Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 071, 27 pages      arXiv:2110.12995      https://doi.org/10.3842/SIGMA.2024.071

Tracking Control for $(x,u)$-Flat Systems by Quasi-Static Feedback of Classical States

Conrad Gstöttner, Bernd Kolar and Markus Schöberl
Institute of Automatic Control and Control Systems Technology, Johannes Kepler University Linz, Austria

Received November 03, 2023, in final form July 22, 2024; Published online July 31, 2024

Abstract
It is well known that for flat systems the tracking control problem can be solved by utilizing a linearizing quasi-static feedback of generalized states. If measurements (or estimates) of a so-called generalized Brunovský state are available, a linear, decoupled and asymptotically stable tracking error dynamics can be achieved. However, from a practical point of view, it is often desirable to achieve the same tracking error dynamics by feedback of a classical state instead of a generalized one. This is due to the fact that the components of a classical state typically correspond to measurable physical quantities, whereas a generalized Brunovský state often contains higher order time derivatives of the (fictitious) flat output which are not directly accessible by measurements. In this paper, a systematic solution for the tracking control problem based on quasi-static feedback and measurements of classical states only is derived for the subclass of $(x,u)$-flat systems.

Key words: flatness; tracking control; nonlinear control.

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References

  1. Chetverikov V.N., Controllability of flat systems, Differ. Equ. 43 (2007), 1558-1568.
  2. Delaleau E., Fliess M., An algebraic interpretation of the structure algorithm with an application to feedback decoupling, IFAC Proc. Vol. 25 (1992), 179-184.
  3. Delaleau E., Fliess M., Algorithme de structure, filtrations et découplage, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), 101-106.
  4. Delaleau E., Pereira da Silva P.S., Filtrations in feedback synthesis. I. Systems and feedbacks, Forum Math. 10 (1998), 147-174.
  5. Delaleau E., Pereira da Silva P.S., Filtrations in feedback synthesis. II. Input-output decoupling and disturbance decoupling, Forum Math. 10 (1998), 259-276.
  6. Delaleau E., Rudolph J., Decoupling and linearization by quasi-static feedback of generalized states, in Proceedings of 3rd European Control Conference (Rome, 1995), 1995, 1069-1074.
  7. Delaleau E., Rudolph J., Control of flat systems by quasi-static feedback of generalized states, Internat. J. Control 71 (1998), 745-765.
  8. Di Benedetto M.D., Grizzle J.W., Moog C.H., Rank invariants of nonlinear systems, SIAM J. Control Optim. 27 (1989), 658-672.
  9. Fliess M., Lévine J., Martin P., Rouchon P., On differentially flat nonlinear systems, IFAC Proc. Vol. 25 (1992), 408-412.
  10. Fliess M., Lévine J., Martin P., Rouchon P., Sur les systèmes non linéaires différentiellement plats, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), 619-624.
  11. Fliess M., Lévine J., Martin P., Rouchon P., Flatness and defect of non-linear systems: introductory theory and examples, Internat. J. Control 61 (1995), 1327-1361.
  12. Fliess M., Lévine J., Martin P., Rouchon P., A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems, IEEE Trans. Automat. Control 44 (1999), 922-937.
  13. Grizzle J.W., Di Benedetto M.D., Moog C.H., Computing the differential output rank of a nonlinear system, in Proceedings of 26th IEEE Conference on Decision and Control, IEEE, 1987, 142-145.
  14. Gstöttner C., Kolar B., Schöberl M., On the linearization of flat two-input systems by prolongations and applications to control design, IFAC-PapersOnLine 53 (2020), 5479-5486, arXiv:1910.07215.
  15. Gstöttner C., Kolar B., Schöberl M., A structurally flat triangular form based on the extended chained form, Internat. J. Control 95 (2022), 1144-1163, arXiv:2007.09935.
  16. Gstöttner C., Kolar B., Schöberl M., A flat system possessing no $(x, u)$-flat output, IEEE Control Syst. Lett. 7 (2023), 1033-1038, arXiv:2206.03845.
  17. Gstöttner C., Kolar B., Schöberl M., Necessary and sufficient conditions for the linearisability of two-input systems by a two-dimensional endogenous dynamic feedback, Internat. J. Control 96 (2023), 800-821, arXiv:2106.14722.
  18. Hagenmeyer V., Delaleau E., Exact feedforward linearization based on differential flatness, Internat. J. Control 76 (2003), 537-556.
  19. Hagenmeyer V., Delaleau E., Robustness analysis of exact feedforward linearization based on differential flatness, Automatica J. IFAC 39 (2003), 1941-1946.
  20. Isidori A., Nonlinear control systems, 3rd ed., Comm. Control Engrg. Ser.,Springer, Berlin, 1995.
  21. Kolar B., Contributions to the differential geometric analysis and control of flat systems, Ph.D. Thesis, Johannes Kepler Universität Linz, Aachen, 2016.
  22. Kolar B., Rams H., Schlacher K., Time-optimal flatness based control of a gantry crane, Control Eng. Pract. 60 (2017), 18-27.
  23. Kolar B., Schöberl M., Schlacher K., Properties of flat systems with regard to the parameterization of the system variables by the flat output, IFAC-PapersOnLine 49 (2016), 814-819.
  24. Kugi A., Kiefer T., Nichtlineare Trajektorienfolgeregelung für einen Laborhelikopter, e+i Elektrotechnik und Informationstechnik 122 (2005), 300-307.
  25. Martin P., Contribution à l'étude des systèmes différentiellement plats, Ph.D. Thesis, École Nationale Supérieure des Mines de Paris, 1992.
  26. Martin P., An intrinsic sufficient condition for regular decoupling, Systems Control Lett. 20 (1993), 383-391.
  27. Martin P., Murray R., Rouchon P., Flat systems, in Plenary Lectures and Mini-Courses, Proceedings of 4th European Control Conference (ECC), 1997, 211-264.
  28. Moog C.H., Nonlinear decoupling and structure at infinity, Math. Control Signals Systems 1 (1988), 257-268.
  29. Nicolau F., Respondek W., Flatness of multi-input control-affine systems linearizable via one-fold prolongation, SIAM J. Control Optim. 55 (2017), 3171-3203.
  30. Nicolau F., Respondek W., Normal forms for multi-input flat systems of minimal differential weight, Internat. J. Robust Nonlinear Control 29 (2019), 3139-3162.
  31. Nijmeijer H., Right-invertibility for a class of nonlinear control systems: a geometric approach, Systems Control Lett. 7 (1986), 125-132.
  32. Nijmeijer H., Respondek W., Dynamic input-output decoupling of nonlinear control systems, IEEE Trans. Automat. Control 33 (1988), 1065-1070.
  33. Nijmeijer H., Schumacher J.M., Zeros at infinity for affine nonlinear control systems, IEEE Trans. Automat. Control 30 (1985), 566-573.
  34. Respondek W., Right and left invertibility of nonlinear control systems, in Nonlinear Controllability and Optimal Control, Monogr. Textbooks Pure Appl. Math., Vol. 133, Dekker, New York, 1990, 133-176.
  35. Rudolph J., Well-formed dynamics under quasi-static state feedback, in Geometry in Nonlinear Control and Differential Inclusions (Warsaw, 1993), Banach Center Publ., Vol. 32, Polish Academy of Sciences, Institute of Mathematics, Warsaw, 1995, 349-360.
  36. Rudolph J., Flatness-based control: An introduction, Shaker Verlag, Düren, 2021.
  37. Rudolph J., Delaleau E., Some examples and remarks on quasi-static feedback of generalized states, Automatica J. IFAC 34 (1998), 993-999.
  38. Spong M., Vidyasagar M., Robot dynamics and control, Wiley, 1989.

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