(684d) Economic Nonlinear Model Predictive Control with a Path-Following Approach

Nonlinear Model Predictive Control (NMPC) with economic objective has been an active research field in recent years [1]. Especially its performance and closed-loop stability analysis have received much attention. The idea of economic NMPC is to integrate real time optimization (RTO) and supervisory layers in the process control hierarchy into a single economic NMPC layer in order to simultaneously optimize and control the process. Papers [2] and [3] provide tutorial and survey on recent developments.  

In this work, we study a sensitivity-based advanced-step NMPC (asNMPC) as described in [4,5] with economic cost function. This type of controller consists of an online and an offline step. In the offline step, a full nonlinear programming problem (NLP) is solved for a predicted starting point in the future, while waiting for the measurement data of this point to become available. After the measurement data is obtained, a fast online correction is calculated based on the optimal sensitivity of the NLP problem (instead of solving the full NLP again). Thus, the computational delay between obtaining the measurements and implementing the computed inputs is significantly reduced.  

When active set changes occur in the sensitivity update, special measures are required to ensure that the sensitivity update approximates the true NLP solution well enough. In [6] some methods are discussed for handling active set changes. Recently [7] proposed a path-following algorithm for the fast sensitivity update. The algorithm is capable of handling active set changes and non-unique-bounded Lagrange multipliers, which arise when the NLP satisfies MFCQ. This path-following algorithm is used in the online step of the asNMPC procedure. In addition to computing the optimal sensitivity along the path by solving a sequence of QPs, the algorithm solves a sequence of linear programming (LP) problems for computing the optimal changes of the Lagrange multipliers along the path.  

In this work, we apply the path-following algorithm in the context of asNMPC for economic NMPC of a large-scale optimization problem consisting of a 41-tray distillation column case study. We employ orthogonal collocation to discretize the dynamic optimization problem and use automatic differentiation to obtain first and second order derivatives. We compare the performance of our method to ideal NMPC in which we solve the full NLP assuming negligible computational delay. We show that our method is able to preserve computational efficiency of the asNMPC in the presence of active set changes.    

References:  

1.Rawlings, J.B., Angeli, D., and Bates, C.N., 2012. Fundamental of economic model predictive control. In proceeding of IEEE 51st Annual Conference on Decision and Control (CDC), pp. 3851-3861.

2.Ellis, M., Durand, H., and Christofides, P.D., 2014. A tutorial review of economic model predictive control methods. Journal of Process Control, Volume 24, pp. 1156-1178.

3.Tran, T., Ling, K-V., and Maciejowski, J.M., 2014. Economic model predictive control â?? a review. In proceeding of The 31st International Symposium on Automation and Robotics in Construction and Mining (ISARC 2014).

4.Biegler, L.T., Yang, X., and Fischer, G.A.G., 2015. Advances in sensitivity-based nonlinear model predictive control and dynamic real-time optimization. Journal of Process control, Vol. 30. Pp. 104-116.

5.Zavala, V.M., and Biegler, L.T., 2009. The advanced-step NMPC controller: optimality, stability, and robustness. Automatica, Vol. 45. Pp. 86-93

6.Yang, X., and Biegler, L.T., 2013. Advanced-multi-step nonlinear model predictive control. Journal of Process Control, Vol. 23. Pp. 1116-1128.

7.Jäschke, J., Yang, X., and Biegler, L.T., 2014. Fast economic model predictive control based on NLP-sensitivities. Journal of Process Control, Vol. 24. Pp. 1260-1272.