Kybernetika 53 no. 3, 530-544, 2017

Bounded-input-bounded-state stabilization of switched processes and periodic asymptotic controllability

Andrea BacciottiDOI: 10.14736/kyb-2017-3-0530

Abstract:

The main result of this paper is a sufficient condition for the existence of periodic switching signals which render asymptotically stable at the origin a linear switched process defined by a pair of $2\times 2$ real matrices. The interest of this result is motivated by the application to the problem of bounded-input-bounded-state (with respect to an external input) stabilization of linear switched processes.

Keywords:

switched processes, asymptotic controllability, bounded-input-bounded-state stability

Classification:

93D20, 93B60

paper.pdf

References:

  1. A. Bacciotti: Periodic open-loop stabilization of planar switched systems. Europ. J. Control 21 (2015), 22-27.   DOI:10.1016/j.ejcon.2015.09.002
  2. A. Bacciotti: Periodic asymptotic controllability of switched systems. Libertas Mathematica (new series) 34 (2014), 23-46.   CrossRef
  3. A. Bacciotti and L. Mazzi: Asymptotic controllability by means of eventually periodic switching rules. SIAM J. Control Optim. 49 (2011), 476-497.   DOI:10.1137/100798260
  4. R. W. Brockett: Finite Dimensional Linear Systems. Wiley, New York 1970.   DOI:10.1137/1.9781611973884
  5. F. Ceragioli: Finite valued feedback laws and piecewise classical solutions. Nonlinear Analysis 65 (2006), 984-998.   DOI:10.1016/j.na.2005.10.030
  6. R. Conti: Asymptotic control. In: Control Theory and Topics in Functional Analysis, International Atomic Energy Agency, Vienna 1976, pp. 329-360.   CrossRef
  7. R. Conti: Linear Differential Equations and Control. Academic Press, London 1976.   CrossRef
  8. H. Lin and P. J. Antsaklis: Stability and stabilization of switched linear systems: a survey of recent results. IEEE Trans. Automat. Control 54 (2009), 308-322.   DOI:10.1109/tac.2008.2012009
  9. Z. Huang, C. Xiang, H. Lin and T. Lee: Necessary and sufficient conditions for regional stabilisability of generic switched linear systems with a pair of planar subsystems. Int. J. Control 83 (2010), 694-715.   DOI:10.1080/00207170903384321
  10. A. Kundu, D. Chatterjee and D. Liberzon: Generalized switching signals for input-to-state stability of switched systems. Automatica 64 (2016), 270-277.   DOI:10.1016/j.automatica.2015.11.027
  11. D. Liberzon: Switching in Systems and Control. Birkhäuser, Boston 2003.   DOI:10.1007/978-1-4612-0017-8
  12. E. D. Sontag: Mathematical Control Theory. Springer-Verlag, New York 1990.   DOI:10.1007/978-1-4684-0374-9
  13. E. D. Sontag: Smooth stabilization implies coprime factorization. IEEE Trans. Automat. Control 34 (1989), 435-443.   DOI:10.1109/9.28018
  14. Z. Sun and S. S. Ge: Switched Linear Systems. Springer-Verlag, London 2005.   DOI:10.1007/1-84628-131-8
  15. J. Tokarzewski: Stability of periodically switched linear systems and the switching frequency. Int. J. Systems Sci. 18 (1987), 698-726.   DOI:10.1080/00207728708964001
  16. L. Vu, D. Chatterjee and D. Liberzon: Input-to-state stability of switched systems and switching adaptive control. Automatica 43 (2007), 639-646.   DOI:10.1016/j.automatica.2006.10.007
  17. Wei Feng and JiFeng Zhang: Input-to-state stability of switched nonlinear systems. Science in China Series F: Information Sciences 51 (2008), 1992-2004.   DOI:10.1007/s11432-008-0161-7
  18. M. Wicks, P. Peleties and R. A. DeCarlo: Switched controller synthesis for the quadratic stabilization of a pair of unstable linear systems. Europ. J. Control 4 (1998), 140-147.   DOI:10.1016/s0947-3580(98)70108-6
  19. J. L. Willems: Stability Theory of Dynamical Systems. Nelson, London 1970.   CrossRef
  20. X. Xu and P. J. Antsaklis: Stabilization of second-order LTI switched systems. Int. J. Control 73 (2000), 1261-1279.   DOI:10.1080/002071700421664
  21. G. Yang and D. Liberzon: Input-to-state stability for switched systems with unstable subsystems: A hybrid Lyapunov construction. In: Proc. IEEE Conference on Decision and Control 2014 (2015), pp. 6240-6245.   DOI:10.1109/cdc.2014.7040367
  22. T. Yoshizawa: Stability Theory by Liapunov's Second Method. Publications of the Mathematical Society of Japan No. 9, 1966.   CrossRef
  23. Yue-E Wang, Xi-Ming Sun, Peng Shi and Jun Zhao: Input-to-State stability of switched nonlinear systems with time delays under asynchronous switching. IEEE Trans. Cybernetics 43 (2013), 2261-2265.   DOI:10.1109/tcyb.2013.2240679
  24. Yue-E Wang, Xi-Ming Sun, Wei Wang and Jun Zhao: Stability properties of switched nonlinear delay systems with synchronous or asynchronous switching. Asian J. Control 17 (2015), 1187-1195.   DOI:10.1002/asjc.964