Home Flachheitsanalyse nichtlinearer Systeme aus algebraischer Perspektive
Article
Licensed
Unlicensed Requires Authentication

Flachheitsanalyse nichtlinearer Systeme aus algebraischer Perspektive

  • Klemens Fritzsche

    Dr.-Ing. Klemens Fritzsche, B.Sc. ist wissenschaftlicher Mitarbeiter am Institut für Regelungs- und Steuerungstheorie an der Fakultät Elektrotechnik und Informationstechnik der TU Dresden. Arbeitsgebiete: Regler- und Beobachterentwurf für nichtlineare Systeme.

    EMAIL logo
Published/Copyright: September 4, 2025

Zusammenfassung

Für die Flachheitsanalyse werden verallgemeinerte Jacobi-Matrizen genutzt, deren Einträge nichtkommutative Ore-Polynome darstellen. Basierend auf einer Verallgemeinerung des Satzes von der Umkehrabbildung für dynamische Systeme wird ein bekanntes Flachheitskriterium neu eingeordnet und ein Werkzeug zur Verifikation vorgestellt. Durch Beispiele veranschaulichte Schwächen des Zugangs bezüglich der Flachheitsanalyse werden diskutiert, was die vielfach belegte Nützlichkeit zur Bestimmung flacher Ausgänge jedoch nicht trübt. Die dafür nötige Vorgehensweise wird beschrieben und auf die Berechnung flacher Eingänge übertragen.

Abstract

For the flatness analysis, generalized Jacobians are used, whose entries represent noncommutative Ore polynomials. Based on a generalization of the inverse function theorem for dynamical systems, a well-known flatness condition is reclassified, and a tool for its verification is presented. Supported by examples, this perspective reveals weaknesses in the condition which are discussed. Nevertheless, the approach remains useful for determining flat outputs. The procedure is described and transferred to the computation of flat inputs.


Korrespondenzautor: Klemens Fritzsche, Fakultät Elektrotechnik und Informationstechnik, Institut für Regelungs- und Steuerungstheorie, Technische Universität Dresden, 01062 Dresden, Germany, E-mail: 

Über den Autor / die Autorin

Klemens Fritzsche

Dr.-Ing. Klemens Fritzsche, B.Sc. ist wissenschaftlicher Mitarbeiter am Institut für Regelungs- und Steuerungstheorie an der Fakultät Elektrotechnik und Informationstechnik der TU Dresden. Arbeitsgebiete: Regler- und Beobachterentwurf für nichtlineare Systeme.

Danksagung

Der Autor dankt Prof. Klaus Röbenack für wertvolle Diskussionen und seine Unterstützung. Zudem gilt sein Dank den anonymen Gutachtern für die kritische Durchsicht und ihre hilfreichen Anmerkungen.

  1. Research ethics: Not applicable.

  2. Informed consent: Not applicable.

  3. Author contributions: The author has accepted responsibility for the entire content of this manuscript and approved its submission.

  4. Use of Large Language Models, AI and Machine Learning Tools: None declared.

  5. Conflict of interest: The author states no conflict of interest.

  6. Research funding: None declared.

  7. Data availability: Not applicable.

Literatur

[1] M. Fliess, J. Lévine, P. Martin, and P. Rouchon, “Flatness and defect of non-linear systems: introductory theory and examples,” Int. J. Control, vol. 61, no. 6, pp. 1327–1361, 1995. https://doi.org/10.1080/00207179508921959.Search in Google Scholar

[2] P. Martin, R. M. Murray, and P. Rouchon, “Flat systems, equivalence and feedback,” in Advances in the Control of Nonlinear Systems. Lecture Notes in Control and Information Sciences, vol. 264, A. Baños, F. Lamnabhi-Lagarrigue, and F. J. Montoya, Eds., London, Springer, 2001, pp. 5–32.10.1007/BFb0110377Search in Google Scholar

[3] R. Rothfuß, J. Rudolph, and M. Zeitz, “Flachheit: ein neuer Zugang zur Steuerung und Regelung nichtlinearer Systeme,” at – Automatisierungstechnik, vol. 45, no. 11, pp. 517–525, 1997. https://doi.org/10.1524/auto.1997.45.11.517.Search in Google Scholar

[4] M. Fliess, J. Lévine, P. Martin, and P. Rouchon, “Some open questions related to flat nonlinear systems,” in Open Problems in Mathematical Systems and Control Theory. Communications and Control Engineering, V. Blondel, Ed., et al.., London, Springer, 1999, pp. 99–103.10.1007/978-1-4471-0807-8_21Search in Google Scholar

[5] J. Rudolph, Beiträge zur flachheitsbasierten Folgeregelung linearer und nichtlinearer Systeme endlicher und unendlicher Dimension, Aachen, Shaker Verlag, 2003.Search in Google Scholar

[6] S. Waldherr and M. Zeitz, “Conditions for the existence of a flat input,” Int. J. Control, vol. 81, no. 3, pp. 439–443, 2008. https://doi.org/10.1080/00207170701561443.Search in Google Scholar

[7] S. Waldherr and M. Zeitz, “Flat inputs in the MIMO case,” IFAC Proc. Vol., vol. 43, no. 14, pp. 695–700, 2010. https://doi.org/10.3182/20100901-3-it-2016.00147.Search in Google Scholar

[8] K. Fritzsche, Y. Guo, and K. Röbenack, “Canonical form observers for non-integrable nonlinear single-output systems using flat inputs and dynamic compensators,” in Proc. 25th International Conference on System Theory, Control and Computing (ICSTCC), 2021, pp. 31–38.10.1109/ICSTCC52150.2021.9607105Search in Google Scholar

[9] K. Fritzsche, Y. Guo, and K. Röbenack, “Nonlinear control of non-observable non-flat MIMO state space systems using flat inputs,” in Proc. 23rd International Conference on System Theory, Control and Computing (ICSTCC), 2019, pp. 173–179.10.1109/ICSTCC.2019.8886157Search in Google Scholar

[10] K. Fritzsche, K. Röbenack, and Y. Guo, “Flat input based canonical form observers for non-integrable nonlinear systems,” Syst. Theory Control Comput. J., vol. 2, no. 1, pp. 13–21, 2022. https://doi.org/10.52846/stccj.2022.2.1.31.Search in Google Scholar

[11] J. O. A. Limaverde Filho, E. L. F. Fortaleza, and M. C. M. M. de Campos, “A derivative-free nonlinear Kalman filtering approach using flat inputs,” Int. J. Control, vol. 95, no. 11, pp. 2900–2910, 2022. https://doi.org/10.1080/00207179.2021.1941264.Search in Google Scholar

[12] F. Nicolau, W. Respondek, and J.-P. Barbot, “Flat inputs: theory and applications,” SIAM J. Control Optim., vol. 58, no. 6, pp. 3293–3321, 2020. https://doi.org/10.1137/19m127776x.Search in Google Scholar

[13] R. Schenkendorf and M. Mangold, “Parameter identification for ordinary and delay differential equations by using flat inputs,” Theor. Found. Chem. Eng., vol. 48, no. 5, pp. 594–607, 2014. https://doi.org/10.1134/s0040579514050224.Search in Google Scholar

[14] J.-F. Stumper, F. Svaricek, and R. Kennel, “Trajectory tracking control with flat inputs and a dynamic compensator,” in Proc. 2009 European Control Conference (ECC), Budapest, Ungarn, 23.–26. August 2009, 2009, pp. 248–253.10.23919/ECC.2009.7074412Search in Google Scholar

[15] K. Fritzsche, Algebraische Flachheitsanalyse nichtlinearer Systeme, Aachen, Shaker Verlag, 2024.Search in Google Scholar

[16] C. Gstöttner, B. Kolar, and M. Schöberl, “Necessary and sufficient conditions for the linearisability of two-input systems by a two-dimensional endogenous dynamic feedback,” Int. J. Control, vol. 96, no. 3, pp. 800–821, 2023. https://doi.org/10.1080/00207179.2021.2015542.Search in Google Scholar

[17] F. Nicolau and W. Respondek, “Flatness of multi-input control-affine systems linearizable via one-fold prolongation,” SIAM J. Control Optim., vol. 55, no. 5, pp. 3171–3203, 2017. https://doi.org/10.1137/140999463.Search in Google Scholar

[18] F. Nicolau and W. Respondek, “Two-input control-affine systems linearizable via one-fold prolongation and their flatness,” Eur. J. Control, vol. 28, pp. 20–37, 2016. https://doi.org/10.1016/j.ejcon.2015.11.001.Search in Google Scholar

[19] M. Franke, Über die Konstruktion flacher Ausgänge für nichtlineare Systeme und zur Polzuweisung durch statische Ausgangsrückführungen, Berlin, Logos Verlag, 2013.Search in Google Scholar

[20] M. Franke and K. Röbenack, “On the computation of flat outputs for nonlinear control systems,” in Proc. 2013 European Control Conference (ECC), Zürich, Schweiz, 2013, pp. 167–172.10.23919/ECC.2013.6669771Search in Google Scholar

[21] K. Fritzsche, M. Franke, C. Knoll, and K. Röbenack, “Zur systematischen Bestimmung flacher Ausgänge nichtlinearer Mehrgrößensysteme,” at – Automatisierungstechnik, vol. 64, no. 12, pp. 948–960, 2016. https://doi.org/10.1515/auto-2016-0079.Search in Google Scholar

[22] J. Lévine, Analysis and Control of Nonlinear Systems, Berlin, Heidelberg, Springer, 2009.10.1007/978-3-642-00839-9_13Search in Google Scholar

[23] J. Lévine, “On necessary and sufficient conditions for differential flatness,” Appl. Algebra Eng. Commun. Comput., vol. 22, no. 1, pp. 47–90, 2011. https://doi.org/10.1007/s00200-010-0137-x.Search in Google Scholar

[24] G. Verhoeven and F. Antritter, “Ein Werkzeug zur automatisierten Flachheitsanalyse nichtlinearer Systeme,” at – Automatisierungstechnik, vol. 61, no. 1, pp. 60–71, 2013. https://doi.org/10.1524/auto.2013.0003.Search in Google Scholar

[25] M. Fliess, J. Lévine, P. Martin, and P. Rouchon, “A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems,” IEEE Trans. Autom. Control, vol. 44, no. 5, pp. 992–937, 1999.10.1109/9.763209Search in Google Scholar

[26] J. Descusse and C. H. Moog, “Decoupling with dynamic compensation for strong invertible affine non-linear systems,” Int. J. Control, vol. 42, no. 6, pp. 1387–1398, 1985. https://doi.org/10.1080/00207178508933432.Search in Google Scholar

[27] S. Singh, “Decoupling of invertible nonlinear systems with state feedback and precompensation,” IEEE Trans. Autom. Control, vol. 25, no. 6, pp. 1237–1239, 1980. https://doi.org/10.1109/tac.1980.1102546.Search in Google Scholar

[28] E. Delaleau and J. Rudolph, “Control of flat systems by quasi-static feedback of generalized states,” Int. J. Control, vol. 71, no. 5, pp. 745–765, 1998. https://doi.org/10.1080/002071798221551.Search in Google Scholar

[29] A. J. Krener and A. Isidori, “Linearization by output injection and nonlinear observers,” Syst. Control Lett., vol. 3, no. 1, pp. 47–52, 1983. https://doi.org/10.1016/0167-6911(83)90037-3.Search in Google Scholar

[30] R. Marino, “Adaptive observers for single output nonlinear systems,” IEEE Trans. Autom. Control, vol. 35, no. 9, pp. 1054–1058, 1990. https://doi.org/10.1109/9.58536.Search in Google Scholar

[31] K. Fritzsche and K. Röbenack, “On a generalized flat input definition and physical realizability,” IFAC-PapersOnLine, vol. 53, no. 2, pp. 5994–5999, 2020. https://doi.org/10.1016/j.ifacol.2020.12.1661.Search in Google Scholar

[32] G. Conte, C. H. Moog, and A. M. Perdon, Algebraic Methods for Nonlinear Control Systems – Theory and Applications, London, Springer, 2007.10.1007/978-1-84628-595-0Search in Google Scholar

[33] A. Ilchmann, “Algebraic theory of time-varying linear systems: a survey,” IFAC Proc. Vol., vol. 38, no. 1, pp. 1185–1191, 2005. https://doi.org/10.3182/20050703-6-cz-1902.00199.Search in Google Scholar

[34] Y. Zheng, J. C. Willems, and C. Zhang, “A polynomial approach to nonlinear system controllability,” IEEE Trans. Autom. Control, vol. 46, no. 11, pp. 1782–1788, 2001. https://doi.org/10.1109/9.964691.Search in Google Scholar

[35] P. M. Cohn, Free Rings and their Relations, London, Academic Press, 1985.Search in Google Scholar

[36] J. Middeke, A Computational View on Normal Forms of Matrices of Ore Polynomials, Ph.D. thesis, Österreich, Johannes Kepler Universität Linz, 2011.Search in Google Scholar

[37] B. Beckermann, H. Cheng, and G. Labahn, “Fraction-free row reduction of matrices of Ore polynomials,” J. Symb. Comput., vol. 41, no. 5, pp. 513–543, 2006. https://doi.org/10.1016/j.jsc.2005.10.002.Search in Google Scholar

[38] K. Fritzsche and K. Röbenack, “Unimodular completions and orthogonal complements of matrices over univariate Ore extensions,” SIAM J. Matrix Anal. Appl., vol. 44, no. 1, pp. 128–155, 2023. https://doi.org/10.1137/22m1477027.Search in Google Scholar

[39] J. Lévine and D. V. Nguyen, “Flat output characterization for linear systems using polynomial matrices,” Syst. Control Lett., vol. 48, no. 1, pp. 69–75, 2003. https://doi.org/10.1016/s0167-6911(02)00257-8.Search in Google Scholar

[40] K. Reinschke, Lineare Regelungs-und Steuerungstheorie, 2nd ed. Berlin, Heidelberg, Springer, 2014.10.1007/978-3-642-40960-8Search in Google Scholar

[41] K. Sato, “Algebraic controllability and observability of nonlinear differential algebraic systems with geometric index one,” SICE J. Control Meas. Syst. Integr., vol. 7, no. 5, pp. 283–290, 2014. https://doi.org/10.9746/jcmsi.7.283.Search in Google Scholar

[42] M. van Nieuwstadt, M. Rathinam, and R. M. Murray, “Differential flatness and absolute equivalence of nonlinear control systems,” SIAM J. Control Optim., vol. 36, no. 4, pp. 1225–1239, 1998. https://doi.org/10.1137/s0363012995274027.Search in Google Scholar

[43] B. Charlet, J. Lévine, and R. Marino, “On dynamic feedback linearization,” Syst. Control Lett., vol. 13, pp. 143–151, 1989. https://doi.org/10.1016/0167-6911(89)90031-5.Search in Google Scholar

[44] K. Fritzsche and K. Röbenack, “On the computation of differentially flat inputs,” in Proc. 22nd International Conference on System Theory, Control and Computing (ICSTCC), 2018, pp. 12–19.10.1109/ICSTCC.2018.8540663Search in Google Scholar

[45] S. A. Abramov, H. Q. Le, and Z. Li, “Univariate Ore polynomial rings in computer algebra,” J. Math. Sci., vol. 131, no. 5, pp. 5885–5903, 2005. https://doi.org/10.1007/s10958-005-0449-8.Search in Google Scholar

[46] M. Fliess, J. Lévine, and P. Rouchon, “Index of an implicit time-varying linear differential equation: a noncommutative linear algebraic approach,” Linear Algebra Appl., vol. 186, pp. 59–71, 1993. https://doi.org/10.1016/0024-3795(93)90285-v.Search in Google Scholar

[47] K. Fritzsche and K. Röbenack, “On hyper-regularity and unimodularity of Ore polynomial matrices,” Int. J. Appl. Math. Comput. Sci., vol. 28, no. 3, pp. 583–594, 2018. https://doi.org/10.2478/amcs-2018-0045.Search in Google Scholar

Erhalten: 2024-11-11
Angenommen: 2025-05-26
Online erschienen: 2025-09-04
Erschienen im Druck: 2025-09-25

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 12.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/auto-2024-0156/pdf
Scroll to top button