An introduction to symbolic dynamics and coding
An introduction to symbolic dynamics and coding
SIAM Journal on Matrix Analysis and Applications
Introduction to Automata Theory, Languages and Computability
Introduction to Automata Theory, Languages and Computability
Computationally Efficient Approximations of the Joint Spectral Radius
SIAM Journal on Matrix Analysis and Applications
Distributed Lyapunov Functions in Analysis of Graph Models of Software
HSCC '08 Proceedings of the 11th international workshop on Hybrid Systems: Computation and Control
Joint Spectral Characteristics of Matrices: A Conic Programming Approach
SIAM Journal on Matrix Analysis and Applications
Uniform stabilization of discrete-time switched and Markovian jump linear systems
Automatica (Journal of IFAC)
Fast Methods for Computing the $p$-Radius of Matrices
SIAM Journal on Scientific Computing
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We study the problem of approximating the joint spectral radius (JSR) of a finite set of matrices. Our approach is based on the analysis of the underlying switched linear system via inequalities imposed between 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, maximum/minimum-of-quadratics Lyapunov functions. We characterize all path-complete graphs consisting of two nodes on an alphabet of two matrices and compare their performance. For the general case of any set of n x n matrices we propose semidefinite programs of modest size that approximate the JSR within a multiplicative factor of 1/4√n of the true value. We establish a notion of duality among path-complete graphs and a constructive converse Lyapunov theorem for maximum/minimum-of-quadratics Lyapunov functions.