Joint Spectral Radius and Path-Complete Graph Lyapunov Functions
Author(s)
Ahmadi, Amir Ali; Parrilo, Pablo A.; Roozbehani, Mardavij; Jungers, Raphael M.
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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.
Date issued
2014-02Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science; Massachusetts Institute of Technology. Laboratory for Information and Decision SystemsJournal
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
ISSN
0363-0129
1095-7138