Linear time-invariant systems play significant role in the control field. A number of methods have been published for identification of the deterministic part of a process. However, identification of the stochastic part has had much less attention. This paper deals with estimation of covariance matrices of the noise entering a linear system. The process and measurement noise covariance matrices are tuning parameters of the Kalman filter, and they affect the quality of the state estimation. The noise covariance matrices are generally not known, and their estimation from the measured data is a challenging task. This paper introduces a method for estimation of the noise covariance matrices using Bayesian approach along with Monte Carlo numerical methods. Performance of the approach is tested on various systems and noise properties. The second part of the paper compares Monte Carlo approach with the recently published methods. The speed of convergence is compared with the Cramér-Rao bounds.
The performance of Kalman filter depends directly on the noise covariances, which are usually not known and need to be estimated. Several estimation algorithms have been published in past decades, but the measure of estimation quality is missing. The Cramér-Rao bounds represent limitation of quality of parameter estimation that can be obtained from given data. In this article, The Cramér-Rao bounds for noise covariance estimation of linear time-invariant stochastic system will be derived. Two different linear system models will be considered. Further, the performance of earlier published methods will be discussed according to the Cramér-Rao bounds. The analogy between the Cramér-Rao bounds and the Riccati equation will be pointed out.
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