×
Wang, Zidong; Liu, Yurong; Wei, Guoliang; Liu, Xiaohui

A note on control of a class of discrete-time stochastic systems with distributed delays and nonlinear disturbances. (English) Zbl 1194.93134

Automatica 46, No. 3, 543-548 (2010).
Summary: This paper is concerned with the state feedback control problem for a class of discrete-time stochastic systems involving sector nonlinearities and mixed time-delays. The mixed time-delays comprise both discrete and distributed delays, and the sector nonlinearities appear in the system states and all delayed states. The distributed time-delays in the discrete-time domain are first defined and then a special matrix inequality is developed to handle the distributed time-delays within an algebraic framework. An effective linear matrix inequality (LMI) approach is proposed to design the state feedback controllers such that, for all admissible nonlinearities and time-delays, the overall closed-loop system is asymptotically stable in the mean square sense. Sufficient conditions are established for the nonlinear stochastic time-delay systems to be asymptotically stable in the mean square sense, and then the explicit expression of the desired controller gains is derived. A numerical example is provided to show the usefulness and effectiveness of the proposed design method.

Cited in 47 Documents

MSC:

93C55 Discrete-time control/observation systems
93E15 Stochastic stability in control theory

Keywords:

discrete-time nonlinear stochastic system; mixed time delays; Lyapunov-Krasovskii functional; linear matrix inequality
PDF BibTeX XML Cite
\textit{Z. Wang} et al., Automatica 46, No. 3, 543--548 (2010; Zbl 1194.93134)
Full Text: DOI Link

References:

[1] Berman, N.; Shaked, U., \(H_\infty\) control for discrete-time nonlinear stochastic systems, IEEE Transactions on Automatic Control, 51, 6, 1041-1046 (2006) · Zbl 1366.93616
[2] Boukas, E. K.; Liu, Z.-K., Deterministic and stochastic time-delay systems (2002), Birkhauser: Birkhauser Boston · Zbl 0998.93041
[3] Fridman, E.; Orlov, Y., Exponential stability of linear distributed parameter systems with time-varying delays, Automatica, 45, 2, 194-201 (2009) · Zbl 1154.93404
[4] Fridman, E.; Tsodik, G., \(H_\infty\) control of distributed and discrete delay systems via discretized Lyapunov functional, European Journal of Control, 15, 1, 84-96 (2009) · Zbl 1298.93149
[5] Gao, H.; Chen, T., New results on stability of discrete-time systems with time-varying state delay, IEEE Transactions on Automatic Control, 52, 2, 328-334 (2007) · Zbl 1366.39011
[6] Gao, H.; Lam, J.; Wang, C., Robust energy-to-peak filter design for stochastic time-delay systems, Systems & Control Letters, 55, 2, 101-111 (2006) · Zbl 1129.93538
[7] Han, Q.-L., Absolute stability of time-delay systems with sector-bounded nonlinearity, Automatica, 41, 12, 2171-2176 (2005) · Zbl 1100.93519
[8] Ho, D. W.C., Robust fuzzy design for nonlinear uncertain stochastic systems via sliding-mode control, IEEE Transactions on Fuzzy Systems, 15, 3, 350-358 (2007)
[9] Khalil, H. K., Nonlinear systems (1996), Prentice-Hall: Prentice-Hall Upper Saddle River, NJ · Zbl 0626.34052
[10] Kolmanovskii, V.; Myshkis, A. D., Introduction to the theory and applications of functional differential equations (1999), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0917.34001
[11] Kolmanovskii, V.; Shaikhet, L., About one application of the general method of Lyapunov functionals construction, International Journal of Robust and Nonlinear Control, 13, 9, 805-818 (2003) · Zbl 1075.93017
[12] Lam, J.; Gao, H.; Xu, S.; Wang, C., \(H_\infty\) and \(L_2 / L_\infty\) model reduction for system input with sector nonlinearities, Journal of Optimization Theory and Applications, 125, 1, 137-155 (2005) · Zbl 1062.93020
[13] Liu, Y.; Wang, Z.; Liang, J.; Liu, X., Synchronization and state estimation for discrete-time complex networks with distributed delays, IEEE Transactions on Systems, Man and Cybernetics, Part B, 38, 5, 1314-1325 (2008)
[14] (McEneaney, W. M.; Yin, G.; Zhang, Q., Stochastic analysis, control, optimization and applications. Stochastic analysis, control, optimization and applications, Systems and control: Foundations and applications series (1999), Birkhauser: Birkhauser Boston, Cambridge, MA)
[15] Niculescu, S.-I., (Delay effects on stability: A robust control approach. Delay effects on stability: A robust control approach, Lecture notes in control and information sciences, Vol. 269 (2001), Springer-Verlag: Springer-Verlag London) · Zbl 0997.93001
[16] Niu, Y.; Ho, D. W.C., Design of sliding mode control for nonlinear stochastic systems subject to actuator nonlinearity, IEE Proceedings Control Theory & Applications, 153, 6, 737-744 (2006)
[17] Shaikhet, L., Stability of systems of stochastic linear difference equations with varying delays, Theory of Stochastic Processes, 4(20), 1-2, 258-273 (1998) · Zbl 0938.93059
[18] Wang, Z.; Ho, D. W.C.; Liu, X., A note on the robust stability of uncertain stochastic fuzzy systems with time-delays, IEEE Transactions on Systems, Man and Cybernetics, Part A, 34, 4, 570-576 (2004)
[19] (Wonham, W. M.; Reid, A. B., Random differential equations in control theory. Random differential equations in control theory, Probabilistic methods in applied mathematics, Vol. 2 (1970), Academic: Academic New York) · Zbl 0235.93025
[20] Xie, L.; Fridman, E.; Shaked, U., Robust \(H_\infty\) control of distributed delay systems with application to combustion control, IEEE Transactions on Automatic Control, 46, 12, 1930-1935 (2001) · Zbl 1017.93038
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.