A convergent approximation of the optimal parameter estimator

Continuous time linear stochastic systems with unknown bilinear parameters are considered. A specific approximation to the optimal nonlinear filter used as a recursive parameter estimator is derived by retaining third-order moments and using a Gaussian approximation for higher-order moments. With probability one, the specific approximation is proven to converge to a minimum of the likelihood function. The proof uses the ordinary differential equation technique and requires that the slow system is bounded on finite time intervals and the fixed-parameter fast system is asymptotically stable. The fixed parameter fast system is proven asymptotically stable if the parameter update gain is small enough. Essentially, the specific approximation is asymptotically equivalent to the recursive prediction error method, thus inheriting its asymptotic rate of convergence. A numerical simulation for a simple example indicates that the specific approximation has better transient response than other commonly used parameter estimators