Graph Lyapunov function for switching stabilization and distributed computation

Abstract

This paper studies stabilization of discrete-time switched linear systems (SLSs) using the notion of graph control Lyapunov functions (GCLFs). A GCLF is a set of Lyapunov functions defined on a weighted digraph, where each Lyapunov function is represented by a node in the digraph and there is a Lyapunov inequality associated with each subgraph consisting of a node and its out-neighbors. The weight of a directed edge indicates the decay or growth rate of the Lyapunov functions. It is proved that an SLS is switching stabilizable if and only if there exists a GCLF. The main benefits of GCLFs are reduced computational cost and conservatism for stabilizability tests. Besides, we show that the proposed GCLF framework unifies several control Lyapunov functions and the related stabilization theorems. Moreover, we propose a distributed algorithm to evaluate the stabilizability with reduced computational costs by taking benefits of the graph structure of GCLFs. Several examples are given to demonstrate the efficiency of the algorithm.