Structure-Preserving Model Reduction of Port-Hamiltonian Systems

Numerical simulation is an essential tool for the analysis, design and control of complex physical systems. Many of today’s technical systems are complex multi-physics systems, i.e. the energy exchange between different physical domains must be taken into account (e.g. in electromotors).  The port-Hamiltonian systems paradigm provides an energy-based framework for the modeling and control of such systems. The geometric description via (Stokes-)Dirac structures allows us to easily interconnect systems of different physical domains. By exploiting its inherent system characteristics such as passivity, the modeling in port-Hamiltonian form also facilitates the subsequent controller design. If controllers are formulated as port-Hamiltonian systems as well, they can be designed to shape the energetic behaviour of the coupled system consisting of plant and controller. This approach, for instance, paved the way for a paradigm shift in robotics towards safe interaction and human-robot collaboration.

However, depending on the physical system at hand and the desired accuracy of its model, systems in high state-space dimension may arise in the modeling stage which are computationally infeasible for simulation or real-time control. Model reduction addresses this issue by generating reduced-order models which approximate the original model with respect to predefined goals. For the reduction of port-Hamiltonian systems, the aim is not only to achieve a high approximation quality, but also to preserve the port-Hamiltonian structure during reduction, so that the advantages mentioned can also be exploited for the reduced model. These are competing objectives: with conventional reduction methods, a considerable part of the degrees of freedom is given up for the preservation of the structure, which are then no longer available for a possible increase of the approximation quality.

Therefore, the aim of this project is to introduce new degrees of freedom in the structure-preserving model reduction of port-Hamiltonian systems and thus to transfer the outcomes of the last years in the reduction of general state space models to the class of port-Hamiltonian systems. Furthermore, global, rigorous error bounds for the model reduction error shall be transferred and extended.