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Delayindependent stability criteria under arbitrary switching of a class of switched nonlinear timedelay systems
 Marwen Kermani^{1}Email author and
 Anis Sakly^{1}
https://doi.org/10.1186/s1366201505601
© Kermani and Sakly 2015
 Received: 2 February 2015
 Accepted: 30 June 2015
 Published: 22 July 2015
Abstract
This paper addresses the stability problem of a class of switched nonlinear timedelay systems modeled by delay differential equations. Indeed, by transforming the system representation under the arrow form, using a constructed Lyapunov function, the aggregation techniques, the BorneGentina practical stability criterion associated with the Mmatrix properties, new delayindependent conditions to test the global asymptotic stability of the considered systems are established. In addition, these stability conditions are extended to be generalized for switched nonlinear systems with multiple delays. Note that the results obtained are explicit, they are simple to use, and they allow us to avoid the problem of searching a common Lyapunov function. Finally, an example is provided, with numerical simulations, to demonstrate the effectiveness of the proposed method.
Keywords
 continuous switched nonlinear systems time delays
 global asymptotic stability
 BorneGentina criterion
 common Lyapunov function
 arrow form state matrix
 arbitrary switching
1 Introduction
Switched systems are a class of important hybrid systems which consist of a finite number of subsystems that are governed by differential or difference equations and a switching law which defines a specific subsystem being activated during a certain interval of time. Due to the physical properties or various environmental factors, many realworld systems can be modeled as switched systems such as computer science, autonomous transmission systems, computer disc drivers, control systems, electrical engineering and technology, automotive industry, air traffic management, chemical systems, power systems and communication networks, and other applications [1–9]. On the other hand, considerable efforts have been made as regards the analysis and the design of switched systems. There are still many open and challenging issues remaining to be tackled, despite great successes reported during the past several decades. Among those research topics, stability analysis and stabilization have attracted most attention [1–4, 10–41]. Hence, several methods have been proposed for these matters. It is commonly recognized that there are mainly three basic types of problems considering the stability and the stabilization issues of switched systems [10–12]: (i) guaranteeing of asymptotical stability of the switched system with arbitrary switching; (ii) identification of the limited but useful class of stabilizing switching laws; and (iii) construction of asymptotically stabilizing switching signals. Specifically, the stability analysis under arbitrary switching problem (i) which will be focused on in this work deals with the case that all subsystems are stable. This problem seems trivial, but it is fundamental and important [10, 13–17], since we can find many examples where all subsystems are stable but inappropriate switching rules can make the whole system unstable. In addition, stability under arbitrary switching is a desirable property of switched systems due to its practical importance and also it allows us to consider higher control specifications for the system. For this problem, it is well known that the existence of a common Lyapunov function for individual systems guarantees stability of the switched system under arbitrary switching [16, 19]. Therefore, this method is usually very difficult to apply even for continuoustime switched linear systems [18, 19]; however, it becomes more complicated for switched nonlinear systems. Yet, some attempts are presented to construct a common Lyapunov function for nonlinear switched systems [20, 21].
On the other hand, time delay is a common phenomenon encountered in various practical and engineering systems [42, 43] such as chemical processes, nuclear reactors, models of lasers, electrical systems, aircraft stabilization, biological systems, and systems with lossless transmission lines; and most of them appear in the form of timevarying delay. It is a wellknown fact that the presence of delays is an inherent feature of many physical processes, the big sources of instability and poor performances in switched systems. Thus, it is important to investigate the stability analysis problem for switched delay systems [22–24, 27, 28, 30–41]. It is noted that current methods of the analysis and design for timedelay systems can be classified into two categories: delayindependent criteria and delaydependent ones. In this work, in view of a delayindependent analysis, we expect to aid in studying stability analysis of switched systems under an arbitrary switching law.
Presently, the most important consideration in the analysis of switched systems is their stability. Recently, many researchers focused on switched timedelay systems. Indeed, the stability analysis problem of switched timedelay systems has attracted a lot of attention from many researchers [32, 35–40]. However, the presence of delays makes this problem much more complicated. Thus, the main approach for stability analysis under arbitrary switching relies on the use of a LyapunovKrasovskii functional and the LMI approach for constructing a common Lyapunov function [40]. In fact, getting such a function becomes more complicated even for switched linear systems. Consequently, few results have been obtained for continuoustime switched nonlinear timedelay systems [40].
Motivated by these mentioned shortcomings for the existing results in this framework as well in the sense of various methods that can be employed in this paper, we address this challenging problem. Indeed, based on the construction of a common Lyapunov function as well as the use of the BorneGentina practical stability criterion [22–26, 44–57] associated with the Mmatrix properties [58, 59], new delayindependent sufficient stability conditions for continuoustime switched nonlinear timedelay systems under arbitrary switching are established. Subsequently, these obtained results are extended to be generalized for continuoustime switched nonlinear systems with multiple delays. Note that these proposed results can guarantee stability under arbitrary switching and allow us to avoid searching of a common Lyapunov function, which is very difficult in this case.
Within the frame of studying the stability analysis, this approach was introduced in [44, 45] for continuoustimedelay systems and in our previous work [22, 24] for discretetime switched timedelay systems.
This paper is organized as follows. Section 2 formulates the problem and presents some definitions. The main results of this paper are given in Section 3. Section 4 is devoted to the derivation of new delayindependent conditions for the asymptotic stability of a class of switched nonlinear systems defined by differential equations. Then this result is extended for switched systems with multiple delays in Section 5. A numerical example is provided to illustrate the design results in Section 6. Finally, concluding remarks are given in Section 7.
Notations
The notations in this paper are fairly standard. If not explicitly stated, matrices are assumed to have compatible dimensions. I is an identity matrix with appropriate dimension. Let \(\Re^{n}\) denote an n dimensional linear vector space over the reals; \(\Vert \cdot \Vert \) stands for the Euclidean norm of vectors. For any \(u = ( u_{i} )_{1 \le i \le n}, v = ( v_{i} )_{1 \le i \le n} \in \Re^{n}\) we define the scalar product of the vector u and v as \(\langle u,v \rangle = \sum_{i = 1}^{n} u_{i}v_{i}\). Denote by \(\lambda ( M )\) the set of eigenvalues of the matrix \(M = ( m_{i,j} )_{1 \le i,j \le n}\), \(M^{T}\) is its transpose and \(M^{  1}\) its inverse and we denote \(M^{*} = ( m_{{i,j}}^{*} )_{1 \le i,j \le n}\) with \(m_{{i,j}}^{*} = m_{i,j}\) if \(i = j\) and \(m_{{i,j}}^{*} = \vert m_{i,j} \vert \) if \(i \ne j\) and \(\vert M \vert = \vert m_{i,j} \vert \), \(\forall i,j\).
In the sequel, we denote \(( x ( t ),t ) = (\cdot )\).
2 Preliminaries and problem formulation
Before addressing the main results, some definitions are first introduced.
Definition 1
The equilibrium point of system (1) is said to be uniformly asymptotically stable if for any \(\varepsilon > 0\), there is a \(\delta ( \varepsilon ) > 0\) such that \(\max_{  h \le t \le 0}\Vert \phi ( t ) \Vert < \delta\) implies \(\Vert x ( t,\phi ) \Vert \le \varepsilon\), \(t \ge 0\). For arbitrary switching \(\sigma ( t )\), there is also a \(\delta '\) such that \(\max_{  h \le t \le 0}\Vert \phi ( t ) \Vert < \delta '\) implies \(\Vert x ( t,\phi ) \Vert \to 0\) as \(t \to \infty\).
Now, the following lemma and criterion are preliminarily presented, which will play important roles in our further derivation.
Kotelyanski lemma
[58]
The real parts of the eigenvalues of the matrix A, with nonnegative offdiagonal elements, are less than a real number μ if and only if all those of the matrix M; \(M = \mu I_{n}  A\) are positive, with \(I_{n}\) the n identity matrix.
BorneGentina practical stability criterion
[30]
Let consider the autonomous nonlinear continuous process described in state space by \(\dot{x} = A (\cdot )x\); \(A (\cdot )\) is an \(n \times n\) matrix, \(A (\cdot ) = \{ a_{ij} \}_{1 \le i,j \le n}\). If the overvaluing matrix \(M ( A (\cdot ) )\) has its nonconstant elements isolated in only one row, the verification of the Kotelyanski condition enables one to conclude to the stability of the initial system.
The following theorem of the Mmatrix properties is required.
Theorem 1
[44]

The principal minors of A are positive:$$ ( A )\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 1& 2&\cdots& j \\ 1& 2&\cdots& j \end{array}\displaystyle \right ) > 0, \quad \forall j = 1,\ldots,n. $$(2)

For any positive real numbers \(\eta = ( \eta_{1},\ldots,\eta_{n} )^{T}\) the algebraic equations \(Ax = \eta\) have a positive solution \(w = ( w_{1},\ldots,w_{n} )\).
Remark 1
A continuoustime system characterized by a matrix A is stable if the matrix A is the opposite of an Mmatrix.
Here the definition of a pseudoovervaluing matrix is given, which will be used in the subsequent analysis.
Definition 2

The offdiagonal elements of the matrix \(T_{c} (\cdot )\) are nonnegative.

When all the real parts of the eigenvalues of \(T_{c} (\cdot )\) are negative, this matrix is the opposite of an Mmatrix. It allows an inverse whose elements are all nonpositive.

When we denote by \(\lambda_{M} < 0\) the real part of the eigenvalue of maximum real part of \(T_{c} (\cdot )\), it comes the inequalities: \(\operatorname{Re} ( \lambda_{M_{C} (\cdot )} ) \le \lambda_{M}\).

When its inverse is an irreducible matrix, then \(T_{c} (\cdot )\) admits an eigenvector \(u ( x,t )\) relative to the eigenvalues \(\lambda_{M}\) and whose components are strictly positive.

In addition, if we suppose now that the nonconstant elements are isolated in only one row of \(T_{c} (\cdot )\). Then the main eigenvector u related to the main eigenvalue \(\lambda_{M}\) is a constant vector.
Remark 2
In the case that the nonlinear elements of the matrix \(A (\cdot )\) are isolated in only one row, the conditions of Theorem 1, Definition 2 and Remark 1 still valid.
Assumption 1
We suppose that the nonlinear elements of the pseudoovervaluing matrix \(T_{c} (\cdot )\) are isolated in only the last row.
3 Main results
Theorem 2
Proof
Let \(w \in \Re^{n}\) with components (\(w_{m} > 0\), \(\forall m = 1,\ldots,n \)) and \(x ( t ) \in \Re^{n}\) is the state vector.
It is clear that \(V ( t = 0 ) \ge 0\).
4 Stability of switched nonlinear systems modeled by delay differential equations
Theorem 3
Proof
Corollary 1
Proof
The proof is completed. □
5 Stability analysis of switched systems with multiple delays
In this part, a generalization of the previous results will be given to a class of switched nonlinear systems with multiple delays.
Generalizing the Lyapunov function introduced in (5), it is easy to obtain the following sufficient stability conditions for system (46).
Theorem 4
Proof
Theorem 5
Next, Theorem 5 can be simplified to the following delaydependent stability conditions.
Corollary 2
Proof
6 Simulation results
The two delays are \(h_{1} > 0\) and \(h_{2} > 0\).
Note that, due to the complexity and the important number of the subsystems, it is difficult to find a common Lyapunov function for all the subsystems. Then we cannot guarantee stability of this switched system under the arbitrary switching sequence (1).
Therefore, Figure 3, Figure 4, and Figure 5 allow one to conclude that the switched system converges to zero. This implies that the system given in this example is globally asymptotically stable, which demonstrates the effectiveness of the proposed method.
This example shows that the obtained stability conditions are sufficient and very close to be necessary; on the other hand, the proposed results make it possible to avoid searching a common Lyapunov function, which is a very difficult matter in this case.
7 Conclusion
In this paper, a new approach for the stability analysis problem of a class of continuoustime switched nonlinear timedelay systems under arbitrary switched rules has been developed. By introducing a new constructed common Lyapunov function, the application of the BorneGentina criterion, the Mmatrix properties, the aggregation techniques, and the vector norms notion. New delayindependent stability conditions under arbitrary switching are deduced. In addition, these obtained conditions are extended to be generalized for switched nonlinear systems with multiple delays. Compared with the existing results, the benefit of this method is that it can avoid the research of a common Lyapunov function which is usually very difficult, or even not possible. A numerical example is given to demonstrate the applicability of the proposed approach.
Note that these proposed results could be further used as a constructive solution to the problems of state and static output feedback stabilization.
The limit of this paper is that it has been confined to the boundaries of numerical examples. It would be beneficial to extend the research further so as to include real systems.
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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