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

Conrad Gstöttner, Bernd Kolar, Markus Schöberl

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.

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

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 -flat systems.

Paper Structure

This paper contains 14 sections, 6 theorems, 73 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Geometric framework, flatness and quasi-static feedback of classical states
  5. Exact feedback linearization of flat systems
  6. Exact feedback linearization of $\boldsymbol{(x,u)}$-flat systems
  7. Examples
  8. Academic example
  9. 3D gantry crane
  10. Tracking control design for $\boldsymbol{(x,u)}$-flat systems
  11. Examples continued
  12. Academic example
  13. 3D gantry crane
  14. Conclusions

Key Result

Lemma 2.3

Consider a set of smooth functions $g^1,\dots,g^k$ as well as another smooth function $h$ which are all defined on the same manifold. The following conditions are equivalent: Furthermore, if the differentials $\mathrm{d} g^1,\dots,\mathrm{d} g^k$ are linearly independent, then the function $\psi$ is unique.

Figures (1)

  • Figure 1: Schematic diagram of a 3D gantry crane.

Theorems & Definitions (19)

  • Remark 1.1
  • Example 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • ...and 9 more