Graph Control Lypaunov Function for Switched Linear Systems

摘要

The goal of this thesis is to study stabilization of discrete-time switched linear systems (SLSs) and controlled switched linear systems (CSLSs). To analyze stabilizability of SLSs and CSLSs, we introduce the notion of graph control Lyapunov functions (GCLFs), which is a graph theoretic approach to standard Lyapunov theorems. The GCLF is a set of Lyapunov functions which satisfy Lyapunov inequalities associated with a weighted directed graph (digraph). Each Lyapunov function represents each node in the digraph, and each Lyapunov inequality represents a subgraph consisting of edges connecting a node and its out-neighbors (directed rooted tree). The weight of each directed edge indicates the decay or growth rate of the Lyapunov functions from the tail to the head of the edge. It is proved that a SLS is switching stabilizable if and only if there exists a GCLF. The GCLF is an extension of recently developed graph Lyapunov functions for stability of SLSs under arbitrary switchings to stabilization of SLS under controlled switchings. We prove that GCLFs unify several existing control Lyapunov functions and related stabilization theorems. As a special class of GCLFs, we also study periodic control Lyapunov functions (PCLFs) whose value decreases periodically instead of at each time step as in the classical control Lyapunov functions. The PCLF is a GCLF with a single node and a self-loop. Using PCLFs, we develop stabilizability and control design conditions for SLSs and CSLSs. The PCLF approach is less conservative than existing results in that they apply to a larger class of SLSs and CSLSs. Computational algorithms are developed to find GCLFs/PCLFs and check stabilizability of SLSs (CSLSs).