Optimal filtering for bilinear system states and its application to terpolymerization process identification.

*(English)*Zbl 1034.60044The optimal nonlinear finite-dimensional filter exists and can be obtained in a closed form in the case of linear state and observation equations with the help of the linear Kalman-Bucy filter equations. However, the optimal nonlinear finite-dimensional filter can also be obtained in some other cases. A classification of the “general situation” cases where the optimal nonlinear finite-dimensional filter exists is given by S. S.-T. Yau [J. Math. Syst. Estim. Control 4, No. 2, 181–203 (1994; Zbl 0811.93059)].

The authors of this paper study a relatively simple case of polynomial system states, where the optimal nonlinear finite-dimensional filter can be obtained in a closed form. In the case of a bilinear state equation the corresponding filtering equations are derived. A similar filtering problem was considered for cubic polynomial states and linear observations by the authors [Appl. Math. E-Notes 2, 36–44 (2002; Zbl 0992.49018)]. The possibility to solve the optimal filtering problem for an arbitrary polynomial state and linear observations is discussed. The proposed optimal filter for bilinear system states and linear observations is applied to find solution of an identification problem for the terpolymerization process [see B. A. Ogunnaike, Int. J. Control 59, No. 3, 711–729 (1994; Zbl 0800.93100) for more details]. Numerical simulations are conducted for the optimal filter for bilinear system states, the optimal linear filter available for the linearized model, and the mixed filter designed as a combination of those filters. The simulation results show an advantage of the optimal bilinear filter in comparison to the other filters.

The authors of this paper study a relatively simple case of polynomial system states, where the optimal nonlinear finite-dimensional filter can be obtained in a closed form. In the case of a bilinear state equation the corresponding filtering equations are derived. A similar filtering problem was considered for cubic polynomial states and linear observations by the authors [Appl. Math. E-Notes 2, 36–44 (2002; Zbl 0992.49018)]. The possibility to solve the optimal filtering problem for an arbitrary polynomial state and linear observations is discussed. The proposed optimal filter for bilinear system states and linear observations is applied to find solution of an identification problem for the terpolymerization process [see B. A. Ogunnaike, Int. J. Control 59, No. 3, 711–729 (1994; Zbl 0800.93100) for more details]. Numerical simulations are conducted for the optimal filter for bilinear system states, the optimal linear filter available for the linearized model, and the mixed filter designed as a combination of those filters. The simulation results show an advantage of the optimal bilinear filter in comparison to the other filters.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

##### MSC:

60G35 | Signal detection and filtering (aspects of stochastic processes) |

93E11 | Filtering in stochastic control theory |

62M20 | Inference from stochastic processes and prediction |

62P30 | Applications of statistics in engineering and industry; control charts |