(194h) Infinite Horizon NMPC Via Time Transformation and Orthogonal Collocation

In the chemical industry, nonlinear model predictive control (NMPC) has proven highly valuable for effectively managing nonlinear MIMO systems while accommodating operational constraints. When dealing with finite predictive horizons, maintaining the stability and recursive feasibility of NMPC requires the utilization of a terminal cost function and/or terminal constraints [1]. However, computing these terminal properties potentially poses a formidable challenge, particularly in the context of nonlinear dynamic processes. Motivated by this significant challenge, the main aim of this study is to introduce a novel formulation of infinite-horizon NMPC through time transformation.

Previously, an infinite-horizon NMPC was developed and demonstrated for open-loop stable systems with a moving horizon and an adaptive discretization of the sampling times [2]. In this presentation, we introduce an alternate approach within a conventional moving horizon approach, where an infinite-horizon time transformation only for the final sampling time in the horizon. An equality terminal constraint is enforced for the closed-loop system to reach the (steady state) setpoint at the end of this infinite horizon, which is easily shown to ensure recursive feasibility. The numerical solution approach involves simultaneous nonlinear optimization and full discretization using orthogonal collocation on finite elements. The key advantage of this approach lies in solving the optimization problem of the proposed NMPC as a boundary value problem. Consequently, numerical stability is ensured through the theory of dichotomies for system of differential equations, thus eliminating the need for an open-loop stable requirement.

A potential issue with the time-transformation approach is the possibility of the setpoint becoming unreachable at the end of the infinite horizon due to the limited number of control actions at the beginning of the predictive horizon. To address this challenge, a sensitivity analysis is conducted when the terminal setpoint is unreachable, allowing for an extension of the number of control actions at the beginning of the infinite predictive horizon. As a result, this infinite horizon extension leads to a straightforward approach to design stable, robust and highly effective NMPC controllers.

To validate the efficacy of the proposed NMPC formulation, several case studies including a Continuous Stirred Tank Reactor (CSTR) and a tray-by-tray distillation column will beanalyzed and presented. These case studies will demonstrate the practical application and robustness of the developed approach in real-world scenarios.

References

[1] D. Limón, T. Alamo, F. Salas, and E. F. Camacho, “On the stability of constrained MPC without terminal constraint,” IEEE transactions on automatic control, vol. 51, no. 5, pp. 832–836, 2006.

[2] L. Würth and W. Marquardt, “Infinite-Horizon Continuous-Time NMPC via Time Transformation,” IEEE Transactions on Automatic Control, vol. 59, no. 9, pp. 2543–2548, Sep. 2014, doi: 10.1109/TAC.2014.2308605.