## Optimal Control

The last two decades witnessed a rapidly growing demand in the field of mathematical modelling, simulation, and optimization with respect to delivering trustworthy quantitative computational results in real-time, most notably in model-based optimizing feedback control of processes. Applications comprise challenging problems with high scientific, economical, ecological, and societal impact including industrial process control, autonomous driving, robotics, and microgrid control. This research groups has paired this growing demand with a considerable body of research for mathematical and computational methods that deliver sub-millisecond feedback rates for large-scale applications. The decisive factor of this scientific progress was a paradigm shift from a black-box coupling of numerical simulation and numerical optimization methods towards a highly integrated and dovetailed approach that is best understood mathematically from a homotopy vantage point focusing on parametric optimal control problems.

These problems are parametrically dependent on the current (estimate of) the process state and are approximately solved by successive linearization, coupled with structure exploiting linear algebra and differentiable simulation methods, amounting to specifically tailored variants of inexact Sequential Quadratic Programming (iSQP). For instance, to reach the minimal amount of feedback delay, which measures the time between arrival of the current process state and the actual feedback of the optimizing control to the process, the Real-Time Iteration (RTI) and its Multi-Level Iteration (MLI) variants, both developed in this research group, exploit that the system linearization is independent of the information of the current system state.

Since the algorithmic constituents of the MLI such as simulation, linearization, structure exploiting linear algebra, and optimization are so intricately interwoven, the appropriate modularization is a highly challenging endeavor and shall be guided by the mathematical homotopy paradigm. Moreover, the developed mathematical methods allow for a high degree of parallelization opportunities resulting from the MLI hierarchy, Multiple Shooting discretizations, and in particular for robustification with scenario tree approaches, which was pioneered in the DFG project Structure Exploitation for Scenario-Tree NMPC and MHE (BO 864/15-1) within the project cluster Optimizing Control for Uncertain Systems.

Parallel methods for MLI have been described in scientific publications, but are currently lacking in software implementations of MLI. We aim to close this gap between both theory and software in order to continue to provide the research community with the fastest methods for robust optimizing feedback control.

Software Packages:

MUSCOD-II MLI/STMLI ParaOCP