear vector space over the reals with the 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, Rn×n the space of n×n matrices with real entries. R+ is the set of positive real numbers. Ik1k2 denotes the set of integers k1,k1+1,k1+2,...,k2 and In is the identity matrix with appropriate dimension.


## 2. Problem Statement and Preliminaries

### 2.1. Problem Statement

Consider the following uncertain discrete-time switched nonlinear system with time varying delays described by(1)xk+1=Aσk.xk+Dσk.xk-τσkkxθ=ϕθs=-r2,-r2+1,…,-1where xk∈Rn denotes the state vector, σk:0,∞→I1L is the switching signal which is a piecewise constant function, L is the number of subsystems, and σk=i implies that the i-th subsystem is activated. τik, i=1,2,...,L are the time varying delays satisfying 0≤r1≤τik≤r2, where r1 and r2 are constant scalars.Therefore, the switched system is composed ofL subsystems which are expressed as(2)xk+1=Ai.xt+Di.xk-τk,i∈I1Lwhere Ai.i∈I1L and Di.i∈I1L are matrices with appropriate dimensions having nonlinear elements.Assuming that all subsystems are uncertain of polytopic type which can be described as(3)Ai.=∑p=1LpμipkAip.,i∈I1L(4)Di.=∑q=1LqλiqkDiq.,i∈I1Lwhere Aip., p∈I1Lp and Diq.q∈I1Lq are, respectively, the vertex matrices denoting the extreme points of the polytope Ai.,i∈I1L and Di.,i∈I1L, Lp is the number of the vertex matrices Ai., Lq is the number of the vertex matrices Di., and the weighting factors μipkp∈I1Lp, λiqkq∈I1Lq are unknown polytopic uncertainties parameters for each i∈I1L belonging to μik:∑p=1Lpμipk=1, μipk≥0 and λiqk: ∑q=1Nqλiqk=1, λiqk≥0.


### 2.2. Preliminaries

Now, the following lemmas, criterion, remark, and definitions are preliminarily presented for later development.Lemma 1 (see [32]).
The matrixA. is said to be an M-matrix if the following properties are verified: (i)
All the eigenvalues ofA. have a positive real part.(ii)
The real eigenvalues are positives.(iii)
The principal minors ofA. are positive:(5)A.12⋯j12⋯j>0,∀j∈I1n.(iv)
For any positive vectorx=x1,...,xnT the algebraic equation A.x admits a positive solution w=w1,...,wnT.In Kotelyanski lemma [33], the real parts of the eigenvalues of the matrix A., with nonnegative off-diagonal elements, are less than a real number μ if and only if all those of matrix M., where M.=μIn-A., are positive, with In the n identity matrix.In this case, all the principal minors of matrix-A. are positive. Then, the Kotelyanski lemma permits deducing on stability properties of the system given by A..Now, we will introduce the discrete-time Borne and Gentina practical stability criterion and the pseudo-overvaluing matrix.In discrete-time Borne and Gentina practical stability criterion [30, 31], let us consider the discrete-time nonlinear system xk+1=A.xk and the overvaluing matrix TM.=aj,k∀j,k=1,...,n. If the nonlinearities are isolated in either one row or one column of TM., the verification of the Kotelyanski condition would enable us to achieve stability conclusion of the original system characterized by A.. The Kotelyanski lemma applied to the overvaluing matrix obtained by the use of the regular vector norm pw=w1,w2,...wn with w=w1,w2,...wn leads to the following sufficient conditions of asymptotic stability of original system: In-TM.12⋯j12⋯j>0j=1,2,...,n.Definition 2 (see [28]).
The matrixTM. is the comparison matrix of the system given by a matrix A. with respect to the vector norm p if the inequality (6)pxk+1≤TM.pxk∀x∈E and k>0 is verified for each corresponding component. Then, the stability of the comparison system zk+1=TM.zk with the initial conditions such as z0=px0.In this case, the following properties are satisfying:(i)
If all the elements ofTM. are nonnegative, it is assumed that the eigenvalue of TM., the biggest in module, is both real and positive and is called main eigenvalue of TM..(ii)
If all the elements of matrixTM. are nonnegative, it is assumed that the principal minors of In-TM. are all positive, the spectral radius of TM. is inferior to the unit, and all the elements of In-TM.-1 are nonnegative.(iii)
WhenTM. is an irreducible matrix, the main eigenvector of TM. is the same as of TM.T and all their elements are nonnegative.Remark 3.
A discrete-time system given by a matrixA. is stable if matrix In-TM. verified the Kotelyanski conditions; in this case In-TM. is an M-matrix.


## 2.1. Problem Statement

Consider the following uncertain discrete-time switched nonlinear system with time varying delays described by(1)xk+1=Aσk.xk+Dσk.xk-τσkkxθ=ϕθs=-r2,-r2+1,…,-1where xk∈Rn denotes the state vector, σk:0,∞→I1L is the switching signal which is a piecewise constant function, L is the number of subsystems, and σk=i implies that the i-th subsystem is activated. τik, i=1,2,...,L are the time varying delays satisfying 0≤r1≤τik≤r2, where r1 and r2 are constant scalars.Therefore, the switched system is composed ofL subsystems which are expressed as(2)xk+1=Ai.xt+Di.xk-τk,i∈I1Lwhere Ai.i∈I1L and Di.i∈I1L are matrices with appropriate dimensions having nonlinear elements.Assuming that all subsystems are uncertain of polytopic type which can be described as(3)Ai.=∑p=1LpμipkAip.,i∈I1L(4)Di.=∑q=1LqλiqkDiq.,i∈I1Lwhere Aip., p∈I1Lp and Diq.q∈I1Lq are, respectively, the vertex matrices denoting the extreme points of the polytope Ai.,i∈I1L and Di.,i∈I1L, Lp is the number of the vertex matrices Ai., Lq is the number of the vertex matrices Di., and the weighting factors μipkp∈I1Lp, λiqkq∈I1Lq are unknown polytopic uncertainties parameters for each i∈I1L belonging to μik:∑p=1Lpμipk=1, μipk≥0 and λiqk: ∑q=1Nqλiqk=1, λiqk≥0.


## 2.2. Preliminaries

Now, the following lemmas, criterion, remark, and definitions are preliminarily presented for later development.Lemma 1 (see [32]).
The matrixA. is said to be an M-matrix if the following properties are verified: (i)
All the eigenvalues ofA. have a positive real part.(ii)
The real eigenvalues are positives.(iii)
The principal minors ofA. are positive:(5)A.12⋯j12⋯j>0,∀j∈I1n.(iv)
For any positive vectorx=x1,...,xnT the algebraic equation A.x admits a positive solution w=w1,...,wnT.In Kotelyanski lemma [33], the real parts of the eigenvalues of the matrix A., with nonnegative off-diagonal elements, are less than a real number μ if and only if all those of matrix M., where M.=μIn-A., are positive, with In the n identity matrix.In this case, all the principal minors of matrix-A. are positive. Then, the Kotelyanski lemma permits deducing on stability properties of the system given by A..Now, we will introduce the discrete-time Borne and Gentina practical stability criterion and the pseudo-overvaluing matrix.In discrete-time Borne and Gentina practical stability criterion [30, 31], let us consider the discrete-time nonlinear system xk+1=A.xk and the overvaluing matrix TM.=aj,k∀j,k=1,...,n. If the nonlinearities are isolated in either one row or one column of TM., the verification of the Kotelyanski condition would enable us to achieve stability conclusion of the original system characterized by A.. The Kotelyanski lemma applied to the overvaluing matrix obtained by the use of the regular vector norm pw=w1,w2,...wn with w=w1,w2,...wn leads to the following sufficient conditions of asymptotic stability of original system: In-TM.12⋯j12⋯j>0j=1,2,...,n.Definition 2 (see [28]).
The matrixTM. is the comparison matrix of the system given by a matrix A. with respect to the vector norm p if the inequality (6)pxk+1≤TM.pxk∀x∈E and k>0 is verified for each corresponding component. Then, the stability of the comparison system zk+1=TM.zk with the initial conditions such as z0=px0.In this case, the following properties are satisfying:(i)
If all the elements ofTM. are nonnegative, it is assumed that the eigenvalue of TM., the biggest in module, is both real and positive and is called main eigenvalue of TM..(ii)
If all the elements of matrixTM. are nonnegative, it is assumed that the principal minors of In-TM. are all positive, the spectral radius of TM. is inferior to the unit, and all the elements of In-TM.-1 are nonnegative.(iii)
WhenTM. is an irreducible matrix, the main eigenvector of TM. is the same as of TM.T and all their elements are nonnegative.Remark 3.
A discrete-time system given by a matrixA. is stable if matrix In-TM. verified the Kotelyanski conditions; in this case In-TM. is an M-matrix.


## 3. Main Results

In this subsection, we present our first result on robust stability analysis for system (1).Theorem 4.
System (1) is robustly asymptotically stable under arbitrary switching rule σk and all admissible uncertainties (3) and (4) if In-TM. is an M-matrix, where(7)TM.=maxi∈I1Lp∈I1LpAip.+r2-r1+1maxi∈I1Lq∈I1Lqsup.Diq.Proof.
Let us consider system (1) for all admissible uncertainties (3) and (4); let w∈Rn with components wp>0,∀p=1,...,n and xk∈Rn be the state vector.
Choose the common Lyapunov-Krasovskii functional:(8)Vxk,k=V1xk,k+V2xk,k+V3xk,kwhere(9)V1xk,k=xk,w(10)V2xk,k=∑j=k-τσkkk-1DMxj,w(11)V3xk,k=∑m=-r2+2-r1+1∑d=k+m+1k-1DMxl,wwith(12)DM=maxi∈I1Lq∈I1Lqsup.Diq..We can easily verify that Vxk,k≥0.
Hence, we get that the difference of the Lyapunov functionalV1xk,k alongside the trajectories of system (1) has the following form: (13)ΔV1xk,k=xk+1,w-xk,w=Aσk.xk+Dσk.xk-τσkk,w-xk,w≤Aσk.xk+Dσk.xk-τσkk,w-xk,w=Aσk.xk,w+Dσk.xk-τσkk,w-xk,w≤AM.xk,w+DMxk-τσkk,w-xk,wwith (14)AM.=maxi∈I1Lp∈I1LpAip..Moreover, ΔV2xk,k is given as follows: (15)ΔV2xk,k=∑j=k+1-τσk+1k+1kDMxj,w-∑j=k-τσkkk-1DMxj,w=∑j=k+1-τσk+1k+1k-r1DMxj,w+DMxk,w-DMxk-τσkk,w+∑j=k+1-r1k-1DMxj,w-∑j=k+1-τσkkk-1DMxj,w.Since τσkk≥r1 one can have (16)∑j=k+1-r1k-1DMxj,w-∑j=k+1-τσkkk-1DMxj,w≤0.From (15), we have (17)ΔV2xk,k≤∑j=k+1-τσk+1k+1k-r1DMxj,w+DMxk,w-DMxk-τσkk,w.Finally, ΔV3xk,k is estimated as(18)ΔV3xk,k=∑m=-r2+2-r1+1∑l=k+mkDMxl,w-∑m=-r2+2-r1+1∑l=k+m+1k-1DMxl,w=∑m=-r2+2-r1+1∑l=k+mk-1DMxl,w+DMxk,w-∑m=-r2+2-r1+1∑l=k+mk-1DMxl,w-∑m=-r2+2-rm+1∑l=k+mk-1DMxk+m-1,w=∑m=-r2+2-r1+1DMxk,w-∑m=-r2+2-r1+1DMxk+m-1,w=r2-r1DMxk,w-∑m=k+1+r2k-r1DMxm,w.Since τσkk≤r2, we have (19)∑j=k+1-τσk+1k+1k-r1DMxj,w-∑j=k+1-r2k-r1DMxj,w≤0.Thus, it follows from (17) and (18) that(20)ΔV3xk,k+ΔV2xk,k≤r2-r1DMxk,w+DMxk,w-DMxk-τσkk,w.Combining (13) and (20) gives(21)ΔVxk,k≤AM.xk,w+DMr2-r1+1xk,w-xk,w=AM.+r2-r1+1DM-Inxk,w=TM.-Inxk,wwhere TM. is given in (4).
On the other hand, we assume thatIn-TM. is an M-matrix. Indeed, we can give a vector ρ∈R+∗nρp∈R+∗p=1,...,n such that In-TM.Tw=ρ, ∀w∈R+∗n.
Thus, we get(22)In-TM.xk,w=In-TM.Tw,xk=ρ,xk.Then we have (23)TM.-Inxk,w=-ρ,xk.Finally, we obtain (24)Δvk≤TM.-Inxk,w≤-∑p=1nρpxpk<0.The proof is completed.According to the Lyapunov theory, robust stability condition of system (1), under switching law σk=i∈I1L by Theorem 4, is immediately derived.


## 4. Application to Discrete-Time Uncertain Switched Nonlinear Systems with Time Varying Delays Defined by Difference Equations

In the sequel, to illustrate the effectiveness of the obtained results, as application of discrete-time uncertain switched nonlinear systems with time varying delays modeled by difference equations will be proposed.Let us consider the following functional difference equation for all the subsystems:(25)Si:yk+n+∑p=1Lpμipk∑m=0n-1aipn-m.yk+m+∑q=1Lqλiqk∑m=0n-1diqn-m.yk+m-τik=0where μipkp∈I1Lp and λiqkq∈I1Lq denote the polytopic uncertain parameters as defined in (3) and (4) for each i∈I1L. yk∈R is the output and n is the subsystem order. aipn-m. and diqn-m. are nonlinear coefficients.Consider the following change of variable:(26)xmk+1=xm+1k,m=1,…,n-1.Combining (25) and (26) leads to (27)xnk+1=-∑p=1Lpμipk∑m=0n-1aipn-m.xm+1k-∑q=1Lqλiqk∑m=0n-1dipn-m.xm+1k-τik.From (27), subsystems Si, i∈I1L will be given under matrix representation as follows: (28)xk+1=∑p=1LlμipkAip.xk+∑q=1LqλiqkDiq.xk-τikxθ=ϕθs=-r2,-r2+1,…,-1(29)Aip.=01⋯000⋱⋮⋮⋮⋱1-aipn.-aipn-1.⋯-aip1.,∀i∈I1L,∀p∈I1Lp(30)Diq.=00⋯000⋱⋮⋮⋮⋱0-diqn.-diqn-1.⋯-diq1.∀i∈I1L,∀q∈I1Lq.According to the switching signal σk, the resultant switched system will be given by(31)xk+1=Aσk.xk+Dσk.xk-τσkkxθ=ϕθs=-r2,-r2+1,…,-1.The regular basis changeP permits characterizing the dynamics of subsystems Si by the change of coordinate defined by (32)zk=Pxkwith (33)P=11…10α1α2…αn-10α12α22…αn-12⋮⋮⋮…⋮0α1n-1α2n-1…αn-1n-11αj≠αq∀j,q.Combining (30), (31), and (32) leads to the following state representation: (34)zk+1=∑p=1LpμipkEip.zk+∑q=1LqλiqkFiq.zk-τikzθ=Pϕθs=-r2,…,-1,0where (35)Eip.=P-1Aip.P=α10⋯0β10⋱⋱⋮⋮⋮⋱⋱0⋮0⋯0αn-1βn-1γip1.⋯⋯γipn-1.γipn..The elements of the vertex matrix Eip. are defined by (36)γipj.=-Gipαj,∀j=1,…,n-1γipn.=-aip1.-∑j=1n-1αjwhere (37)GAip.s,.=sn+∑m=0n-1aipn-m.smand(38)βj=∏q=1q≠jn-1αj-αq-1,∀j=1,…,n-1.The vertex matrix Fiq.is given by (39)Fiq.=P-1Diq.P=0n-1,n-1⋯0n-1,1δiq1.⋯δiqn-1.δiqn.with(40)δiqj.=-NDiq.αj,∀j=1,…,n-1δiqn.=-diq1.and (41)NDiq.s,.=∑m=0n-1diqn-m.sm.Thus, the matrix Ti. is given as follows: (42)Ti.=α10⋯0β10⋱⋱⋮⋮⋮⋱⋱0⋮0⋯0αn-1βn-1ti1.⋯⋯tin-1.tin.,∀i∈I1Lwith(43)ti,p,qj.=γipj.+r2-r1+1sup.δiqj.,j=1,…,n-1ti,p,qn.=γipn.+r2-r1+1sup.δiqn..Finally, the comparison matrix TM. of system (31) will be given as follows: (44)TM.=α10⋯0β10⋱⋱⋮⋮⋮⋱⋱0⋮0⋯0αn-1βn-1t-n.⋯⋯t-n.t-n.with (45)t-n.=maxi∈I1Lp∈I1Lpq∈I1Lqti,p,qn.t-j.=maxi∈I1Lp∈I1Lpq∈I1Lqti,p,qj.,j=1,…,n-1.Now, based on the Borne and Gentina practical stability criterion, we are in a position to give sufficient robust stability conditions of system (31) that are illustrated in the following theorem.Theorem 5.
System (31) is robustly asymptotically stable, under arbitrary switching signal σk and all admissible uncertainties (3) and (4), if there exist αjj=1,...,n-1, αj≠αq,∀j≠q, satisfying the following conditions: (46)i1-αj>0,j=1,…n-1(47)ii1-t-n.-∑j=1n-1t-j.βj1-αj-1>0Proof.
For an arbitrary choiceαj<1j=1,...,n-1, αj≠αq,∀j≠q and from (44), it is clear that all the elements of TM. are isolated in one row and positive. The verification of the Borne and Gentina practical stability criterion enables concluding to the stability of the original system (31).
Therefore, it comes the following sufficient robust asymptotic stability conditions:(48)In-TM.12⋯j12⋯j>0j=1,…,n.It is clear that, for j=1,...n-1, condition (46) is verified as follows: 0<αj<1. The last condition j=n gives us (49)det⁡In-TM.=1-α10⋯0-β10⋱⋱⋮⋮⋮⋱⋱0⋮0⋯01-αn-1-βn-1-t-1.⋯⋯-t-n-1.1-t-n..This implies 1-t-n.-∑j=1n-1t-j.βj1-αj-1>0.
This completes this proof.To simplify the use of the obtained stability conditions, Theorem5 can be reduced to Corollary 6.Corollary 6.
If system (31) is robustly asymptotically stable under arbitrary switching σk and all admissible uncertainties (3) and (4), the following conditions are satisfied ∀αj∈01j=1,...,n-1αj≠αq∀j≠q, for each i∈I1L, p∈I1Lp, and q∈I1Lq:(50)iβjGAip.αj,.+r2-r1+1sup.NDiq.αj,.<0(51)iiGAip.s=1,.+r2-r1+1sup.NDiq.s=1,.>0(52)iiiγipn.+r2-r1+1sup.δiqn.>0Proof (see [9]).
By (47), we can obtain the following result:(53)maxi∈I1Lp∈I1Lpq∈I1Lqγipn.+r2-r1+1sup.δiqn.+∑j=1n-1maxi∈I1Lp∈I1Lpq∈I1Lqγipj.+r2-r1+1sup.δiqj.βj1-αj-1<1.This implies (54)1-maxi∈I1Lp∈I1Lpq∈I1Lqγipn.+r2-r1+1sup.δiqn.-∑j=1n-1maxi∈I1Lp∈I1Lpq∈I1Lqγipj.+r2-r1+1sup.δiqj.βj1-αj-1>0.It is easy to deduce that the relation below is more restrictive than (54) (55)1-γipn.+r2-r1+1sup.δiqn.-∑j=1n-1γipj.+r2-r1+1sup.δiqj.βj1-αj-1>0.Therefore, In order to deduce the stability conditions of system (31), we consider (50) and (52) to find (51).
Thus, we deduce from (50), (52), and (55) that(56)1+aip1.+r2-r1+1sup.diq1.+∑j=1n-1αj+∑j=1n-111-αjs-αjGAip.s,.+r2-r1+1sup.NDiq.s,.Hss=αj>0with (57)Hs=∏j=1n-1s-αj.To complete this proof, let us first observe that(58)GAip.s,.+r2-r1+1sup.NDiq.s,.Hs=s+aip1.+r2-r1+1sup.diq1.+∑j=1n-1αj+∑j=1n-1s-αjGAip.s,.+r2-r1+1sup.NDiq.s,.1-αjHss=αj.From (56), (57), and (58), we have(59)GAip.s,.+r2-r1+1sup.NDiq.s,.Hss=1>0.By a simple verification, we obtain (60)Hs=1=∏j=1n-11-αj>0∀αj∈01which yields(61)GAip.s=1,.+r2-r1+1sup.NDiq.s=1,.>0.This ends the proof.


## 5. Extension Results for Uncertain Switched Nonlinear Systems with Multiple Varying Delays

This subsection is aimed at generalizing the previously reached results to uncertain switched nonlinear systems with multiple varying delays systems.Considering a class of uncertain switched nonlinear systems with multiple varying delays formed byL subsystems given by(62)xk+1=Aσk.xk+∑l=1hDσk,l.xk-τl,σkkxθ=ϕθθ=-max1≤l≤mr2,l,-max1≤l≤mr2,l+1,…,-1where σk is the switching signal given in (1), Ai.∈Rn×n and Dl,i.∈Rn×nl=1,...h are matrices of appropriate dimensions with nonlinear elements of appropriate dimensions. τi,lk, i∈I1L and l∈I1h are the time varying delays satisfying 0≤r1,l≤τi,lk≤r2,l, where r1,l and r2,ll∈I1h are constant scalars.We assume thatAi. and Dl,i. are uncertain of polytopic type described by (63)Ai.=∑p=1LpμipkAip.,i∈I1L(64)Di,l.=∑q=1Lq,lλiq,lkDiq,l.,i∈I1L.Now, it became easier to establish the following sufficient stability conditions for system (62) by generalizing the common Lyapunov-Krasovskii functional (8).Theorem 7.
System (62) is globally robustly asymptotically stable under arbitrary switching σk and all admissible uncertainties (63) and (64) if In-Tl,M. is an M-matrix, where(65)TM,l.=maxi∈I1Lp∈I1LpAip.+r2-r1+1maxi∈I1Lq∈I1Lqsup.∑l=1hDiq,l.with r2=maxl∈I1hτ2,l and r1=minl∈I1hτ1,l.Proof.
It suffices to choose the following Lyapunov functionVxk,k and follow the same steps as given in the proof of Theorem 5.(66)Vxk,k=V1xk,k+V2xk,k+V3xk,kwhere(67)V1xk,k=xk,w(68)V2xk,k=∑l=1h∑j=k-τl,σkk-1DM,lxj,w(69)V3xk,k=∑l=1h∑j=k-τl,σkk-1∑m=-rl,2+2-r1,l+1∑d=k+m+1k-1DM,lxl,wwith(70)DM,l=maxi∈I1Lq∈I1Lq,lsup.Diq,l..Hence, we will apply this result to determine delay-dependent stability criteria for systems described by a set of difference equations with multiple time varying delays:(71)Si:yk+n+∑p=1Lpμipk∑m=0n-1aipn-m.yk+m+∑l=1h∑q=1Lq,lλiq,lk∑m=0n-1diq,ln-m.yk+m-τi,lk=0where yk∈R is the output and aipn-m. and diq,ln-m.are nonlinear coefficients.
From (26) and taking into consideration the switched rule signal σk, the resulting studied switched nonlinear system can be described by the following state space representation:(72)xk+1=Aσk.xk+∑l=1hDσk,l.xk-τl,σkkxθ=ϕθθ=-maxl∈I1hr2,l,-maxl∈I1hr2,l+1,…,-1.The matrix Diq,l. is represented by (73)Diq,l.=00⋯000⋱⋮⋮⋮⋱0-diq,ln.-diq,ln-1.⋯-diq,l1.and Aip. is given in (29).
Therefore, the new polynomialNDl,iq.s,. for i∈I1l, l∈I1h and q∈I1Lq is defined by(74)NDl,iq.s,.=∑m=0n-1dl,iqm.sm.Then, according to (30), system (72) will be represented in the arrow form, ∀i∈I1l, ∀p∈I1Lp, ∀l∈I1h, and ∀q∈I1Lq as follows:(75)zk+1=∑p=1LpμipkEip.zk+∑l=1h∑q=1LqλiqkFiq,l.zk-τi,lkzθ=Pϕθθ=-maxl∈I1hr2,l,-maxl∈I1hr2,l+1,…,-1where Fip. is given in (35) and Eiq,l. is represented by (76)Eiq,l.=P-1Dl,i.P=0n-1,n-1⋯0n-1,1δiq,l1.⋯δiq,ln-1.δiq,ln.with (77)δiq,lj.=-NDiq.s,.,∀j=1,…,n-1,l=1,…,hδiq,ln.=-diq,l1..Consequently, matrix Tipq,l., ∀i∈I1L, ∀p∈I1Lp, ∀q∈I1Lq, and ∀l∈I1h, is given as follows:(78)Tipq,l.=α10⋯0β10⋱⋱⋮⋮⋮⋱⋱0⋮0⋯0αn-1βn-1tipq,l1.⋯⋯tipq,ln-1.tipq,ln.(79)tipq,lj.=γipj.+r2-r1+1sup.∑l=1hδiq,lj​.,j=1,…,n-1tipq,ln.=γipn.+r2-r1+1sup.∑l=1hδiq,ln​..Finally, the comparison matrix TM. of system (72) is given by(80)TM.=α10⋯0β10⋱⋱⋮⋮⋮⋱⋱0⋮0⋯0αn-1βn-1t-n.⋯⋯t-n.t-n.where (81)t-n.=maxi∈I1Lp∈I1Lpq∈I1Lqtipq,ln.t-j.=maxi∈I1Lp∈I1Lpq∈I1Lqtipq,lj.,j=1,…,n-1.Thus, using the special form of system (72) we can announce the following Theorem.Theorem 8.
System (72) is globally robustly asymptotically stable, for any arbitrary switching σk and all admissible uncertainties (63) and (64), if there exist αjj=1,...,n-1, αj≠αq,∀j≠q, such as (82)i1-αj>0,j=1,…n-1(83)ii1-t-n.-∑j=1n-1t-j.βj1-αj-1>0.Next, Theorem8 can be simplified to the following corollary.Corollary 9.
System (72) is globally robustly asymptotically stable under arbitrary switching rule σk and all admissible uncertainties (63) and (64), if there exist αj∈R-j=1,...,n-1, αj≠αq, ∀j≠q and ∀i∈I1L, ∀p∈I1Lp, ∀q∈I1Lq, ∀l∈I1h, such as (84)iβjGAip.αj,.+r2-r1+1sup.∑l=1hNDiq,l.αj,.<0(85)iiGAip.s=1,.+r2-r1+1sup.∑l=1hNDiq.s=1,.>0(86)iiiγipn.+r2-r1+1sup.∑l=1hδiq,ln​.>0


## 6. Illustrative Examples

Example 1.
Let us consider system (72) with three subsystems, where the randomly switched model is given as (87)A11.=01-0.45+0.08f.0.95-0.43Φ.,A12.=01-0.6+0.08f.0.9-0.4Φ.,A21.=01-0.4+0.085f.1-0.5Φ.,A22.=01-0.5+0.07f.1.2-0.8Φ.,A31.=01-0.55+0.085f.0.9-0.55Φ.,A32.=01-0.6+0.075f.1-0.6Φ.D11,1.=00-0.012+0.02ψ.0.04-0.03ψ.,D11,2.=00-0.018+0.015ψ.0.037-0.035ψ.D12,1.=00-0.01+0.02ψ.0.04-0.023ψ.,D12,2.=00-0.01+0.025ψ.0.037-0.04ψ.,D21,1.=00-0.015+0.02ψ.0.05-0.03ψ.,D21,2.=00-0.01+0.017ψ.0.04-0.035ψ.,D22,1.=00-0.013+0.01ψ.0.03-0.01ψ.,D22,2.=00-0.02+0.0184ψ.0.04-0.0357ψ.D31,1.=00-0.015+0.025ψ.0.045-0.024ψ.,D31,2.=00-0.02+0.03ψ.0.03-0.045ψ.,D32,1.=00-0.007+0.017ψ.0.021-0.043ψ.andD32,2.=00-0.013+0.04ψ.0.03-0.02ψ.with f.Φ. and ψ. being general nonlinear functions.Hence, we suppose thatψ.∈E0.8,1,1.2 and the corresponding delay functions are listed as follows: r1,1k=4+sin2kπ/2, r1,2k=3+sin2kπ/2, r2,1k=4+sin2kπ/2, r2,2k=2+sin2kπ/2r3,1k=3+sin2kπ/2, r3,2k=2+sin2kπ/2k=0,1,2.....Thus, by Corollary9, with α=0.2, we obtain the following stability conditions: (88)Φ.<-2.04+0.5f.Φ.>-0.54+0.17f.Φ.<1.254.Due to those inequalities, the robust stability domain given by the nonlinear Φ. relative to the nonlinear f. is illustrated in Figure 1.Figure 1
Stability domain for Example1 obtained from Corollary 9.By fixing the sampling time atTe=0.1s, consider the random switching sequence depicted in Figure 2; the initial conditions are ϕk=-0.80.7T, the nonlinearities f.=4.5, φ.=0.5, and ψ.=1, and the incertitude parameters are chosen as μ11k=μ21k=μ31k=0.6 and λ11k=λ21k=λ31k=0.7. The simulation results, respectively, given in Figures 3 and 4 hint about the state responses of the system and state norm where the switching signal is randomly generated.Figure 2
Random switching signal for system given in Example1.Figure 3
The state responses of the system given in Example1.Figure 4
The state norm of the system given in Example1.Noting that, due to the complexity of this randomly switched system alongside the important number of the subsystems, it is hard to find a common Lyapunov function, then, we cannot guarantee stability of this switched system under randomly switching that values our proposed approach.Example 2 (see [27, 34]).
Consider a randomly switched system represented by a set of differential equations given as follows:(89)x¨t+∑p=12μiptaipx˙t+∑p=12μiptφipxxxt+∑q=12λiqtbiqx˙t-τit+∑q=12λiqtciqxt-τit=0,where aip, biq, and ciq are parameters and i∈1,2.
By the Taylor-Young formula,x¨k=xk+2-2xk+1+xk/Te2 and x˙k=xk+1-xk/Te, where Te is the sampling time.
According to (25), all subsystems can be represented under the following matrix representation: xk+1=Ai.xk+Dixk-τik, where xk is the state vector and the matrices Ai.=∑p=12μipkAip. and Di=∑q=12λiqkDiq are given by(90)Aip.=01-1+aipTeφipxx2-aipTeandDiq=00biqTe-ciqTe2-biqTe.Let us consider the following particular values: a11=8b11=0.01c11=0.2, a12=7.5b12=0.017c12=0.35, a21=7.2b21=0.013c21=0.25, and a21=8.5b21=0.015c21=0.3. The sampling time Te is fixed at 0.2s and the time varying delays functions are τ1k=1+3cos2kπ/2, k=0,1,2.... and τ2k=1+2cos2kπ/2, k=0,1,2.....By Corollary6, with α=0.2, the following stability conditions can be deduced:(i)
φ11min∗.=0.616<φ11x/x<φ11max∗.=1.02.(ii)
φ12min∗.=0.65<φ12x/x<φ12max∗.=1.03.(iii)
φ21min∗.=0.66<φ21x/x<φ21max∗.=1.027.(iv)
φ22min∗.=0.59<φ22x/x<φ22max∗.=1.0282.According to the previous stability conditions and with a particular choice of functionsφ11x/x=0.6, φ12x/x=0.8, φ21x/x=0.9, and φ22x/x=1 and by considering the random switching sequence given in Figure 5, the vector-valued initial function ϕs=10.6T and the incertitude parameters μ11k=μ21k=0.6 andλ11k=λ21k=0.7. Then, the simulation results are obtained as follows: a typical result plotted in Figures 6 and 7 shows that the system state and the norm state converge to zero, where the switching signal is randomly generated. Thus, the simulation affirms the theoretical results.Figure 5
Random switching signal for system given in Example2.Figure 6
The state responses of the system given in Example2.Figure 7
The state norm of the system given in Example2.


## 7. Conclusion

In this paper, we have interestingly addressed the stability issue for a class of discrete-time switched nonlinear systems with time varying delays and with polytopic uncertainties. New sufficient stability conditions have been yielded by constructing a new common Lyapunov-Krasovskii functional alongside resorting to theM-matrix properties, Borne-Gentina practical stability criterion, the aggregation techniques, and the vector norms notion. Numerical simulations are given to illustrate the effectiveness of our results. It is expected that the idea and the technique in this paper will be worth of use for the future research works on that filed.



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