(434h) Improving Computational Efficiency of Multi-Stage NMPC Using an Adaptive Horizon | AIChE

# (434h) Improving Computational Efficiency of Multi-Stage NMPC Using an Adaptive Horizon

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Harvard John A. Paulson School of Engineering and
Norwegian University of Science and Technology
Model predictive control (MPC) is a model-based control strategy that re-optimizes over future state predictions using current state information to obtain optimal control actions (Rawlings and Mayne, 2009). In the case where we have a nonlinear model with respect to the states and control input variables, then it becomes nonlinear MPC (NMPC). This requires at every re-optimization step to solve a nonlinear program with an increased complexity that may cost a significant computational effort. Furthermore, the nonlinear models are built from several parameters that may have some associated uncertainty. A nominal NMPC will exhibit a reduced control performance especially with constraint violations if there exists a significant plant-model mismatch. As a result, robust NMPC approaches have been studied to formulate robust optimization problems that handle multiple scenarios of parameter realizations (Campo and Morari, 1987). One such approach is the multi-stage NMPC.

The multi-stage NMPC represents an optimization with uncertainty problem based on a scenario tree evolution of the uncertain parameter realizations into the prediction horizon (Lucia et al., 2013). At each NMPC iteration, a multi-scenario nonlinear program is formulated that minimizes the expected cost subject to model equations, state, and input bounds for each scenario. The problem allows recourse actions justified by presence of feedback, by enforcing non-anticipativity constraints. The multi-stage problem is robust against plant-model mismatch, but it increases the problem size significantly since the number of scenarios is dependent on the number of parameter realizations, and the prediction horizon. This robustness is at the expense of an increased computational cost, hence increased computational delay. A strategy to limit the growth of the scenario tree and the problem size is to implement a robust horizon where branching stops (Sergio Lucia et al., 2013). Further, to minimize the number of scenarios and limit computational cost, a common heuristic of selecting three realizations {max, nominal, min} for each parameter has been implemented (S. Lucia et al., 2013; Martí et al., 2015; Lucia et al., 2016; Jang et al., 2016). However, selecting a few parameter realizations may not be a good representative of the uncertainty set leading to unnecessary conservativeness.

This work is an extension of Mdoe et al. (2021) but is motivated by the possibility of obtaining the uncertain parameter realizations from statistical analysis of historical data (Krishnamoorthy et al., 2018; Thombre et al., 2020). The uncertain parameter set contains discrete points corresponding to the most critical parameter realizations inside a convex hull that the multi-stage NMPC must satisfy. Therefore, the aim of this work is to reduce the problem size further by rather investigating on limiting the prediction horizon. We aim at automatically updating the prediction horizon at each multi-stage NMPC iteration such that the closed loop system is stabilizing, hence improving the computational efficiency. The adaptive horizon algorithm will continuously update the minimum possible prediction horizon that is stabilizing in all possible scenarios, thus minimizing computational delay (Mdoe et al., 2021). This control strategy is termed as the adaptive horizon multi-stage NMPC.

Previously in Mdoe et al. (2021), where the idea of adaptive multi-stage NMPC was first introduced, a brief sketch on its closed-loop stability property was presented. Another contribution of this work is to rigorously establish stability and recursive feasibility properties of the adaptive horizon multi-stage NMPC. Under assumptions of availability full state feedback information and relaxed formulation of multi-stage NMPC with a robust horizon, the adaptive horizon multi-stage NMPC was found to be recursively feasible and input-to-state practically (ISpS) stable for all possible cases of horizon update. The closed loop performance of the controller was tested on two numerical problems: a cooled CSTR system (Klatt and Engell, 1998), and a quad-tank system (Raff et al., 2006). It was found that the proposed controller had a reduced computational cost per iteration without any loss of robustness when compared to the original multi-stage NMPC with a robust horizon.

References

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Jang, H., Lee, J.H., Biegler, L.T., 2016. A robust NMPC scheme for semi-batch polymerization reactors. IFAC-PapersOnLine, 11th IFAC Symposium on Dynamics and Control of Process SystemsIncluding Biosystems DYCOPS-CAB 2016 49, 37â€“42. https://doi.org/10.1016/j.ifacol.2016.07.213

Klatt, K.-U., Engell, S., 1998. Gain-scheduling trajectory control of a continuous stirred tank reactor. Computers & Chemical Engineering 22, 491â€“502. https://doi.org/10.1016/S0098-1354(97)00261-5

Krishnamoorthy, D., Thombre, M., Skogestad, S., JÃ¤schke, J., 2018. Data-driven Scenario Selection for Multistage Robust Model Predictive Control. IFAC-PapersOnLine, 6th IFAC Conference on Nonlinear Model Predictive Control NMPC 2018 51, 462â€“468. https://doi.org/10.1016/j.ifacol.2018.11.046

Lucia, Sergio, 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, 1306â€“1319. https://doi.org/10.1016/j.jprocont.2013.08.008

Lucia, S., Schliemann-Bullinger, M., Findeisen, R., Bullinger, E., 2016. A Set-Based Optimal Control Approach for Pharmacokinetic/Pharmacodynamic Drug Dosage Design. IFAC-PapersOnLine, 11th IFAC Symposium on Dynamics and Control of Process SystemsIncluding Biosystems DYCOPS-CAB 2016 49, 797â€“802. https://doi.org/10.1016/j.ifacol.2016.07.286

Lucia, S., Subramanian, S., Engell, S., 2013. Non-conservative robust Nonlinear Model Predictive Control via scenario decomposition, in: 2013 IEEE International Conference on Control Applications (CCA). Presented at the 2013 IEEE International Conference on Control Applications (CCA), pp. 586â€“591. https://doi.org/10.1109/CCA.2013.6662813

MartÃ­, R., Lucia, S., Sarabia, D., Paulen, R., Engell, S., de Prada, C., 2015. Improving scenario decomposition algorithms for robust nonlinear model predictive control. Computers & Chemical Engineering 79, 30â€“45. https://doi.org/10.1016/j.compchemeng.2015.04.024

Mdoe, Z., Krishnamoorthy, D., Jaschke, J., 2021. Adaptive Horizon Multistage Nonlinear Model Predictive Control, in: 2021 American Control Conference (ACC). Presented at the 2021 American Control Conference (ACC), pp. 2088â€“2093. https://doi.org/10.23919/ACC50511.2021.9483183

Raff, T., Huber, S., Nagy, Z.K., Allgower, F., 2006. Nonlinear model predictive control of a four tank system: An experimental stability study, in: 2006 IEEE Conference on Computer Aided Control System Design, 2006 IEEE International Conference on Control Applications, 2006 IEEE International Symposium on Intelligent Control. Presented at the 2006 IEEE Conference on Computer Aided Control System Design, 2006 IEEE International Conference on Control Applications, 2006 IEEE International Symposium on Intelligent Control, pp. 237â€“242. https://doi.org/10.1109/CACSD-CCA-ISIC.2006.4776652

Rawlings, J.B., Mayne, D.Q., 2009. Model Predictive Control Theory and Design. Nob Hill Pub, Llc, Madison, Wis.

Thombre, M., Mdoe, Z., JÃ¤schke, J., 2020. Data-Driven Robust Optimal Operation of Thermal Energy Storage in Industrial Clusters. Processes 8, 194.