In this paper we deal with the stabilizability property for discrete-time switched linear systems. A recent necessary and sufficient characterization of stabilizability, based on set theory, is considered as the reference for comparing the computation-oriented sufficient conditions. The classical BMI conditions based on Lyapunov-Metzler inequalities are considered and extended. Novel LMI conditions for stabilizability, derived from the geometric ones, are presented that permit to combine generality with computational affordability. For the different conditions, the geometrical interpretations are provided and the induced stabilizing switching laws are given. The relations and the implications between the stabilizability conditions are analyzed to infer and compare their conservatism and their complexity.Index Terms-Switched systems, stabilizability, LMI.M. Fiacchini is with GIPSA-lab,

A class of Lyapunov functions is proposed for discrete-time linear systems interconnected with a cone bounded nonlinearity. Using these functions, we propose sufficient conditions for the global stability analysis, in terms of linear matrix inequalities (LMI), only taking the bounded sector condition into account. Unlike frameworks based on the Lur'e-type function, the additional assumptions about the derivative or discrete variation of the nonlinearity are not necessary. Hence, a wider range of cone bounded nonlinearities can be covered. We also show that there is a link between global stability LMI conditions based on this new Lyapunov function and a transfer function of an auxiliary system being strictly positive real. In addition, the novel function is considered in the local stability analysis problem of discrete-time Lur'e systems subject to a saturating feedback. A convex optimization problem based on sufficient LMI conditions is formulated to maximize an estimate of the basin of attraction. Another specificity of this new Lyapunov function is the fact that the estimate is composed of disconnected sets. Numerical examples reveal the effectiveness of this new Lyapunov function in providing a less conservative estimate with respect to the quadratic function.

In this paper, the stabilizability of discrete-time linear switched systems is considered. Several sufficient conditions for stabilizability are proposed in the literature, but no necessary and sufficient. The main contributions are the necessary and sufficient conditions for stabilizability based on set-theory and the characterization of a universal class of Lyapunov functions. An algorithm for computing the Lyapunov functions and a procedure to design the stabilizing switching control law are provided, based on such conditions. Moreover a sufficient condition for non-stabilizability for switched system is presented. Several academic examples are given to illustrate the efficiency of the proposed results. In particular, a Lyapunov function is obtained for a system for which the Lyapunov-Metzler condition for stabilizability does not hold.