Joint Spectral Radius and Path-Complete Graph Lyapunov Functions
Name
Ahmadi-2014-Joint spectral radiu.pdf
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1.28 MB
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Adobe PDF
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Author(s)
Ahmadi, Amir Ali
Parrilo, Pablo A.
Roozbehani, Mardavij
Jungers, Raphael M.
Date Issued
February 2014
Journal
SIAM Journal on Control and Optimization
Publisher
Society for Industrial and Applied Mathematics
Citation
Ahmadi, Amir Ali, Raphaël M. Jungers, Pablo A. Parrilo, and Mardavij Roozbehani. “Joint Spectral Radius and Path-Complete Graph Lyapunov Functions.” SIAM J. Control Optim. 52, no. 1 (January 2014): 687–717. © 2014, Society for Industrial and Applied Mathematics
Version
Final published version
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/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.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Laboratory for Information and Decision Systems
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1137/110855272
