In the past, the dynamics of nonlinear systems were often investigated using either minimal models, from which a qualitative understanding of the influences of relevant system parameters can be obtained, or parameter studies of accurate, but high-dimensional and computationally expensive models, e.g. from a finite element discretization (FEM). In recent years, significant progress has been made in the dynamics community in the systematic reduction of accurate, high-dimensional models to low-dimensional reduced models. These reduced models on the basis of invariant manifolds represent the slow dynamics of interest for the analysis qualitatively, so that the advantages of the previous approaches can be combined. The methods have been and are being actively developed in the academic community, but have not yet been adopted in development processes in industry. Research is being conducted at the Chair of Applied Mechanics – Dynamics to make these methodological approaches accessible for industrial use.

In the last twenty years, Koopman operator theory has established itself in the field of dynamics as an alternative to classical modeling by means of (nonlinear) differential equations. Koopman operator theory is the theoretical foundation for a wide variety of data-driven methods such as Dynamic Mode Decomposition (DMD), Extended Dynamic Mode Decomposition (EDMD), Sparse Identification of Nonliner Dynamics (SINDy), to name just a few prominent examples. The basic promise is that any autonomous system can be described by a (possibly infinite-dimensional) linear model. This is interesting for control problems, for example, where a linear model with a medium dimension can be solved more easily in real time than a smaller but non-linear model, and (small) errors in the model description can be compensated for by the control system. However, there are two problems with a purely data-driven model for controlling a system: many systems - from quadcopters to nuclear power plants - cannot be operated without control, so a chicken-and-egg problem arises for generating the data-driven model; secondly, data is primarily generated where the system can be operated safely with control, as measurement data is primarily available in these states. In the event of a major disturbance, a data-driven model must therefore extrapolate from its training range, which can lead to major errors and system failure.

Research at the professorship is concerned with developing methods for the linearization of models that already exist in the form of differential equations. Details of Koopman operator theory can be understood as a systematic method for finding a coordinate transformation that transforms the system to a linear or bilinear system. This can be the starting point for efficient control based on a linear or bilinear model, which is later adaptively improved with further measurements of the controlled system.

Following system analysis, system optimization is a core task of engineers from all disciplines. The Chair of Applied Mechanics - Dynamics deals with issues concerning the optimization of specific dynamic mechanical and mechatronic systems, also and especially in the context of the profiling initiative 2025 “Engineering of Cyber Physical Systems”. One example of this is the holistic consideration and optimization of the energy efficiency of bipedal robots - a complex, controlled, non-linear mechatronic system - for which Prof. Römer conducted research at KIT before his appointment to TU Bergakademie Freiberg.