(238a) The Autocovariance Least-Squares Method for Batch Processes: Application to Experimental Chemical Systems

Heat flow calorimetry is frequently used to monitor and control batch and semibatch polymerization reactions^[1–5].This technique requires the determination of the overall heat transfer coefficient between the reaction medium and the jacket (UA), which is a function of not only the reactor setup (geometry and wall material) and operating conditions (stirring and cooling fluid flow rate), but also of the properties of the reaction medium that can change throughout the polymerizations, as the viscosity can increase drastically and fouling at the reactor wall can occur. Different approaches for the estimation of UA have been proposed in the literature, for example, temperature oscillation calorimetry (OC)^[6], Luenberger observer (LO)^[7], extended Kalman filter (EKF)^[8], cascaded observers (CO)^[9] and Unscented Kalman filter (UKF)^[10]. These techniques have presented the following issues: difficulty to be extended to industrial applications (e.g., OC), disregard of the stochastic nature of the system (e.g., LO), complex calibration procedure (e.g., CO), and the presence of negative estimates during the process (e.g., EKF and UKF). On the other hand, Moving Horizon Estimator (MHE) is a more robust estimator that can incorporate restrictions to its states and handle nonlinear models^[11]. Unlike a recursive-based estimator (e.g., EKF and UKF), the MHE is based on the solution of an optimization problem at each time step.  To obtain accurate estimates, the covariance matrices of the process (Q) and of the measurement noises (R) must be provided to specify the statistics of such estimators. These covariances are typically determined by trial and error, but for real applications, the covariance calculation can be a challenging problem. The estimation of the covariances can be performed by several techniques for batch and semibatch processes, including the linearized and Monte Carlo approach^[12], the direct optimization^[13], the direct and sensitivity method^[14]. Recently, the Autocovariance Least-Squares (ALS) method has emerged as a viable solution to estimate process and measurement noise covariances for continuous systems (linear and nonlinear) for simulated and experimental data^[15–17].  This presentation will discuss the extension of the ALS technique to address batch processes and the application of this method to laboratory chemical systems.   

First, some important aspects of covariance and state estimation will be introduced; then, three experimental systems will be presented: 1) the hydrolysis of acetic anhydride carried out with excess of water in three different non-adiabatic vessels: (a) a cylindrical plastic vessel, (b) a volumetric flask, and (c) a thermal bottle or Dewar flask^[18]; 2) the in-line monitoring of a batch emulsion polymerization reactor (without initiator), heated to the reaction temperature by three different temperature trajectories (step, ramp and arc); 3) the vinyl acetate emulsion polymerization in a semibatch reactor, where the reactions were started as a conventional batch emulsion polymerization, and then after 20 min, monomer additions were performed with 10 min intervals^[19]. Finally, the problem of the simultaneous online estimation of states and parameters in these systems, using EKF, UKF and MHE with statistics defined by ALS, will be discussed in terms of the accuracy of the estimated variables. Preliminary results associated with the hydrolysis of acetic anhydride indicated that the UKF is more effective than the EKF to monitor the reactors under different conditions with heuristic diagonal covariances^[20]. This work will show that high-quality estimates for this and the other experimental systems in focus are obtained when the statistics of the state estimators are systematically determined by ALS, especially for the implementation of the MHE in systems under repeated perturbations.   

References

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[20]    F. D. Rincon, W. H. Hirota, C. Sayer, G. A. C. Le Roux, and R. Giudici, “Parameter estimation via Kalman filters and Luenberger-like observer,” In preparation.