NMPC using Pontryagin's Minimum Principle-Application to a two-phase semi-batch hydroformylation reactor under uncertainty

- A newly-proposed PMP-method is utilized for solving shrinking horizon NMPC problems of constrained semi-batch processes.
- The computationally demanding matrix factorizations for large prediction horizons can be replaced by the recursion of the co-states with PMP.
- PMP-based NMPC is found to be faster than the DSM-based NMPC, especially in the beginning of the batch.
- 9% increase in the desired product and 26% reduction in the batch time is achieved with the closed-loop operation using PMP-based NMPC.

Nonlinear model predictive control (NMPC) is an important tool for the real-time optimization of batch and semi-batch processes. Direct methods are often the methods of choice to solve the corresponding optimal control problems, in particular for large-scale problems. However, the matrix factorizations associated with large prediction horizons can be computationally demanding. In contrast, indirect methods can be competitive for smaller-scale problems. Furthermore, the interplay between states and co-states in the context of Pontryagin's Minimum Principle (PMP) might turn out to be computationally quite efficient.This work proposes to use an indirect solution technique in the context of shrinking-horizon NMPC. In particular, the technique deals with path constraints via indirect adjoining, which allows meeting active path constraints explicitly at each iteration. Uncertainties are handled by the introduction of time-varying backoff terms for the path constraints. The resulting NMPC algorithm is applied to a two-phase semi-batch reactor for the hydroformylation of 1-dodecene in the presence of uncertainty, and its performance is compared to that of NMPC that uses a direct simultaneous optimization method. The results show that the proposed algorithm (i) can enforce feasible operation for different uncertainty realizations both within batch or from batch to batch, and (ii) is significantly faster than direct simultaneous NMPC, especially at the beginning of the batch. In addition, a modification of the PMP-based NMPC scheme is proposed to enforce active constraints via tracking.