TY - GEN

T1 - Analysis of the joint spectral radius via Lyapunov functions on path-complete graphs

AU - Ahmadi, Amir Ali

AU - Jungers, Raphaël M.

AU - Parrilo, Pablo A.

AU - Roozbehani, Mardavij

N1 - Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.

PY - 2011

Y1 - 2011

N2 - 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 × 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.

AB - 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 × 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.

KW - Finite automata

KW - Joint spectral radius

KW - Lyapunov methods

KW - Semidefinite programming

KW - Stability of switched systems

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U2 - 10.1145/1967701.1967706

DO - 10.1145/1967701.1967706

M3 - Conference contribution

AN - SCOPUS:79956010261

SN - 9781450306294

T3 - HSCC'11 - Proceedings of the 2011 ACM/SIGBED Hybrid Systems: Computation and Control

SP - 13

EP - 22

BT - HSCC'11 - Proceedings of the 2011 ACM/SIGBED Hybrid Systems

T2 - 14th ACM International Conference on Hybrid Systems: Computation and Control, HSCC 2011

Y2 - 12 April 2011 through 14 April 2011

ER -