Maximum Likelihood Identification of an Ornstein-Uhlenbeck Model and Its CRLB

This paper applies Maximum Likelihood Estimation (MLE) to the identification of a stochastic error model of a gyroscope. The error model used for illustration features an Ornstein-Uhlenbeck process with an unknown time constant driven by a process noise with unknown variance, and a white measurement noise also with unknown variance. As the setup of MLE, the likelihood function ($L F)$ is derived in the steady-state Kalman filter framework and is defined in reference to the parameters of the Kalman filter gain and innovation variance. The resulting log-likelihood function (LLF) is a quadratic function of the measurements, facilitating the evaluation of the Cramér-Rao Lower Bound (CRLB) and makes it possible to confirm the statistical efficiency, i.e., optimality, of the ML estimator presented in this paper.