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

Tuning a state estimator for a linear state space model requires knowledge of the characteristics of the independent disturbances entering the states and the measurements. In Odelson, Rajamani, and Rawlings (2006), 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 stochastic disturbance structure that affects the states. The disturbance structure also provides information about the minimum number of disturbances affecting the state. This minimum number is usually unknown and must be determined from data. A semidefinite optimization problem is solved to estimate the disturbance structure and the covariances of the noises entering the system.