(139c) Advanced-Multi-Step Strategies for Economic-Based Nonlinear Model Predictive Control

Authors: 
Yang, X., Carnegie Mellon University
Fischer, G. A. G., Petróleo Brasileiro S.A. – Petrobras
Biegler, L. T., Carnegie Mellon University



Nonlinear Model Predictive Control (NMPC) has been widely used in process industry due to its ability to handle variable bounds and multi-input-multi-output systems. However, since solution time of NLP problems being solved is always comparable to sampling time of the process, computational delay exists and may lead to deterioration of controller performance and system stability. Moreover, scales of the NLP problems might be so large that solution time exceeds one sampling time, leading to incorrect manipulated variables. We propose an advanced-multi-step NMPC (amsNMPC) method, which not only eliminates computational delay, but also provides suboptimal controls as solution time exceeds one sampling time. It is based on nonlinear programming (NLP) sensitivity. The basic idea is to divide calculations into two parts: the part on-line and the part in the background. In the background, NLP problems are solved with actual states as initial conditions at a certain frequency, while NLP sensitivity and optimal control of the most recently solved NLP are used to update control on-line once the actual corresponding state measurement is achieved.

A serial approach and a parallel approach are developed given different number of processors. The serial approach applies one processor, and solves the NLP problems every multiple sampling times. The Karush-Kuhn-Tucker (KKT) matrix of the most recently solved NLP is used repeatedly but with update until the next NLP is solved. The parallel approach solves an NLP problem every sampling time by assigning the NLP to a free processor. Multiple processors are applied to assure that there is always a processor available for a new NLP problem. A new KKT matrix is obtained every sampling time, so no update of KKT matrix is needed. Since manipulated variables are not achieved by solving NLPs with corresponding actual states as initial conditions, but by updating existing optimal solutions of NLPs with NLP sensitivity, which does backsolve and takes negligible amount of time, the computational delay can be avoided.

We extend this approach to deal with state constrained problems as well as economics-based objective functions. Here, we have modified and analyzed the original amsNMPC method for robust stability. In order for the controller to have higher tolerance on disturbances, we also apply soft constraints to state variables by adding slack variables to the lower and upper bounds and adding exact penalty terms to the objective. This allows us to develop a Lyapunov stability analysis to prove both the nominal stability and robust stability. Moreover, this theory is extended to consider objective functions for economically oriented amsNMPC. Finally, the resulting approach is demonstrated on large-scale process applications.