Joint Spectral Radius and Path-Complete Graph Lyapunov Functions

Abstract

Abstract. We introduce the framework of path-complete graph Lyapunov functions for approximation of the joint spectral radius. The approach is based on the analysis of the underlying switched system via inequalities imposed among multiple Lyapunov functions associated to a labeled directed graph. Inspired by concepts in automata theory and symbolic dynamics, we define a class of graphs called path-complete graphs, and show that any such graph gives rise to a method for proving stability of the switched system. This enables us to derive several asymptotically tight hierarchies of semidefinite programming relaxations that unify and generalize many existing techniques such as common quadratic, common sum of squares, path-dependent quadratic, and maximum/minimumof-quadratics Lyapunov functions. We compare the quality of approximation obtained by certain classes of path-complete graphs including a family of dual graphs and all path-complete graphs with two nodes on an alphabet of two matrices. We derive approximation guarantees for several families of path-complete graphs, such as the De Bruijn graphs. This provides worst-case perfomance bounds for path-dependent quadratic Lyapunov functions and a constructive converse Lyapunov theorem for maximum/minimum-of-quadratics Lyapunov functions.

On Path-Complete Lyapunov Functions: Geometry and Comparison

“…This relates to the setting of [1], where Path-Complete Lyapunov functions with quadratic pieces are used for the approximation of the exponential growth rate, a.k.a. the joint spectral radius [23], [35], of switching systems.…”

“…In order to further analyze such tools, Ahmadi et al recently introduced the concept of Path-Complete Lyapunov functions [1]. There, multiple Lyapunov functions such as the one of [11] mentioned above are represented by directed and labeled graphs, see Figure 1.…”

“…Fundamental properties of a multiple Lyapunov function represented by a graph G can be deduced from the properties of that graph. In particular, in [1], [24] the authors provide a characterization of the set of graphs that represent stability certificates for switching systems on M modes. This property, known as Path-Completeness, leads to the concept of PathComplete Lyapunov functions (see Definition II.3 below).…”

“…Several challenges exist in the study of Path-Complete Lyapunov functions, in particular for matters related to how these certificates compare with each other with respect to their conservativeness 2 . In this paper we first ask a natural question which aims at revealing the 1 In a more general setting, the labels on the edges can be taken as finite sequences of modes. Our results extend there through the socalled expanded graph [1, Definition 2.1].…”

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

“…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].…”