Stabilizability of linear switching systems

A complete characterization of stabilizability for linear switching systems is not available in the literature. In this paper, we show that the asymptotic stabilizability of linear switching systems is equivalent to the existence of a hybrid Lyapunov function for the controlled system, for a suitable control strategy. Further, we prove that asymptotic stabilizability of a switching system with minimum dwell time, is equivalent to Input to State Stability (ISS) of the controlled switching system, with a stabilizing control law. We then derive some structural reductions of the hybrid state space, which allow a decomposition of the original problem into simpler subproblems. The relationships between this approach and the well-known Kalman decomposition of linear dynamic control systems are explored.

[1]  Franco Blanchini.  The gain scheduling and the robust state feedback stabilization problems , 2000, IEEE Trans. Autom. Control..

[2]  Eduardo Sontag.  Smooth stabilization implies coprime factorization , 1989, IEEE Transactions on Automatic Control.

[3]  Clyde F. Martin,et al.  A Converse Lyapunov Theorem for a Class of Dynamical Systems which Undergo Switching , 1999, IEEE Transactions on Automatic Control.

[4]  Kennan T. Smith,et al.  Linear Topological Spaces , 1966 .

[5]  A. Morse.  Supervisory control of families of linear set-point controllers Part I. Exact matching , 1996, IEEE Trans. Autom. Control..

[6]  D. Liberzon.  ISS and integral-ISS disturbance attenuation with bounded controls , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[7]  Maria Domenica Di Benedetto,et al.  Computation of maximal safe sets for switching systems , 2004, IEEE Transactions on Automatic Control.

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

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

[10]  Daniel Liberzon,et al.  Switching in Systems and Control , 2003, Systems & Control: Foundations & Applications.

[11]  M. D. Di Benedetto,et al.  Digital idle speed control of automotive engines: A safety problem for hybrid systems , 2006 .

[12]  John Lygeros,et al.  Synthesizing Controllers for Nonlinear Hybrid Systems , 1998, HSCC.

[13]  S. Ge,et al.  Switched Linear Systems: Control and Design , 2005 .

[14]  Franco Blanchini,et al.  Set invariance in control , 1999, Autom..

[15]  Shuzhi Sam Ge,et al.  Switched Linear Systems , 2005 .

[16]  John Lygeros,et al.  Controllers for reachability specifications for hybrid systems , 1999, Autom..

[17]  A. J. van der Schaft,et al.  Equivalence of switching linear systems by bisimulation , 2006 .

[18]  A.J. van der Schaft,et al.  The maximal controlled invariant set of switched linear systems , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[19]  J. L. Mancilla-Aguilar,et al.  On converse Lyapunov theorems for ISS and iISS switched nonlinear systems , 2001 .