Optimal Risk-Sensitive Filtering for System Stochastic of Second and Third Degree

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

Application of risk-sensitive optimal filtering equations to excitable noise system

“…A concept of the stochastic risk-sensitive estimator, introduced more recently by McEneaney [23], regards a dynamic system where f (x) is a nonlinear function with linear observations and presence of parameter ǫ multiplying diffusion term in both equations (state and observations). The Fitz Hugh Nagumo model [30] contains third degree terms and some parameters multiplying the diffusion term, in the state equations, which supports similarity with the risk sensitive statement problem presented in [4], reason by which it is carried out the application of this methodology. The closed-form for risk-sensitive filtering equations was obtained in [2].…”

“…The vector A 0 (t) represents other disturbances affecting the process. The solution is given for the filtering equations, (see [4]) the estimator is the solution oḟ…”

“…This highly non linear response to perturbations makes the dynamics of the forced Fitz Hugh-Nagumo system nontrivial. Regarding system of equations (15) are obtained the optimal stochastic risk-sensitive filtering equations for systems of third degree [4] and they are compared with optimal polynomial filtering equations [8] as you can see in the following section.…”

“…After this, numerous theories over optimal filtering equations have been developed for different real cases. Linear systems with delays in the time [7], polynomial systems [4], in linear case, with Poisson's noise [6], with white noise as diffusion term in the state and observations equation [2], [3] are some of the grate diversity of these. On the other hand, we can find chaotic systems with induction of external noise [27] and stochastic resonance [10].…”