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Exponential stabilizability and observability at the target imply semiglobal exponential stabilizability by templated output feedback

Vincent Andrieu, Lucas Brivadis, Jean-Paul Gauthier, Ludovic Sacchelli, Ulysse Serres

Abstract

For nonlinear analytic control systems, we introduce a new paradigm for dynamic output feedback stabilization. We propose to periodically sample the usual observer based control law, and to reshape it so that it coincides with a ''control template'' on each time period. By choosing a control template making the system observable, we prove that this method allows to bypass the uniform observability assumption that is used in most nonlinear separation principles. We prove the genericity of control templates by adapting a universality theorem of Sussmann.

Exponential stabilizability and observability at the target imply semiglobal exponential stabilizability by templated output feedback

Abstract

For nonlinear analytic control systems, we introduce a new paradigm for dynamic output feedback stabilization. We propose to periodically sample the usual observer based control law, and to reshape it so that it coincides with a ''control template'' on each time period. By choosing a control template making the system observable, we prove that this method allows to bypass the uniform observability assumption that is used in most nonlinear separation principles. We prove the genericity of control templates by adapting a universality theorem of Sussmann.
Paper Structure (14 sections, 26 theorems, 90 equations, 1 figure)

This paper contains 14 sections, 26 theorems, 90 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Templated output feedback
  3. Problem statement and main assumptions
  4. Hybrid dynamic output feedback design
  5. Main results
  6. Output feedback stabilization theorem
  7. Well-posedness
  8. High-gain observer with control templates
  9. Stabilization by means of a templated state feedback
  10. Stabilization by means of templated output feedback
  11. Universality theorem
  12. Conclusion and perspective
  13. Additional technical lemmas
  14. Proofs of Lemmas \ref{['lem:analytic']}, \ref{['lem:hyp12']} and \ref{['Ass_HG']}

Key Result

Lemma 2

If system syst is observable over a compact set $\mathcal{K}_x\subset\mathbb{R}^n$ in some positive time $\tau$ for some input $u\in C^\omega([0, T], \mathbb{R}^m)$, then there exists $k\in\mathbb{N}$ such that syst is also differentially observable over $\mathcal{K}_x$ of order $k$.

Figures (1)

  • Figure 1: Trajectory of the input $u=\mu\mathsf{R} v^*$ applied to the system when using the templated output feedback strategy of \ref{['eq:closed-loop']}. To lighten the notations, here we write $\hat{x}:= \phi(z, \Delta, \mu, \mathsf{R})$. The illustration corresponds to the case of a one-dimensional input. After each jump, $u(\tau_i)=\lambda(\hat{x}(\tau_i))$. Then, over each time-interval, the input follows the shape of the control template $v^*$. Note that the input is also rescaled over each interval by $|\lambda(\hat{x}(\tau_i))|$, in order to guarantee that it remains close to $\lambda(\hat{x})$ and $u\to0$ as $\hat{x}\to0$.

Theorems & Definitions (53)

  • Definition 1: Observability) (see, e.g., MR1862985
  • Lemma 2
  • Lemma 3
  • Definition 4: Control template
  • Lemma 5
  • Theorem 6: Output feedback stabilization theorem
  • Theorem 7: Universality theorem
  • Lemma 8: bernard2019observer
  • Proposition 9
  • Lemma 10
  • ...and 43 more