Abstract
The risk-sensitive filtering design problem with respect to the exponential mean-square cost criterion is con-sidered for stochastic Gaussian systems with polynomial of second and third degree drift terms and intensity parameters multiplying diffusion terms in the state and observations equations. The closed-form optimal fil-tering equations are obtained using quadratic value functions as solutions to the corresponding Focker- Plank-Kolmogorov equation. The performance of the obtained risk-sensitive filtering equations for stochastic polynomial systems of second and third degree is verified in a numerical example against the optimal po-lynomial filtering equations (and extended Kalman-Bucy for system polynomial of second degree), through comparing the exponential mean-square cost criterion values. The simulation results reveal strong advan-tages in favor of the designed risk-sensitive equations for some values of the intensity parameters.
Highlights
IntroductionUndefined parameters in the value function are calculated through ordinary differential equations composed by collecting terms corresponding to each power of the state-dependent polynomial in the nonlinear parabolic PDE equations
Since the linear optimal filter was obtained by Kalman and Bucy (60’s), numerous works are based on it, see for example [1,2,3,4,5], of the variety of all those
Undefined parameters in the value function are calculated through ordinary differential equations composed by collecting terms corresponding to each power of the state-dependent polynomial in the nonlinear parabolic PDE equations
Summary
Undefined parameters in the value function are calculated through ordinary differential equations composed by collecting terms corresponding to each power of the state-dependent polynomial in the nonlinear parabolic PDE equations. This procedure leads to the obtention of the optimal risk-sensitive filtering equations. Polynomial filtering equations (and extended KalmanBucy for systems of second degree), through comparing the exponential mean-square cost criterion values in finite horizon time. Tables of the criterion values and graphs of the simulations are included This exponential mean-square cost criterion function contains the parameter which appear in the dynamic system, which leads to a more robust solution. This work is organized as follows: filtering problem statement, optimal risk-sensitive filtering for stochastic system of second degree, optimal risk-sensitive filtering for stochastic system of third degree, application for systems of second degree, application for systems of third degree, conclusions and references
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