M-Matrix-Based Robust Stability and Stabilization Criteria for Uncertain Switched Nonlinear Systems with Multiple Time-Varying Delays

This paper focuses on the robust stability and the memory feedback stabilization problems for a class of uncertain switched nonlinear systems with multiple time-varying delays. Especially, the considered time delays depend on the subsystem number. Based on a novel common Lyapunov functional, the aggregation techniques, and the Borne and Gentina criterion, new sufficient robust stability and stabilization conditions under arbitrary switching are established. Compared with existing results, the proposed criteria are explicit, simple to use, and obtained without finding a common Lyapunov function for all subsystems through linear matrix inequalities, considered very difficult in this situation. Moreover, compared with the memoryless one, the developed controller guarantees the robust stability of the corresponding closed-loop system with more performance by minimizing the effect of the delays in the system dynamics. Finally, two numerical simulation examples are shown to prove the practical utility and the effectiveness of the proposed theories.


Introduction
Switched systems constitute an important class of hybrid systems, which can be described by a family of subsystems and a rule that orchestrates the switching amongst them [1].
Recently, switched systems have attracted considerable attention, and some valuable results have been achieved . Among these research topics, stability analysis, stabilization, and control design of switched systems under arbitrary switching are fundamental issues in the design and the analysis of such systems. is kind of switching strategy lies in the fact that the stability of each autonomous or closed-loop subsystem does not necessarily imply the stability of the corresponding switched system. In this framework, it is well known that the existence of a common Lyapunov function (CLF) for all the subsystems through the linear matrix inequalities (LMIs) is a sufficient condition for such systems to be asymptotically stable under arbitrary switching [3]. However, this function is very difficult to find even for switched linear systems [3]. erefore, this task becomes more and more compiled when switched nonlinear systems are involved [5].
Frequently, to avoid the conservatism related to the existence of a CLF, some attention has been devoted to considering switched systems under restricted switching. Although many interesting results have been proposed for this alternative, such as the dwell time approach [7] and the multiple Lyapunov function [6], stability under arbitrary switching remains more suitable for real systems. In fact, it offers more effectiveness for control design along with stability preserved.
As is well known, time-delay is usually often encountered in many engineering processes, which is considered in many recent studies [4, 13-20, 25-31, 33-37].
us, the presence of this phenomenon can affect the dynamic characteristics of systems, and it leads to the degradation of the system performance. Besides, when practical systems with errors or external disturbances are modeling, uncertainties parameters are frequently included. In this context, two types of uncertainties exist in the literature, which are mainly polytopic uncertainties and norm bounded. Indeed, one of the most significant exigencies for a control system is robustness [38,39]. erefore, from a practical viewpoint, it is necessary to investigate switched time-varying delay systems with extra uncertain parameters. In this regard, many of the uncertain systems can be approximated by systems with polytopic uncertainties.
In recent years, switched nonlinear time-varying delay systems have received a major interest, and many significant results have been established [4, 13-20, 25-31, 33]. us, from the switching strategies, the existing results can be classified into two categories, which are, respectively, restrictive switching and arbitrary switching. In fact, stability analysis and stabilization under restrictive switching have been investigated mainly based on the Lyapunov-Krasovskii functional (LKF) and the average dwell time approach [15]. For example, in [15], the robust stability and the control design problems for switched nonlinear systems have been investigated by using the average dwell time approach. e work in [34] addresses state feedback controllers design for switched nonlinear time-delay systems. Furthermore, the stability analysis of switched nonlinear systems has been investigated in [17] by employing the trajectory-based comparison method.
On the other side, the stability analysis and stabilization of switched time-delay systems under arbitrary switching have been studied based on the common Lyapunov-Krasovskii functional (CLKF) [14] for all the subsystems. Despite the difficulty related to the application of this method for switched nonlinear systems, some results exist for this framework. For instance, in [18], the adaptive control problem for switched nonlinear systems has been presented based on the adaptive backstepping technique and the CLF approach. In addition, in [20], the stabilization problem for switched nonlinear systems has been investigated based on the Metzler matrices. Moreover, the work in [19] deals with the stability analysis of switched nonlinear interconnected systems based on the vector Lyapunov approach and M-matrix theory. e authors in [25,28] have focused on the stability analysis of switched nonlinear systems by using the aggregation techniques and the M-matrix theory. Furthermore, by including the Takagi-Sugeno (TS) fuzzy model as a powerful approximation tool of the initial nonlinear system, based on the aggregation techniques, algebraic stability criterion for TS Fuzzy switched systems were proposed in [30,31].
It should be noted that all the aforementioned works for feedback stabilization have considered memoryless state feedback controllers. However, this kind of controllers cannot have an effect on the time-delay systems, since it does not introduce the past state information of the systems. In [26], a memory state feedback controller for time-varying delay switched systems has been considered. Indeed, it has been verified that this kind of controller had better immunity to reduce the influence of delay in system dynamics.
From a practical point of view, switched dynamical systems can be affected by mode depending time-varying delays. However, due to its complexities, this kind of systems is less considered [20,40].
To the best of our knowledge, the robust stability analysis and the memory state feedback controller design for uncertain switched nonlinear systems with mode depending multiple time-varying delays under arbitrary switching have not been studied yet, which are the subject of this work.
Motivated by this consideration, in this paper, new robust stability criteria and memory feedback controller design under random switching for a class of uncertain switched nonlinear systems with multiple time-varying delays have been established. Indeed, based on a CLF, the aggregation techniques [41], and the Borne-Gentina criterion [41], new robust stability conditions for the considered autonomous systems are given. Besides, the obtained results are extended to develop a memory state feedback controller through the pole assignment control for the closed-loop corresponding switched systems. e main contributions of this paper are emphasized as follows: (1) ere are no results to address switched nonlinear systems with uncertain parameters and mode-dependent multiple time-varying delays. Out of research interest, novel stability analysis and feedback controller design under arbitrary switching for more general kinds of switched nonlinear systems will be presented. (2) Compared to the existing criterion for switched systems under arbitrary switching, by using the aggregation techniques the difficulty related to the existence of a CLF through the LMIs approach can be avoided. (3) Contrary to searching a CLF through the LMIs approach considering a hard task in this investigation, the developed stability and stabilization criteria are explicit and simple to use. (4) Although there are some studies on memory state feedback control, the memory state feedback controller has not been involved for switched nonlinear systems with multiple time-varying delays. In addition, the developed controller has an explicit form, and it allows stabilizing the resulting closed-loop systems under arbitrary switching without any computations over LMIs constraints. e rest of the paper is organized as follows: Section 2 gives the problem statement and some definitions. In Section 3, the main results are presented. Section 4 focuses on the application of the main results to switched nonlinear systems modeled by differential equations. In Section 5, some simulation examples are provided to illustrate the 2 Mathematical Problems in Engineering effectiveness of the proposed approach. Finally, some conclusions are addressed in Section 6.
Notations. roughout this paper, I n is an identity matrix, R n denotes the n-column vectors, and ‖.‖ denotes the Euclidean norm. In addition, for any given vectors v � (v l ) 1 ≤ l ≤ n , w � (w l ) 1 ≤ l ≤ n ∈ R n , the scalar product of vectors u and v is defined as〈v, w〉 � n l�1 v l w l . e sign function is defined as For a given matrix A, λ(A) denotes the set of its eigenvalues and A T and A − 1 denote its transpose and inverse, respectively. We de-

Problem Statement.
Consider the following switched nonlinear system with multiple time-varying delays given by } is the switching signal, and σ(t) � i ∈ N means that the i th subsystem is active with N being the number of subsystems. A i (.), D l,i (.), and B i (.) are matrices which have nonlinear elements with appropriate dimensions, and ϕ(t) is the continuous vector valued function specifying the initial state of the system. r l,i (t) denotes the time-varying delay functions which satisfy where τ and d are two constant scalars. Assume that all subsystems are uncertain of polytopic type, which are represented as where A ip (.), p ∈ P, and D l,iq (.) q ∈ Q are, respectively, the vertex matrices denoting the extreme points of the polytopes A i (.) and D i (.). Pis the number of the vertex matrices A i (.), Q is the number of the vertex matrices D l,i (.) and the weighting factors μ ip (t), λ l,iq (t) are polytopic uncertainties parameters belonging toμ ip (t): P p�1 μ ip (t) � 1 μ ip (t): P p�1 μ ip (t) � 1, μ ip (t) ≥0, and λ l,iq (t): Q q�1 λ l,iq (t) � 1, λ l,iq (t)≥ 0.

Preliminaries.
In the sequel, we introduce some lemmas, definitions, and criteria, which play important roles in deducing our main results.
Lemma 1 (see [40]). e matrix A � (a ij ) 1 ≤ i,j ≤ n is called an M − matrix if the following conditions are satisfied: (ii) All the successive principal minors of A are positive: (iii) For any positive vector x � (x 1 , . . . , x n ) T , the system of equations A(.)x has a positive solution Definition 1. (see [41]). e matrix T mc (.) is said to be the pseudo-overvaluing matrix of the system given by . , x n ] T , if the next inequality is satisfied: where D + denotes the right-hand derivative operator.

Assumption 1.
In what follows, we assumed that all the nonlinear elements T mc (.) are separated in the last row.
Lemma 2 (see [41]). If T mc (.)is the pseudo-overvaluing matrix of the system: _ x � A(.)x, then it verifies the following properties:

constant vector
Lemma 3 (see [41]). e application of the Kotelyanski lemma [42] to the pseudo-overvaluing matrix T mc (.)is relative to the system: _ x � A(.)x; A(.) � (a ij (.)) 1 ≤ i,j ≤ n allows deducing the stability of the corresponding system, if T mc (.) is the opposite of an M − matrix, which implies that all the successive principal minors have alternated signs with the first being negative: . . ⋮ a n,1 (.) a n,2 (.) . . . a n,n (.) Mathematical Problems in Engineering 3

Stability Analysis.
In this section, we investigate sufficient delay-dependent stability conditions for the autonomous system (1).
and d is given in (3).
Define the following Lyapunov functional for the autonomous system (1): with where e right derivative of V(t) along the trajectory of system (1) yields to where en, where A mc (.) � max i∈ N p∈ P which implies

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From (15) and (17), we obtain where is T mc (.) given in (9), knowing that e main eigenvector v(t, x(t))of T mc (.) relative to the main eigenvalue λ m is constant. We assume that T mc (.) is the opposite of an M − matrix. erefore, we can find a vector ω ∈ R * n + (ω h ∈ R * + h � 1, . . . , n) satisfying the following relation(− T mc (.)) T ] � ω, ∀] ∈ R * n + . us, it easy to follow that Substituting (21) into (19) leads to erefore, it can be established that is completes the proof of eorem 1.
□ Remark 1. eorem 1 gives the main results of the stability analysis for the autonomous system (1) under σ(t) � i ∈ N and all admissible uncertainties (4) and (5). e conditions presented in eorem 1 will be simplified by applying the Borne-Gentina criterion in eorem 3 and Corollary 1.

Memory State Feedback Design.
In this section, we consider the following memory state feedback controller: (23) where K i (.) and L l,i (.), i ∈ N, are nonlinear controller gains to be determined. e resulting closed-loop switched system composed from (1) and (23) is represented by In what follows, we present our result for the memory state feedback control of system (1). (1) is robustly stabilizable via controller (23) under σ(t) � i ∈ N, for all admissible uncertainly parameters μ ip (t) and λ iq (t) for each p ∈ P and q ∈ Q, such that the closed-loop switched system (5) is robustly asymptotically globally stable, if there exist matrices K ip (.) and L l,iq (.), l ∈ L, with appropriate dimensions, satisfying that

Theorem 2. System
and d is introduced in (3).
Proof. We assume that there exist matrices K ip (.) and L l,iq (.), ∀i ∈ N, p ∈ P, q ∈ Q, and l ∈ L satisfying that T mc (.) is the opposite of an M − matrix. According to the proof of eorem (1), system (1) is robustly asymptotically stabilizable via controller (23) under σ(t) � i ∈ N and all admissible uncertainties (4) and (5). e proof of eorem 2 is completed.

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Remark 2.
eorem 2 gives the main results for the stabilization of the control system (1) via controller (23). In the sequel, by applying the Borne-Gentina criterion, this result will be applied in eorem 4 to develop a memory feedback controller via the pole assignment control to stabilize the corresponding closed-loop system under σ(t) � i ∈ N and all admissible uncertainties (4) and (5).

Application to Switched Systems Modeling via Differential Equations
In this subsection, we apply the previously reached results for a class of switched nonlinear systems modeled by a set of differential equations. Considering a class of uncertain switched nonlinear systems with multiple time-varying delays formed by N subsystems, each subsystem S i , i ∈ N is given by the following differential equation: where y(t) ∈ R n , a h ip (.), and d h iq,l (.) are nonlinear coefficients, ∀i ∈ N, p ∈ P, q ∈ Q, l ∈ L, and (h � 1, . . . , n − 1). u(t) ∈ R is the control input. r i,l (t) denotes the timevarying delays satisfying that 0 ≤ r i,l (t) ≤ τ and | _ r i,l (t)| ≤ d < 1 where τ and d are given, respectively, in (2) and (3).
Consider the following change of variable: Due to (27), relation (26) becomes or under matrix form, we obtain the following state representation: where x(t) is the state vector, whose components are x h (t), h � 1, . . . , n.
e vertex matrices A ip (.), D l,iq (.), and B i (.) are given as follows: 6 Mathematical Problems in Engineering where a h ip (.) is a coefficient of the instantaneous characteristic polynomial G A ip (.) (s) of matrix A ip (.) given by and Assume that all subsystems are uncertain of polytopic type, which can be described as Considering the switched rule given in (1), the switched control system will be represented as Finally, according to the controller (23), the closed-loop system will be represented by A change of base for system (35) into the arrow matrix form [31] allows that where z(t) � Px(t) is the new state vector and P is the corresponding passage matrix given by with α j , j � 1, . . . , n − 1 being distinct arbitrary constant parameters. e vertex matrices in the arrow form E ip (.) and F l,iq (.) are given by Mathematical Problems in Engineering 7 with e elements of E ip (.)are given as follows: 40) and the elements of F l,iq (.) are Taking into account the previous relations, the matrix T l,ipq (.) is given by Finally, the common pseudo-overvaluing matrix T mc (.) of system (35) can be deduced such as where

Stability Conditions for Continuous-Time Uncertain Switched Nonlinear Systems with Multiple Time-Varying
Delays. In this subsection, we give some sufficient stability conditions for the autonomous system (34). (34) is robustly globally asymptotically stable under σ(t) � i ∈ N and admissible uncertainties (4) and (5), if there exist α h < 0 (h � 1, . . . , n − 1), α h ≠ α q , ∀h ≠ q, such that the following condition is satisfied:

Theorem 3. e autonomous system
Proof. e application of the Borne-Gentina criterion to T mc (.) yields to the following stability conditions for the autonomous system (34): where Δ h is the h th principal minor of T mc (.). erefore, for h � 1, . . . , n − 1, the first condition in eorem 3 is verified such that α h ∈ R * − . Finally, for h � n, the last condition is verified as follows: at is, e division of this previous condition by eorem 3 can be simplified to Corollary 1. (5), if there exist α h (h � 1, . . . , n − 1) < 0 such that α h ≠ α q , ∀h ≠ q, and the inequalities below are satisfied:
erefore, the n th principal minor of T mc (.) is calculated as follows: is proof is complete.

Memory Feedback Stabilization for Uncertain Switched Nonlinear Systems with Time-Varying Delays.
In this subsection, a new memory feedback stabilization for the control system (34) via the pole assignment control is given in eorem 4.

Theorem 4.
Let all n poles p 1 , . . . , p n of system (34) be imposed as real, distinct, and negative. en, the control system (34) is stabilizing via control law (23), such that the corresponding closed-loop, switched system is robustly globally asymptotically stable under σ(t) � i ∈ N and admissible uncertainties (4) and (5), if the following conditions are satisfied: Proof. For p h � α h are real and negative h � 1, . . . , n − 1, the Borne-Gentina criterion yields to the following stabilization conditions: Since the new dynamic of the system permits concluding that overall t j (.) � 0 for j � 1, . . . , n − 1 and t n (.) � p n t n (.) � p n , thus (55) becomes

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and the system is stable since p h < 0 h � 1, . . . , n.

Illustrative Examples
In this section, two numerical examples are introduced to demonstrate the theoretical results.
From Corollary 1, with α � − 1, we obtain the following robust stability conditions:   Figures 2-4 where the switched signal given in Figure 5 is randomly generated. Figures 2 and 3, we observe that the considered system is robustly asymptotically stable under randomly switching and any admissible uncertainties (4) and (5), which proves the effectiveness of the result given in Corollary 1.

Remark 5.
e considered system in Example 1 is subject to uncertain complex nonlinear dynamics and mode depending on multiple time-varying delays. However, it is very difficult to find a CLF for the system under consideration in Example 1.

Remark 6.
e result given in Corollary 1 can construct an alternative to searching a CLF through the LMIs approach for studying robust stability under arbitrary switching.
Indeed, in [32], the authors introduced a simple linear example without time-delay and uncertainty for which a CLF does not exist.
All the subsystems can be represented under matrix representation such as Consider the following controller:

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All the closed-loop subsystems will be represented as follows: where e time-varying delay functions are  (67) All the vertex matrices will be represented under the arrow form such as For the following pole placement p 1 � − 1 and p 2 � − 2, by eorem 4, we obtain the following robust stabilization conditions: e simulation results for fixed initial points ϕ(t) � [2 − 1] T are given in Figures 6 -8, respectively, which show the state responses, the state trajectory, and the state's norm of the system given in Example 1 where the switching mode given in Figure 9 is randomly generated. e simulation results reveal that the state trajectories closed-loop system controlled by the memory state feedback controller are converging to zero, and the closed-loop system is robustly asymptotically stable where the switching signal is randomly generated.

Remark 7.
e developed memory state feedback controller given in eorem 4 can reduce the effect of the delays especially for switched systems with multiple time-varying delays and it guaranteed to the considering system more performance and immunity to the delays as well as the uncertainties compared with the memoryless controller.
Remark 8. Form eorem 4, we obtain the robust stability of a closed-loop system given in Example 2 where the switching signal is randomly generated and for any admissible uncertainties (4) and (5). In fact, the result given in eorem 4 can be an alternative to find a CLF through the LMI approach.

Conclusion
is paper has investigated new robust stability and stabilization criteria under arbitrary switching for a class of uncertain switched nonlinear systems. e systems under consideration are subject to multiple time-varying delays and polytopic-type parameter uncertainty. e proposed results are obtained by using a novel CLF, the Borne-Gentina criterion, and the aggregation techniques. Compared to the existing results in this area, the developed criteria are explicit, are simple to use, and can construct an interesting alternative to find a CLF through the LMI approach, considered a hard task in this case.
Future research will extend the results of this paper to switched stochastic systems with time-varying delays and actuator saturation.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.