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On Piecewise Quadratic Control-Lyapunov Functions for Switched Linear Systems

Summary: On Piecewise Quadratic Control-Lyapunov Functions
for Switched Linear Systems
Wei Zhang, Alessandro Abate, Michael P. Vitus and Jianghai Hu
Abstract-- In this paper, we prove that a discrete-time
switched linear system is exponentially stabilizable if and only
if there exists a stationary hybrid-control law that consists
of a homogeneous switching-control law and a piecewise-
linear continuous-control law under which the closed-loop
system has a piecewise quadratic Lyapunov function. Such
a converse control-Lyapunov function theorem justifies many
of the earlier controller-synthesis methods that have adopted
piecewise-quadratic Lyapunov functions and piecewise-linear
continuous-control laws for convenience or heuristic reasons.
Furthermore, several important properties of the proposed
stabilizing control law are derived and their connections to
other existing controllers studied in the literature are discussed.
The stabilization problem of switched systems, especially
autonomous switched linear systems, is receiving increasing
research attention in recent years ([1], [2]). Many existing


Source: Abate, Alessandro - Faculty of Mechanical, Maritime and Materials Engineering, Technische Universiteit Delft
Zhang, Wei - Department of Electrical and Computer Engineering, Ohio State University


Collections: Computer Technologies and Information Sciences; Engineering