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Learning Safety-Compatible Observers for Unknown Systems

Juho Bae, Daegyeong Roh, Han-Lim Choi

TL;DR

A delta ISS Lyapunov function that serves as the robustness certificate is learned that renders the observer safety-compatible and can be consumed by certificate-based safe controllers so that, when the controller tolerates bounded estimation error, the controller's certificate remains valid under output feedback.

Abstract

This paper presents a data-driven approach for jointly learning a robust full-state observer and its robustness certificate for systems with unknown dynamics. Leveraging incremental input-to-state stability (delta ISS) notions, we jointly learn a delta ISS Lyapunov function that serves as the robustness certificate and prove practical convergence of the estimation error under standard fidelity assumptions on the learned models. This renders the observer safety-compatible: they can be consumed by certificate-based safe controllers so that, when the controller tolerates bounded estimation error, the controller's certificate remains valid under output feedback. We further extend the approach to interconnected systems via the small-gain theorem, yielding a distributed observer design framework. We validate the approach on a variety of nonlinear systems.

Learning Safety-Compatible Observers for Unknown Systems

TL;DR

A delta ISS Lyapunov function that serves as the robustness certificate is learned that renders the observer safety-compatible and can be consumed by certificate-based safe controllers so that, when the controller tolerates bounded estimation error, the controller's certificate remains valid under output feedback.

Abstract

This paper presents a data-driven approach for jointly learning a robust full-state observer and its robustness certificate for systems with unknown dynamics. Leveraging incremental input-to-state stability (delta ISS) notions, we jointly learn a delta ISS Lyapunov function that serves as the robustness certificate and prove practical convergence of the estimation error under standard fidelity assumptions on the learned models. This renders the observer safety-compatible: they can be consumed by certificate-based safe controllers so that, when the controller tolerates bounded estimation error, the controller's certificate remains valid under output feedback. We further extend the approach to interconnected systems via the small-gain theorem, yielding a distributed observer design framework. We validate the approach on a variety of nonlinear systems.

Paper Structure

This paper contains 17 sections, 4 theorems, 41 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Incremental input--to--state stability
  4. Robust observers
  5. Hard--constrained neural networks
  6. Problem Statement
  7. Joint Learning of Dynamics, Observer, and $\delta$ISS Lyapunov Function
  8. Dynamics, output injection, and $\delta$ISS Lyapunov function
  9. Decreasing condition of $\delta$ISS Lyapunov function
  10. Loss function
  11. Performance guarantees
  12. Numerical Demonstration
  13. Uncontrolled Lorenz attractor
  14. Bicycle path--following with an open--loop input
  15. Interconnected FitzHugh-Nagumo system
  16. ...and 2 more sections

Key Result

Theorem 3

Consider the following interconnection of two subsystems Here the input to the first subsystem is seen as $(x_b,u_a)$, and $(x_a,u_b)$ to the second. Denote by $x_{a}\left(t,\xi_{a},x_{b},u_a\right)$ (resp. $x_{b}\left(t,\xi_{b},x_{a},u_b\right)$) the solution of the first (resp. second) subsystem with initial state $\xi_a$ (resp. $\xi_b$) and input $(x_b If there exists $\rho \in \mathcal{K}_{\i

Figures (4)

  • Figure 1: Two-step procedure. An unconstrained embedded injection $\tilde{L}$ is projected to satisfy $\nabla V \tilde{L} \leq \Lambda$, yielding a contracting pair $(L_1^*, L_2^*)$. A consistency loss then drives $L_1^*, L_2^* \rightarrow L^*$.
  • Figure 2: Observer results for Lorenz oscillator with noisy measurement $y + d_y$ with $d_y \sim \mathcal{N}(0,1)$.
  • Figure 3: Observer results for bicycle path--following dynamics with noises $d_y \sim \mathcal{N}(0, 0.05)$ and $d_u \sim \mathcal{N}(0, 0.05)$ entering $y$ and $u$, respectively.
  • Figure 4: Distributed observer results on the interconnected FHN system with noisy measurements $y_a+d_{y_a}$ and $y_b+d_{y_b}$ ($d_{y_a},d_{y_b}\!\sim\!\mathcal{N}(0,0.1)$) and a broadcast input $u+d_u$ with common noise $d_u\!\sim\!\mathcal{N}(0,0.5)$.

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Theorem 3: angeli2002
  • Definition 4: Robust full--state observer, sontag1997
  • Theorem 5: angeli2002
  • Proposition 6
  • proof
  • Theorem 7
  • proof
  • Remark 1
  • ...and 2 more