Variance reduction in Covariance Based Realization Algorithm with application to closed-loop data

The Covariance Based Realization Algorithm (CoBRA), one branch of subspace methods, enables the estimation of multivariable models using a large number of data points due to the use of finite size covariance matrices. In addition, the covariance pre-processing allows CoBRA to ignore any (high-order) noise dynamics and focus on the estimation of the low-order deterministic model. However, subspace methods and CoBRA in particular are not maximum-likelihood methods. In this paper, an in-depth study on the statistical behavior of the noise effects is conducted. An approach is provided to reduce the variance of an estimate obtained by CoBRA via the choice of optimal row and column weighting matrices. For closed-loop implementation of CoBRA, a two stage procedure is proposed with the estimate on an intermediate instrument. At the first stage, a least-length perturbation of a scalar accuracy function is used to obtain an analytic solution for the optimal weighting matrices. The resulting instruments produced by the first stage are used for the identification at the second stage to extract the plant model. Simulation examples are given to illustrate the efficiency of the proposed two stage CoBRA method on the basis of closed-loop data and compared with other subspace methods.