A comparative performance evaluation of nonlinear observers for a fed-batch evaporative crystallization process

Abstract

Different nonlinear observers are compared throughout this work where they are part of an NMPC framework used to control a fed-batch crystallization process . We study which observer-optimizer pair offers the best control performance while maintaining adequate computational burden so that a posterior real-time implementation is feasible. At the same time, the relationship between state estimation accuracy and control performance is covered. Along the way we distinguish between stochastic and deterministic observers and compare which class is more suitable for our case study. The observers we make use of are: the moving horizon estimator (MHE), a nonlinear version of a Luenberger observer (extended Luenberger observer, ELO) and nonlinear variants of the Kalman filter such as extended Kalman filter(EKF), unscented Kalman filter (UKF) and ensemble Kalman filter (EnKF). Special variants of UKF and EKF that make use of a non constant system covariance matrix, which according to some literature is suitable to describe uncertainty distribution in batch processes, are also included in the analysis. The analysis focuses on how four main error sources such as unmeasured disturbances, uncertain initial conditions, model mismatch, and stochastic disturbances may impact observer estimation accuracy as well as their repercussion on control effectiveness and consequently on process performance. Results show that unmeasured disturbances are the most detrimental to observer and process performance in our case study. In spite of this finding, we present a methodology to tackle and solve this problem. All the analysis is first made under an open-loop configuration and then moves onto a closed-loop setup. All testing is based on computer simulations of the crystallization process. The evaluation criterion is based on the magnitude of a normalized root-mean squared error throughout 50 batch runs. The results are then used to identify if a link between estimation accuracy and control performance exists. The computational burden is also evaluated along 50 batch simulations, and is measured on the basis of CPU time required by every observer at every estimation stage.