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A Structurally Flat Triangular Form for Three-Input Systems

Georg Hartl, Conrad Gstöttner, Markus Schöberl

Abstract

We present a broadly applicable structurally flat triangular form for x-flat control-affine systems with three inputs. Building on recent results for the derivative structure of flat outputs, we define the triangular form together with regularity conditions that guarantee structural flatness, and derive necessary and sufficient conditions for a system with a given x-flat output to be static feedback equivalent to this form. Further, we present sufficient conditions under which general x-flat three-input systems can be rendered static feedback equivalent to the proposed triangular form after a finite number of input prolongations.

A Structurally Flat Triangular Form for Three-Input Systems

Abstract

We present a broadly applicable structurally flat triangular form for x-flat control-affine systems with three inputs. Building on recent results for the derivative structure of flat outputs, we define the triangular form together with regularity conditions that guarantee structural flatness, and derive necessary and sufficient conditions for a system with a given x-flat output to be static feedback equivalent to this form. Further, we present sufficient conditions under which general x-flat three-input systems can be rendered static feedback equivalent to the proposed triangular form after a finite number of input prolongations.

Paper Structure

This paper contains 11 sections, 3 theorems, 57 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation and Terminology
  3. Differential Flatness
  4. Known Results
  5. Main Results
  6. A General Flat Triangular Form for Three-Input Systems
  7. Equivalence to the Triangular Form by Prolongations
  8. Examples
  9. Conclusion
  10. Proof of Theorem \ref{['thm:SFE_to_GTF3']}
  11. Proof of Corollary \ref{['cor:prolong']}

Key Result

Theorem 1

Consider a system of the form eq:f_xu_affine_three_inputs that admits an $x$-flat output~eq:x_flat_output_three_inputs, characterized by the relative degrees $K$ and the multi-index $R$. First, the flat-output components can always be rearranged such that Then, after a possible relabeling of the input components, let $\hat{u}^1 = \varphi^1_{[k^1]}(x,u)$ replace $u^1$. Under this input transformat

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Remark 1
  • Definition 3
  • Remark 2
  • Theorem 2
  • Corollary 1
  • Remark 3
  • Remark 4
  • ...and 2 more