Flatness based system decompositions

FWF Project P32151: 07/2019-06/2023

Principal Investigator (PI): Markus Schöberl

Collaborators: Conrad Gstöttner, Johannes Diwold

Publications:

C. Gstöttner, B. Kolar, M. Schöberl: On the Linearization of Flat Two-Input Systems by Prolongations and Applications to Control Design, 21st IFAC World Congress, Berlin, Germany,  IFAC-PapersOnline 53(2), pp. 5479-5486, 2020, arXiv

C. Gstöttner, B. Kolar, M. Schöberl: On a Flat Triangular Form Based on the Extended Chained Form, 24th International Symposium on Mathematical Theory of Networks and Systems (MTNS), IFAC-PapersOnLine 54(9), pp. 245-252, 2021, arXiv

 

C. Gstöttner, B. Kolar, M. Schöberl: A Structurally Flat Triangular Form Based on the Extended Chained Form, International Journal of Control, 2020, IJC (open access), arXiv

J. Diwold, B. Kolar, M. Schöberl: A normal form for two-input forward-flat nonlinear discrete-time systems, International Journal of System Science, 2021, IJSS (open access)  arXiv

C. Gstöttner, B. Kolar, M. Schöberl: A Finite Test for the Linearizability of Two-Input Systems by a Two-Dimensional Endogenous Dynamic Feedback, European Control Conference ECC, (accepted), 2021, arXiv

J. Diwold, B. Kolar,  M. Schöberl: A Trajectory-Based Approach to Discrete-Time Flatness, IEEE Control Systems Letters, vol 6, pp. 289-294, 2022, arXiv

B. Kolar, J. Diwold, M. Schöberl: Zur Theorie und Anwendung der Flachheit nichtlinearer zeitdiskreter Systeme, at - Automatisierungstechnik 69(7), pp. 574-584, at (open access) 2021

B. Kolar, M. Schöberl, J. Diwold: Differential-Geometric Decomposition of Flat Nonlinear Discrete-Time Systems, Automatica (accepted) 2021, arXiv

Submitted Publications:

B. Kolar, J. Diwold, M. Schöberl: Necessary and Sufficient Conditions for Difference Flatness, 2019, preprint

Preprints:

C. Gstöttner, B. Kolar, M. Schöberl, Necessary and Sufficient Conditions for the Linearizability of Two-Input Systems by a Two-Dimensional Endogenous Dynamic Feedback, 2021, arXiv

Talks:

On the Linearization of Flat Two-Input Systems by Prolongations and Applications to Control Design, 21st IFAC World Congress, Berlin, Germany, July 2020, virtual online talk (C. Gstöttner)

A Finite Test for the Linearizability of Two-Input Systems by a Two-Dimensional Endogenous Dynamic Feedback, European Control Conference ECC, Rotterdam, Netherlands, July, 2021, virtual online talk (C. Gstöttner)

Executive Summary:

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.