Numerical methos for robust nonlinear MPC with internal feedback optimization

• Applicant: Professor Dr. Moritz Diehl
• Subject Area: Automation, Mechatronics, Control Systems, Intelligent Technical Systems, Robotics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 560056112

While model predictive control (MPC) is a powerful method for feedback control, in its standard form, nominal MPC, it is not aware of the uncertainty of the employed model. This can lead to constraint violation which, especially in safety-critical applications, is not tolerable. By augmenting the MPC problem with a model of uncertainty, robust MPC is able to adapt the planned trajectories accordingly, allowing it to guarantee constraint satisfaction. Robust MPC can be separated into open- and closed-loop approaches. The predictions in open-loop robust MPC are based on a fixed control trajectory, but this leads to unrealistically conservative uncertainty predictions, impeding the performance of the MPC policy. In contrast, closed-loop robust MPC includes future feedback laws in its predictions, allowing for a significant reduction of the predictive uncertainty. In consequence, it can recover performance similar to that of nominal MPC, but with robustness guarantees. However, closed-loop robust MPC problems are typically troubled by high computational costs, which is a barrier to their adoption in real-time control. In order to enable the use of robust MPC in a real-time setting, this computational bottleneck needs to be overcome. During the first funding phase, we already made significant progress in this direction. Contributions in particular include the algorithms SIRO and zoRO which target ellipsoidal-tube robust MPC problems with resp. without internal feedback optimization. Both are able to significantly reduce the computational complexity compared to standard algorithms. For the second funding phase, we intend to build upon these successes. While zoRO has reached a mature state (including an efficient open-source implementation), the algorithm SIRO is still in continued development. Thus, the first objective will be to widen the applicability of SIRO, in order to develop a reliably converging method for robust MPC with internal feedback optimization. Further, the first funding phase has focussed on ellipsoidal tubes for the uncertainty set representation. These can struggle to adequately handle nonlinear transformations, which the computationally expensive scenario tress handle more naturally. Thus, the second objective is to merge the advantages of both approaches by combining them in the form of ellipsoid trees. Finally, the path towards computationally tractable formulations includes approximations of exactly robustified formulations. Therefore, the third objective is to develop a clear theoretical understanding of the consequences of approximation. Overall, the main outcome will be theoretically well-founded formulations and algorithms for real-time capable closed-loop robust MPC, made available to the research community in the form of open source software.

• DFG Programme: Research Grants