A convergent approximation of the continuous-time optimal parameter estimator

Continuous-time linear stochastic systems that are bilinear in the state and 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 proved to converge to a minimum of the likelihood function. The proof uses the ordinary differential equation technique and requires that the trajectories of the slow system be bounded on finite time intervals and that the fixed parameter fast system by asymptotically stable. The fixed parameter fast system is proved to be asymptotically stable if the parameter update gain is small enough. Essentially, the specific approximation is asympotically 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 convergent parameter estimators