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

Improved results on exponential stability of discrete-time switched delay systems

Xiang Xie 1, , Honglei Xu 2, , Xinming Cheng 3,, and  Yilun Yu 1,

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.

Keywords: Discrete-time switched delay systems, stability analysis, mode-dependent average dwell time.
Mathematics Subject Classification: Primary:58F15, 58F17;Secondary:53C3.
Citation: Xiang Xie, Honglei Xu, Xinming Cheng, Yilun Yu. Improved results on exponential stability of discrete-time switched delay systems. Discrete and 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.

[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. 

[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. 

[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. 

[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. 

[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. 

[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. 

[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.

Figure 1.  State trajectories of switched system under ADT switching with $\tau _a = 2$
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Figure 2.  State trajectories of switched system under MDADT switching with $\tau _{a1} = 1, \tau _{a2} = 1, \tau _{a3} = 1, \tau _{a4} = 2$
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Table 1.  Computation Results For The Switched Delay System (9) Under Two Different Switching Schemes
Switching SchemesADT SwitchingMDADT Switching
Criteria for controller designTheorem 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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Switching SchemesADT SwitchingMDADT Switching
Criteria for controller designTheorem 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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