Ahmadi, Amir Ali; Jungers, Raphaël M.; Parrilo, Pablo A.; Roozbehani, Mardavij Joint spectral radius and path-complete graph Lyapunov functions. (English) Zbl 1292.93093 SIAM J. Control Optim. 52, No. 1, 687-717 (2014). Summary: 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. Cited in 1 ReviewCited in 31 Documents MSC: 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory 93B60 Eigenvalue problems 93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) 65Q10 Numerical methods for difference equations 37C75 Stability theory for smooth dynamical systems 68Q45 Formal languages and automata 90C22 Semidefinite programming 93D30 Lyapunov and storage functions 05C90 Applications of graph theory Keywords:joint spectral radius; stability of switched systems; linear difference inclusions; finite automata; Lyapunov methods; semidefinite programming Software:SeDuMi; JSR PDFBibTeX XMLCite \textit{A. A. Ahmadi} et al., SIAM J. Control Optim. 52, No. 1, 687--717 (2014; Zbl 1292.93093) Full Text: DOI arXiv Link