Identification of Disturbance Covariances Using Maximum Likelihood Estimation

Disturbance model identification is necessary both for estimator design and controller performance monitoring. Here we present a maximum likelihood estimation (MLE) method to identify process and measurement noise covariances. By writing the outputs in terms of the process and measurement noises, we form a normal distribution for the sequence of measurements. The variance of this distribution is a function of the unknown noise covariances, and the likelihood is optimized with respect to these covariances. We show that a solution to this problem exists. By comparing the first order conditions to those of the autocovariance-least squares (ALS) method, we derive necessary conditions for uniqueness. Several numerical methods, including utilizing the sparsity of the solution, are presented and demonstrated to decrease the computational time for the problem. Simulations are used to compare the MLE method to several existing methods: the ALS method, an alternate MLE method based on the innovations, and an expectation maximization method. Although the solving the MLE problem is considerably slower than solving the ALS problem, the MLE solution is shown to maximize the likelihood compared to the ALS problem. ∗The authors gratefully acknowledge the financial support of the industrial members of the TexasWisconsin Modeling and Control Consortium and NSF through grant #CTS-1159088 †[email protected] ‡[email protected]