Nonlinear filter of stochastic approximation type using empirical distribution of residuals

State estimation of dynamical systems is one of the most important problems in control systems engineering. The paper is concerned with a nonlinear filter of stochastic approximation type for state estimation of nonlinear dynamical systems and its modifications to improve the convergence property. First, the asymptotic variance of the estimation error is derived for the nonlinear filter of stochastic approximation type. Then it is shown that a suitable nonlinear transformation function minimizes the variance. Since the optimal transformation function requires the knowledge of the probability distribution of observation noise, a construction method of the asymptotically optimal transformation function is also provided. Numerical simulation results illustrate the usefulness of the proposed idea