(239b) Unconstrained Nonlinear State Estimation Applied to PMMA and LLDPE Polymerization Processes

State estimation is a fundamental problem in most fields of science and engineering. Specific to polymer processes, estimation is widely used: to infer accurate values from noisy sensor data, as a soft sensor to predict unmeasurable product quality variables such as the melt index or polymer density, and to estimate kinetic parameters to obtain accurate dynamic process models. Dynamic process models are used to infer reactor operating conditions like occurrence of segregation and for model predictive control for achieving effective grade transition. It is well known that the Kalman filter provides an optimal solution to the state estimation problem for linear stochastic systems in the presence of Gaussian disturbances and measurement noise. However, most polymer processes exhibit highly nonlinear dynamics. The design of estimators to address nonlinear problems has lead to the development of numerous nonlinear estimators which include the Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF), Moving Horizon Estimation (MHE), Particle filters (PF), Ensemble Kalman filter (EnKF), recursive and non-recursive nonlinear dynamic data reconciliation (RNDDR and NDDR) [1-4]. In this work, two simulated reactors are used to evaluate various unconstrained state estimation techniques (EKF, UKF, PF, EnKF). First, estimation algorithms are applied onto a lab scale methyl methacrylate reactor [5] to differentiate estimator performance in different scenarios of plant and measurement noise. Based on this study, it is established that the UKF eliminates the linearization error prevalent in the case of the EKF. It is also shown that in certain cases of plant operation, the posterior of the estimated states are non-Gaussian, in which case the particle filter (unlike the EKF and UKF) is able to capture the statistics of the posterior distribution upto a much higher moment. It is important to use the full state information to generate a single measurement at each time step (in the Gaussian case, the mode is same as the mean and is the optimal estimate). Tuning of the EKF and UKF based on the process to measurement noise, and tuning of the PF based on plots of the weights to apriori estimates is investigated. Next, these techniques are applied on a simulated industrial scale gas phase polyethylene (LLDPE) reactor [6]. A reduced order model [7] is used to infer the melt index and polymer density of the large scale simulated plant model. This realistic scenario is used to identify issues in applying estimation techniques. It is shown that the particle filter fails in this case when the process noise is very high as the likelihood function fails to weight the particles. Possible solutions to this issue are discussed. The unscented particle filter, which generates the importance distribution differently, is used to improve filter performance. Further, design of the maximum likelihood function to represent the measurement distribution accurately in particle filters is discussed. The EnKF is shown to provide good estimation results on the LLDPE reactor. [1] S. Arulampalam, S. Maskell, N. Gordon, T. Clapp, A tutorial on particle filters for on-line non-linear/non-Gaussian Bayesian tracking, IEEE Transactions on Signal Processing, 30, 174-189. [2] F. Daum, Nonlinear filters: beyond the Kalman filter, IEEE Aerospace and Electronic Systems Magazine 20 (8) (2005), pp. 57?69. [3] J.H. Lee and N.L. Ricker , Extended Kalman filter based nonlinear model predictive control. Industrial & Engineering Chemical Research 33 (1994), pp. 1530?1541. [4] S.C. Patwardhan, J. Prakash, and S.L. Shah (2007). Soft sensing and state estimation: Review and recent trends. Proc. of IFAC Conf. on Cost Effective Automation in Networked Product Development and Manufacturing. [5] Silva, A.B. and Flores, A.T. (1999). Effect of process design/operation on the steady-state operability of a methyl methacrylate polymerization reactor. Ind. Eng. Chem. Res., 38, 4790-4804. [6] K.B. McAuley, J.F. MacGregor, and A.E. Hamielec, 1990. A kinetic model for industrial gas phase ethylene copolymerization. A.I.Ch.E. Journal 36, pp. 837?850. [7] K.B. McAuley and J.F. MacGregor, Nonlinear product property control in industrial gas-phase polyethylene reactors. AIChE J. 39 5 (1993), pp. 855?866.