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Improved results on exponential stability of discrete-time switched delay systems
1. | School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China |
2. | Department of Mathematics and Statistics, Curtin University, Perth, WA 6845, Australia |
3. | School of Information Science and Engineering, Central South University, Changsha, Hunan 410083, China |
In this paper, we study the exponential stability problem of discrete-time switched delay systems. Combining a multiple Lyapunov function method with a mode-dependent average dwell time technique, we develop novel sufficient conditions for exponential stability of the switched delay systems expressed by a set of numerically solvable linear matrix inequalities. Finally, numerical examples are presented to illustrate less conservativeness of the obtained results.
References:
[1] |
M. S. Branicky,
Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans. Automat. Control, 43 (1998), 475-482.
doi: 10.1109/9.664150. |
[2] |
Q. Chen, L. Yu and W. Zhang,
Delay-dependent output feedback guaranteed cost control for uncertain discrete-time systems with multiple time-varying delays, Control Theory & Applications, 1 (2007), 97-103.
doi: 10.1049/iet-cta:20050443. |
[3] |
J. Daafouz, P. Riedinger and C. Lung,
Stability analysis and control synthesis for switched systems: A switched Lyapunov function approach, IEEE Trans. Automat. Control, 47 (2002), 1883-1887.
doi: 10.1109/TAC.2002.804474. |
[4] |
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. |
[5] |
D. Liberzon,
Switching in Systems and Control, Springer, 2003.
doi: 10.1007/978-1-4612-0017-8. |
[6] |
D. Liberzon, J. P. Hespanha and A. S. Morse,
Stability of switched systems: A Lie-algebraic condition, Systems & Control Letters, 37 (1999), 117-122.
doi: 10.1016/S0167-6911(99)00012-2. |
[7] |
D. Liberzon and S. Morse,
Basic problems in stability and design of switched systems, IEEE Control Syst., 19 (1999), 59-70.
doi: 10.1109/37.793443. |
[8] |
H. Lin and P. J. Antsaklis,
Stability and stabilizability of switched linear systems: A survey of recent results, IEEE Trans. Automat. Control, 54 (2009), 308-322.
doi: 10.1109/TAC.2008.2012009. |
[9] |
W. Michiels, V. V. Assche and S. I. Niculescu,
Stabilization of time-delay systems with a controlled time-varying delay and applications, IEEE Trans. Automat. Control, 50 (2005), 493-504.
doi: 10.1109/TAC.2005.844723. |
[10] |
S. Pettersson,
Synthesis of switched linear systems, Proceedings on 42nd IEEE Conference on Decision and Control, (2003), 5283-5288.
doi: 10.1109/CDC.2003.1272477. |
[11] |
X. Sun, J. Zhao and D. J. Hill,
Stability and $l_2$-gain analysis for switched delay systems: A delay-dependent method, Automatica, 42 (2006), 1769-1774.
doi: 10.1016/j.automatica.2006.05.007. |
[12] |
M. Wicks and R. Decarlo,
Solution of coupled Lyapunov equations for the stabilization of multimodal linear systems, Proceedings of the American Control Conference, 1997 (1997), 1709-1713.
doi: 10.1109/ACC.1997.610876. |
[13] |
X. Xie, H. Xu and R. Zhang, Exponential stabilization of impulsive switched systems with time delays using guaranteed cost control Abstract and Applied Analysis, 2014 (2014), Art. ID 126836, 8 pp.
doi: 10.1155/2014/126836. |
[14] |
H. Xu, X. Liu and K. L. Teo,
Delay independent stability criteria of impulsive switched systems with time-invariant delays, Math. Comput. Model., 47 (2008), 372-379.
doi: 10.1016/j.mcm.2007.04.011. |
[15] |
H. Xu, X. Liu and K. L. Teo, Robust stability analysis of guaranteed cost control for impulsive switched systems, IEEE Trans. Syst., Man, and Cyber.-Part B, 35 (2008), 1419-1422. Google Scholar |
[16] |
H. Xu, X. Xie and L. Shi, An MDADT-based approach for l2-gain analysis of discrete-time switched delay systems Mathematical Problems in Engineering, 2016 (2016), Art. ID 1673959, 8 pp.
doi: 10.1155/2016/1673959. |
[17] |
H. Yan, Robust exponential stability and l2-gain for switched discrete-time nonlinear cascade systems, 33rd Chinese Control Conference, (2014), 4198-4203. Google Scholar |
[18] |
G. Zhai, X. Chen and H. Lin, Stability and l2 gain analysis of discrete-time switched systems, Transactions of the Institute of Systems, Control and Information engineers, 15 (2002), 117-125. Google Scholar |
[19] |
G. Zhai, B. Hu, K. Yasuda and A. N. Michel,
Stability analysis of switched systems with stable and unstable subsystems: An average dwell time approach, American Control Conference, 1 (2000), 200-204.
doi: 10.1109/ACC.2000.878825. |
[20] |
L. Zhang, E. K. Boukas and P. Shi,
Exponential H∞ filtering for uncertain discrete-time switched linear systems with average dwell time: A μ-dependent approach', International Journal of Robust and Nonlinear Control, 18 (2008), 1188-1207.
doi: 10.1002/rnc.1276. |
[21] |
W. Zhang and L. Yu,
Stability analysis for discrete-time switched time-delay system, Automatica, 45 (2009), 2265-2271.
doi: 10.1016/j.automatica.2009.05.027. |
[22] |
X. Zhao, L. Zhang, P. Shi and M. Liu,
Stability and stabilization of switched linear systems with mode-dependent average dwell time, IEEE Trans. Automat. Control, 57 (2012), 1809-1815.
doi: 10.1109/TAC.2011.2178629. |
show all references
References:
[1] |
M. S. Branicky,
Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans. Automat. Control, 43 (1998), 475-482.
doi: 10.1109/9.664150. |
[2] |
Q. Chen, L. Yu and W. Zhang,
Delay-dependent output feedback guaranteed cost control for uncertain discrete-time systems with multiple time-varying delays, Control Theory & Applications, 1 (2007), 97-103.
doi: 10.1049/iet-cta:20050443. |
[3] |
J. Daafouz, P. Riedinger and C. Lung,
Stability analysis and control synthesis for switched systems: A switched Lyapunov function approach, IEEE Trans. Automat. Control, 47 (2002), 1883-1887.
doi: 10.1109/TAC.2002.804474. |
[4] |
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. |
[5] |
D. Liberzon,
Switching in Systems and Control, Springer, 2003.
doi: 10.1007/978-1-4612-0017-8. |
[6] |
D. Liberzon, J. P. Hespanha and A. S. Morse,
Stability of switched systems: A Lie-algebraic condition, Systems & Control Letters, 37 (1999), 117-122.
doi: 10.1016/S0167-6911(99)00012-2. |
[7] |
D. Liberzon and S. Morse,
Basic problems in stability and design of switched systems, IEEE Control Syst., 19 (1999), 59-70.
doi: 10.1109/37.793443. |
[8] |
H. Lin and P. J. Antsaklis,
Stability and stabilizability of switched linear systems: A survey of recent results, IEEE Trans. Automat. Control, 54 (2009), 308-322.
doi: 10.1109/TAC.2008.2012009. |
[9] |
W. Michiels, V. V. Assche and S. I. Niculescu,
Stabilization of time-delay systems with a controlled time-varying delay and applications, IEEE Trans. Automat. Control, 50 (2005), 493-504.
doi: 10.1109/TAC.2005.844723. |
[10] |
S. Pettersson,
Synthesis of switched linear systems, Proceedings on 42nd IEEE Conference on Decision and Control, (2003), 5283-5288.
doi: 10.1109/CDC.2003.1272477. |
[11] |
X. Sun, J. Zhao and D. J. Hill,
Stability and $l_2$-gain analysis for switched delay systems: A delay-dependent method, Automatica, 42 (2006), 1769-1774.
doi: 10.1016/j.automatica.2006.05.007. |
[12] |
M. Wicks and R. Decarlo,
Solution of coupled Lyapunov equations for the stabilization of multimodal linear systems, Proceedings of the American Control Conference, 1997 (1997), 1709-1713.
doi: 10.1109/ACC.1997.610876. |
[13] |
X. Xie, H. Xu and R. Zhang, Exponential stabilization of impulsive switched systems with time delays using guaranteed cost control Abstract and Applied Analysis, 2014 (2014), Art. ID 126836, 8 pp.
doi: 10.1155/2014/126836. |
[14] |
H. Xu, X. Liu and K. L. Teo,
Delay independent stability criteria of impulsive switched systems with time-invariant delays, Math. Comput. Model., 47 (2008), 372-379.
doi: 10.1016/j.mcm.2007.04.011. |
[15] |
H. Xu, X. Liu and K. L. Teo, Robust stability analysis of guaranteed cost control for impulsive switched systems, IEEE Trans. Syst., Man, and Cyber.-Part B, 35 (2008), 1419-1422. Google Scholar |
[16] |
H. Xu, X. Xie and L. Shi, An MDADT-based approach for l2-gain analysis of discrete-time switched delay systems Mathematical Problems in Engineering, 2016 (2016), Art. ID 1673959, 8 pp.
doi: 10.1155/2016/1673959. |
[17] |
H. Yan, Robust exponential stability and l2-gain for switched discrete-time nonlinear cascade systems, 33rd Chinese Control Conference, (2014), 4198-4203. Google Scholar |
[18] |
G. Zhai, X. Chen and H. Lin, Stability and l2 gain analysis of discrete-time switched systems, Transactions of the Institute of Systems, Control and Information engineers, 15 (2002), 117-125. Google Scholar |
[19] |
G. Zhai, B. Hu, K. Yasuda and A. N. Michel,
Stability analysis of switched systems with stable and unstable subsystems: An average dwell time approach, American Control Conference, 1 (2000), 200-204.
doi: 10.1109/ACC.2000.878825. |
[20] |
L. Zhang, E. K. Boukas and P. Shi,
Exponential H∞ filtering for uncertain discrete-time switched linear systems with average dwell time: A μ-dependent approach', International Journal of Robust and Nonlinear Control, 18 (2008), 1188-1207.
doi: 10.1002/rnc.1276. |
[21] |
W. Zhang and L. Yu,
Stability analysis for discrete-time switched time-delay system, Automatica, 45 (2009), 2265-2271.
doi: 10.1016/j.automatica.2009.05.027. |
[22] |
X. Zhao, L. Zhang, P. Shi and M. Liu,
Stability and stabilization of switched linear systems with mode-dependent average dwell time, IEEE Trans. Automat. Control, 57 (2012), 1809-1815.
doi: 10.1109/TAC.2011.2178629. |


Switching Schemes | ADT Switching | MDADT Switching |
Criteria for controller design | Theorem 1 in [19] | Theorem 1 in this paper |
Switching signals |
Switching Schemes | ADT Switching | MDADT Switching |
Criteria for controller design | Theorem 1 in [19] | Theorem 1 in this paper |
Switching signals |
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