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

This paper provides new sufficient conditions on robust asymptotic stability for a class of uncertain discrete-time switchednonlinear 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.Themain 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 themerit and effectiveness of the proposed approach.


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][2][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,.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][23][24][25][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][10][11][12][13][27][28][29][30][31] and M -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.Section 2 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 , we denote the transpose by   .Let R denote the field of real numbers and R  denote an  dimensional linear vector space over the reals with the norm ‖.‖.For any  = (  ) 1≤≤ , V = (V  ) 1≤≤ ∈ R  , we define the scalar product of the vectors  and V as ⟨, V⟩ = ∑  =1   V  , R × the space of  ×  matrices with real entries.R + is the set of positive real numbers.
Assuming that all subsystems are uncertain of polytopic type which can be described as where   (.),  ∈ [ 2.2.Preliminaries.Now, the following lemmas, criterion, remark, and definitions are preliminarily presented for later development.
In Kotelyanski lemma [33], the real parts of the eigenvalues of the matrix (.), with nonnegative off-diagonal elements, are less than a real number  if and only if all those of matrix (.), where (.) =   − (.), are positive, with   the  identity matrix.
In this case, all the principal minors of matrix (−(.))are positive.Then, the Kotelyanski lemma permits deducing on stability properties of the system given by (.).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 ( + 1) = (.)()and the overvaluing matrix   (.) = {| , |} ∀,  = 1, ..., .If the nonlinearities are isolated in either one row or one column of   (.), the verification of the Kotelyanski condition would enable us to achieve stability conclusion of the original system characterized by (.).The Kotelyanski lemma applied to the overvaluing matrix obtained by the use of the regular vector norm leads to the following sufficient conditions of asymptotic stability of original system: Definition 2 (see [28]).The matrix   (.) is the comparison matrix of the system given by a matrix (.) with respect to the vector norm  if the inequality  ( ( + 1)) ≤   (.)  ( ()) ∀ ∈  and  > 0 is verified for each corresponding component.Then, the stability of the comparison system ( + 1) =   (.)() with the initial conditions such as  0 = ( 0 ).
In this case, the following properties are satisfying: (i) If all the elements of   (.) are nonnegative, it is assumed that the eigenvalue of   (.), the biggest in module, is both real and positive and is called main eigenvalue of   (.).
(ii) If all the elements of matrix   (.) are nonnegative, it is assumed that the principal minors of (  −   (.)) are all positive, the spectral radius of   (.) is inferior to the unit, and all the elements of (  −   (.)) −1 are nonnegative.
(iii) When   (.) is an irreducible matrix, the main eigenvector of   (.) is the same as of (  (.))  and all their elements are nonnegative.

Main Results
In this subsection, we present our first result on robust stability analysis for system (1).

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: where   ()  ∈  [1   ] and   ()  ∈  [1   ] denote the polytopic uncertain parameters as defined in (3) and ( 4) for each  ∈  [1 ].() ∈ R is the output and  is the subsystem order. −  (.) and  −  (.) are nonlinear coefficients.
Consider the following change of variable: Combining ( 25) and ( 26) leads to From ( 27), subsystems   ,  ∈  [1 ] will be given under matrix representation as follows: According to the switching signal (), the resultant switched system will be given by The regular basis change P permits characterizing the dynamics of subsystems   by the change of coordinate defined by with Mathematical Problems in Engineering Combining (30), (31), and (32) leads to the following state representation: where The elements of the vertex matrix   (.) are defined by where and The vertex matrix   (.)is given by with and Thus, the matrix   (.) is given as follows: Finally, the comparison matrix   (.) of system (31) will be given as follows: with (  ,, (.)) ,  = 1, . . .,  − 1.
(45) 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.4), if there exist   ( = 1, ...,  − 1),   ̸ =   , ∀ ̸ = , satisfying the following conditions: Proof.For an arbitrary choice |  | < 1 ( = 1, ...,  − 1),   ̸ =   , ∀ ̸ =  and from (44), it is clear that all the elements of   (.) 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).

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.We assume that   (.) and  , (.) are uncertain of polytopic type described by 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 () and all admissible uncertainties ( 63) and ( 64) if (  −  , (.)) is an  − matrix, where with Proof.It suffices to choose the following Lyapunov function ((), ) and follow the same steps as given in the proof of Theorem 5.
(81) Thus, using the special form of system (72) we can announce the following Theorem.
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.
According to (25), all subsystems can be represented under the following matrix representation: ( + 1) =   (.)()+  (−  ()), where () is the state vector and the matrices   (.) = ∑  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.

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 the  − 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.

( i )
All the eigenvalues of (.) have a positive real part.(ii) The real eigenvalues are positives.(iii) The principal minors of (.) are positive:

Figure 3 :Figure 4 :
Figure 3: The state responses of the system given in Example 1.

Figure 6 :Figure 7 :
Figure 6: The state responses of the system given in Example 2.