Covariance and State Estimation of Weakly Observable Systems: Application to Polymerization Processes

Physical models for polymerization may be overly complex considering the available measurements, and they may contain many unobservable and weakly observable modes. Overly complex structures lead to ill-conditioned or singular problems for disturbance variance estimation. Ill conditioning leads to unrealistic data demands for reliable covariance estimates and state estimates. The goal of this paper is to build nonlinear state estimators for weakly observable systems, with focus on polymerization processes. State estimation requires knowledge about the noise statistics affecting the states and the measurements. These noise statistics are usually unknown and need to be estimated from operating data. We introduce a linear time-varying autocovariance least-squares (LTV-ALS) technique to estimate the noise covariances for nonlinear systems using autocorrelations of the data at different time lags. To reduce or eliminate the ill-conditioning problem, we design a reduced-order extended Kalman filter (EKF) to estimate only the strongly observable system states. This reduced filter, which is based on the Schmidt-Kalman filter, is used to perform the estimation of noise covariances by the LTV-ALS technique. Results of the implementation of the proposed method on a large-dimensional ethylene copolymerization example show that better conditioned state and covariance estimation problems can be obtained. We also show that high-quality state estimates can be obtained after the specification of the noise statistics of EKF estimators by ALS.