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Stability analysis for linear delayed systems via an optimally dividing delay interval approach. (English) Zbl 1227.93092
Automatica 47, No. 9, 2126-2129 (2011); corrigendum ibid. 50, No. 6, 1739-1740 (2014).
Summary: This paper addresses the problem of stability for linear systems with time-varying delay. A novel augmented Lyapunov-Krasovskii functional is constructed by using the idea of optimally dividing the delay interval $$[0,\tau (t)]$$ into some variable sub-intervals and using the line integral technology. Using the novel augmented functional, a new delay-dependent stability criterion is proposed for linear systems with time-varying delay. The gain is that this stability criterion can lead to much less conservative stability results compared to other methods for linear systems with delay. Two numerical examples are provided to verify the effectiveness of the proposed criteria.

##### MSC:
 93D09 Robust stability 93C05 Linear systems in control theory 93C15 Control/observation systems governed by ordinary differential equations
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##### References:
 [1] Ariba, Y., & Gouaisbaut, F. (2008). Construction of Lyapunov-Krasovskii functional for time-varying delay systems. In 47th IEEE conference on decision and control (CDC’08). · Zbl 1190.93076 [2] Briat, C. (2008). Robust control and observation of LPV time-delay systems. Ph.D. thesis, INP-Grenoble. France. [3] Fridman, E.; Shaked, U; Liu, K., New conditions for delay-derivative-dependent stability, Automatica, 45, 2723-2727, (2009) · Zbl 1180.93080 [4] Gouaisbaut, F., & Peaucelle, D. (2006). Delay-dependent stability analysis of linear time delay systems. In the sixth IFAC workshop on time-delay systems (TDS’06). · Zbl 1293.93589 [5] Gu, K.; Kharitonov, V.; Chen, J., Stability of time-delay systems, (2003), Birkhäuser Cambridge, MA · Zbl 1039.34067 [6] Han, Q.-L., A discrete delay decomposition approach to stability of linear retarded and neutral systems, Automatica, 45, 517-524, (2009) · Zbl 1158.93385 [7] He, Y.; Wang, Q.; Xie, L.; Lin, C., Further improvement of free-weighting matrices technique for systems with time-varying delay, IEEE transactions on automatic control, 52, 293-299, (2007) · Zbl 1366.34097 [8] Kao, C.-Y.; Rantzer, A., Stability analysis of systems with uncertain time-varying delays, Automatica, 43, 959-970, (2007) · Zbl 1282.93203 [9] Liu, Z.; Zhang, H., Delay-dependent stability for systems with fast-varying neutral-type delay via a PTVD compensation, ACTA automatica SINICA, 36, 147-152, (2010) [10] Nelder, J.A.; Mead, R., A simplex method for function minimization, Computer journal, 7, 308-313, (1965) · Zbl 0229.65053 [11] Park, P.; Ko, J., Stability and robust stability for systems with a time-varying delay, Automatica, 43, 1855-1858, (2007) · Zbl 1120.93043 [12] Shao, H., New delay-dependent stability criteria for systems with interval delay, Automatica, 45, 744-749, (2009) · Zbl 1168.93387 [13] Sun, J.; Liu, G.P.; Chen, J.; Rees, D., Improved delay-range-dependent stability criteria for linear systems with time-varying delays, Automatica, 46, 466-470, (2010) · Zbl 1205.93139
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