×

zbMATH — the first resource for mathematics

Exponential stability and exponential stabilization of singularly perturbed stochastic systems with time-varying delay. (English) Zbl 1207.93112
Summary: The problems of exponential stability and exponential stabilization for linear singularly perturbed stochastic systems with time-varying delay are investigated. First, an appropriate Lyapunov functional is introduced to establish an improved delay-dependent stability criterion. By applying free-weighting matrix technique and by equivalently eliminating time-varying delay through the idea of convex combination, a less conservative sufficient condition for exponential stability in mean square is obtained in terms of \(\varepsilon \)-dependent Linear Matrix Inequalities (LMIs). It is shown that if this set of LMIs for \(\varepsilon =0\) are feasible then the system is exponentially stable in mean square for sufficiently small \(\varepsilon \geqslant 0\). Furthermore, it is shown that if a certain matrix variable in this set of LMIs is chosen to be a special form and the resulting LMIs are feasible for \(\varepsilon =0\), then the system is \(\varepsilon \)-uniformly exponentially stable for all sufficiently small \(\varepsilon \geqslant 0\). Based on the stability criteria, an \(\varepsilon \)-independent state-feedback controller that stabilizes the system for sufficiently small \(\varepsilon \geqslant 0\) is derived. Finally, numerical examples are presented, which show our results are effective and useful.

MSC:
93E15 Stochastic stability in control theory
93E03 Stochastic systems in control theory (general)
93D30 Lyapunov and storage functions
93C70 Time-scale analysis and singular perturbations in control/observation systems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Gu, Stability of Time-Delay System (2003) · doi:10.1007/978-1-4612-0039-0
[2] Boukas, Deterministic and Stochastic Time Delay Systems (2002) · doi:10.1007/978-1-4612-0077-2
[3] Mahmoud, Dissipativity results for linear singular time-delay systems, International Journal of Innovative Computing, Information and Control 4 (11) pp 2833– (2008)
[4] Basin, Optimal linear filtering for systems with multiple state and observation delays, International Journal of Innovative Computing, Information and Control 3 (5) pp 1309– (2007)
[5] Wang, Exponential stability analysis for discrete-time switched linear systems with time-delay, International Journal of Innovative Computing, Information and Control 3 (6B) pp 1557– (2007)
[6] Chen, Delay-dependent robust stabilization and H control of uncertain stochastic systems with time-varying delay, IMA Journal of Mathematical Control and Information 21 (3) pp 345– (2004) · Zbl 1053.93012
[7] Chen, Delay-dependent exponential stability of uncertain stochastic systems with multiple delays: an LMI approach, Systems and Control Letters 54 (6) pp 547– (2005) · Zbl 1129.93547
[8] Fridman, New Lyapunov-Krasovskii functionals for stability of linear retard and neutral type systems, Systems and Control Letters 43 (4) pp 309– (2001) · Zbl 0974.93028
[9] Mao, Stochastic Differential Equations and Applications (1997)
[10] He, Augmented Lyapunov functional and delay-dependent stability criteria for neutral systems, International Journal of Robust and Nonlinear Control 15 (18) pp 923– (2005) · Zbl 1124.34049
[11] He, Further improvement of free-weighting matrices technique for systems with time-varying delay, IEEE Transactions on Automatic Control 52 (2) pp 293– (2007) · Zbl 1366.34097
[12] Vasil’eva, The Boundary Function Method for Singular Perturbation Problems (1995) · doi:10.1137/1.9781611970784
[13] Saberi, Quadratic-type Lyapunov functions for singularly perturbed systems, IEEE Transactions on Automatic Control 29 (6) pp 542– (1984) · Zbl 0538.93049
[14] Corless, On the exponential stability of singularly perturbed systems, SIAM Journal on Control and Optimization 30 (6) pp 1338– (1992) · Zbl 0763.34040
[15] Christofides, Singular perturbations and input-to-state stability, IEEE Transactions on Automatic Control 41 (11) pp 1645– (1996) · Zbl 0864.93086
[16] Socha, Exponential stability of singularly perturbed stochastic systems, IEEE Transactions on Automatic Control 45 (3) pp 576– (2000) · Zbl 0987.34054
[17] Tang C Basar T Stochastic stability of singularly perturbed nonlinear systems 399 404
[18] Luse, Multivariable singularly perturbed feedback systems with time delay, IEEE Transactions on Automatic Control 32 (11) pp 990– (1987) · Zbl 0628.93053
[19] Shao, Stability of time-delay singularly perturbed systems, IEE Proceedings, Control Theory and Applications 142 (2) pp 111– (1995) · Zbl 0822.93062
[20] Pan, Stability bound of multiple time delay singularly perturbed systems, Electronic Letters 32 (14) pp 1327– (1996)
[21] Dragan, Exponential stability for singularly perturbed systems with state delays, Electronic Journal of Qualitative Theory of Differential Equations 6 pp 1– (2000) · Zbl 0971.34063
[22] Dragan, Exponential stability for a class of singularly perturbed ItĂ´ differential equations, Electronic Journal of Qualitative Theory of Differential Equations 7 pp 1– (2000) · Zbl 0971.34034
[23] Tian, The expoential asymptotic stability of singularly perturbed delay differential equations with a bounded lay, Journal of Mathematical Analysis and Applications 270 (1) pp 143– (2002)
[24] Liu, Exponential stability of singularly perturbed systems with time delay, Applicable Analysis 82 (2) pp 117– (2003) · Zbl 1044.34031
[25] Glizer, H control of linear singularly perturbed systems with small state delay, Journal of Mathematical Analysis and Applications 250 (1) pp 49– (2000) · Zbl 0965.93049
[26] Glizer, On stabilization of nonstandard singularly perturbed systems with small delays in state and control, IEEE Transactions on Automatic Control 49 (6) pp 1012– (2004) · Zbl 1365.93313
[27] Fridman, Effects of small delays on stability of singularly perturbed systems, Automatica 38 (5) pp 897– (2002) · Zbl 1014.93025
[28] Fridman, Stability of singularly perturbed differential-differences systems: a LMI approach, Dynamics of Continuous, Discrete and Impulsive Systems 9 (2) pp 201– (2002) · Zbl 1012.34074
[29] Chen, Exponential stability of singularly perturbed Stochastic Systems with Delay, Dynamics of Continuous, Discrete and Impulsive Systems, Series B: Applications and Algorithms 12 (5-6) pp 701– (2005)
[30] Liu H Boukas E-K Sun F H stabilization of Markovian jumping singularly perturbed delayed systems 2245 2250
[31] Haidar, Exponential stability of singular systems with multiple time-varying delays, Automatica 45 (2) pp 539– (2009) · Zbl 1158.93347
[32] Fridman, Stability of linear descriptor systems with delay: a Lyapunov-based approach, Journal of Mathematical Analysis and Applications 273 (1) pp 24– (2002) · Zbl 1032.34069
[33] Boyd, Linear Matrix Inequalities in System and Control Theory (1994) · Zbl 0816.93004 · doi:10.1137/1.9781611970777
[34] Zhu, Delay-dependent robust stability criteria for two classes of uncertain singular time-delay systems, IEEE Transactions on Automatic Control 52 (5) pp 880– (2007) · Zbl 1366.93478
[35] Xu, An improved characterization of bounded realness for singular delay systems and its applications, International Journal of Robust and Nonlinear Control 18 (3) pp 263– (2008) · Zbl 1284.93117
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.