(359e) Advanced-Step Multistage Nonlinear Model Predictive Control

Model predictive control (MPC) has been a mature technique used to control and optimization of chemical processes, especially in dealing with Multiple-Input-Multiple-Output (MIMO) and interactions between variables and constraints [1]. Its nonlinear counterpart, Nonlinear Model Predictive Control (NMPC), has received growing attentions recently due to its efficacy of representing the nonlinear dynamics for a wide state space. However, various sources of uncertainty have deleterious effects on the performance of NMPC and it becomes necessary to robustify against these uncertainties.

We distinguish the sources of uncertainty into two categories: the first type is due to uncertain model parameters and the second one is unmeasured disturbance. The first type uncertainty assumes to be realized within one sampling time despite staying unknown for the current step, while the second type uncertainty requires more than one sampling time to resolve, or will never be resolved, even given longer time. Different approaches should be used to deal with two types of uncertainty: a scenario tree structure [3] captures the evolution of type 1 uncertainty, and robust sensitivity-based method [5] needs to be applied to consider type 2 uncertainty.

In contrast, competing robust NMPC approaches suffer from the following disadvantages. Min-max NMPC [2] seeks to optimize the worst-case scenario and also satisfy constraints for all the cases, which is very conservative. Tube-based NMPC [4], an extension of tube-based MPC, incorporates an ancillary controller that enforces the trajectory of the uncertain system to stay inside a tube which centers around the nominal trajectory. However, tube-based NMPC fails to address the optimal control performance under the influence of uncertainty. Ideal multistage NMPC [3] models the uncertainty evolution as a multistage scenario tree, which takes future control actions as the recourse variables to counteract the effect from uncertainty. However, the inevitably increased optimization problem size presents a difficult task to implement online.

In this talk, we propose parallelizable advanced-step multistage NMPC (as-msNMPC) as a real-time implementable robust NMPC scheme that interactively addresses both sources of uncertainty. This framework precomputes a set of solutions based on predicted state information with advanced-step fashion [6], and updates the control action online with NLP sensitivity, which are two orders of magnitude faster than solving NLP from scratch. The as-msNMPC framework has been applied to a CSTR and a distillation case study.

References:

[1] Grne, L., & Pannek, J. (2013). Nonlinear model predictive control: theory and algorithms.

[2] Lazar, M., De La Peña, D. M., Heemels, W. P. M. H., & Alamo, T. (2006). Min-max nonlinear model predictive control with guaranteed input-tostate stability. In 17th Symposium on Mathematical Theory for Networks and Systems. Kyoto, Japan.

[3] 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.

[4] Mayne, D. Q., Kerrigan, E. C., Van Wyk, E. J., & Falugi, P. (2011). Tube‐based robust nonlinear model predictive control. International Journal of Robust and Nonlinear Control, 21(11), 1341-1353.

[5] Pirnay, H., López-Negrete, R., & Biegler, L. T. (2012). Optimal sensitivity based on IPOPT. Mathematical Programming Computation, 4(4), 307-331.

[6] Zavala, V. M., & Biegler, L. T. (2009). The advanced-step NMPC controller: Optimality, stability and robustness. Automatica, 45(1), 86-93.