Sub-optimal risk-sensitive filtering for third degree polynomial stochastic systems

The risk-sensitive filter design problem with respect to the exponential mean-square criterion is considered for stochastic Gaussian systems with polynomial drift terms and intensity parameters multiplying diffusion terms in the state and observations equations. The closed-form suboptimal filtering algorithm is obtained by linearizing a nonlinear third degree polynomial system at the operating point and reducing the original problem to the optimal filter design for a first degree polynomial system. The reduced filtering problem is solved using quadratic value functions as solutions to the corresponding Fokker-Planck-Kolmogorov equation. The performance of the obtained risk-sensitive filter for stochastic third degree polynomial systems is verified in a numerical example against the mean-square optimal third degree polynomial filter and extended Kalman-Bucy filter, through comparing the exponential mean-square criteria values. The simulation results reveal strong advantages in favor of the designed risk-sensitive algorithm for large values of the intensity parameters.