 
Summary: Exponential Stabilization of
DiscreteTime Switched Linear Systems
Wei Zhang a,, Alessandro Abate b
, Jianghai Hu a
, Michael P. Vitus b
a
Department of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47906, USA
b
Department of Aeronautics and Astronautics, Stanford University, CA 94305, USA
Abstract
This paper studies the exponential stabilization problem for discretetime switched linear systems based on a controlLyapunov
function approach. It is proved that a switched linear system is exponentially stabilizable if and only if there exists a piecewise
quadratic controlLyapunov function. Such a converse controlLyapunov function theorem justifies many of the earlier synthesis
methods that have adopted piecewisequadratic Lyapunov functions for convenience or heuristic reasons. In addition, it is also
proved that if a switched linear system is exponentially stabilizable, then it must be stabilizable by a stationary suboptimal
policy of a related switched LQR problem. Motivated by some recent results of the switched LQR problem, an efficient
algorithm is proposed, which is guaranteed to yield a controlLyapunov function and a stabilizing policy whenever the system
is exponentially stabilizable.
Key words: Switched systems, Piecewise quadratic Lyapunov functions, Switching stabilization, Optimal control,
ControlLyapunov functions.
