(642b) Estimation of Noise Covariances and Disturbance Structure from Data Using Least Squares with Optimal Weighting

For the linear time-invariant state space model, it is well known that the minimum variance state estimator is the Kalman filter when there are no constraints on the states. To define the state estimator, however, information about the noise covariances corrupting the state and the measurements is needed. Let the model be xk+1 = Axk + Gwk, yk = Cxk + vk, and the covariances of noises wk and vk be Qw and Rv respectively. If complete knowledge about the deterministic part of the model i.e. A, B, C is assumed, then the Kalman filter or for that matter any state estimator would require the knowledge of stochastic part of the model i.e. G, Qw , Rv . In [1] the estimation of the covariances Qw,Rv from data is shown where knowledge about G is assumed. The estimation technique in [1] is based on the autocorrelations between the innovations at different times (Autocovariance Least Squares or ALS) [2,3]. The result is a simple least squares problem to be solved for Qw and Rv.

The correct weighting is needed for the least squares estimate of Qw, Rv to have minimum variance. This weighted least squares estimate has the lowest variance among the class of all linear estimators. Here we present the theoretical weighting for the minimum variance ALS technique to estimate Qw and Rv from data [4]. The weighting can also be estimated from data if sufficient data are available. Simulations are shown to further illustrate the reduced variance of the estimates as compared to the unweighted ALS estimates. An incorrect weighting was used in [5].

The G matrix shapes the disturbance wk entering the state. Usually only a few independent disturbances affect the states. This would imply a tall G matrix with more rows than columns. In the absence of any knowledge about G, an incorrect assumption that G=I is often made. The choice G=I gives a non-unique ALS estimate for Qw and Rv when there are fewer measurements than number of states. Here we combine the ALS technique with semidefinite programming (SDP) to estimate the minimum number of disturbances that affect the state [6]. An estimate of G is then made using singular value decompositon. Physical chemical process examples are presented to demonstrate the utility of this technique.

[1] Brian J. Odelson, Murali R. Rajamani, and James B. Rawlings. A new autocovariance least-squares method for estimating noise covariances. Automatica, 42(2):303308, February 2006.

[2] R.K. Mehra. On the identification of variances and adaptive Kalman filtering. IEEE Trans. Auto. Cont., 15(12):175184, 1970.

[3] C. Neethling and P. Young. Comments on "Identification of optimum filter steady-state gain for systems with unknown noise covariances". IEEE Trans. Auto. Cont., 19(5):623625, 1974.

[4] Ghazal A. Ghazal and Heinz Neudecker. On second-order and fourth-order moments of jointly distributed random matrices: a survey. Linear Algebra Appl., 321:6193, 2000.

[5] D.P. Dee, S.E. Cohn, A. Dalcher, and M. Ghil. An efficient algorithm for estimating noise covariances in distributed systems. IEEE Trans. Auto. Cont., 30(11):10571065, 1985.

[6] Maryam Fazel. Matrix Rank Minimization with Applications. PhD thesis, Dept. of Elec. Eng., Stanford University, 2002.