Journal
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
Volume 52, Issue 1, Pages 687-717Publisher
SIAM PUBLICATIONS
DOI: 10.1137/110855272
Keywords
joint spectral radius; stability of switched systems; linear difference inclusions; finite automata; Lyapunov methods; semidefinite programming
Categories
Funding
- NSF [DMS-0757207, CPS-1135843]
- AFOSR MURI [07688-1]
- Communaute Francaise de Belgique-Actions de Recherche Concertees
- Belgian Programme on Interuniversity Attraction Poles
- Div Of Electrical, Commun & Cyber Sys
- Directorate For Engineering [1135843] Funding Source: National Science Foundation
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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/minimum-of-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 performance bounds for path-dependent quadratic Lyapunov functions and a constructive converse Lyapunov theorem for maximum/minimum-of-quadratics Lyapunov functions.
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