Flatness based system decompositions
FWF Project P32151: 07/2019-06/2023
Principal Investigator (PI): Markus Schöberl
B. Kolar, M. Schöberl, J. Diwold: Differential-Geometric Decomposition of Flat Nonlinear Discrete-Time Systems, 2019, preprint
B. Kolar, J. Diwold, M. Schöberl: Necessary and Sufficient Conditions for Difference Flatness, 2019, preprint
C. Gstöttner, B. Kolar, M. Schöberl: On the Linearization of Flat Two-Input Systems by Prolongations and Applications to Control Design, 2019, preprint
In this project we suggest to apply differential geometric methods including exterior algebra for the system theoretic analysis of nonlinear continuous-time and discrete-time systems with respect to their flatness properties. The main intention of the proposed research project is to further elaborate on the generation and the extension of suitable normal forms and system decompositions which facilitate the check for flatness. A key ingredient of the suggested approach is the construction of a sequence of subsystems by a gradual reduction of the original system that simplifies the flatness analysis. It is not clear if a complete and algorithmically verifiable solution of the flatness problem can ever be achieved - nevertheless, to enlarge the system class for which necessary or at least checkable non-trivial sufficient conditions can be obtained is the main goal of this research project. The following main tasks shall be addressed.
Firstly, in the case of ordinary differential equations (ODEs) we wish to analyze the properties of a generalization of an affine input structure, called a partial affine input (PAI) structure, in order to improve the flatness analysis based on the gradual reduction of control systems. In this context the results of Nicolau and Respondek (scientific partner) regarding exact linearization via one-fold prolongation shall be examined in the light of PAI structures. Furthermore, the connection of the partial affine input representation formed by explicit ODEs with an implicit triangular structure proposed by the applicant shall be investigated to further improve the concept of gradual reduction. Additionally, we wish to analyze the concept of orbital flatness (including state dependent time-reparameterizations) with respect to the implicit triangular decomposition in a Pfaffian system representation.
Secondly, for sampled data systems the concept of difference flatness shall be examined by investigating the possibilities of applying a gradual reduction process similar as in the continuous-time case. Additionally, already available results for a special subclass of flat systems from the case of ODEs shall be transferred to the case of nonlinear difference equations. A further research question is to investigate connections between properties of the flat parameterization and structural properties of the system. Moreover, a criterion similar to the well-known ruled manifold test for nonlinear continuous-time systems shall be developed in the discrete-time case, based on the promising observation that in the composition of the flat parameterization with the system equations there occur no non-shifted variables.