Stability of Discrete-Time Systems with Time-Varying Delay: Delay Decomposition Approach

Authors

  • Sreten B. Stojanovic University of Nis, Faculty of Technology Serbia, 16000 Leskovac, Bulevar oslobodjenja 124
  • Dragutin L.J. Debeljkovic University of Belgrade, Faculty of Mechanical Engineering Serbia, 11120 Beograd, Kraljice Marije 16
  • Nebojsa Dimitrijevic University of Belgrade, Faculty of Mechanical Engineering Serbia, 11120 Beograd, Kraljice Marije 16

Keywords:

time-delay systems, interval time-varying delay, asymptotic stability, delay-dependent stability, Lyapunov-Krasovskii methods

Abstract

This article deals with the problem of obtaining delay-dependent stability conditions for a class of discrete-time systems with interval time-varying delay. Using the decomposition the delay interval into two unequal subintervals by tuning parameter α, a new interval delay-dependent Lyapunov-Krasovskii functional is constructed to derive novel delay-dependent stability conditions which are expressed in terms of linear matrix inequalities. This leads to reduction of conservatism in terms of the upper bounds of the maximum time-delay. The numerical examples show that the obtained result is less conservative than some existing ones in the literature.

References

E.K. Boukas, Discrete-time systems with time-varying time delay: stability and stabilizability, Mathematical Problems in Engineering, Article ID 42489:1-10, 2006.

K.F. Chen and I-K Fong, Stability of discrete-time uncertain systems with a time-varying state delay, Proc. IMechE, Part I: J. Systems and Control Engineering, 222: 493-500, 2008.

E. Fridman and U. Shaked, A descriptor system approach to H∞ control of linear time-delay systems, IEEE Transactions on Automatic Control, 47(2): 253-270, 2002. http://dx.doi.org/10.1109/9.983353

E. Fridman and U. Shaked, H∞ control of linear state delay descriptor systems: an LMI approach, Linear Algebra and its Applications, 351-352: 271-302, 2002. http://dx.doi.org/10.1016/S0024-3795(01)00563-8

E. Fridman, New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems, Systems and Control Letters, 43: 309-319, 2001. http://dx.doi.org/10.1016/S0167-6911(01)00114-1

E. Fridman and U. Shaked, Delay-Dependent H∞ Control of Uncertain Discrete Delay Systems, European Journal of Control, 11: 29-37, 2005. http://dx.doi.org/10.3166/ejc.11.29-37

E. Fridman and U. Shaked, Stability and guaranteed cost control of uncertain discrete delay systems, International Journal of Control, 78(4): 235-246, 2005. http://dx.doi.org/10.1080/00207170500041472

H. Gao and T. Chen, New results on stability of discrete-time systems with time-varying state delay, IEEE Transactions on Automatic Control, 52: 328-334, 2007. http://dx.doi.org/10.1109/TAC.2006.890320

H. Gao, J. Lam and Y. Wang, Delay-dependent output-feedback stabilization of discretetime systems with time-varying state delay, IEE Proc.: Control Theory Applications, 151(6): 691-698, 2004. http://dx.doi.org/10.1049/ip-cta:20040822

Q.L. Han and D. Yue, Absolute stability of Lur'e systems with time-varying delay, IET Control Theory, 1(3): 854-859, 2007. http://dx.doi.org/10.1049/iet-cta:20060213

Y. He, M. Wu, J.H. She and G.P. Liu, Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type uncertainties, IEEE Transactions on Automatic Control, 49: 828-832, 2004. http://dx.doi.org/10.1109/TAC.2004.828317

X. Jiang, Q.L. Han and X.H. Yu, Stability criteria for linear discrete-time systems with interval-like time-varying delay, Proc. American Control Conference, New Orleans, USA, 2817-2822, 2005.

Y.S. Lee and W.H. Kwon, Delay-dependent robust stabilization of uncertain discrete-time state-delayed systems, Proc. 15thIFAC World Congr., 15(1): Barcelona, Spain 2002.

V. Leite and M. Miranda, Robust Stabilization of Discrete-Time Systems with Time-Varying Delay: An LMI Approach, Mathematical Problems in Engineering, 2008: Article ID 876509, 15 pages, 2008.

X.G. Liu, R.R. Martin, M. Wu and M.L. Tang, Delay-dependent robust stabilization of discrete-time systems with time-varying delay, IEE Proc.: Control Theory and Applications, 153(6): 689-702, 2006. http://dx.doi.org/10.1049/ip-cta:20050223

M.S. Mahmood, Robust control and filtering for time-delay systems, Marcel-Dekker, New York, 2000.

Y.S. Moon, P. Park and W.H. Kwon, Robust stabilization of uncertain input-delayed systems using reduction method, Automatica, 37: 307-312, 2001. http://dx.doi.org/10.1016/S0005-1098(00)00145-X

P. Park and J.W. Ko Stability and robust stability for systems with a time-varying delay, Automatica, 43: 1855-1858, 2007. http://dx.doi.org/10.1016/j.automatica.2007.02.022

E. Shustin and E. Fridman, On delay-derivative-dependent stability of systems with fastvarying delays, Automatica, 43: 1649-1655, 2007. http://dx.doi.org/10.1016/j.automatica.2007.02.009

M. Wu, Y. He, J.H. She and G.P. Liu, Delay-dependent criteria for robust stability of time-varying delay systems, Automatica, 40: 1435-1439, 2004. http://dx.doi.org/10.1016/j.automatica.2004.03.004

S. Xu and J. Lam, Improved Delay-dependent Stability Criteria for time-delay systems, IEEE Transactions on Automatic Control, 50(3): 384-387, 2005. http://dx.doi.org/10.1109/TAC.2005.843873

S. Xu, J. Lam and Y. Zou, Improved conditions for delay-dependent robust stability and stabilization of uncertain discrete time-delay systems, Asian Journal of Control, 7(3): 344- 348, 2005. http://dx.doi.org/10.1111/j.1934-6093.2005.tb00244.x

D. Yue and Q.L. Han, A delay-dependent stability criterion of neutral systems and its application to a partial element equivalent circuit model, IEEE Transactions on Circuits and Systems-II, 51(12): 685-689, 2004. http://dx.doi.org/10.1109/TCSII.2004.837286

D. Yue, E. Tian and Y. Zhang, A piecewise analysis method to stability analysis of linear continuous/discrete systems with time-varying delay, International Journal of Robust and Nonlinear Control, 19: 1493-1518, 2009. http://dx.doi.org/10.1002/rnc.1399

X.M. Zhang and Q.L. Han, A delay decomposition to delay-dependent stability for linear systems with time-varying delays, International Journal of robust and nonlinear control, 19: 1922-1930, 2009. http://dx.doi.org/10.1002/rnc.1413

X.L. Zhu and G.H. Yang, New results of stability analysis for systems with time-varying delay, International Journal of robust and nonlinear control 20: 596-606, 2010.

Published

2014-09-16

Issue

Vol. 7 No. 4 (2012): International Journal of Computers Communications & Control (November)

Section

Articles

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