Exploiting nonlinear port-Hamiltonian structures for optimization-based and data-driven control

• Applicants: Professor Dr.-Ing. Timm Faulwasser; Professor Dr. Manuel Schaller; Professor Dr. Karl Worthmann
• Subject Area: Automation, Mechatronics, Control Systems, Intelligent Technical Systems, Robotics ; Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 519323897

Port-Hamiltonian (pH) systems are ubiquitous in engineering. Application domains encompass mechanics, electrical circuits, thermodynamics, and multi-energy systems to name but a few. Key advantages of the pH perspective are balance equations for physical quantities like energy and entropy, and closedness of pH systems under power-conserving interconnections. This project concerns optimal control of nonlinear port-Hamiltonian systems. We consider dissipative, reversible, and irreversible pH systems and we investigate physically-motivated optimal control problems (OCPs), e.g., state transition with minimal energy supply and/or entropy growth. We exploit the port-Hamiltonian structure intertwined in dynamics and cost functions to thoroughly analyze and to efficiently solve such OCPs. To this end, we proceed in three main steps. First, we analyze port-Hamiltonian OCPs using dissipativity-based methods invoking the nonlinear balance equations for energy and entropy. Furthermore, we consider multi-objective OCPs in view of minimizing the supplied energy and the generated entropy while simultaneously maximizing a produced quantity of interest. Finally, we broaden the analysis towards optimal control of cyclo-passive port-Hamiltonian systems. Second, we tailor Newton homotopy methods to solve the intrinsically-motivated but singular port-Hamiltonian OCPs. To this end, we approximate the solution by a sequence of regularized problems. Further, we construct preconditioners consistent with the pH structure to efficiently conduct the Newton step. Third, we propose novel variants of SINDy (Sparse Identification of Nonlinear Dynamics) and eDMD (extended Dynamic Mode Decomposition) tailored to nonlinear pH systems: On the one hand, we aim to reduce data requirements by leveraging the pH structures. On the other hand, we ensure that key quantities, e.g., the energy balance, are properly taken into account in the data-driven surrogate models while providing rigorous bounds on the approximation error. In conclusion, we analyze physically-motivated OCPs of pH systems and their numerical solution. We develop novel algorithms to efficiently control nonlinear pH systems, e.g., w.r.t. the required energy supply or data requirements, while providing guarantees on stability, numerical efficiency, and approximation accuracy of data-driven surrogate models.

• DFG Programme: Research Grants
• International Connection: Colombia, France
• Cooperation Partners: Professor Alejandro Garces Ruiz; Professor Dr. Bernhard Maschke