Robust H∞ control for a class of nonlinear stochastic systems with mixed time delay

This paper is concerned with the problem of robust H∞ control for a class of uncertain nonlinear Itô‐type stochastic systems with mixed time delays. The parameter uncertainties are assumed to be norm bounded, the mixed time delays comprise both the discrete and distributed delays, and the sector nonlinearities appear in both the system states and delayed states. The problem addressed is the design of a linear state feedback controller such that, in the simultaneous presence of parameter uncertainties, system nonlinearities and mixed time delays, the resulting closed‐loop system is asymptotically stable in the mean square and also achieves a prescribed H∞ disturbance rejection attenuation level. By using the Lyapunov stability theory and the Itô differential rule, some new techniques are developed to derive the sufficient conditions guaranteeing the existence of the desired feedback controllers. A unified linear matrix inequality is proposed to deal with the problem under consideration and a numerical example is exploited to show the usefulness of the results obtained. Copyright © 2007 John Wiley & Sons, Ltd.

[1]  R. Khasminskii.  Stochastic Stability of Differential Equations , 1980 .

[2]  A. Skorokhod.  Asymptotic Methods in the Theory of Stochastic Differential Equations , 2008 .

[3]  Seung-Jean Kim,et al.  A state-space approach to analysis of almost periodic nonlinear systems with sector nonlinearities , 1999, IEEE Trans. Autom. Control..

[4]  Kolmanovskii,et al.  Introduction to the Theory and Applications of Functional Differential Equations , 1999 .

[5]  Lihua Xie,et al.  Decentralized stabilization of a class of interconnected stochastic nonlinear systems , 2000, IEEE Trans. Autom. Control..

[6]  Ruth J. Williams,et al.  Stabilization of stochastic nonlinear systems driven by noise of unknown covariance , 1998, Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207).

[7]  Miroslav Krstic,et al.  Stabilization of stochastic nonlinear systems driven by noise of unknown covariance , 2001, IEEE Trans. Autom. Control..

[8]  Shengyuan Xu,et al.  Robust H∞ control for uncertain stochastic systems with state delay , 2002, IEEE Trans. Autom. Control..

[9]  El-Kebir Boukas,et al.  Deterministic and Stochastic Time-Delay Systems , 2002 .

[10]  Charalambos D. Charalambous.  Stochastic nonlinear minimax dynamic games with noisy measurements , 2003, IEEE Trans. Autom. Control..

[11]  James Lam,et al.  Nonlinear filtering for state delayed systems with Markovian switching , 2003, IEEE Trans. Signal Process..

[12]  Yuli Fu,et al.  BIBO stabilization of stochastic delay systems with uncertainty , 2003, IEEE Trans. Autom. Control..

[13]  Daniel W. C. Ho,et al.  Filtering on nonlinear time-delay stochastic systems , 2003, Autom..

[14]  Yungang Liu,et al.  Output feedback control design for strict-feedback stochastic nonlinear systems under a risk-sensitive cost , 2003, IEEE Trans. Autom. Control..

[15]  Daniel W. C. Ho,et al.  A note on the robust stability of uncertain stochastic fuzzy systems with time-delays , 2004, IEEE Trans. Syst. Man Cybern. Part A.

[16]  U. Shaked,et al.  H∞ Control for Discrete-Time Nonlinear Stochastic Systems , 2004 .

[17]  Qing-Long Han,et al.  Absolute stability of time-delay systems with sector-bounded nonlinearity , 2005, Autom..

[18]  Huijun Gao,et al.  On H∞ Performance Analysis for Continuous-Time Stochastic Systems with Polytopic Uncertainties , 2005 .

[19]  Huijun Gao,et al.  $${\cal{H}}_{\infty}$$ and $${\cal{L}}_{\bf 2}/{\cal{L}}_{\infty}$$Model Reduction for System Input with Sector Nonlinearities , 2005 .

[20]  James Lam,et al.  Robust integral sliding mode control for uncertain stochastic systems with time-varying delay , 2005, Autom..

[21]  U. Shaked,et al.  H/sub /spl infin// control for discrete-time nonlinear stochastic systems , 2006, IEEE Transactions on Automatic Control.

[22]  Jianqiang Yi,et al.  Worst case control of uncertain jumping systems with multi-state and input delay information , 2006, Inf. Sci..

[23]  James Lam,et al.  Robust energy-to-peak filter design for stochastic time-delay systems , 2006, Syst. Control. Lett..

[24]  Peng Shi,et al.  Robust filtering for jumping systems with mode-dependent delays , 2006, Signal Process..

[25]  Zidong Wang,et al.  Robust stability analysis of generalized neural networks with discrete and distributed time delays , 2006 .