Improved Robust Stability and Stabilization Conditions of Discrete-time Delayed System
The problem of robust stability and robust stabilization for a class of discrete-time uncertain systems with time delay is investigated. Based on Tchebychev inequality, by constructing a new augmented Lyapunov function, some improved sufficient conditions ensuring exponential stability and stabilization are established. These conditions are expressed in the forms of linear matrix inequalities (LMIs), whose feasibility can be easily checked by using Matlab LMI Toolbox. Compared with some previous results derived in the literature, the new obtained criteria have less conservatism. Two numerical examples are provided to demonstrate the improvement and effectiveness of the proposed method.
[1] Y. Liu, Z. Wang and X. Liu. Robust stability of discrete-time stochastic
neural networks with time-varying delays. Neurocomputing, 71 (2008)
823-833.
[2] V. Singh. A new criterion for global robust stability of interval delayed
neural networks. Journal of Computational and Applied Mathematics, 221
(2008) 219-225.
[3] P. Li et al., Delay-dependent robust BIBO stabilization ..., Chaos, Solitons
and Fractals (2007), doi:10.1016/j.chaos.2007.08.059.
[4] K. Lee, B. Huang. Robust H2 optimal filtering for continuous-time
stochastic systems with polytopic parameter uncertainty. Automatica, 44
(2008) 2686-2690.
[5] C. Chen, C. Lee. Robust stability of homogeneous large-scale bilinear
systems with time delays and uncertainties, J. Process Contr. (2009),
doi:10.1016/j.jprocont.2009.03.001.
[6] B. Liu, K. Teo and X. Liu. Robust exponential stabilization for largescale
uncertain impulsive systems with coupling time-delays, Nonlinear
Analysis. 68 (2008) 1169-1183.
[7] Z. Xiang, R. Wang. Robust L∞ reliable control for uncertain nonlinear
switched systems with time delay. Applied Mathematics and Computation,
210 (2009) 202-210.
[8] B. Lee, J. Lee. Delay-dependent stability sriteria for discrete-time delay
systems. Proceedings of 1999 American Control Conference, (1999) 319-
320.
[9] H. Gao, C. Wang. A delay-dependent approach to robust H∞ filtering
for uncertain discrete-time state-delayed systems. IEEE Transactions on
Signal Processing, 52 (2004) 1631-1640.
[10] J. Yoneyama, T. Tsuchiya. New delay-dependent conditions on robust
stability and stabilisation for discrete-time systems with time-delay.
International Journal of Systems Science, 39 (2008) 1033-1040.
[11] W. Chen, Z. Guan and X. Lu. Delaydependent guaranteed cost control
for discrete-time systems with delay. IEE Proceedings Control Theory and
Applications, 150 (2003) 412-416.
[12] E. Fridman, U. Shaked. Stability and guaranteed cost control of uncertain
discrete delay systems. International Journal of Control, 78 (2005)
235-246.
[13] T. Lee, U. Radovic. General decentralized stabilization of large-scale
linear continuous and discrete time-delay systems. International Journal
of Control, 46 (1987) 2127-2140.
[14] B. Boyd, et al. Linear matrix inequalities in systems and control theory.
Philadelphia (PA): SIAM, 1994.
[15] Y. Liu, Z. Wang and X. Liu, Robust stability of discrete-time stochastic
neural networks with time-varying delays, Neurocomputing, 71 (2008)
823-833.
[16] M. Luo et al., Robust stability analysis for discrete-time
stochastic neural networks ..., Appl. Math. Comput. (2009),
doi:10.1016/j.amc.2008.12.084.
[1] Y. Liu, Z. Wang and X. Liu. Robust stability of discrete-time stochastic
neural networks with time-varying delays. Neurocomputing, 71 (2008)
823-833.
[2] V. Singh. A new criterion for global robust stability of interval delayed
neural networks. Journal of Computational and Applied Mathematics, 221
(2008) 219-225.
[3] P. Li et al., Delay-dependent robust BIBO stabilization ..., Chaos, Solitons
and Fractals (2007), doi:10.1016/j.chaos.2007.08.059.
[4] K. Lee, B. Huang. Robust H2 optimal filtering for continuous-time
stochastic systems with polytopic parameter uncertainty. Automatica, 44
(2008) 2686-2690.
[5] C. Chen, C. Lee. Robust stability of homogeneous large-scale bilinear
systems with time delays and uncertainties, J. Process Contr. (2009),
doi:10.1016/j.jprocont.2009.03.001.
[6] B. Liu, K. Teo and X. Liu. Robust exponential stabilization for largescale
uncertain impulsive systems with coupling time-delays, Nonlinear
Analysis. 68 (2008) 1169-1183.
[7] Z. Xiang, R. Wang. Robust L∞ reliable control for uncertain nonlinear
switched systems with time delay. Applied Mathematics and Computation,
210 (2009) 202-210.
[8] B. Lee, J. Lee. Delay-dependent stability sriteria for discrete-time delay
systems. Proceedings of 1999 American Control Conference, (1999) 319-
320.
[9] H. Gao, C. Wang. A delay-dependent approach to robust H∞ filtering
for uncertain discrete-time state-delayed systems. IEEE Transactions on
Signal Processing, 52 (2004) 1631-1640.
[10] J. Yoneyama, T. Tsuchiya. New delay-dependent conditions on robust
stability and stabilisation for discrete-time systems with time-delay.
International Journal of Systems Science, 39 (2008) 1033-1040.
[11] W. Chen, Z. Guan and X. Lu. Delaydependent guaranteed cost control
for discrete-time systems with delay. IEE Proceedings Control Theory and
Applications, 150 (2003) 412-416.
[12] E. Fridman, U. Shaked. Stability and guaranteed cost control of uncertain
discrete delay systems. International Journal of Control, 78 (2005)
235-246.
[13] T. Lee, U. Radovic. General decentralized stabilization of large-scale
linear continuous and discrete time-delay systems. International Journal
of Control, 46 (1987) 2127-2140.
[14] B. Boyd, et al. Linear matrix inequalities in systems and control theory.
Philadelphia (PA): SIAM, 1994.
[15] Y. Liu, Z. Wang and X. Liu, Robust stability of discrete-time stochastic
neural networks with time-varying delays, Neurocomputing, 71 (2008)
823-833.
[16] M. Luo et al., Robust stability analysis for discrete-time
stochastic neural networks ..., Appl. Math. Comput. (2009),
doi:10.1016/j.amc.2008.12.084.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:56260", author = "Zixin Liu", title = "Improved Robust Stability and Stabilization Conditions of Discrete-time Delayed System", abstract = "The problem of robust stability and robust stabilization for a class of discrete-time uncertain systems with time delay is investigated. Based on Tchebychev inequality, by constructing a new augmented Lyapunov function, some improved sufficient conditions ensuring exponential stability and stabilization are established. These conditions are expressed in the forms of linear matrix inequalities (LMIs), whose feasibility can be easily checked by using Matlab LMI Toolbox. Compared with some previous results derived in the literature, the new obtained criteria have less conservatism. Two numerical examples are provided to demonstrate the improvement and effectiveness of the proposed method.
", keywords = "Robust stabilization, robust stability, discrete-time system, time delay.", volume = "4", number = "7", pages = "906-7", }
