Switching Rule for the Exponential Stability and Stabilization of Switched Linear Systems with Interval Time-varying Delays

Authors: Kreangkri Ratchagit

Abstract:

This paper is concerned with exponential stability and stabilization of switched linear systems with interval time-varying delays. The time delay is any continuous function belonging to a given interval, in which the lower bound of delay is not restricted to zero. By constructing a suitable augmented Lyapunov-Krasovskii functional combined with Leibniz-Newton-s formula, a switching rule for the exponential stability and stabilization of switched linear systems with interval time-varying delays and new delay-dependent sufficient conditions for the exponential stability and stabilization of the systems are first established in terms of LMIs. Numerical examples are included to illustrate the effectiveness of the results.

Keywords: Switching design, exponential stability and stabilization, switched linear systems, interval delay, Lyapunov function, linear matrix inequalities.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1080187

References:

[1] V.N. Phat and P.T. Nam (2007), Exponential stability and stabilization of uncertain linear time-varying systems using parameter dependent Lyapunov function. Int. J. of Control, 80, 1333-1341.
[2] V.N. Phat and P. Niamsup, Stability analysis for a class of functional differential equations and applications. Nonlinear Analysis: Theory, Methods & Applications 71(2009), 6265-6275.
[3] V.N. Phat, T. Bormat and P. Niamsup, Switching design for exponential stability of a class of nonlinear hybrid time-delay systems, Nonlinear Analysis: Hybrid Systems, 3(2009), 1-10.
[4] Y.J. Sun, Global stabilizability of uncertain systems with time-varying delays via dynamic observer-based output feedback, Linear Algebra and its Applications, 353(2002), 91-105.
[5] O.M. Kwon and J.H. Park, Delay-range-dependent stabilization of uncertain dynamic systems with interval time-varying delays, Applied Math. Conputation, 208(2009), 58-68.
[6] H. Shao, New delay-dependent stability criteria for systems with interval delay, Automatica, 45(2009), 744-749.
[7] J. Sun, G.P. Liu, J. Chen and D. Rees, Improved delay-range-dependent stability criteria for linear systems with time-varying delays, Automatica, 46(2010), 466-470.
[8] W. Zhang, X. Cai and Z. Han, Robust stability criteria for systems with interval time-varying delay and nonlinear perturbations, J. Comput. Appl. Math., 234 (2010), 174-180.
[9] V.N. Phat, Robust stability and stabilizability of uncertain linear hybrid systems with state delays, IEEE Trans. CAS II, 52(2005), 94-98.
[10] V.N. Phat and P.T. Nam (2007), Exponential stability and stabilization of uncertain linear time-varying systems using parameter dependent Lyapunov function. Int. J. of Control, 80, 1333-1341.
[11] V.N. Phat and P. Niamsup, Stability analysis for a class of functional differential equations and applications. Nonlinear Analysis: Theory, Methods & Applications 71(2009), 6265-6275.
[12] V.N. Phat and P.T. Nam (2007), Exponential stability and stabilization of uncertain linear time-varying systems using parameter dependent Lyapunov function. Int. J. of Control, 80, 1333-1341.
[13] V.N. Phat and P. Niamsup, Stability analysis for a class of functional differential equations and applications. Nonlinear Analysis: Theory, Methods & Applications 71(2009), 6265-6275.
[14] V.N. Phat, Y. Khongtham, and K. Ratchagit, LMI approach to exponential stability of linear systems with interval time-varying delays. Linear Algebra and its Applications 436(2012), 243-251.
[15] K. Ratchagit and V.N. Phat, Stability criterion for discrete-time systems, J. Ineq. Appl., 2010(2010), 1-6.
[16] F. Uhlig, A recurring theorem about pairs of quadratic forms and extensions, Linear Algebra Appl., 25(1979), 219-237.
[17] K. Gu, An integral inequality in the stability problem of time delay systems, in: IEEE Control Systems Society and Proceedings of IEEE Conference on Decision and Control, IEEE Publisher, New York, 2000.
[18] Y. Wang, L. Xie and C.E. de SOUZA, Robust control of a class of uncertain nonlinear systems. Syst. Control Lett., 1991992), 139-149.
[19] S. Boyd, L.El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994.
[20] J.K. Hale and S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.