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November  2011, 16(4): 1185-1195. doi: 10.3934/dcdsb.2011.16.1185

## Necessary and sufficient conditions for stability of impulsive switched linear systems

 1 Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845 2 Department of Mathematics and Statistics, Curtin University of Technology, GPO Box U 1987, Perth, W.A. 6845 3 School of Information Science and Engineering, Central South University, Changsha, 410083, China

Received  October 2010 Revised  June 2011 Published  August 2011

This paper addresses fundamental stability problems of impulsive switched linear systems, featuring given impulsive switching time intervals and switching rules. First, based on the state dynamical behaviors, we construct a new state transition-like matrix, called an impulsive-type state transition (IST) matrix. Then, based on the IST matrix and Lyapunov stability theory, necessary and sufficient conditions for the uniform stability, uniform asymptotic stability, and exponential stability of impulsive switched linear systems are established. These stability conditions require the testing on the IST matrix of the impulsive switched linear systems. The results can be reduced to those for switched linear systems without impulsive effects, and also to those for impulsive linear systems without switchings.
Citation: Honglei Xu, Kok Lay Teo, Weihua Gui. Necessary and sufficient conditions for stability of impulsive switched linear systems. Discrete and Continuous Dynamical Systems - B, 2011, 16 (4) : 1185-1195. doi: 10.3934/dcdsb.2011.16.1185
##### References:
 [1] F. H. F. Leung, P. K. S. Tam and C. K. Li, The control of switching DC-DC converters - a general LWR problem, Industrial Electronics, IEEE Transactions on, 38 (1991), 65-71. [2] Z. Doulgeri and G. Iliadis, Stability of a contact task for a robotic arm modelled as a switched system, Control Theory & Applications, IET, 1 (2007), 844-853. doi: 10.1049/iet-cta:20060191. [3] M. Petreczky, Reachability of linear switched systems: Differential geometric approach, Systems & Control Letters, 55 (2006), 112-118. doi: 10.1016/j.sysconle.2005.06.001. [4] Z. Sun and S. S. Ge, Analysis and synthesis of switched linear control systems, Automatica J. IFAC, 41 (2005), 181-195. doi: 10.1016/j.automatica.2004.09.015. [5] Z. Sun, S. S. Ge and T. H. Lee, Controllability and reachability criteria for switched linear systems, Automatica J. IFAC, 38 (2002), 775-786. doi: 10.1016/S0005-1098(01)00267-9. [6] G. Xie and L. Wang, Controllability and stabilizability of switched linear systems, Systems & Control Letters, 48 (2003), 135-155. doi: 10.1016/S0167-6911(02)00288-8. [7] J. Ezzine and A. H. Haddad, Controllability and observability of hybrid systems, International Journal of Control, 49 (1989), 2045-2055. [8] J. Daafouz, P. Riedinger and C. Iung, Stability analysis and control synthesis for switched systems: A switched Lyapunov function approach, IEEE Transactions on Automatic Control, 47 (2002), 1883-1887. doi: 10.1109/TAC.2002.804474. [9] R. A. Decarlo, M. S. Branicky, S. Pettersson and B. Lennartson, Perspectives and results on the stability and stabilizability of hybrid systems, Proceedings of the IEEE, 88 (2000), 1069-1082. doi: 10.1109/5.871309. [10] L. Hai and P. J. Antsaklis, Stability and stabilizability of switched linear systems: A survey of recent results, IEEE Transactions on Automatic Control, 54 (2009), 308-322. doi: 10.1109/TAC.2008.2012009. [11] D. Liberzon, "Switching in Systems and Control," Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2003. [12] H. Xu, X. Liu and K. L. Teo, Delay independent stability criteria of impulsive switched systems with time-invariant delays, Mathematical and Computer Modelling, 47 (2008), 372-379. doi: 10.1016/j.mcm.2007.04.011. [13] H. Xu, K. L. Teo and X. Liu, Robust stability analysis of guaranteed cost control for impulsive switched systems, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 38 (2008), 1419-1422. doi: 10.1109/TSMCB.2008.925747. [14] D. Zheng, "Linear System Theory," Tsinghua University Press, Beijing, 1990. [15] Z. Sun and S. S. Ge, "Switched Linear Systems," Springer-Verlag, London, 2005. [16] Q. Lu and G. Jiang, The dynamics of a prey-predator model with impulsive state feedback control, Discrete and Continuous Dynamical Systems - Series B, 6 (2006), 1301-1320. doi: 10.3934/dcdsb.2006.6.1301. [17] A. Anguraj and T. Paul, Existence and uniqueness of nonlinear impulsive integro-differential equations, Discrete and Continuous Dynamical Systems - Series B, 6 (2006), 1191-1198. doi: 10.3934/dcdsb.2006.6.1191.

show all references

##### References:
 [1] F. H. F. Leung, P. K. S. Tam and C. K. Li, The control of switching DC-DC converters - a general LWR problem, Industrial Electronics, IEEE Transactions on, 38 (1991), 65-71. [2] Z. Doulgeri and G. Iliadis, Stability of a contact task for a robotic arm modelled as a switched system, Control Theory & Applications, IET, 1 (2007), 844-853. doi: 10.1049/iet-cta:20060191. [3] M. Petreczky, Reachability of linear switched systems: Differential geometric approach, Systems & Control Letters, 55 (2006), 112-118. doi: 10.1016/j.sysconle.2005.06.001. [4] Z. Sun and S. S. Ge, Analysis and synthesis of switched linear control systems, Automatica J. IFAC, 41 (2005), 181-195. doi: 10.1016/j.automatica.2004.09.015. [5] Z. Sun, S. S. Ge and T. H. Lee, Controllability and reachability criteria for switched linear systems, Automatica J. IFAC, 38 (2002), 775-786. doi: 10.1016/S0005-1098(01)00267-9. [6] G. Xie and L. Wang, Controllability and stabilizability of switched linear systems, Systems & Control Letters, 48 (2003), 135-155. doi: 10.1016/S0167-6911(02)00288-8. [7] J. Ezzine and A. H. Haddad, Controllability and observability of hybrid systems, International Journal of Control, 49 (1989), 2045-2055. [8] J. Daafouz, P. Riedinger and C. Iung, Stability analysis and control synthesis for switched systems: A switched Lyapunov function approach, IEEE Transactions on Automatic Control, 47 (2002), 1883-1887. doi: 10.1109/TAC.2002.804474. [9] R. A. Decarlo, M. S. Branicky, S. Pettersson and B. Lennartson, Perspectives and results on the stability and stabilizability of hybrid systems, Proceedings of the IEEE, 88 (2000), 1069-1082. doi: 10.1109/5.871309. [10] L. Hai and P. J. Antsaklis, Stability and stabilizability of switched linear systems: A survey of recent results, IEEE Transactions on Automatic Control, 54 (2009), 308-322. doi: 10.1109/TAC.2008.2012009. [11] D. Liberzon, "Switching in Systems and Control," Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2003. [12] H. Xu, X. Liu and K. L. Teo, Delay independent stability criteria of impulsive switched systems with time-invariant delays, Mathematical and Computer Modelling, 47 (2008), 372-379. doi: 10.1016/j.mcm.2007.04.011. [13] H. Xu, K. L. Teo and X. Liu, Robust stability analysis of guaranteed cost control for impulsive switched systems, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 38 (2008), 1419-1422. doi: 10.1109/TSMCB.2008.925747. [14] D. Zheng, "Linear System Theory," Tsinghua University Press, Beijing, 1990. [15] Z. Sun and S. S. Ge, "Switched Linear Systems," Springer-Verlag, London, 2005. [16] Q. Lu and G. Jiang, The dynamics of a prey-predator model with impulsive state feedback control, Discrete and Continuous Dynamical Systems - Series B, 6 (2006), 1301-1320. doi: 10.3934/dcdsb.2006.6.1301. [17] A. Anguraj and T. Paul, Existence and uniqueness of nonlinear impulsive integro-differential equations, Discrete and Continuous Dynamical Systems - Series B, 6 (2006), 1191-1198. doi: 10.3934/dcdsb.2006.6.1191.
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