Volume 3, Issue 2, December 2019, Page: 20-28
On Delay-Range-Dependent and Delay-Rate-Dependent Stability for Delayed Systems
Yuan He, Deparment of Automation, Beijing Institute of Petro-chemical Technology, Beijing, China
Jintian Hu, Deparment of Automation, Beijing Institute of Petro-chemical Technology, Beijing, China
Shuxia Wang, Deparment of Mathematics and Physics, Beijing Institute of Petro-chemical Technology, Beijing, China
Liansheng Zhang, Deparment of Mathematics and Physics, Beijing Institute of Petro-chemical Technology, Beijing, China
Received: Sep. 27, 2019;       Accepted: Oct. 22, 2019;       Published: Oct. 28, 2019
DOI: 10.11648/j.cse.20190302.11      View  19      Downloads  5
Abstract
It is well known that the phenomena of time delays are frequently encountered in many process and various control systems. The presence of delays can have an effect on system stability and performance, so ignoring them may lead to design flaws and incorrect analysis conclusions. Hence, the stability problem for time-delayed systems has received considerable attention in recent years. This brief focuses on the stability analysis for a class of delayed linear systems. Firstly, we construct a novel augmented Lyapunov-Krasovskii functional (LKF) which includes the lower, the upper bounds of the delay and the delay itself. Secondly, utilizing some integral inequalities and the reciprocally convex combination lemma, we obtain less conservative stability criteria formulated in form of linear matrix inequalities (LMIs). Finally, numerical examples are provided to show the effectiveness of the proposed method.
Keywords
Time Delay, Lyapunov-Krasovskii Functional (LKF), Linear Matrix Inequalities (LMIs)
To cite this article
Yuan He, Jintian Hu, Shuxia Wang, Liansheng Zhang, On Delay-Range-Dependent and Delay-Rate-Dependent Stability for Delayed Systems, Control Science and Engineering. Vol. 3, No. 2, 2019, pp. 20-28. doi: 10.11648/j.cse.20190302.11
Copyright
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Reference
[1]
E. Fridman and U. Shaked. An improved stabilization method for linear time-delay systems. IEEE Transactions on Automatic Control, 2002; 47 (11): 1931-1937.
[2]
J. Sun, G. P. Liu and J. Chen. Delay-dependent stability and stabilization of neutral time-delay systems. International Journal of Robust and Nonlinear Control, 2009; 19 (12): 1364-1375.
[3]
O. M. Kwon, E. J. Cha. New stability criteria for linear system with interval time-varying state delay. Journal of Electrical Engineering & Technology, 2011; 6 (5): 713-722.
[4]
T. Wang, A. G. Wu. Improved delay-range-dependent stability criteria for continuous linear system with time-varying delay. Proceedings of the 25th Chinese Control and Decision Conference, 2013; 182-187.
[5]
J. H. Kim. Note on stability of linear systems with time-varying delay. Automatica, 2011; 47 (9): 2118-2121.
[6]
K. Gu, An integral inequality in the stability problem of time-delay systems, Proceedings of 39th IEEE conference on decision and control, 2000; 2805-2810.
[7]
Q. L. Han and K. Gu. Stability of linear systems with time-varying delay: a generalized discretized Lyapunov function approach. Asian Journal of Control, 2001; 13 (3): 170-180.
[8]
W. Lee, P. Park. Second-order reciprocally convex approach to stability of systems with interval time-varying delays. Applied Mathematics and Computation, 2014; 229 (1): 245–253.
[9]
S. Xu and J. Lam. On equivalence and efficiency of certain stability criteria for time-delay systems. IEEE Transactions on Automatic Control, 2007; 52 (1): 95-101.
[10]
M. Wu, Y. He, J. H. She, and G. P. Liu. Delay-dependent criteria for robust stability of time-varying delay systems. Automatica, 2004; 40 (8): 1435-1439.
[11]
P. L. Liu. Further improvement on delay-range-dependent stability results for linear systems with interval time-varying delays. ISA Trans, 2013; 52 (6): 725-729.
[12]
Y. He, Q. G. Wang, L. Xie, and C. Lin. Further improvement of free-weighting matrices technique for systems with time-varying delay. IEEE Transactions on Automatic Control, 2007; 52 (2): 293-299.
[13]
Y. He, Q. G. Wang, C. Lin, and M. Wu, Delay-range-dependent stability for systems with time-varying delay. Automatica, 2007; 43 (2): 371-376.
[14]
P. Park and J. W. Ko. Stability and robust stability for systems with time-varying delay. Automatica, 2007; 43 (10): 1855-1858.
[15]
X. Jiang and Q. L. Han. New stability criteria for linear systems with interval time-varying delay. Automatica, 2008; 44 (10): 2680-2685.
[16]
X. Jiang, Q. L. Han, S. Liu, and A. Xue. A new H∞ stabilization criterion for networked control systems. IEEE Transactions on Automatic Control, 2008; 53 (4): 1025-1032.
[17]
H. Shao. New delay-dependent stability criteria for systems with interval delay. Automatica, 2009; 45 (3): 744-749.
[18]
J. Sun, G. P. Liu, J. Chen, D. Rees. Improved delay-range-dependent stability criteria for linear systems with time-varying delays. Automatica, 2010; 46 (2): 466-470.
[19]
X. L. Zhu, Y. Wang and G. H. Yang. New stability criteria for continuous time systems with interval time-varying delay. IET Control Theory Appl., 2010; 4 (6): 1101-1107.
[20]
M. Tang, Y. W. Wang and C. Y. Wen. Improved delay-range-dependent stability criteria for linear systems with interval time-varying delays. IET Control Theory Appl., 2012; 6 (6): 868-873.
[21]
A. Seuret, F. Gouaisbaut. Wirtinger-based integral inequality: Application to time-delay systems. Automatica, 2013; 49 (9): 2860-2866.
[22]
K. Liu, & E. Fridman, Wirtinger’s inequality and Lyapunov-based sampled-data stabilization. Automatica, 2012; 48 (1): 102-108.
[23]
P. Park, W. Jeong, C. Jeong. Reciprocally convex approach to stability of systems with time-varying delays, Automatica, 2011; 47 (1): 235-238.
[24]
J. An et al., A novel approach to delay-fractional-dependent stability criterion for linear systems with interval delay, ISA Trans., 2014; 53 (1): 210-219.
[25]
J. H. Kim. Further improvement of Jensen inequality and application to stability of time-delayed systems. Automatica, 2016; 64 (1), 121-125.
[26]
Éva Gyurkovics, A note on Wirtinger-type integral inequalities for time-delay systems, Automatica, 2015; 61 (1): 44-46.
[27]
A. Seuret, & F. Gouaisbaut, (2017). Delay-dependent reciprocally convex combination lemma. http://hal.archives-ouvertes.fr/hal-01257670/.
[28]
C. K. Zhang, Y. He, L. Jiang, M. Wu, Q.-G. Wang, An extended reciprocally convex matrix inequality for stability analysis of systems with time-varying delay, Automatica. 2017; 85 (2): 481-485.
[29]
X. M. Zhang, Q. L. Han, A. Seuret, and F. Gouaisbaut, An improved reciprocally convex inequality and an augmented Lyapunov-Krasovskii functional for stability of linear systems with time-varying delay, Automatica, 2017; 84 (1): 221-226.
[30]
LS Zhang, L He, YD Song. New Results on Stability Analysis of Delayed Systems Derived from Extended Wirtinger’s Integral Inequality, Neurocomputing, 2018; 283 (1): 98-106.
[31]
T. H. Lee, J. H. Park, A novel Lyapunov functional for stability of time-varying delay systems via matrix-refined-function, Automatica, 2017; 80 (1): 239-247.
[32]
H.-B. Zeng, Y. He, M. Wu, & J. She. Free-matrix-based integral inequality for stability analysis of systems with time-varying delay. IEEE Transactions on Automatic Control, 2015; 60 (6): 2768-2772.
Browse journals by subject