The operation of a process in batch mode brings forth several distinguishing characteristics as opposed to continuous operation. Specifically, batch processes are characterized by a specific beginning and end time over the course of which the process variables often change by significant amounts. Additionally, the process variables at the end of the batch need not be at equilibrium, but rather correspond to desirable product properties. Both of these features have significant impact on the way batch processes are modeled, monitored and controlled. Another difficulty with the control of batch processes is that the nonlinearity and time varying characteristics cannot be ignored as the operating point is changing continuously. On the other hand the acceptable product quality is a tight range that makes the application of a high performance control methodology necessary. Predictive controllers are capable of doing this important job as long as a reliable model of the process is available. For nonlinear batch processes this has usually meant the use of fundamental nonlinear models embedded within an optimization. The problem with this approach is that the modeling effort is large, the computation time is large and the solvers are complex. In this paper we present an approach using Latent Variable models that capture the benefits of these nonlinear MPC's without implementation difficulties mentioned above.
Owing to the significant variation of the process variables over the batch duration and the fact that the process trajectory need not evolve around equilibrium points, traditional step test models identified around the nominal equilibrium point are generally inadequate to model batch processes. Several research efforts have focused on utilization of past batch data in building low dimensional models that adequately capture the process dynamics and have good predictive capability. The most promising among them are the Latent Variables Models based on PCA and PLS proposed by Nomikos and MacGregor (1994, 1995) and subsequent variations of these. These models are linear and hence lead to simple and fast on-line algorithms, but yet the capture the nonlinear behavior of batch processes by modeling the time varying covariance structure among all variables throughout the duration of the batch.
There are two classes of model predictive algorithms for batch processes. The first is the high level problem of controlling the final batch product properties, and the second is the lower level problem of trajectory tracking. Latent variable models have been used to effectively treat the former (high level) problem through model predictive mid-course correction approaches (e.g. Flores-Cerrillo and MacGregor (2002), Yabuki and MacGregor (1997)). Flores-Cerrillo and MacGregor (2005) recently proposed a MPC algorithm for the lower level trajectory tracking problem based on PCA models.
In the Latent Variable MPC developed in this study several different latent variable modeling approaches are used to obtain models that better describes the nonlinearity and time varying characteristics of the process. The control methodology is also elaborated to yield better performance and to be capable of rejecting nonstationary disturbances.
The approach selected in this study is to develop a Principal Component Analysis (PCA) on the historical data gathered from past batches with an existing feedback control system plus a few batches with closed-loop designed experiments, and then to implement the control based on the predictions made by this model. Closed-loop identification is almost essential with batch processes to ensure that the product quality is not degraded. Since the data of a batch process is distributed in three dimensions (batches variables batch time), a suitable rearrangement of this cube of data set is required prior to developing a PCA model. Two data unfolding approaches are used: 1) Batch-wise unfolding and 2) Variable-wise unfolding with time lagging. These two methods have different characteristics that make each one appropriate for special conditions. Batch-wise unfolding shows better ability to explain the time varying properties of the process, but the dimensionality problem becomes evident in the algorithm as batch proceeds. On the other hand, column-wise time-lag unfolding is suitable for the conditions when fewer runs are allowed for data generation, but time varying characteristics are not well explained. In order to compensate the aforementioned problems, different phases along the batch are distinguished and multiphase modeling approach is performed. The MPC control formulation is developed to provide simultaneously for trajectory tracking and nonstationary disturbance rejection. Two control formulations are developed. In the first one the control optimization performed in the latent variable space and the manipulated variables then computed from the optimal scores using the PCA model. In the second one the control optimization is performed directly in the space of the manipulated variables.
The resulting algorithms are tested on a case study. The aim is to control a batch reactor temperature by manipulating the set point of the cascade controller built on the jacket temperature. The objective function is minimizing the deviation of the process output from the set point trajectory as well as penalizing the control movements. The two control formulations yield similar performances, but each of them has its own benefits. As an instance, the first algorithm has better performance and is more consistent with the concept of multivariate statistical modeling and it is faster than the second method. However, the second one is more familiar to the control community and it is easy to incorporate hard constraints on the manipulated variable trajectories.
References
[1] Flores-Cerrillo, J., MacGregor, J. F., (2005), Latent Variable MPC for trajectory tracking in batch processes, J. Process Cont. 15, 651-663.
[2] Flores-Cerrillo, J., MacGregor, J. F., (2002), Control of particle size distributions in emulsion semibatch polymerization using midcourse correction policies, Ind. Eng. Res. 41, 1805–1814.
[3] Nomikos P., MacGregor J. F., (1994), “Monitoring batch processes using multiway principal component anakysis”, AIChE Journal 40, 1361-1375.
[4] Nomikos P., MacGregor J. F., (1995), “Multiway partial least squares in monitoring batch processes”, Chemometrics and intelligent laboratory systems 30, 97-108.
[5] Yabuki, Y., MacGregor, J. F., (1997), Product quality control in semibatch reactors using midcourse correction policies, Ind. Eng. Chem. Res. 36, 1268–1275.