Analysis of the Joint Spectral Radius Via Lyapunov Functions on Path-Complete Graphs
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Parrilo_Analysis of the joint.pdf
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Author(s)
Ahmadi, Amir Ali
Jungers, Raphael M.
Parrilo, Pablo A.
Roozbehani, Mardavij
Date Issued
April 2011
Journal
Proceedings of the 14th international conference on Hybrid systems: computation and control (HSCC '11)
Publisher
Association for Computing Machinery (ACM)
Citation
Amir Ali Ahmadi, Rapha\&\#235;l Jungers, Pablo A. Parrilo, and Mardavij Roozbehani. 2011. Analysis of the joint spectral radius via lyapunov functions on path-complete graphs. In Proceedings of the 14th international conference on Hybrid systems: computation and control (HSCC '11). ACM, New York, NY, USA, 13-22.
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Author's final manuscript
Abstract
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/[superscript 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.
MIT Department
Massachusetts Institute of Technology. Laboratory for Information and Decision Systems
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DOI of Published Version
https://doi.org/10.1145/1967701.1967706
