The optimal estimation of the state vector x for nonlinear systems with the drift term f (x,t ) subject to additive white Gaussian disturbances was obtained by Kushner [1]. Unfortunately, there are a very few known examples of nonlinear systems for which the general result can be reduced to a finite-dimensional, closed-form solution in terms of a finite number of conditional moments. The most famous example, the Kalman-Bucy filter, is the case of linear state and observation equations. Other examples of the closed-form optimal filters for nonlinear systems include the case of the state vector that can take only a finite number of admissible states [2], and the case of the drift term that satisfies a certain Riccati equation while the measurements are linear [3].
In this paper, we report several new special cases for which the closed form optimal filter equations has been found for nonlinear systems [4-6]:
(a) f (x,t ) is a polynomial and the measurements are linear an invertible – the case of directly measurable states.
(b) f (x1,x2,t ), x = [ x1,x2 ]T is a polynomial that defines the dynamics of the subset of states x1 (nonlinear states) and depends on linear states x2 , which are independent of x1 (a one way coupling between nonlinear and linear states). The measurements are still required to depend linearly on states, but only x1 must be directly measurable. The simultaneous state and parameter estimation in linear systems can be shown to belong to this class of the problems.
(c) The case (a) where the intensity of Gaussian disturbances is a polynomial function of states.
We describe the method used to derive the optimal filters for the described special cases and give examples illustrating the application of our results to specific polynomial systems.
References:
1. H. J. Kushner, “On differential equations satisfied by conditional probability densities of Markov processes,” SIAM J. Control, vol. 2, pp. 106–119, 1964.
2. W. Wonham, “Some applications of stochastic differential equations to nonlinear filtering,” SIAM J. Control, vol. 2, pp. 347–369, 1965.
3. V. E. Benes, “Exact finite-dimensional filters for certain diffusions with nonlinear drift,” Stochastics, vol. 5, pp. 65–92, 1981.
4. M. V. Basin, “Optimal filtering for polynomial system states,” J. of Dynamic Systems, Measurement, and Control, vol. 125, pp. 123–125, 2003.
5. M. V. Basin and M. Skliar, “Optimal filtering for partially measured polynomial system states,” Proc. ACC, pp. 4022–4027, 2005.
6. M. V. Basin, J. Perez, and M. Skliar, “Optimal filtering for polynomial system states with polynomial multiplicative noise,” Int. J. Robust and Nonlinear Control, vol. 16, pp. 287–298, 2006.