Maximum likelihood recursive state estimation: Exact and robust filtering equations with MSE stability

This paper revisits Mortensen (1968) and proposes a novel statistical method for maximum likelihood (ML) recursive state estimation in a general continuous-time stochastic systems. The main feature of the new method lies on a fine interconnection between complete-information loglikelihood of the stochastic systems under consideration and the value function of the corresponding optimal control problem of the ML estimation. This appealing feature is notably lack in the recursive minimum energy estimation Hijab(1980) and Krener(2003). Extending the results of Surya(2024) to continuous time, distributional identities of the score function and information matrix of state are derived. Using these identities and perturbation method, the exact filtering equations are established. They represent the dynamics of ML estimator (MLE) and gain matrix taking a general form of Riccati equation, which admits an unique positive definite solution with monotone decreasing upper bound. These properties are used to show that the MLE has mean-squared-error stability under uniform observability of the systems. The results agree with the Kalman and Bucy (1961) filters for linear systems and generalize Mortensen (1968), Hijab (1980), Benes (1981) and Krener (2003) for nonlinear systems. A new form of Cramer-Rao lower bound is derived for the covariance matrix of state estimation errors. It leads to a robust formulation of the optimal filter having the least covariance matrix of estimation errors. The Runge-Kutta 4th order method is used for numerical solution of the optimal filtering equations under time-discretization of the stochastic systems. Particle filtering is developed to compare against the performance of the ML estimator. Numerical studies show that the exact filtering equations provide faster and more accurate estimation than the particle filtering.