(371w) Two-Time Scale Unscented Kalman Filters for Nonlinear State Estimation with Time-Delayed Measurements. Application to Polymer Processes

Measurement and control of polymer processes is a very challenging task due to their high nonlinearity, strong relationship between quality variables and process conditions, and the absence of reliable on-line sensors. In these processes many important variables cannot be measured on-line or can only be measured at low sampling frequencies. In this context, state estimation becomes an important step for the proper implementation of control systems.

A particular feature affecting the control of polymer processes is that critical quality variables are usually measured off-line, or on-line but with a considerable analysis time [1]. This introduces significant delays in the availability of the measurements, which have to be taken into account by a proper state estimation method. The problem of state estimation in nonlinear processes has been covered extensively in the past. The most common approach is the extended Kalman filter (EKF). However, this strategy may present problems in the case of highly nonlinear systems such as polymer processes [2]. A relatively new method known as the Unscented Kalman Filter (UKF) has been developed for nonlinear processes. This method is based on the unscented transform technique, a mechanism for propagating the mean and covariance through a nonlinear transformation [3].

Filters are run normally on the basis of measurements that arrive sequentially. Various approaches for the treatment of time delayed measurements have been followed in linear Kalman filters through the optimal way [2]. However, these procedures have not been applied yet to the UKF or other non-linear filters.

In this work, we develop a two-time scale approach to deal with time-delayed measurements using the UKF. The technique is applied to three variants of the UKF: the basic UKF, the Unscented Recursive Nonlinear Dynamic Data Reconciliation (URNDDR) and the Reformulated Constrained Unscented Kalman Filter (RCUKF) [4,5]. The polymer process selected as case study is the copolymerization of methyl methacrylate and vinyl acetate in a continuous stirred tank reactor [6]. In this study, it is assumed that the reactor temperature and the total conversion are on-line measurements with sample time of five minutes, while the weight average molecular weight (Mw), the number average molecular weight (Mn) and the copolymer composition are measured off-line with a delay of thirty minutes. For each simulation run, the estimators are implemented.

The computational load and the square root error were employed to evaluate the performance of the three UKF filters for their on-line application. The values of the performance indexes, obtained by simulation studies, indicate that URNDDR and RCUKF give more accurate estimates at the expense of increasing the computational load. Furthermore, the influence of errors in initial estimations, measurement delays and the interruption of measurements were also analyzed. Results show that the convergence time, when errors in initial estimations are present, decreases if constraints are imposed on variable states. For the three analyzed estimators, it can also be seen that the quality of the estimation decreases when measurements are interrupted and that the two-time scale procedure allows processing delayed measurements satisfactorily giving adequate results in terms of errors.

References

[1]Fonseca, G.; Dubé, M.; Penlidis, A. (2009), A Critical Overview of Sensors for Monitoring Polymerizations. Macromolecular Reaction Engineering, 3, 327-373.

[2] Simon, D. (2006), Optimal State Estimation, Kalman, H∞, and Nonlinear Approaches. Wiley-Interscience. New Yersey.

[3] Julier, J.; Uhlmann J. K. (1997), A New Extension of the Kalman Filter to Nonlinear Systems. Proc. SPIE, 3068, 182.

[4] Vachhani, P.; Narasimhan, S.; Rengaswamy, R. (2006), Robust and Reliable Estimation via Unscented Recursive Nonlinear Dynamic Data Reconciliation. Journal of Process Control, 16, 1075-1086.

[5] Kolås, S.; Foss, B.A.; Schei, T.S. (2009), Constrained Nonlinear State Estimation based on the UKF Approach. Computers and Chemical Engineering, 33, 1386-1401.

[6] Congalidis, J. P.; Richards, J. ; Ray, W. H. (1989), Feedforward and Feedback Control of a Solution Copolimerization Reactor. AIChE Journal, 35, 891-907.