The identification of accurate disturbance models from data has applica- tion both to estimator design and controller performance monitoring. Knowl- edge of the process and measurement noise covariances allows the calculation of the optimal Kalman filter gain. In controller performance monitoring, an accurate disturbance model is necessary to calculate a theoretical bench- mark for the performance. For linear models with white noise disturbances, existing methods for disturbance model identification include subspace iden- tification and autocovariance least-squares (ALS). Rather than finding the process and measurement noise matrices, subspace ID determines the op- timal Kalman gain and innovation variance (in addition to the state space model) [1]. ALS uses the autocovariances of the L-innovations to form a least-squares problem for the noise covariances [2]. As an alternative to these methods, a maximum likelihood method is proposed to find the process and measurement noise covariances.
To form the maximum likelihood problem, the entire set of measurements (or L-innovations) is written as a linear combination of the white noises af- fecting this system. This measurement signal then has a multivariate normal distribution with a known mean and unknown variance. The unknown vari- ance is expressed in terms of the process and measurement noise covariances, and the likelihood is also expressed in terms of these noise covariance ma- trices. Maximizing the likelihood (or log likelihood) becomes a nonlinear optimization problem for the noise covariance matrices. The existence and uniqueness of this solution are discussed. Whereas ALS requires knowledge of the unknown noise variances to optimally weight the least-squares problem, the maximum likelihood method does not have any such requirement [2].
While the size of the measurement signal makes the problem computa- tionally demanding, the symmetry and sparsity of the problem aid in the numerical optimization. By solving for the square-root of each noise covari- ance matrix, the optimal covariance matrices are guaranteed to be symmetric and positive semi-definite, and the problem can be treated as unconstrained. Evaluating the objective function is computationally demanding because it requires calculating the log determinant and inverse of the large covariance matrix for the signal. However, because this covariance matrix is created from banded and block-diagonal matrices, it has a sparse structure. By exploiting this sparsity, the computational effort in computing the log determinant and the inverse is greatly decreased.
Simulations demonstrate the effectiveness of the maximum likelihood es- timation problem in finding the process and measurement noise covariances
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for low-dimensional systems. These covariances can be used to calculate the optimal estimator gain and performance benchmarks. The maximum like- lihood method is compared to existing approaches, and fruitful avenues of future research are discussed.

References

[1] S. J. Qin. An overview of subspace identification. Comput. Chem. Eng.,
30:1502â??1513, 2006.
[2] M. R. Rajamani and J. B. Rawlings. Estimation of the disturbance struc- ture from data using semidefinite programming and optimal weighting. Automatica, 45:142â??148, 2009.
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