Jamal Daafouz

Brief paper: On the algebraic characterization of invariant sets of switched linear systems

In this paper, a suitable LaSalle principle for continuous-time linear switched systems is used to characterize invariant sets and their associated switching laws. An algorithm to determine algebraically these invariants is proposed. The main novelty of our approach is that we require no dwell time conditions on the switching laws. By not focusing on restricted control classes we are able to describe the asymptotic properties of the considered switched systems. Observability analysis of a flying capacitor converter is proposed as an illustration.

Stabilization of continuous-time singularly perturbed switched systems

In this article, stability of continuous-time switched linear systems in the singular perturbation form is investigated. We show that the stability of slow and fast switched subsystems is not a sufficient condition for stability of the corresponding two-time scale switched system, under an arbitrary switching law. Thus, LMI conditions to design a state-feedback control law stabilizing continuous-time singularly perturbed switched linear systems are proposed.

Stability Analysis of Singularly Perturbed Switched Linear Systems

This note is concerned with the stability of planar linear singularly perturbed switched systems. We propose a characterization of the stability properties of such multiple time scale switched systems as the perturbation parameter goes to zero. We also study transitions as this parameter varies and we restrict their number and nature. Finally, we compare the results obtained in this way with the Tikhonov-type results for differential inclusions available in the literature.

On stability analysis of linear discrete-time switched systems using quadratic Lyapunov functions

The paper deals with the stability properties of linear discrete-time switched systems with polytopic sets of dynamics. The most classical and viable way of studying the uniform asymptotic stability of such a system is to check for the existence of a quadratic Lyapunov function. It is known from the literature that letting the Lyapunov function depend on the time-varying dynamic improves the chance that a quadratic Lyapunov function exists. We prove that the dependence on the dynamic can be actually assumed to be linear, with no prejudice on the effectiveness of the method. Moreover, we show that no gain in the sensibility is obtained if we allow the Lyapunov function to depend on the time as well. We conclude by showing that Lyapunov quadratic stability is a strictly stronger notion that uniform asymptotic stability.

Suboptimal switched controls in context of singular arcs

In this paper suboptimal control in the case of hybrid systems is addressed. After a brief recall of necessary conditions on the optimal hybrid trajectory, it is shown that many hybrid optimal problems cannot have an hybrid solution. Nevertheless, we show in context of switched systems that there is some possibility to take advantage of the study of a convex embedding problem for which a solution exists. Indeed the presence of singular arcs in this solution explains why the original switched system has no solution and how suboptimal chattering solutions can be found.

Stabilization of Discrete-Time Nonlinear Systems Subject to Input Saturations: A New Lyapunov Function Class

Abstract This paper addresses the problem of stabilization of discrete-time systems including a cone-bounded nonlinearity and a saturating actuator. In the sense of Lyapunov stability, we introduce a new candidate Lyapunov function which takes nonlinearity behavior into account. The local stability criterion is formulated as a set of Bilinear Matrix Inequalities (BMI) conditions. We present an optimization problem in order to guarantee the closed-loop stability aiming the largest basin of attraction, which may be nonconvex, and/or, nonconnected. Furthermore, a simple iterative algorithm is proposed in order to solve our BMI problem. Some numerical examples are presented to highlight the relevance of the new Lyapunov function in regard to the classical quadratic function.

Bumpless transfer for discrete-time switched systems

A bumpless transfer method for discrete-time switched linear systems is presented. It is based on an additional controller which is activated at the switching time for reducing the control discontinuities. Dwell time conditions to guarantee the stability of the closed-loop system are provided. Simulation tests on the Eisenhüttenstadt hot strip mill of ArcelorMittal are shown.

Characterization of stability transitions and practical stability of planar singularly perturbed linear switched systems

This paper is concerned with the stability of planar linear singularly perturbed switched systems in continuous time. Based on a necessary and sufficient stability condition, we characterize all possible stability transitions for this class of switched systems and we propose a practical stability result. We answer the questions related to what happen as ∈, the singular perturbation parameter, grows and how many times the system can change its stability behavior (asymptotic stability, stability, instability) and which transitions are possible. Moreover, we analyze practical stability from the viewpoint of Tikhonov approach and in particular based on existing results obtained in the context of differential inclusions. We show that these approaches can be applied to singularly perturbed switched systems allowing to prove practical stability in some cases. This kind of stability focuses on the behavior of the system on compact time-intervals as ∈ tends to 0 (in particular, it does not ensure the asymptotic stability towards the origin). It is therefore different from the stability criteria where ∈ is fixed (arbitrarily small) and the asymptotic behavior for large times is considered. For planar systems, it turns out that when practical stability can be deduced from Tikhonov-type results, then global uniform asymptotic stability (for ∈ > 0 small) holds true. It is an open question whether this is still true for higher dimensional singularly perturbed switched systems.

Stability of planar singularly perturbed switched systems

This paper is concerned with the stability analysis of planar linear singularly perturbed switched systems. We show that this class of switched systems has always a stability behavior common to all switched systems corresponding to small values of the singular perturbation parameter. Moreover, we propose necessary and sufficient conditions for the asymptotic stability.

On Performance Analysis of Time Delay Systems using a Switched System Approach

Abstract The presence of a bounded time-varying delay in discrete-time controlled system implies a deterioration of the performance (stability and guaranteed cost). This paper studies controllers design for such a system, leading to a stabilized controlled system and to an upper bound of guaranteed cost as small as possible. The two contexts of state or only output availability are considered. These controllers are obtained via optimization algorithms using Linear Matrix Inequalities and Lyapunov-Krasovskii functions approach. An example illustrates the main results.