Joint Spectral Radius and Path-Complete Graph Lyapunov Functions
Massachusetts Institute of Technology（麻省理工学院）；Université catholique de Louvain（天主教鲁汶大学）
Amir Ali Ahmadi, Raphaël Jungers, Pablo A. Parrilo, Mardavij Roozbehani
备注：
To appear in SIAM Journal on Control and Optimization. Version 2 has gone through two major rounds of revision. In particular, a section on the performance of our algorithm on application-motivated problems has been added and a more comprehensive literature review is presented
发表信息：
SIAM J. Control and Optimization, Vol. 52, No. 1, pp. 687-717, 2014
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, 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 provide approximation guarantees for several families of path-complete graphs, such as the De Bruijn graphs, establishing as a byproduct a constructive converse Lyapunov theorem for maximum/minimum-of-quadratics Lyapunov functions.
Automated Synthesis of Lyapunov Functions for Multi-Agent Systems under Jointly Connected Topology
ICTEAM Institute UCLouvain（UCLouvain ICTEAM研究所）；Sch. Automat. Sci. Elect. Engineering Beihang University（北京航空航天大学自动化科学与电气工程学院）；Sch. Mechanical Engineering Wuhan University of Science and Technology（武汉科技大学机械工程学院）
