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Sampled-data funnel control and its use for safe continual learning

Lukas Lanza, Dario Dennstädt, Karl Worthmann, Philipp Schmitz, Gökçen Devlet Şen, Stephan Trenn, Manuel Schaller

TL;DR

A novel sampled-data output-feedback controller for nonlinear systems of arbitrary relative degree that ensures reference tracking within prescribed error bounds and capability to serve as a safety filter for various learning-based controller designs, enabling the use of learning techniques in safety-critical applications is proposed.

Abstract

We propose a novel sampled-data output-feedback controller for nonlinear systems of arbitrary relative degree that ensures reference tracking within prescribed error bounds. We provide explicit bounds on the maximum input signal and the required uniform sampling time. A key strength of this approach is its capability to serve as a safety filter for various learning-based controller designs, enabling the use of learning techniques in safety-critical applications. We illustrate its versatility by integrating it with two different controllers: a reinforcement learning controller and a non-parametric predictive controller based on Willems et al.'s fundamental lemma. Numerical simulations illustrate effectiveness of the combined controller design.

Sampled-data funnel control and its use for safe continual learning

TL;DR

A novel sampled-data output-feedback controller for nonlinear systems of arbitrary relative degree that ensures reference tracking within prescribed error bounds and capability to serve as a safety filter for various learning-based controller designs, enabling the use of learning techniques in safety-critical applications is proposed.

Abstract

We propose a novel sampled-data output-feedback controller for nonlinear systems of arbitrary relative degree that ensures reference tracking within prescribed error bounds. We provide explicit bounds on the maximum input signal and the required uniform sampling time. A key strength of this approach is its capability to serve as a safety filter for various learning-based controller designs, enabling the use of learning techniques in safety-critical applications. We illustrate its versatility by integrating it with two different controllers: a reinforcement learning controller and a non-parametric predictive controller based on Willems et al.'s fundamental lemma. Numerical simulations illustrate effectiveness of the combined controller design.
Paper Structure (12 sections, 7 theorems, 46 equations, 12 figures, 2 algorithms)

This paper contains 12 sections, 7 theorems, 46 equations, 12 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Control objective, system class, and preliminary results
  3. System class
  4. Auxiliary results
  5. Sampled-data feedback controller
  6. Numerical example: pure ZoH feedback
  7. Safeguarded data-based control
  8. Data-driven MPC using Willems' fundamental lemma
  9. Reinforcement Learning: Q-table control
  10. Nonlinear example: Van der Pol oscillator
  11. Conclusion and future work
  12. Proofs of auxiliary results

Key Result

Lemma 2.1

Let $y_{\mathop{\mathrm{ref}}\limits}\in W^{r,\infty}(\mathds{R}_{\geq0},\mathds{R}^m)$, $\varphi \in \mathcal{G}$, and $y^0\in\mathcal{C}^{r-1}([-\sigma,0],\mathds{R}^m)$ with $\chi(y^0)\in\mathcal{D}_0^r$ be given. Then there exist constants $\varepsilon_k,\mu_k>0$ defined in eq:ve_mu_gam such tha for all $t\in [0,\delta)$ and for all $k=1,\ldots,r-1$.

Figures (12)

  • Figure 1: Error evolution in a funnel $\mathcal{F}_{\varphi}$ with boundary $1/\varphi(t)$; the figure is based on BergLe18a, edited for present purpose.
  • Figure 2: Mass-on-car system. The figure is based on SeifBlaj13BergIlch21.
  • Figure 3: Outputs, reference, and error tolerance.
  • Figure 4: Controls.
  • Figure 5: Schematic structure of the combined controller.
  • ...and 7 more figures

Theorems & Definitions (18)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 3.1
  • Remark 3.1
  • Remark 3.2
  • Theorem 5.1
  • proof
  • Remark 5.1
  • ...and 8 more