(564e) Fast Robust Nonlinear Model Predictive Control Based on Sensitivity Update

As an advanced optimal control technique, Model Predictive Control (MPC) has been widely used in process control industry mainly because it is naturally suited to deal with constraints and multiple-input-multiple-output [1]. Its nonlinear counterpart, nonlinear model predictive control (NMPC), is getting more attention because it accurately captures the nonlinear effect of process dynamics. However, the system performance deteriorates under the influence of uncertainty. The current robust NMPC techniques are often shy of applications due to either model conservativeness or formidable computational effort. To improve this situation, Lucia et al. [2] introduced a multistage nonlinear model predictive control framework to serve as a promising non-conservative robust NMPC scheme. Multistage NMPC models the uncertainty evolution explicitly with a truncated scenario tree whose structure exploits the degrees of freedom by allowing future control inputs to adjust to future available information.

Multistage NMPC outperforms standard NMPC by robustly satisfying constraints and lowering the objective cost on average [3]. However, the price of the performance advancement with a multistage scheme is paid with a significant increase of CPU seconds. In order to reduce computational time, the sensitivity-based fast NMPC update algorithm is used to solve the parametric nonlinear program (PNLP) so that the update time is almost negligible compared to iterative solution procedures [4]. Allowing for non-unique multipliers, the path-following algorithm [5] solves parametric QP sub-problems and a jump step to capture the change of active sets.

In this talk, we first examine the performance of multistage NMPC compared with standard NMPC and two other robust NMPC approaches (min-max NMPC and back-off). Min-max optimizes the worst-case scenario while satisfying constraints for all the scenarios. Robust NMPC with back-off constraints tightens the nominal constraints with back-off terms, which are calculated from Monte Carlo simulations. A nonlinear CSTR example with a setpoint tracking objective has been explored with two separate uncertain parameters: activation energy and inlet concentration. Under both cases, the performance of multistage NMPC is superior than standard NMPC and robust NMPC approaches. Also, multistage scheme of different number of robust horizons is investigated and results are discussed. In addition, several probability distributions of the uncertain parameter are considered, where multistage formulation shows a clear advantage over other approaches. Lastly, additional case studies demonstrate that the sensitivity-based update can generate a rapid solution of multistage NMPC problems.

References:

[1] Grne, Lars, and Jrgen Pannek. "Nonlinear Model Predictive Control: Theory and Algorithms." (2013).

[2] Lucia, Sergio, Tiago Finkler, and Sebastian Engell. "Multi-stage nonlinear model predictive control applied to a semi-batch polymerization reactor under uncertainty." Journal of Process Control 23.9 (2013): 1306-1319.

[3] Lucia, Sergio, and Sebastian Engell. "Potential and Limitations of Multi-stage Nonlinear Model Predictive Control." IFAC-PapersOnLine 48.8 (2015): 1015-1020.

[4] Wolf, Inga J., and Wolfgang Marquardt. "Fast NMPC schemes for regulatory and economic NMPC–A review." Journal of Process Control 44 (2016): 162-183.

[5] Kungurtsev, Vyacheslav, and Johannes Jäschke. "A Predictor-Corrector Path-Following Algorithm for Dual-Degenerate Parametric Optimization Problems." SIAM Journal on Optimization 27.1 (2017): 538-564.