On the Stabilizability of Discrete-Time Switched Linear Systems: Novel Conditions and Comparisons

Abstract

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,

Output feedback stabilization of switching discrete-time linear systems with parameter uncertainties

“…Matlab LMI toolbox, we get that both LMI (31) and LMI (18) are feasible. Note that the solvability of (18) Table 2: Observer-based controllers for the proposed LMI design applied to system (38) The simulation results corresponding to these observer-based controller gains obtained by solving LMIs (31) are given in Figure 1.…”

“…As compared with the other conditions presented in this paper, following the comment in subsection 4.2.3, if we use the gridding method, we must solve conditions (18) by scaling the parameters i , γ i , µ i via the change of variables…”

“…Note that the switching instants in Figure 1 In order to boost comparisons between the proposed LMI conditions (31), (18) and (36), we add to the previous example parameter uncertainties as follows: Table 2, and for the same step of discretization. The results are summarized in Table 3.…”

“…Among the existing methods, we have: dwell-time and average dwell-time approaches for stability analysis and stabilization problems [21,43]; approaches based on a specific class of switching laws [4], [23] and under arbitrary switching sequences [3]; sliding mode technique [39]; algebraic approach [2]; Lyapunov-Metzler approach 15 [14,18]; input-output approach [31].…”

Stabilization and control Lyapunov functions for language constrained discrete-time switched linear systems

“…Concerning the problem of stabilizability of switched system, it is known that convex Lyapunov functions lead to conservative results, and nonconvex ones must be considered, see [6]. Nonconvex Lyapunov functions induced by the union of ellipsoids are employed in [7,11,26,12,8], while more general homogeneous functions have been considered in [9].…”