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Robust integral sliding mode control for uncertain stochastic systems with time-varying delay. (English) Zbl 1093.93027
Summary: This paper is concerned with sliding mode control for uncertain stochastic systems with time-varying delay. Both time-varying parameter uncertainties and an unknown nonlinear function may appear in the controlled system. An integral sliding surface is first constructed. Then, by means of linear matrix inequalities (LMIs), a sufficient condition is derived to guarantee the global stochastic stability of the stochastic dynamics in the specified switching surface for all admissible uncertainties. The synthesized sliding mode controller guarantees the reachability of the specified sliding surface. Finally, a simulation example is presented to illustrate the proposed method.

##### MSC:
 93E03 General theory of stochastic systems 93B12 Variable structure systems
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##### References:
 [1] Chang, K. -Y.; Chang, W. -J.: Variable structure controller design with H$\infty$norm and variance constraints for stochastic model reference systems. IEE Proceedings: control theory and applications 146, 511-516 (1999) [2] Chang, K. -Y.; Wang, W. -J.: H$\infty$norm constraint and variance control for stochastic uncertain large-scale system via sliding mode concept. IEEE transactions on circuits and systems--I: fundamental theory and applications 46, 1275-1280 (1999) · Zbl 0962.93027 [3] Chang, K. -Y.; Wang, W. -J.: Robust covariance control for perturbed stochastic multivariable system via variable structure control. Systems & control letters 37, 323-328 (1999) · Zbl 0948.93008 [4] Choi, H. H.: A new method for variable structure control system design: A linear matrix inequality approach. Automatica 33, 2089-2092 (1997) · Zbl 0911.93022 [5] El-Khazali, R.: Variable structure robust control of uncertain time-delay systems. Automatica 34, 327-332 (1998) · Zbl 0965.93025 [6] Higham, D.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM review 43, 525-546 (2001) · Zbl 0979.65007 [7] Kolmanovskii, V.; Myshkis, A.: Applied theory of functional differential equations. (1992) · Zbl 0917.34001 [8] Li, X.; Decarlo, R. A.: Robust sliding mode control of uncertain time-delay systems. International journal of control 76, 1296-1305 (2003) · Zbl 1036.93005 [9] Mao, X.: Stability of stochastic differential equations with respect to semimartingales. (1991) · Zbl 0724.60059 [10] Mao, X.; Koroleva, N.; Rodkina, A.: Robust stability of uncertain stochastic differential delay systems. Systems & control letters 35, 325-336 (1998) · Zbl 0909.93054 [11] Niu, Y.; Lam, J.; Wang, X.; Ho, D. W. C.: Sliding mode control for nonlinear state-delayed systems using neural network approximation. IEE Proceedings: control theory and applications 150, 233-239 (2003) [12] Niu, Y.; Lam, J.; Wang, X.; Ho, D. W. C.: Observer-based sliding mode control for nonlinear state-delayed systems. International journal of systems science 35, 139-150 (2004) · Zbl 1059.93025 [13] Tan, C. P.; Edwards, C.: An LMI approach for designing sliding mode observers. International journal of control 74, 1559-1568 (2001) · Zbl 1101.93304 [14] Verriest, E. I.; Florchinger, P.: Stability of stochastic systems with uncertain time delay. Systems & control letters 24, 41-47 (1995) · Zbl 0867.34065 [15] Wang, Z.; Qiao, H.; Burnham, K. J.: On the stabilization of bilinear uncertain time-delay stochastic systems with Markovian jumping parameters. IEEE transactions on automatic control 47, 640-646 (2002) [16] Xu, S.; Chen, T.: Reduced-order H$\infty$filtering for stochastic systems. IEEE transactions on signal processing 50, 2998-3007 (2002) [17] Xu, S.; Chen, T.: Robust H$\infty$control for uncertain stochastic systems with state delay. IEEE transactions on automatic control 47, 2089-2094 (2002)