Flatness based system decompositions

The analysis and control of systems is advantageously based on mathematical models of the underlying (physical) processes. Dynamical processes can be described by so-called differential equations, where ordinary differential equations can be classified according to their linear or nonlinear structure. The analysis of the latter system class is a challenging field of research in systems and control theory, and especially regarding the controller design the class of nonlinear systems is much more complicated than linear systems. In this project we want to contribute here. In particular we want to derive systematic tools for the analysis of nonlinear systems which facilitate the design of feedforward and feedback control strategies. With mathematical methods we want to perform system analysis, i.e. characterize formal properties of the dynamical systems, and in particular we want to analyze a system property called flatness. In this research project we strive for the development of methods that allow for a verification of the flatness property of nonlinear systems. Flatness is a very useful concept and allows to control complex nonlinear systems in a systematic and efficient way. Roughly speaking, the flatness property allows to parameterize the system solutions by a set of arbitrary functions, which is a very pleasing property for nonlinear systems. However, there is no systematic test which can be applied to check a system for the flatness property efficiently. Moreover, using digital control units naturally leads to the concept of difference equations in contrast to differential equations. For these systems we also want to work towards tools that are useful in the context of flatness. In industrial applications often the linearization with respect to a set-point is carried out in order to apply linear methods. New approaches and developments that directly deal with the nonlinear system have of course benefits compared to these approximate methods, as they in general increase the performance, the efficiency, the robustness, and they are beneficial with respect to energy consumption, to mention but a few.

The analysis and control of systems is advantageously based on mathematical models of the underlying (physical) processes. Dynamical processes can be described by so-called differential equations, where ordinary differential equations can be classified according to their linear or nonlinear structure. The analysis of the latter system class is a challenging field of research in systems and control theory, and especially regarding the controller design the class of nonlinear systems is much more complicated than linear systems. In this project we have derived systematic tools for the analysis of nonlinear systems which facilitate the design of feedforward and feedback control strategies. In this research project we have developed methods that allow for a verification of the flatness property of nonlinear systems. Flatness is a very useful concept and allows to control complex nonlinear systems in a systematic and efficient way. Roughly speaking, the flatness property allows to parameterize the system solutions by a set of arbitrary functions, which is a very pleasing property for nonlinear systems. However, there is no systematic test which can be applied to check a system for the flatness property efficiently. For important classes of systems we have derived such systematic tests. Moreover, using digital control units naturally leads to the concept of difference equations in contrast to differential equations. For these systems we also have worked towards tools that are useful in the context of flatness, and we have succeeded in providing conditions and algorithms for testing the flatness property for a very large class of nonlinear discrete-time systems. For the so-called flatness-based tracking control, we have shown that, in contrast to previous approaches, it is possible to design the control systematically using the sensor signals available from the problem and that the estimation of fictitious sensor signals is unnecessary. In industrial applications often a linearization with respect to a set-point is carried out in order to apply linear methods. New findings, in particular those on flatness-based tracking control, that directly deal with the nonlinear system have of course benefits compared to these approximate methods, as they in general increase the performance, the efficiency, the robustness, and are beneficial with respect to energy consumption, to mention but a few.