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Filtering on nonlinear time-delay stochastic systems. (English) Zbl 1010.93099
This paper considers the filtering problem for a general class of nonlinear time-delay stochastic systems. The main goal is to design a full-order filter such that the dynamics of the estimation error is guaranteed to be stochastically exponentially ultimately bounded in the mean square. Both filter analysis and synthesis problems are considered. Sufficient conditions are proposed for the existence of desired exponential filters, which are expressed in terms of the solutions to algebraic Riccati inequalities involving scalar parameters. The explicit characterization of the desired filters is also derived. The method relies not on optimization theory but on Lyapunov-type stochastic stability results. A simulation example is given to illustrate the design procedures and performances of the proposed method.

##### MSC:
 93E11 Filtering in stochastic control 93C23 Systems governed by functional-differential equations 93C10 Nonlinear control systems 93E15 Stochastic stability
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##### References:
 [1] Bucy, R. S.; Joseph, P. D.: Filtering for stochastic processes with applications to guidance. (1968) · Zbl 0174.21903 [2] Gelb, A.: Applied optimal estimation. (1974) [3] Hsiao, F.; Pan, S.: Robust Kalman filter synthesis for uncertain multiple time-delay stochastic systems. Journal of dynamic system measures and control 118, 803-808 (1996) · Zbl 0866.93097 [4] Hsieh, C.; Skelton, R. E.: All covariance controllers for linear discrete time systems. IEEE transactions on automatic control 35, 908-915 (1990) · Zbl 0719.93075 [5] Ito, K.; Rozovskii, B.: Approximation of the kushner equation for nonlinear filtering. Society for industrial and applied mathematics journal on control optimization 38, 893-915 (2000) · Zbl 0952.93126 [6] Jazwinski, A. H.: Stochastic processes and filtering theory. (1970) · Zbl 0203.50101 [7] Khargonekar, P. P.; Petersen, I. R.; Zhou, K.: Robust stabilization of uncertain linear systemquadratic stabilizability and H$\infty$control theory. IEEE transactions on automatic control 35, 356-361 (1990) · Zbl 0707.93060 [8] Mahalanabis, A. K.; Farooq, M.: A second-order method for state estimation of nonlinear dynamical systems. International journal of control 14, 631-639 (1971) · Zbl 0224.93043 [9] Mahmoud, M. S.; Al-Muthairi, N. F.; Bingulac, S.: Robust Kalman filtering for continuous time-lag systems. Systems and control letters 38, 309-319 (1999) · Zbl 0986.93068 [10] Mao, M.: Robustness of exponential stability of stochastic differential delay equations. IEEE transactions on automatic control 41, 442-447 (1996) · Zbl 0851.93074 [11] Mao, X.: Stochastic differential equations and applications. (1997) · Zbl 0892.60057 [12] Niculescu, S.I., Verriest, E.I., Dugard, L., & Dion, J.M. (1998). Stability and robust stability of time-delay systems: a guided tour. In L. Dugard et al. (Eds.), Stability and control of time-delay systems, Lecture Notes in Control and Information Science, Vol. 228 (pp. 1-71). Berlin: Springer. · Zbl 0914.93002 [13] Saberi, A., Sannuti, P., & Chen, B.M. (1995). H2optimal control. Series in Systems and Control Engineering. London: Prentice-Hall International. · Zbl 0876.93001 [14] Scherzinger, B. M.; Kwong, R. H.: Estimation and control of discrete time stochastic systems having cone-bounded nonlinearities. International journal of control 36, 33-52 (1982) · Zbl 0516.93051 [15] Tarn, T. -J.; Rasis, Y.: Observers for nonlinear stochastic systems. IEEE transactions on signal processing 21, 441-448 (1976) · Zbl 0332.93075 [16] Wang, Z.; Huang, B.; Unbehauen, H.: Robust H$\infty$observer design of linear state delayed systems with parametric uncertaintythe discrete-time case. Automatica 35, 1161-1167 (1999) · Zbl 1041.93514 [17] Wang, Z.; Huang, B.; Unbehauen, H.: Robust H$\infty$observer design of linear time-delay systems with parametric uncertainty. Systems and control letters 42, 303-312 (2001) · Zbl 0985.93005 [18] Xie, L.; Soh, Y. C.: Robust Kalman filtering for uncertain systems. Systems and control letters 22, 123-129 (1994) · Zbl 0792.93118 [19] Yavin, Y.: Numerical studies in nonlinear filtering. Lecture notes in control and information sciences 65 (1985) · Zbl 0551.62067 [20] Yaz, E. (1988). Linear state estimators for non-linear stochastic systems with noisy non-linear observations. International Journal of Control 48, 2465-2475. · Zbl 0850.93791 [21] Yaz, E.; Azemi, A.: Observer design for discrete and continuous non-linear stochastic systems. International journal of systems and science 24, 2289-2302 (1993) · Zbl 0789.93019 [22] Zakai, M.: On the ultimate boundedness of moments associated with solutions of stochastic differential equations. Society for industrial and applied mathematics journal on control 5, 588-593 (1967) · Zbl 0267.60066