**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**30455

##### 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:**
lyapunov function,
linear matrix inequalities,
Switching design,
exponential stability and stabilization,
switched linear systems,
interval delay

**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.