Estimation of the disturbance structure from data using semidefinite programming and optimal weighting

Designing a state estimator for a linear state-space model requires knowledge of the characteristics of the disturbances entering the states and the measurements. In [Odelson, B. J., Rajamani, M. R., & Rawlings, J. B. (2006). A new autocovariance least squares method for estimating noise covariances. Automatica, 42(2), 303–308], the correlations between the innovations data were used to form a least-squares problem to determine the covariances for the disturbances. In this paper we present new and simpler necessary and sufficient conditions for the uniqueness of the covariance estimates. We also formulate the optimal weighting to be used in the least-squares objective in the covariance estimation problem to ensure minimum variance in the estimates. A modification to the above technique is then presented to estimate the number of independent stochastic disturbances affecting the states. This minimum number of disturbances is usually unknown and must be determined from data. A semidefinite optimization problem is solved to estimate the number of independent disturbances entering the system and their covariances.