Direct method for Yau filtering system with nonlinear observations

ABSTRACT For all known finite dimensional filters, one always assumes that the observation terms be degree one polynomials. However, in practice, the observation terms may be nonlinear, e.g. tracking problems. In this paper, we consider the Yau filtering system ( is constant for all i, j) with nonlinear observation terms and arbitrary initial condition. The novelty of the paper lies in (i) the real time computation of the solution of the Duncan-Mortensen-Zakai (DMZ) equation is reduced to the computation of Kolmogorov equation. Based on Gaussian approximation of the initial condition, the Kolmogorov equation can be solved in terms of ordinary differential equations; (ii) For a given probability density function, we give a new and original approach to do Gaussian approximation which is very effective and simple. The direct method developed here can be easily implemented in a real time and memoryless way. Besides, we do not need the controllability and observability assumption. Compared to the extended Kalman filter, our method is much stable and has theoretical proof. The numerical experiments show that the proposed Gaussian approximation method is very effective and our method can track the states very well.

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