An optimal estimator for continuous nonlinear systems with nonlinear dynamics, and nonlinear measurement based on the continuous least square error criterion is derived. The solution is exact, explicit, in closed form and gives recursive formulas of the optimal filter. For the derivation of the filter, the following elements are combined: (i) the least squares (LS) criterion based on statistical-deterministic-likelihood approach to estimation; (ii) the state-dependent coefficient (SDC) form representation of the nonlinear system; and (iii) the calculus of variation. The resulting filter is optimal per sample. The filter’s gains need the solution of a nonsymmetric 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.
Part of the book: Nonlinear Systems