# American Institute of Mathematical Sciences

January  2017, 22(1): 199-208. doi: 10.3934/dcdsb.2017010

## 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

* Corresponding author

Received  August 2015 Revised  May 2016 Published  December 2016

Fund Project: The work is supported by the National Natural Science Foundation of China (11171079) and the Australian Research Council (DP160102819).

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.

Citation: Xiang Xie, Honglei Xu, Xinming Cheng, Yilun Yu. Improved results on exponential stability of discrete-time switched delay systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 199-208. doi: 10.3934/dcdsb.2017010
##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] D. Liberzon, Switching in Systems and Control, Springer, 2003. doi: 10.1007/978-1-4612-0017-8.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] D. Liberzon, Switching in Systems and Control, Springer, 2003. doi: 10.1007/978-1-4612-0017-8.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
State trajectories of switched system under ADT switching with $\tau _a = 2$
State trajectories of switched system under MDADT switching with $\tau _{a1} = 1, \tau _{a2} = 1, \tau _{a3} = 1, \tau _{a4} = 2$
Computation Results For The Switched Delay System (9) Under Two Different Switching Schemes
 Switching Schemes ADT Switching MDADT Switching Criteria for controller design Theorem 1 in [19] Theorem 1 in this paper Switching signals $\tau _a^* = 0.1175$\mu = 1.1$\lambda \le 1.5$ $\tau _{a1}^* = 0.0312$, $\tau _{a2}^* = 0.0409$, $\tau _{a3}^* = 0.0642$, $\tau _{a4}^* = 0.1175$, ${\mu _1} = {\mu _2} = {\mu _3} = {\mu _4} = 1.1$, ${\lambda _1} \le 4.6$, ${\lambda _2} \le 3.2$, ${\lambda _3} \le 2.1$, ${\lambda _4} \le 1.5$
 Switching Schemes ADT Switching MDADT Switching Criteria for controller design Theorem 1 in [19] Theorem 1 in this paper Switching signals $\tau _a^* = 0.1175$\mu = 1.1$\lambda \le 1.5$ $\tau _{a1}^* = 0.0312$, $\tau _{a2}^* = 0.0409$, $\tau _{a3}^* = 0.0642$, $\tau _{a4}^* = 0.1175$, ${\mu _1} = {\mu _2} = {\mu _3} = {\mu _4} = 1.1$, ${\lambda _1} \le 4.6$, ${\lambda _2} \le 3.2$, ${\lambda _3} \le 2.1$, ${\lambda _4} \le 1.5$
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