This paper is concerned with the problem of global asymptotic stability of linear discrete-time systems with interval-like time-varying delay in the state. By utilizing the concept of delay partitioning, a new linear-matrix-inequality-(LMI-) based criterion for the global asymptotic stability of such systems is proposed. The proposed criterion does not involve any free weighting matrices but depends on both the size of delay and partition size. The developed approach is extended to address the problem of global asymptotic stability of state-delayed discrete-time systems with norm-bounded uncertainties. The proposed results are compared with several existing results. 1. Introduction A source of instability for discrete-time systems is time delay which inevitably exists in various engineering systems. In many applications, time delays are unavoidable and must be taken into account in a realistic system design, for instance, chemical processes, echo cancellation, local loop equalization, multipath propagation in mobile communication, array signal processing, congestion analysis and control in high speed networks, neural networks, and long transmission line in pneumatic systems [1–6]. The stability analysis of time-delay systems has received considerable attention during the last two decades [3–7]. According to dependence of delay, the existing stability criteria are generally classified into two categories: delay-dependent criteria and delay-independent criteria. It is well known that delay-independent stability criteria are usually more conservative than the delay-dependent ones, especially if the size of time delay is small [6–10]. Therefore, much attention has been paid in recent years to the study of delay-dependent stability criteria. A number of publications relating to the delay-dependent stability of continuous time-delay systems have appeared (see, e.g., [10–17] and the references cited therein). In contrast, less attention has been paid to studying the problem of stability of discrete time-delay systems. Several delay-dependent criteria for the stability of discrete-time systems have appeared [3, 6, 7, 18, 19]. Reference [7] (see [6] also) presents a novel delay-dependent linear-matrix-inequality-(LMI-) based condition for the global asymptotic stability of linear discrete-time systems with interval-like time-varying delay. The criteria proposed in [6, 7] are less conservative with smaller numerical complexity than [19–23]. The delay partitioning approach has been efficiently applied in [24–30] to the stability analysis of systems with
References
[1]
J. K. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 1977.
[2]
J. Richard, “Time-delay systems: an overview of some recent advances and open problems,” Automatica, vol. 39, no. 10, pp. 1667–1694, 2003.
[3]
H. Shao, “New delay-dependent stability criteria for systems with interval delay,” Automatica, vol. 45, no. 3, pp. 744–749, 2009.
[4]
V. K. R. Kandanvli and H. Kar, “Robust stability of discrete-time state-delayed systems employing generalized overflow nonlinearities,” Nonlinear Analysis: Theory, Methods and Applications, vol. 69, no. 9, pp. 2780–2787, 2008.
[5]
K. F. Chen and I. K. Fong, “Stability analysis and output-feedback stabilisation of discrete-time systems with an interval time-varying state delay,” IET Control Theory and Applications, vol. 4, no. 4, pp. 563–572, 2010.
[6]
H. Shao and Q. Han, “New stability criteria for linear discrete-time systems with interval-like time-varying delays,” IEEE Transactions on Automatic Control, vol. 56, no. 3, pp. 619–625, 2011.
[7]
H. Huang and G. Feng, “Improved approach to delay-dependent stability analysis of discrete-time systems with time-varying delay,” IET Control Theory and Applications, vol. 4, no. 10, pp. 2152–2159, 2010.
[8]
M. S. Mahmoud, F. M. Al-Sunni, and Y. Shi, “Switched discrete-time delay systems: delay-dependent analysis and synthesis,” Circuits, Systems, and Signal Processing, vol. 28, no. 5, pp. 735–761, 2009.
[9]
M. S. Mahmoud and Y. Xia, “Robust stability and stabilization of a class of nonlinear switched discrete-time systems with time-varying delays,” Journal of Optimization Theory and Applications, vol. 143, no. 2, pp. 329–355, 2009.
[10]
S. Xu and J. Lam, “On equivalence and efficiency of certain stability criteria for time-delay systems,” IEEE Transactions on Automatic Control, vol. 52, no. 1, pp. 95–101, 2007.
[11]
E. Fridman and U. Shaked, “A descriptor system approach to control of linear time-delay systems,” IEEE Transactions on Automatic Control, vol. 47, no. 2, pp. 253–270, 2002.
[12]
Q. Han, “On stability of linear neutral systems with mixed time delays: a discretized Lyapunov functional approach,” Automatica, vol. 41, no. 7, pp. 1209–1218, 2005.
[13]
Q. Han, “A discrete delay decomposition approach to stability of linear retarded and neutral systems,” Automatica, vol. 45, no. 2, pp. 517–524, 2009.
[14]
Q. Han, “Improved stability criteria and controller design for linear neutral systems,” Automatica, vol. 45, no. 8, pp. 1948–1952, 2009.
[15]
Q. Han and K. Gu, “Stability of linear systems with time-varying delay: a generalized discretized lyapunov functional approach,” Asian Journal of Control, vol. 3, no. 3, pp. 170–180, 2001.
[16]
X. Jiang and Q. Han, “On control for linear systems with interval time-varying delay,” Automatica, vol. 41, no. 12, pp. 2099–2106, 2005.
[17]
M. Wu, Y. He, J. She, and G. Liu, “Delay-dependent criteria for robust stability of time-varying delay systems,” Automatica, vol. 40, no. 8, pp. 1435–1439, 2004.
[18]
Y. He, M. Wu, G. Liu, and J. She, “Output feedback stabilization for a discrete-time system with a time-varying delay,” IEEE Transactions on Automatic Control, vol. 53, no. 10, pp. 2372–2377, 2008.
[19]
H. Gao and T. Chen, “New results on stability of discrete-time systems with time-varying state delay,” IEEE Transactions on Automatic Control, vol. 52, no. 2, pp. 328–334, 2007.
[20]
H. Gao, J. Lam, C. Wang, and Y. Wang, “Delay-dependent output-feedback stabilisation of discrete-time systems with time varying state delay,” IEE Proceedings—Control Theory and Applications, vol. 151, no. 6, pp. 691–698, 2004.
[21]
E. Fridman and U. Shaked, “Stability and guaranteed cost control of uncertain discrete delay systems,” International Journal of Control, vol. 78, no. 4, pp. 235–246, 2005.
[22]
X. Jiang, Q. Han, and X. Yu, “Stability criteria for linear discrete-time systems with interval-like time-varying delay,” in Proceedings of the American Control Conference (ACC '05), pp. 2817–2822, Portland, Ore, USA, June 2005.
[23]
B. Zhang, S. Xu, and Y. Zou, “Improved stability criterion and its applications in delayed controller design for discrete-time systems,” Automatica, vol. 44, no. 11, pp. 2963–2967, 2008.
[24]
H. Huang and G. Feng, “State estimation of recurrent neural networks with time-varying delay: a novel delay partition approach,” Neurocomputing, vol. 74, no. 5, pp. 792–796, 2011.
[25]
B. Du, J. Lam, Z. Shu, and Z. Wang, “A delay-partitioning projection approach to stability analysis of continuous systems with multiple delay components,” IET Control Theory and Applications, vol. 3, no. 4, pp. 383–390, 2009.
[26]
Y. Zhao, H. Gao, J. Lam, and B. Du, “Stability and stabilization of delayed T-S fuzzy systems: a delay partitioning approach,” IEEE Transactions on Fuzzy Systems, vol. 17, no. 4, pp. 750–762, 2009.
[27]
J. Liu, B. Yao, and Z. Gu, “Delay-dependent filtering for Markovian jump time-delay systems: a piecewise analysis method,” Circuits, Systems, and Signal Processing, vol. 30, no. 6, pp. 1253–1273, 2011.
[28]
L. Wu, J. Lam, X. Yao, and J. Xiong, “Robust guaranteed cost control of discrete-time networked control systems,” Optimal Control Applications and Methods, vol. 32, no. 1, pp. 95–112, 2011.
[29]
P. Kokil, H. Kar, and V. K. R. Kandanvli, “Stability analysis of linear discrete-time systems with interval delay: a delay-partitioning approach,” ISRN Applied Mathematics, vol. 2011, Article ID 624127, 10 pages, 2011.
[30]
X. Meng, J. Lam, B. Du, and H. Gao, “A delay-partitioning approach to the stability analysis of discrete-time systems,” Automatica, vol. 46, no. 3, pp. 610–614, 2010.
[31]
S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, Pa, USA, 1994.
[32]
P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox for Use with Matlab, MATH Works, Natick, Mass, USA, 1995.
