Controllability of switched linear systems

In this note, the controllability of a switched linear system, which is a collection of linear time-invariant systems along with some maps for “switching” among them, is addressed. First, a controllable state set is defined as a basic tool to discuss the controllability condition. Second, it is proved that the controllability of a multi-input system is equivalent to that of a single-input system. Third, a sufficient and necessary condition for controllability of third-order systems is presented. According to the proof of this criterion, we state a conjecture on the controllability condition of -dimensional systems. A numeric example is given to illustrate the result.

[1]  A. Morse.  Supervisory control of families of linear set-point controllers , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[2]  H. Sussmann.  A general theorem on local controllability , 1987 .

[3]  Kumpati S. Narendra,et al.  Adaptive control using multiple models , 1997, IEEE Trans. Autom. Control..

[4]  A. Haddad,et al.  On the Controllability and Observability of Hybrid Systems , 1988, 1988 American Control Conference.

[5]  A. Michel,et al.  Stability analysis of switched systems , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[6]  Yin Zhenyu,et al.  Hybrid controllability of linear switched systems , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[7]  A. Michel,et al.  Stability theory for hybrid dynamical systems , 1998, IEEE Trans. Autom. Control..

[8]  Zheng Da-zhong.  Sufficient and Necessary Condition of Controllability and Reachability of a Class of Linear Switching Systems , 2001 .

[9]  Cheong Boon Soh,et al.  Lyapunov stability of a class of hybrid dynamic systems , 2000, Autom..

[10]  A. Morse,et al.  Basic problems in stability and design of switched systems , 1999 .

[11]  Zhendong Sun,et al.  On reachability and stabilization of switched linear systems , 2001, IEEE Trans. Autom. Control..

[12]  S. M. Williams,et al.  Adaptive frequency domain control of PWM switched power line conditioner , 1990 .

[13]  John N. Tsitsiklis,et al.  Complexity of stability and controllability of elementary hybrid systems , 1999, Autom..

[14]  Raymond A. DeCarlo,et al.  Switched Controller Synthesis for the Quadratic Stabilisation of a Pair of Unstable Linear Systems , 1998, Eur. J. Control.

[15]  M. Branicky.  Multiple Lyapunov functions and other analysis tools for switched and hybrid systems , 1998, IEEE Trans. Autom. Control..

[16]  A. Michel,et al.  Stability theory for hybrid dynamical systems , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.