Contents I. Introduction
The optimal mean-square filtering theory was initiated by Kalman and Bucy for linear stochastic systems, and then continued for nonlinear systems in a variety of papers (see for example [1], [2], [9], [13], [17], [12], [18], [8], [14], [16], [15]). Special conditions are given for the existence, continuity and boundlessness of a drift in the state equation and a linear function in the observation one. A concept of the stochastic risk-sensitive estimator, introduced more recently by McEneaney [11], in regard to a dynamic system including nonlinear drift , linear observations, and intensity parameters multiplying diffusion terms in both, state and observation, equations. Again, the exponential mean-square (EMS) criterion, introduced in [3] for deterministic systems and in [5] for stochastic ones, is used instead of the conventional mean-square criterion to provide a robust estimate, which is less sensitive to parameter variations in noise intensity. Advantage of exponential-quadratic criteria is the robustness of the obtained solution with respect to noise level. Indeed since the solution to the classic LQ problem is independent of noise level, it occurs to be too sensitive to parameter variations in noise intensity. This paper presents a solution to the risk-sensitive filtering problem with respect to the exponential mean-square criterion for stochastic third degree polynomial systems including intensity parameters multiplying diffusion terms in both, state and observation, equations. The closed-form suboptimal filtering algorithm is obtained 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 (affine) system. The reduced filtering problem is solved seeking quadratic value functions as solutions to the corresponding Fokker-Planck-Kolmogorov equation. Undefined parameters in the value functions are calculated through ordinary differential equations composed by collecting terms corresponding to each power of the state-dependent polynomial in the FPK equation. The closed-form risk-sensitive filter equations are explicitly obtained.
