Flat Systems - Geometric Systems Theory and Applications

The analysis and influencing of systems is expediently carried out using mathematical models of the underlying (physical) processes. Dynamic processes can be described with the help of so-called differential equations. Non-linear differential equations in particular pose an enormous challenge with regard to system analysis and control. Control means the targeted influencing of systems by actuators - e.g. forces are applied to an aircraft by engines and thus influence its behavior (hopefully to the desired extent). For non-linear systems, which have the so-called flatness property, the analysis is simplified, but this property is generally very difficult to prove. Essentially, the flatness property means that the system solutions can be parameterized with the help of functions whose time evolution can be assigned freely, which is a remarkable property for non-linear systems. If a digital computer is used to control processes, it can be useful to consider a so-called difference equation instead of a a differential equation. The system property of flatness is also a very interesting and demanding one for difference equations. In this research project, we dedicate ourselves to the task of developing methods that enable the proof of the flatness property for nonlinear systems. Even if the problem cannot be completely solved, it is already a significant gain in knowledge if results can be obtained for certain classes of systems. Research should be carried out for nonlinear differential equations as well as for nonlinear difference equations. Besides the system-theoretical interest in flatness, this concept also offers enormous possibilities for flatness-based control. Here, too, we want to make contributions, for example by investigating which system variables have to be measured for a practical realisation of a so-called trajectory-tracking control and which effects the degrees of freedom occurring in the design have on the controller performance. In industrial applications, non-linear systems are often considered by linearisation around an operating point in order to be able to apply the more powerful methods from linear system theory. New insights, which are now directly based on the non-linear system, naturally offer corresponding advantages compared to the approximate solutions mentioned above, especially in terms of performance, efficiency, energy consumption and robustness, to name but a few.