Mehmet Nur Alpaslan PARLAKÇI

Robust stability of linear uncertain discrete-time systems with interval time-varying delay

This paper presents a robust stability problem for linear uncertain discrete-time systems with interval time-varying delay and norm-bounded uncertainties. First, a necessary and sufficient stability condition is obtained by employing a well-known lifting method and switched system approach for nominal discrete-time delay systems. Both the stability method of checking the characteristic values inside the unit circle and a Lyapunov function-based stability result are taken into consideration. Second, a simple Lyapunov--Krasovskii functional (LKF) is selected, and utilizing a generalized Jensen sum inequality, a sufficient stability condition is presented in the form of linear matrix inequalities. Third, a novel LKF is proposed together with the use of a convexity approach in the LKF. Finally, the proposed method is extended to the case when the system under consideration is subject to norm-bounded uncertainties. Three numerical examples are introduced to illustrate the effectiveness of the proposed approach, along with some numerical comparisons.

Anahtar Kelimeler:

Discrete-time systems, time-varying delay, norm-bounded uncertainties, robust stability, lifting method, Lyapunov--Krasovskii functional, linear matrix inequalities

Robust stability of linear uncertain discrete-time systems with interval time-varying delay

Keywords:

Discrete-time systems, time-varying delay, norm-bounded uncertainties, robust stability, lifting method, Lyapunov--Krasovskii functional, linear matrix inequalities,

PDF

___

K. Gu, V.L. Kharitonov, J. Chen, Stability of Time Delay Systems, Boston, MA, USA, Birkh¨ auser, 2003.
M. Wu, Y. He, J.H. She, Stability Analysis and Robust Control of Time-Delay System, New York, NY, USA, Springer, 2010.
E. Gyurkovics, “Robust control design for discrete time linear uncertain systems with delayed control”, IEE Proceedings D, Vol. 140, pp. 423–428, 1993.
T.J. Su, W.J. Shyr, “Robust D-stability for linear uncertain discrete time delay systems”, IEEE Transactions on Automatic Control, Vol. 39, pp. 425–428, 1994.
V.N. Phat, “Robust stability and stabilizability of uncertain linear hybrid systems with state delays”, IEEE Transactions on Automatic Control, Vol. 52, pp. 94–98, 2005.
C.P. Huang, T.Y. Juang, “Robustness analysis of discrete time delay systems”, International Journal of Systems Science, Vol. 37, pp. 1–7, 2006.
V.J.S. Leite, M.F. Miranda, “Robust stabilization of discrete-time systems with time-varying delay: an LMI approach”, Mathematical Problems in Engineering, pp. 1–15, 2008.
J. Qui, Y. Xia, H. Yang, J. Zhang, “Robust stabilisation for a class of discrete-time systems with time-varying delays via delta operators”, IET Control Theory and Applications, Vol. 2, pp. 87–93, 2008.
B. Zhang, S. Xu, Y. Zou, “Improved stability criterion and its applications in delayed controller design for discretetime systems”, Automatica, Vol. 44, pp. 2963–2967, 2008.
B.T. Anh, N.K. Son, “Robust stability of delay difference systems under fractional perturbations in infinitedimensional spaces”, International Journal of Control, Vol. 83, pp. 498–505, 2010.
M.S. Mahmoud, N.B. Almutairi, “Robust stability and stabilization methods for a class of nonlinear discrete-time delay systems”, Applied Mathematics and Computation, Vol. 215, pp. 4280–4292, 2010.
H. Shao, Q.L. Han, “New stability criteria for linear discrete-time systems with interval-like time-varying delays”, IEEE Transactions on Automatic Control, Vol. 56, pp. 619–625, 2011.
C. Peng, “Improved delay-dependent stabilisation criteria for discrete systems with a new finite sum inequality”, IET Control Theory and Applications, Vol. 6, pp. 448–453, 2012.
X.G. Liu, R.R. Martin, M. Wu, M.L. Tang, “Delay-dependent robust stabilisation of discrete-time systems with time-varying delay”, IEE Proceedings - Control Theory and Applications, Vol. 153, pp. 689–702, 2006.
Y. Xia, G.P. Liu, P. Shi, D. Rees, E.J.C. Thomas, “New stability and stabilization conditions for systems with time-delay”, International Journal of Systems Science, Vol. 38, pp. 17–24, 2007.
H. Gao, T. Chen, “New results on stability of discrete-time systems with time-varying state delay”, IEEE Transactions on Automatic Control, Vol. 52, pp. 328–334, 2007.
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, Vol. 10, pp. 1033–1040, 2008.
D. Yue, E. Tian, 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, Vol. 19, pp. 1493–1518, 2009. K.F. Chen, 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, pp. 563–572, 2010.
X. Meng, J. Lam, B. Du, H. Gao, “A delay-partitioning approach to the stability analysis of discrete-time systems”, Automatica, Vol. 46, pp. 610–614, 2010.
H. Huang, G. Feng, “Improved approach to delay-dependent stability analysis of discrete-time systems with timevarying delay”, IET Control Theory and Applications, Vol. 4, pp. 2152–2159, 2010.
V.K.R. Kandanvli, H. Kar, “Delay-dependent LMI condition for global asymptotic stability of discrete-time uncertain state-delayed systems using quantization/overflow nonlinearities”, International Journal of Robust and Nonlinear Control, Vol. 21, pp. 1611–1622, 2011.
X. Li, H. Gao, “A new model transformation of discrete-time systems with time-varying delay and its application to stability analysis”, IEEE Transactions on Automatic Control, Vol. 56, pp. 2172–2178, 2011.
C.Y. Kao, “On stability of discrete-time LTI systems with varying time delays”, IEEE Transactions on Automatic Control, Vol. 57, pp. 1243–1248, 2012.
K. Ramakrishnan, G. Ray, “Robust stability criteria for a class of uncertain discrete-time systems with time-varying delay”, Applied Mathematical Modelling, Vol. 37, pp. 1468–1479, 2013.
K. Ogata, Discrete-Time Control Systems, Upper Saddle River, NJ, USA, Prentice Hall, 1994.