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Optimal estimator for continuous nonlinear systems with nonlinear dynamics, and nonlinear measurement based on the continuous least square error criterion is derived. This optimal solution is exact, explicit, in closed form and gives recursive formulas of the optimal filter. For derivation of the filter: (i) the LS criterion based on the statistical-deterministic-likelihood approach to estimation; (ii) the SDC form representation of the nonlinear system; and (iii) the calculus of variations; are combined. The resulting filter is optimal per sample. The filter's gains needs the solution of a non-symmetric Differential Matrix Riccati Equation.  The stability of the estimator is investigated. The performances are demonstrated by simulation of the Van der Pol equation with noisy nonlinear measurement, and system driving noise.

Optimal State Estimation of Nonlinear Dynamic Systems