Exponential stabilization of discrete-time switched linear systems

  • Authors:

  • Wei Zhang;Alessandro Abate;Jianghai Hu;Michael P. Vitus

  • Affiliations:

  • Department of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47906, USA;Department of Aeronautics and Astronautics, Stanford University, CA 94305, USA;Department of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47906, USA;Department of Aeronautics and Astronautics, Stanford University, CA 94305, USA

  • Venue:
  • Automatica (Journal of IFAC)
  • Year:
  • 2009

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Abstract

This article studies the exponential stabilization problem for discrete-time switched linear systems based on a control-Lyapunov function approach. It is proved that a switched linear system is exponentially stabilizable if and only if there exists a piecewise quadratic control-Lyapunov function. Such a converse control-Lyapunov function theorem justifies many of the earlier synthesis methods that have adopted piecewise quadratic 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 linear-quadratic regulator (LQR) problem. Motivated by some recent results of the switched LQR problem, an efficient algorithm is proposed, which is guaranteed to yield a control-Lyapunov function and a stabilizing policy whenever the system is exponentially stabilizable.