(635d) A Hybrid Data-Driven/Model-Based Framework for the Dynamic Characterization and Refinement of Model Uncertainty in Batch Systems: Application to Robust Online Optimization and Control

Stochastic optimization has emerged as a powerful means of solving a variety of optimization problems under uncertainty, which originate from different areas including supply-chain optimization, optimal design, robust dynamic optimization/optimal control, process scheduling and so on. However, the formulation of this type of optimization problems requires good quality information on the underlying characteristics of the uncertainty, which can affect the objective function as well as the problem constraints. Therefore, in recent times, the definition of strategies for extracting statistical information from historical economic/process data has progressively gained importance.

For the specific case of robust dynamic optimization/optimal control applied to batch processes, we can envisage the general strategy for â??extracting statistical informationâ? as a two-step (iterative) procedure:

1. First, historical process data (data from previous successful batches) and the process model are used to select a proper set of model parameters to be the uncertain parameters (UPs) and estimate their joint probability distribution (PDF);
2. Then, every â??nâ? batch cycles (n â?¥ 1), the PDF is re-estimated as new data become available as well as the UPs set is updated by adding, removing or replacing parameters.

Notice that this dynamic PDF of the dynamic UPs set comprises an input to the optimization phase of many online optimization/control frameworks. Three specific features make the aforementioned dynamic estimation of PDF and UPs set non-trivial:

1. The available process data are usually measurements of process states and not measurements of the UPs themselves; 
2. Some of the UPs might exhibit temporal dynamics (on a longer time scale than that of a single batch).
3. Proper metrics must be developed to manage the dynamic UPs set.

There are virtually no literature contributions concerning this specific type of problem, even though there exist frameworks for similar problems. In fact, there exist methods for online optimization/control coupled with run-to-run update of the process model ([1], [2] and [3]) but these frameworks allow only constant UPs sets and very often do not explicitly estimate the PDF of the UPs (except for [3], where the PDF is approximated using polynomial chaos expansion). Consequently, the aforementioned dynamic estimation problem appears to be an interesting research challenge.

In this paper, we tackle this problem using an iterative multi-step strategy that we can categorize as a Bayesian approach and involves a combination of Maximum Likelihood Estimation (MLE), Fitting (FG) and Sensitivity Analysis (SA). The methodology comprises two phases, referred to as phase I and phase II. Phase I is executed only before the very first batch cycle and allows estimating an initial UPs set as well as the corresponding PDF by means of a combination of FG, residuals analysis and SA. Necessary input information include historical process data along with the model of the target batch process. Phase II is performed every â??nâ? batches (n â?¥ 1) and aims at updating the UPs set as well as the corresponding PDF, based on fresh process data and the process model. Its underlying logic is complex and combines FG, residuals analysis, SA and MLE in the following way:

1. New statistical information is acquired from the newly available process data via combination of FG and residuals analysis;
2. This information, combined with the results of a proper SA, is used to re-estimate a new set of UPs;
3. MLE is applied to merge the same newly available statistical information with the previous PDF of the UPs towards improved uncertainty characterization (a new PDF). 

In contrast to conventional strategies ([1], [2] and [3]), this new method for â??extracting statistical informationâ?Â allows to dynamically and conveniently select the model parameters to consider as UPs, based on fresh process data. Moreover, it also allows estimating explicitly and dynamically the probability distribution of the UPs, which is necessary input to many robust dynamic optimization/optimal control strategies. Therefore, it is more general and flexible than other conventional strategies.

The new methodology proposed here is combined with a robust dynamic optimization/optimal control framework and applied to a fed-batch version of the Williams Otto process. The results achieved from the case study clearly demonstrate the benefit in applying the new approach, also compared to other conventional strategies ([1], [2] and [3]).

1. Nagy ZK. Model based robust batch-to-batch control of particle size and shape in pharmaceutical crystallization. IFAC Proceedings Volumes. 2009;7:195-200.
2. Chen T, Liu Y, Chen J. An integrated approach to active model adaptation and on-line dynamic optimisation of batch processes. Journal of Process Control. 2013;23:1350â??1359.
3. Mandur J, Budman, H. A Robust algorithm for Run-to-run Optimization of Batch Processes. 10th IFAC International Symposium on Dynamics and Control of Process Systems. 2013:541-546.