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Flatness based system decompositions

FWF Project P32151: 07/2019-06/2023

Principal Investigator (PI): Markus Schöberl

Research Associates: Conrad Gstöttner, Johannes Diwold

Scientific Partner: Bernd Kolar

 

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, opens an external URL in a new window 53(2), pp. 5479-5486, 2020, arXiv, opens an external URL in a new window

C. Gstöttner, B. Kolar, M. Schöberl: A Structurally Flat Triangular Form Based on the Extended Chained Form, International Journal of Control, 2020, IJC, opens an external URL in a new window (open access), arXiv, opens an external URL in a new window

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, opens an external URL in a new window 54(9), pp. 245-252, 2021, arXiv, opens an external URL in a new window

J. Diwold, B. Kolar, M. Schöberl: A normal form for two-input forward-flat nonlinear discrete-time systems, International Journal of Systems Science, 2021, IJSS, opens an external URL in a new window (open access)  arXiv, opens an external URL in a new window

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, opens an external URL in a new window), pp. 970-977, 2021, arXiv, opens an external URL in a new window

B. Kolar, J. Diwold, M. Schöberl: Zur Theorie und Anwendung der Flachheit nichtlinearer zeitdiskreter Systeme, at - Automatisierungstechnik 69(7), pp. 574-584, at, opens an external URL in a new window (open access) 2021

B. Kolar, M. Schöberl, J. Diwold: Differential-Geometric Decomposition of Flat Nonlinear Discrete-Time Systems, Automatica 132, pp. 109828, opens an external URL in a new window, 2021,  arXiv, opens an external URL in a new window

J. Diwold, B. Kolar,  M. Schöberl, Discrete-time Flatness-based Control of a Gantry Crane, Control Engineering Practice, 119, 2022, (CEP, opens an external URL in a new window, open access) arXiv, opens an external URL in a new window

J. Diwold, B. Kolar,  M. Schöberl: A Trajectory-Based Approach to Discrete-Time Flatness, IEEE Control Systems Letters, vol 6, pp. 289-294, 2022, CSL,, opens an external URL in a new window arXiv, opens an external URL in a new window

C. Gstöttner, B. Kolar, M. Schöberl, Necessary and Sufficient Conditions for the Linearisability of Two-Input Systems by a Two-Dimensional Endogenous Dynamic Feedback, International Journal of Control, 2021 (IJC, opens an external URL in a new window), arXiv, opens an external URL in a new window

B. Kolar, J. Diwold, M. Schöberl: Necessary and Sufficient Conditions for Difference Flatness, Transacctions on Automatic Control (TAC, opens an external URL in a new window), accepted, 2022, arXiv, opens an external URL in a new window

F. Nicolau, C. Gstöttner, W. Respondek, Normal Forms for X-Flat Two-Input Control-Affine Systems in Dimension Five, 25th International Symposium on Mathematical Theory of Networks and Systems (MTNS), 2022

B. Kolar, J. Diwold, C. Gstöttner, M. Schöberl, On the Exact Linearization and Control of Flat Discrete-time Systems,  International Journal of Control (accepted), 2022, arXiv, opens an external URL in a new window

J. Diwold, B. Kolar, M. Schöberl. Flatness Analysis for the Sampled-data Model of a Single Mast Stacker Crane, 12th IFAC Symposium on Nonlinear Control Systems (accepted), 2023, arXiv, opens an external URL in a new window

J. Diwold, B. Kolar, M. Schöberl. Discrete-time Flatness-based Controller Design using an Implicit Euler-discretization, 12th IFAC Symposium on Nonlinear Control Systems (accepted), 2023, arXiv, opens an external URL in a new window

Submitted Publications:

none

Preprints:

C. Gstöttner, B. Kolar, M. Schöberl. Control of (x,u)-flat systems by quasi-static feedback of classical states, 2021, arXiv, opens an external URL in a new window

C. Gstöttner, B. Kolar, M. Schöberl, A Flat System Possessing no (x,u)-Flat Output, 2022, arXiv, opens an external URL in a new window

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)

On a geometric test for the flatness of nonlinear difference equations, SIAM Conference on Applied Algebraic Geometry (AG21), Virtual Conference, August 2021 (M. Schöberl)

Zeitdiskrete flachheitsbasierte Regelung eines Brückenkrans, GMA 1.40 Fachausschuss, Anif, September 2021 (J. Diwold)

A Trajectory-Based Approach to Discrete-Time Flatness, 60th IEEE Conference on Decision and Control, Virtual Conference, December 2021 (J. Diwold)

Normal Forms for X-Flat Two-Input Control-Affine Systems in Dimension Five,  25th International Symposium on Mathematical Theory of Networks and Systems (MTNS), 2022 (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.