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Trajectory tracking model-following control using Lyapunov redesign with output time-derivatives to compensate unmatched uncertainties

Niclas Tietze, Kai Wulff, Johann Reger

Abstract

We study trajectory tracking for flat nonlinear systems with unmatched uncertainties using the model-following control (MFC) architecture. We apply state feedback linearisation control for the process and propose a simplified implementation of the model control loop which results in a simple model in Brunovsky-form that represents the nominal feedback linearised dynamics of the nonlinear process. To compensate possibly unmatched model uncertainties, we employ Lyapunov redesign with numeric derivatives of the output. It turns out that for a special initialisation of the model, the MFC reduces to a single-loop control design. We illustrate our results by a numerical example.

Trajectory tracking model-following control using Lyapunov redesign with output time-derivatives to compensate unmatched uncertainties

Abstract

We study trajectory tracking for flat nonlinear systems with unmatched uncertainties using the model-following control (MFC) architecture. We apply state feedback linearisation control for the process and propose a simplified implementation of the model control loop which results in a simple model in Brunovsky-form that represents the nominal feedback linearised dynamics of the nonlinear process. To compensate possibly unmatched model uncertainties, we employ Lyapunov redesign with numeric derivatives of the output. It turns out that for a special initialisation of the model, the MFC reduces to a single-loop control design. We illustrate our results by a numerical example.
Paper Structure (16 sections, 3 theorems, 57 equations, 3 figures)

This paper contains 16 sections, 3 theorems, 57 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem definition
  3. Unmatched uncertainties
  4. Unmatched model uncertainties within the model-following control architecture
  5. State feedback linearisation of the MCL and the PCL
  6. Trajectory tracking
  7. Main result
  8. Control design
  9. Efficient implementation of the MCL
  10. Stability and asymptotic tracking
  11. Special case: matched model uncertainty
  12. Special case: initialising the model on the reference
  13. Illustrative example
  14. Control design
  15. Simulation of the closed loop
  16. ...and 1 more sections

Key Result

Theorem 1

The uncertainty $\phi_\mathrm{u}\equiv 0$ in eq:pertrubation_transformed_unmatched if and only if $\Delta_\mathrm{u}(x) \equiv0$ in eq:perturbation_decomposition.

Figures (3)

  • Figure 1: Blockdiagram of the model-following control (MFC) architecture with nonlinear process model.
  • Figure 2: Blockdiagram of the model-following control (MFC) architecture with model of the linearised process dynamics in Brunovský-form.
  • Figure 3: Simulation of the closed loop with MFC \ref{['eq:example_u']} in blue and the MFC \ref{['eq:u_pcl_matched_perturbation']} in orange. Order from top to bottom: tracking error $y - y_\mathrm{d}$, Lyapunov function $\tilde{V}(\tilde{\xi}_\mathrm{n})$, auxiliary variable $\tilde{w}$, and control effort

Theorems & Definitions (5)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 3