A numerical technique for the stability analysis of linear switched systems

Abstract

SummaryIn this report the ray-gridding approach, a new numerical technique for the stability analysis of linear switched systems is presented. It is based on uniform partitions of the state-space in terms of ray directions which allow refinable families of polytopes of adjustable complexity to be examined for invariance. In this framework the existence of a polyhedral Lyapunov function that is common to a family of asymptotically stable subsystems can be checked efficiently via simple iterative algorithms. The technique can be used to prove the stability of switched linear systems, classes of linear timevarying systems and Linear Differential Inclusions.We also present preliminary results on two other related problems; namely, the existence of stabilising switching sequences for a switched system constructed from a family of unstable linear subsystems and the construction of multiple polyhedral Lyapunov functions.

Stability of Switched Linear Differential Systems

“…3 reduces the computation of quadratic MLFs to the solution of a system of structured LMIs (11)-(12), a straightforward matter for standard LMI solvers. The class of polyhedral Lyapunov functions (PLFs) is universal for linear systems with structured uncertainties; in [37] PLFs are applied to linear switched systems in state space form, and a numerical procedure to overcome the complexity of PLF computations is illustrated, see pp. 1021-1022 ibid.…”

Polytope Lyapunov Functions for Stable and for Stabilizable LSS

“…That is why, other classes of functions have been used in the literature to construct Lyapunov functions: positive polynomials of higher degree, SOS, piecewise-quadratic, piecewise-linear, etc. (see surveys [37,60]). In contrast to quadratic functions, all those classes are dense which implies their universality, i.e., every stable LSS has a Lyapunov function from those classes.…”

“…The first polytope algorithms originated in late eighties with Molchanov and Pyatnitskii [40,41] and Barabanov [5]. Then this method was developed in various directions by Amato, Ambrosino, Ariola, Blanchini, Miani, Julian, Guivant, Desages, Polanski, Shorten, Yfoulis and others (see [1,6,7,9,31,39,43,44,60]). The polytope function can be easily defined by faces of the corresponding level polyhedron P (unit ball):…”

“…The main challenge in the design of the polytope Lyapunov functions is to chose the location of vertices in a proper way. In [39,43,59,60] this is done by placing all vertices on a given system of ray directions. First, one construct a system of rays that actually form an ε-net on the unit sphere in R d .…”

“…In [39,43] this is done by solving LP problems. In [59,60] the authors introduce iterative ray-gridding approach and demonstrate its efficiency in examples of dimensions d = 2 and d = 3. Unfortunately, in higher dimensions the number of vertices grows dramatically, which makes those methods hardly applicable.…”