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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

Honglei Xu 1, , Kok Lay Teo 2, and  Weihua Gui 3,

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.
Keywords: state transition matrix., stability analysis, Impulsive switched linear systems.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Honglei Xu, Kok Lay Teo, Weihua Gui. Necessary and sufficient conditions for stability of impulsive switched linear systems. Discrete & 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, 38 (1991), 65.   Google Scholar

[2]

Z. Doulgeri and G. Iliadis, Stability of a contact task for a robotic arm modelled as a switched system,, Control Theory & Applications, 1 (2007), 844.  doi: 10.1049/iet-cta:20060191.  Google Scholar

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M. Petreczky, Reachability of linear switched systems: Differential geometric approach,, Systems & Control Letters, 55 (2006), 112.  doi: 10.1016/j.sysconle.2005.06.001.  Google Scholar

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Z. Sun, S. S. Ge and T. H. Lee, Controllability and reachability criteria for switched linear systems,, Automatica J. IFAC, 38 (2002), 775.  doi: 10.1016/S0005-1098(01)00267-9.  Google Scholar

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G. Xie and L. Wang, Controllability and stabilizability of switched linear systems,, Systems & Control Letters, 48 (2003), 135.  doi: 10.1016/S0167-6911(02)00288-8.  Google Scholar

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[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.  doi: 10.1109/TAC.2002.804474.  Google Scholar

[9]

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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.  doi: 10.1109/TAC.2008.2012009.  Google Scholar

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D. Liberzon, "Switching in Systems and Control,", Systems & Control: Foundations & Applications, (2003).   Google Scholar

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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.  doi: 10.1016/j.mcm.2007.04.011.  Google Scholar

[13]

H. Xu, K. L. Teo and X. Liu, Robust stability analysis of guaranteed cost control for impulsive switched systems,, IEEE Transactions on Systems, 38 (2008), 1419.  doi: 10.1109/TSMCB.2008.925747.  Google Scholar

[14]

D. Zheng, "Linear System Theory,", Tsinghua University Press, (1990).   Google Scholar

[15]

Z. Sun and S. S. Ge, "Switched Linear Systems,", Springer-Verlag, (2005).   Google Scholar

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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.  doi: 10.3934/dcdsb.2006.6.1301.  Google Scholar

[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.  doi: 10.3934/dcdsb.2006.6.1191.  Google Scholar

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, 38 (1991), 65.   Google Scholar

[2]

Z. Doulgeri and G. Iliadis, Stability of a contact task for a robotic arm modelled as a switched system,, Control Theory & Applications, 1 (2007), 844.  doi: 10.1049/iet-cta:20060191.  Google Scholar

[3]

M. Petreczky, Reachability of linear switched systems: Differential geometric approach,, Systems & Control Letters, 55 (2006), 112.  doi: 10.1016/j.sysconle.2005.06.001.  Google Scholar

[4]

Z. Sun and S. S. Ge, Analysis and synthesis of switched linear control systems,, Automatica J. IFAC, 41 (2005), 181.  doi: 10.1016/j.automatica.2004.09.015.  Google Scholar

[5]

Z. Sun, S. S. Ge and T. H. Lee, Controllability and reachability criteria for switched linear systems,, Automatica J. IFAC, 38 (2002), 775.  doi: 10.1016/S0005-1098(01)00267-9.  Google Scholar

[6]

G. Xie and L. Wang, Controllability and stabilizability of switched linear systems,, Systems & Control Letters, 48 (2003), 135.  doi: 10.1016/S0167-6911(02)00288-8.  Google Scholar

[7]

J. Ezzine and A. H. Haddad, Controllability and observability of hybrid systems,, International Journal of Control, 49 (1989), 2045.   Google Scholar

[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.  doi: 10.1109/TAC.2002.804474.  Google Scholar

[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.  doi: 10.1109/5.871309.  Google Scholar

[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.  doi: 10.1109/TAC.2008.2012009.  Google Scholar

[11]

D. Liberzon, "Switching in Systems and Control,", Systems & Control: Foundations & Applications, (2003).   Google Scholar

[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.  doi: 10.1016/j.mcm.2007.04.011.  Google Scholar

[13]

H. Xu, K. L. Teo and X. Liu, Robust stability analysis of guaranteed cost control for impulsive switched systems,, IEEE Transactions on Systems, 38 (2008), 1419.  doi: 10.1109/TSMCB.2008.925747.  Google Scholar

[14]

D. Zheng, "Linear System Theory,", Tsinghua University Press, (1990).   Google Scholar

[15]

Z. Sun and S. S. Ge, "Switched Linear Systems,", Springer-Verlag, (2005).   Google Scholar

[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.  doi: 10.3934/dcdsb.2006.6.1301.  Google Scholar

[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.  doi: 10.3934/dcdsb.2006.6.1191.  Google Scholar

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