(654d) Data-Based Methods for Control of Batch Processes

Batch and fed-batch processes have found widespread applications in a variety of sectors especially those that are used for the manufacture of high-quality products such as bio-chemicals and polymers. The primary control objective in batch processes is to reach a specified product quality by batch termination. Consequently, batch control strategies are typically designed to reach a desired end-point and can be categorized into trajectory tracking or end-point based approaches. In trajectory tracking approaches, optimal process variable trajectories that terminate at the desired end-point are generated by master controllers or optimizers by solving a dynamic optimization problem (e.g., see Cruickshank et al. (2000)) and subsequently tracked using low level PID or predictive controllers (e.g., see Flores-Cerrillo and MacGregor (2005); Soroush and Kravaris (1993a,b)). With increased availability of computational resources and more efficient optimization algorithms, shrinking horizon end-point based model predictive control (MPC) is becoming a possibility for batch process control. In computing the control action in this approach, a dynamic optimization problem that incorporates the desired end-point (in the objective function and/or constraints) is solved at each sampling instance until batch termination, using plant measurements (or state variable estimates) to update model predictions at each sampling time. A truncated version of the resulting optimal input changes are then implemented directly on the process, the prediction horizon is shifted one sampling instance, and the process is repeated at the next sampling instance.

Despite significant developments in optimization algorithms, practical application of MPC (for trajectory tracking or end-point based control) is restricted due to the on-line computational requirements of repeatedly solving an optimization problem using the full non-linear model. The common workaround to this problem has been to substitute a linearized version of the non-linear model (at the estimated states) at each sampling instance, rendering the optimization problem convex and therefore efficiently solvable (see Mayne (2000) for a review). However, the control performance is limited with this approach due to the strong non-linearities present in batch processes. Recently, Aumi and Mhaskar (2009) proposed a computationally efficient alternative for existing end-point based designs that was based on the concept of ?reverse-time reachability regions? (RTRRs). RTRRs were defined as the set of states from where the desired end-point can be reached by batch termination while satisfying input constraints. By mathematically characterizing RTRRs using the full non-linear model off-line, the on-line computational demands of the RTRR based MPC design were significantly lower compared to end-point based approaches. Specifically, only the immediate control action was required to be computed in order to guarantee end-point reachability (if possible) as opposed to the entire input trajectory. As a result, the online computation time associated with this design when using the non-linear model was shown to be tractable for real-time application.

The increased availability of past process data, however, motivates exploiting the use of data-based modeling approaches in conjunction with nonlinear control tools. For batch systems, identification experiments, such as those in which a pseudo-random binary signal (PRBS) is applied on the process, are often too expensive to justify; however, the availability of measurements from past batches in batch databases can be exploited to improve the achievable level of accuracy. Within the range of process operating conditions in a typical batch database, the process behavior is highly non-linear and time-varying, making conventional system identification approaches, where a single linear model would be estimated, ill-suited for identifying accurate batch process models. Conventional system identification methods are also limited in that only input-output data are used in estimating the model parameters. Although on-line sensors for the quality properties are usually unavailable, every batch process does have frequent observations available on many easily measured process variables from which the process states can be inferred, implying an accurate model could potentially be identified.

Motivated by the above considerations, this work first considers the problem of empirically modeling a non-linear batch process using multiple local linear models. In this modeling approach, the database is first clustered into a number of operating regions, a weighting scheme is devised for the training data that can also be utilized to appropriately weight the local models given an initial condition, and finally, local linear models are estimated simultaneously using partial least squares (PLS) regression. The resulting model is then used to first formulate a trajectory tracking predictive controller and then incorporated within the RTRR based MPC framework in Aumi and Mhaskar (2009). The modeling approach and MPC designs are applied on a fed-batch reactor example and the advantages of using the MPC design over existing options are highlighted.

References

Aumi, S. and Mhaskar, P. (2009). Safe-steering of batch processes. AIChE J., 55:2861-2872.

Cruickshank, S. M., Daugulis, A. J., and McLellan, P. J. (2000). Dynamic modeling and optimal fed-batch feeding strategies for a two-phase partitioning bioreactor. Biotech. & Bioeng., 67:224-233

Flores-Cerrillo, J. and MacGregor, J. F. (2005). Latent variable MPC for trajectory tracking in batch processes. J. Proc. Contr., 15(6):651-663.

Mayne, D. Q. (2000). Non-linear model predictive control: Challenges and opportunities. In Allgower, F. and Zheng, A., editors, Non-Linear Model Predictive Control, pages 23-44. Birkhauser, Basel.

Soroush, M. and Kravaris, C. (1993a). Optimal-design and operation of batch reactors. 1. Theoretical framework. Ind. & Eng. Chem. Res., 32:866-881

Soroush, M. and Kravaris, C. (1993b). Optimal-design and operation of batch reactors. 2. A case-study. Ind. & Eng. Chem. Res., 32:882-893