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

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.


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