# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
A novel approach on stabilization for linear systems with time-varying input delay. (English) Zbl 1243.93103
Summary: This paper presents new results on delay-dependent stability and stabilization for linear systems with interval time-varying delays. Some less conservative delay-dependent criteria for determining the stability of the time-delay systems are obtained in this paper. Based on the stability conditions, we propose a new state transformation technology to facilitate controller designing efficiently and computationally. The method is also applicable to the existing stability conditions reported by now, while the existing technologies may fail to derive computational control procedures from the stability conditions. Finally, some numerical examples well illustrate the effectiveness of the proposed method.
##### MSC:
 93D21 Adaptive or robust stabilization 93C15 Control systems governed by ODE 93C05 Linear control systems
##### Keywords:
linear systems; stability and stabilization; time-delays
##### References:
 [1] Ariba, Y.; Gouaisbaut, F.: An augmented model for robust stability analysis of time-varying delay systems, International journal of control 82, 1616-1626 (2009) · Zbl 1190.93076 · doi:10.1080/00207170802635476 [2] Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V.: Linear matrix inequalities in system and control theory, (1994) [3] Basin, M.; Shi, P.; Calderon-Alvarez, D.: Joint state filtering and parameter estimation for linear time-delay systems, Signal processing 91, 782-792 (2011) · Zbl 1217.94034 · doi:10.1016/j.sigpro.2010.08.011 [4] Dugard, L.; Verriest, E. I.: Stability and control of time-delay systems, (1998) [5] Gu, K.; Kharitonov, V. L.; Chen, J.: Stability of time-delay systems, (2003) [6] Gu, K.; Niculescu, S. I.: Survey on recent results in the stability and control of time-delay systems, Journal of dynamic systems, measurement, and control 124, 158-165 (2003) [7] Hu, L.; Shi, P.; Cao, Y.: Delay-dependent filtering design for time-delay systems with Markovian jumping parameters, International journal of adaptive control and signal processing 21, 434-448 (2007) · Zbl 1120.60043 · doi:10.1002/acs.938 [8] Hale, J. K.; Lunel, S. M. Verduyn: Introduction of functional differential equations, (1993) [9] Han, Q. L.; Gu, K.: Stability of linear systems with time-varying delay: a generalized discretized Lyapunov functional approach, Asian journal of control 3, 170-180 (2001) [10] He, Y.; Wang, Q.; Lin, C.; Wu, M.: Delay-range-dependent stability for systems with time-varying delay, Automatica 43, 371-376 (2007) · Zbl 1111.93073 · doi:10.1016/j.automatica.2006.08.015 [11] Lin, Z.; Fang, H.: On asymptotic stabilizability of linear systems with delayed input, IEEE transactions on automatic control 52, 998-1013 (2007) [12] Lakshmanan, S.; Senthilkumar, T.; Balasubramaniam, P.: Improved results on robust stability of neutral systems with mixed time-varying delays and nonlinear perturbations, Applied mathematical modelling 35, 5355-5368 (2011) · Zbl 1228.93091 · doi:10.1016/j.apm.2011.04.043 [13] Mahmoud, M. S.; Ismail, A.: New results on delay-dependent control of time-delay systems, IEEE transactions on automatic control 50, 95-100 (2005) [14] Nguang, S. K.; Shi, P.; Ding, S.: Delay dependent fault estimation for uncertain time delay nonlinear systems: an LMI approach, International journal of robust and nonlinear control 16, 913-933 (2006) · Zbl 1135.93022 · doi:10.1002/rnc.1116 [15] Park, M. J.; Kwon, O. M.; Park, Ju H.; Lee, S. M.: A new augmented lyapunovckrasovskii functional approach for stability of linear systems with time-varying delays, Applied mathematics and computation 217, 7197-7209 (2011) · Zbl 1219.93106 · doi:10.1016/j.amc.2011.02.006 [16] Richard, J. P.: Time-delay systems: an overview of some recent advances and open problems, Automatica 39, 1667-1694 (2003) · Zbl 1145.93302 · doi:10.1016/S0005-1098(03)00167-5 [17] Shao, H. Y.: New delay-dependent stability criteria for systems with interval delay, Automatica 45, 744-749 (2009) · Zbl 1168.93387 · doi:10.1016/j.automatica.2008.09.010 [18] T. Senthilkumar, P. Balasubramaniam, Delay-dependent robust stabilization and H-infinity control for nonlinear stochastic systems with Markovian jump parameters and interval time-varying delays, Journal of Optimization Theory and Applications. doi:10.1007/s10957-011-9858-7. [19] Shi, P.; Boukas, E. K.; Agarwal, R.: Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delay, IEEE transactions on automatic control 44, 2139-2144 (1999) · Zbl 1078.93575 · doi:10.1109/9.802932 [20] Sun, J.; Liu, G. P.; Chen, J.; Reers, D.: Improved delay-range-dependent stability criteria for linear systems with time-varying delays, Automatica 46, 466-470 (2010) · Zbl 1205.93139 · doi:10.1016/j.automatica.2009.11.002 [21] Shi, P.; Mahmoud, M.; Nguang, S.; Ismail, A.: Robust filtering for jumping systems with mode-dependent delays, Signal processing 86, 140-152 (2006) · Zbl 1163.94387 · doi:10.1016/j.sigpro.2005.05.005 [22] Wang, D.; Wang, W.; Shi, P.: Robust fault detection for switched linear systems with state delays, IEEE transactions on systems, man and cybernetics, part B: cybernetics 39, 800-805 (2009) [23] Xia, Y.; Fu, M.; Shi, P.: Analysis and synthesis of dynamical systems with time-delays, (2009) [24] Xu, S.; Lam, J.: A survey of linear matrix inequality techniques in stability analysis of delay systems, International journal of systems science 39, 1095-1113 (2008) · Zbl 1156.93382 · doi:10.1080/00207720802300370 [25] Xia, Y.; Liu, G. P.; Shi, P.; Rees, D.: Robust delay-dependent sliding mode control for uncertain time-delay systems, International journal of robust and nonlinear control 18, 1142-1161 (2008) [26] Yang, R.; Gao, H.; Shi, P.: Delay-dependent robust H$\infty$ control for uncertain stochastic time-delay systems, International journal of robust and nonlinear control 20, 1852-1865 (2010) · Zbl 1203.93191 · doi:10.1002/rnc.1552 [27] Zhang, X. M.; Wu, M.; She, J. H.; He, Y.: Delay-dependent stabilization of linear systems with time-varying state and input delays, Automatica 41, 1405-1412 (2005) · Zbl 1093.93024 · doi:10.1016/j.automatica.2005.03.009