Contributions to the Differential Geometric Analysis and Control of Flat Systems

This work deals with the differential geometric analysis and the control of flat systems. Flat continuous-time systems possess the property that all system variables can be expressed by a flat output and its time derivatives. Similarly, flat discrete-time systems possess the property that all system variables can be expressed by a flat output and its forward-shifts. Even though flatness is extremely useful for trajectory planning and controller design, a major problem is that both in the continuous-time and the discrete-time case there do not exist efficiently verifiable necessary and sufficient conditions. Accordingly, also the computation of flat outputs is very difficult, and a substantial part of the present work is devoted to this problem. Roughly speaking, the idea of the presented methods is to decompose the system step by step into smaller subsystems, until finally a flat output can be read off.

With regard to continuous-time systems, the map that describes the parameterization of the system variables by the flat output is analyzed in detail. For instance, upper bounds for the order of the highest time derivatives of the flat output are derived. Furthermore, also the Jacobian matrix of this map is studied.

Another topic is the design of flatness based tracking controls for continuous-time systems. An elegant standard approach is based on the exact linearization of the system by means of a quasi-static state feedback, which necessitates that time derivatives of the flat output up to a certain order are measured. However, often it would be desirable to work with measurements of the state variables instead. It is shown that, under certain conditions, it is indeed possible to replace the required time derivatives of the flat output systematically by the state.