Marwen Kermani

Stability and stabilization studies for a class of switched nonlinear systems via vector norms approach

This paper is concerned with the problems of stability analysis and stabilization with a state feedback controller through pole placement for a class of both continuous and discrete-time switched nonlinear systems. These systems are modeled by differential or difference equations. Then, a transformation under the arrow form is employed. Note that, the main contribution in this work is twofold: firstly, based on the construction of an appropriated common Lyapunov function, as well the use of the vector norms notion, the recourse to the Kotelyanski lemma, the M-matrix proprieties, the aggregation techniques and the application of the Borne-Gentina criterion, new sufficient stability conditions under arbitrary switching for the autonomous system are deduced. Secondly, this result is extended for designing a state feedback controller by using pole assignment control, which guarantee that the corresponding closed-loop system is globally asymptotically stable under arbitrary switching. The main novelties features of these obtained results are the explicitness and the simplicity in their application. Moreover, they allow us to avoid the search of a common Lyapunov function which is a difficult matter. Finally, as validation to stabilize a shunt DC motor under variable mechanical loads is performed to demonstrate the effectiveness of the proposed results.

On stability analysis of discrete-time uncertain switched nonlinear time-delay systems

This paper addresses the stability analysis problem for a class of discrete-time switched nonlinear time-delay systems with polytopic uncertainties. These considered systems are characterized by delayed difference nonlinear equations which are given in the state form representation. Then, a transformation under the arrow form is employed. Indeed, by constructing an appropriated common Lyapunov function, and also by resorting to the Kotelyanski lemma and the M-matrix proprieties, new delay-independent stability conditions under arbitrary switching law are deduced. Compared with the existing results of switched systems, those obtained results are formulated in terms of the unknown polytopic uncertain parameters, explicit and easy to apply. Moreover, this method allows us to avoid the search for a common Lyapunov function which is a difficult matter. Finally, a numerical example is presented to illustrate the effectiveness of the proposed approach.

A new stability analysis and stabilization of uncertain switched linear systems based on vector norms approach

In the present paper a new stability analysis and stabilization of continuous-time uncertain switched linear systems is considered. This approach is based on the comparison, the overvaluing principle, the application of Borne-Gentina criterion and the Kotelyanski conditions. The stability conditions issued from vector norms correspond to a vector Lyapunov function. Indeed, the switched system to be controlled will be represented in the Companion form. A comparison system relative to regular vector norms is used in order to get the simple arrow form of the state matrix that yields to a suitable use of Borne-Gentina criterion for the establishment of sufficient conditions as function of the uncertain parameters for global asymptotic stability.

Stability analysis and stabilization of switched linear systems based on vector norms approach

The stability analysis and the stabilization problems for continuous-time switched linear systems are studied in this paper, by investigating a new stability conditions based on the comparison, the overvaluing principle, the application of Borne-Gentina criterion and the Kotelyanski conditions. These stability conditions issued from vector norms correspond to a vector Lyapunov function. Indeed, the switched system to be controlled will be represented in the Companion form. A comparison system relative to a regular vector norm is used in order to get the simple arrow form of the state matrix that yields to a suitable use of Borne-Gentina criterion for the establishment of sufficient conditions for global asymptotic stability.

On the stability analysis of switched nonlinear systems with time varying delay under arbitrary switching

This paper addresses the stability problem for a class of switched nonlinear time varying delay systems modeled by delay differential equations. By transforming the system representation under the arrow form and using a new constructed Lyapunov function, the aggregation techniques, the Borne-Gentina practical stability criterion associated with the properties, new delay-independent stability conditions of the considered systems are established. Compared with the existing results in this area, the obtained result is explicit, simple to use and allows 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.

Delay-Dependent Stability Analysis of TS Fuzzy Switched Time-Delay Systems

This paper proposes a new approach to deal with the problem of stability under arbitrary switching of continuous-time switched time-delay systems represented by TS fuzzy models. The considered class of systems, initially described by delayed differential equations, is first put under a specific state space representation, called arrow form matrix. Then, by constructing a pseudo-overvaluing system, common to all fuzzy submodels and relative to a regular vector norm, we can obtain sufficient asymptotic stability conditions through the application of Borne and Gentina practical stability criterion. The stability criterion, hence obtained, is algebraic, is easy to use, and permits avoiding the problem of existence of a common Lyapunov-Krasovskii functional, considered as a difficult task even for some low-order linear switched systems. Finally, three numerical examples are given to show the effectiveness of the proposed method.

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 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.

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

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.

New criteria for asymptotic stability of a class of discrete-time TS fuzzy switched time-delay systems

Based on vector norms approach, this paper addresses the problem of stability under arbitrary switching of nonlinear discrete-time switched systems with constant time-delay using TS fuzzy models. The idea consists in constructing an overvaluing system, common to all fuzzy subsystems and whose stability permits to conclude to that of the original system.

A new stability approach of fuzzy switched continuous-time nonlinear systems

New sufficient stability conditions for continuous-time TS fuzzy switched nonlinear systems under arbitrary switching law are proposed. This approach is based on the application of the overvaluing principle, the Borne and Gentina criterion associated to the arrow form matrix and the Kotelyanski condition. To demonstrate the effectiveness of the proposed method, a numerical example is provided.