Identifying the Uncertainty Structure Using Maximum Likelihood Estimation

The identification of accurate disturbance models from data has application both to estimator design and controller performance monitoring. Methods to find the disturbance model include maximum likelihood estimation, Bayesian estimation, covariance matching, correlation techniques (such as autocovariance least-squares), and subspace identification methods. Here we formulate a maximum likelihood estimation (MLE) problem for the unknown process and measurement noise covariances. To form the MLE problem, the entire set of measurements is written as a linear combination of the white noises affecting the system. This measurement signal then has a multivariate normal distribution with a known mean and unknown variance. Since the structure of the variance is known, the likelihood is expressed in terms of the unknown process and measurement noise covariance matrices. The MLE problem is a nonlinear optimization problem for these covariances. A solution to this problem is shown to exist. Necessary conditions for uniqueness are shown to be the same as those for the autocovariance least-squares problem. While the size of the measurement signal makes the problem computationally demanding, the symmetry and sparsity of the problem aid in the numerical optimization. Simulations demonstrate the effectiveness of the MLE problem in finding the process and measurement noise covariances for low-dimensional systems. The MLE method is compared to existing approaches, and fruitful avenues of future research are discussed.