Kybernetika 56 no. 4, 727-752, 2020

Controllability of linear impulsive systems -- an eigenvalue approach

Vijayakumar S. Muni and Raju K. GeorgeDOI: 10.14736/kyb-2020-4-0727

Abstract:

This article considers a class of finite-dimensional linear impulsive time-varying systems for which various sufficient and necessary algebraic criteria for complete controllability, including matrix rank conditions are established. The obtained controllability results are further synthesised for the time-invariant case, and under some special conditions on the system parameters, we obtain a Popov-Belevitch-Hautus (PBH)-type rank condition which employs eigenvalues of the system matrix for the investigation of their controllability. Numerical examples are provided that demonstrate--for the linear impulsive systems, null controllability need not imply their complete controllability, unlike for the non-impulsive linear systems.

Keywords:

controllability, eigenvalues, impulses

Classification:

15A18, 34A37, 93B05

paper.pdf

References:

  1. Z. Benzaid and M. Sznaier: Constrained controllability of linear impulse differential systems. IEEE Trans. Automat. Contr. 39(5) (1994), 1064-1066.   DOI: 10.1109/9.284894
  2. R. K. George, A. K. Nandakumaran and A. Arapostathis: A note on controllability of impulsive systems. J. Math. Anal. Appl. 241(2) (2000), 276-283.   DOI: 10.1006/jmaa.1999.6632
  3. Z. H. Guan, T. H. Qian and X. Yu: Controllability and observability of linear time-varying impulsive systems. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 49(8) (2002), 1198-1208.   DOI: 10.1109/TCSI.2002.801261
  4. Z. H. Guan, T. H. Qian and X. Yu: On controllability and observability for a class of impulsive systems. Systems Control Lett. 47(3) (2002), 247-257.   DOI: 10.1016/S0167-6911(02)00204-9
  5. J. Han, Y. Liu, S. Zhao and R. Yang: A note on the controllability and observability for piecewise linear time-varying impulsive systems. Asian J. Control 15(6) (2013), 1867-1870.   DOI: 10.1002/asjc.642
  6. V. Lakshmikantham, D. D. Bainov and P. S. Simeonov: Theory of impulsive differential equations. World Scientific, Singapore 1989.   CrossRef
  7. S. Leela, F. A. McRae and S. Sivasundaram: Controllability of impulsive differential equations. J. Math. Anal. Appl. 177(1) (1993), 24-30.   DOI: 10.1006/jmaa.1993.1240
  8. V. S. Muni and R. K. George: Controllability of semilinear impulsive control systems with multiple time delays in control. IMA J. Math. Control \& Inform. 36(3) (2019), 869-899.   DOI: 10.1093/imamci/dny011
  9. W. J. Terrell: Stability and stabilization: An introduction. Princeton University Press, Princeton, USA 2009.   CrossRef
  10. G. Xie and L. Wang: Controllability and observability of a class of linear impulsive systems. J. Math. Anal. Appl. 304(1) (2005), 336-355.   DOI: 10.1016/j.jmaa.2004.09.028
  11. S. Zhao and J. Sun: Controllability and observability for a class of time-varying impulsive systems. Nonlinear Anal. RWA. 10(3) (2009), 1370-1380.   DOI: 10.1016/j.nonrwa.2008.01.012
  12. S. Zhao and J. Sun: Controllability and observability for impulsive systems in complex fields. Nonlinear Anal. RWA. 11(3) (2010), 1513-1521.   DOI: 10.1016/j.nonrwa.2009.03.009