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Flatness-based control of a Timoshenko beam

Simon Schmidt, Nicole Gehring, Abdurrahman Irscheid

Abstract

The paper presents an approach to flatness-based control design for hyperbolic multi-input systems, building upon the hyperbolic controller form (HCF). The transformation into HCF yields a simplified system representation that considerably facilitates the design of state feedback controllers for trajectory tracking. The proposed concept is demonstrated for a Timoshenko beam and validated through numerical simulations, demonstrating trajectory tracking and closed-loop stability.

Flatness-based control of a Timoshenko beam

Abstract

The paper presents an approach to flatness-based control design for hyperbolic multi-input systems, building upon the hyperbolic controller form (HCF). The transformation into HCF yields a simplified system representation that considerably facilitates the design of state feedback controllers for trajectory tracking. The proposed concept is demonstrated for a Timoshenko beam and validated through numerical simulations, demonstrating trajectory tracking and closed-loop stability.

Paper Structure

This paper contains 9 sections, 2 theorems, 18 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Flatness-based parametrization
  4. Preliminary transformations
  5. Derivation of the parametrization
  6. Hyperbolic Controller Form (HCF)
  7. State feedback tracking controller
  8. Simulation results
  9. Concluding remarks

Key Result

Lemma 1

There exists an invertible transformation that maps the state $\boldsymbol{x}(\cdot,t)$ of eq_sys_x into the state $\boldsymbol{\eta}(\cdot,t)$ of the HCF in eq_HCF.

Figures (4)

  • Figure 3: Schematic of the HCF-based control design.
  • Figure 4: A Timoshenko beam clamped at $z=0$ and actuated at $z=1$.
  • Figure 5: Convergence of the flat output $\boldsymbol{y}(t)$ to its reference $\boldsymbol{y}_{\mathrm{r}}(t)$.
  • Figure 6: Beam displacement $w(z,t)$ during the transition between two stationary solutions.

Theorems & Definitions (7)

  • Remark 1
  • Remark 2
  • Lemma 1: HCF
  • proof
  • Remark 3
  • Theorem 1
  • proof