Nonlinear model predictive control (NMPC) is a model-based dynamic control strategy that reoptimizes a process system with respect to a control objective subject to relevant constraints at every sample time. The NMPC controller offers flexibility to include constraints in online decision making, and a good control performance even when the system drifts far from the desired operating point. Although NMPC has inherent robustness against uncertainty, the property may break when there are significant disturbances leading to infeasibility. As a result, Lucia et al. (2013) proposed a robust NMPC controller known as the multistage NMPC. Multistage NMPC formulates an optimization problem by explicitly considering all possible scenarios along the prediction horizon. The scenarios are determined by propagating a finite set of uncertain parameter realizations using a scenario tree representation. While multistage NMPC ensures no constraint violations, it is very conservative resulting into a loss in performance in the objective. Another drawback is the huge computational effort required to solve a typically larger optimization problem resulting from the exponential growth of the scenario tree.

In order to speed up computation, the sensitivity assisted multistage NMPC (samNMPC) was proposed by Yu (2019). The samNMPC controller strategy is to reduce the size of the scenario tree by only selecting scenarios that are most likely to violate constraints, plus the nominal case. These constraints are termed critical constraints and are determined by cheap sensitivity calculations. Online selection of critical constraints is trivial only when the constraints are monotonically changing, which holds for most process systems having state interval bounds (Holtorf et al., 2019). While the method successfully speeds up the multistage NMPC, has good stability properties, and creates a parallelizable linear structure as shown by Thombre et al. (2021), it remains to be seen how its conservativeness can be reduced.

We present a robust control framework that reduces the conservativeness of the sensitivity assisted multistage NMPC (samNMPC). Process systems may often exhibit strong correlations among the uncertain model parameters. In this case, modeling scenarios independently using the vertices of the hyperrectangle results in a needlessly high conservative controller. Therefore, we propose a way of mitigating conservativeness by mining information from historical operational data of the process system. We use principal component analysis (PCA) to determine new orthogonal unit directions of maximum variance. The approach had been previously proposed by Krishnamoorthy et al. (2018) for multistage NMPC and applied to control thermal energy storage for district-heating by Thombre et al. (2020). We combine data mining and samNMPC employing a trick to retain the monotonicity property of the constraints, vital for critical scenario detection. In order to retain the validity of the monotonicity assumption on the constraints, all the uncertain parameters in the optimal control problem are projected into the new space using linear transformations from the PCA. Then, the critical scenarios are built using the extremes that lie on the axes of the new unit directions and the new origin.

The data-driven samNMPC was tested on an example problem for reference tracking and its control performance benchmarked against samNMPC, ideal multistage NMPC and standard NMPC controllers. The data-driven samNMPC controller exhibited zero constraint violations as the samNMPC but had less conservativeness. That is, the loss (integral tracking error) was significantly reduced in the data-driven samNMPC compared to the original samNMPC. The improvement in the controller tracking performance while enforcing constraint adherence in operation justify the method’s advantage. On top of that, we have found a way to incorporate historical information and retain the online scenario selection heuristic present in the original samNMPC.

References

Holtorf, F., Mitsos, A., & Biegler, L. T. (2019). Multistage NMPC with on-line generated scenario trees: Application to a semi-batch polymerization process. Journal of Process Control, 80, 167-179.

Krishnamoorthy, D., Thombre, M., Skogestad, S., & Jäschke, J. (2018). Data-driven scenario selection for multistage robust model predictive control. IFAC-PapersOnLine, 51(20), 462-468.

Lucia, S., Finkler, T., & Engell, S. (2013). Multi-stage nonlinear model predictive control applied to a semi-batch polymerization reactor under uncertainty. Journal of process control, 23(9), 1306-1319.

Thombre, M., Mdoe, Z., & Jäschke, J. (2020). Data-Driven Robust Optimal Operation of Thermal Energy Storage in Industrial Clusters. Processes, 8(2), 194.

Thombre, M., Yu, Z. J., Jäschke, J., & Biegler, L. T. (2021). Sensitivity-Assisted multistage nonlinear model predictive control: Robustness, stability and computational efficiency. Computers & Chemical Engineering(148), 107269.

Yu, Z. J. (2019). Advances in Decision-making Under Uncertainty with Nonlinear Model Predictive Control. (Doctoral Dissertation, Carnegie Mellon University).