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A Backstepping-KKL observer for a cascade of a nonlinear ODE with a heat equation

Adam Braun, Lucas Brivadis, Jean Auriol

TL;DR

This work addresses state estimation for a cascade system comprising a nonlinear ODE and a one-dimensional heat equation, with the ODE’s output shaping a boundary condition and a boundary measurement taken at the opposite end. The authors fuse an infinite-dimensional KKL observer for the ODE with a backstepping PDE observer by constructing a Backstepping-KKL transformation $T=T_0+\\mathbb{T}$, where $T_0$ propagates along ODE trajectories and cancels the boundary term, while $\\mathbb{T}$ handles the PDE. They prove the existence of $T_0$, establish injectivity of the full transformation on the relevant state space under a differential observability assumption, and show convergence of the observer to the true state provided a left-inverse of $T$ exists; the convergence analysis discusses exponential rates for the transformed error and the practical impact of the inverse $T^{-1}$ on speed. Numerical simulations on two nonlinear examples illustrate effective state reconstruction and highlight trade-offs in choosing the PDE/ODE contraction parameter $\\gamma$. The work provides a first step toward applying KKL observers to nonlinear infinite-dimensional interconnections and suggests extensions to broader classes of linear PDEs in cascade with nonlinear ODEs.

Abstract

We propose an observer design for a cascaded system composed of an arbitrary nonlinear ordinary differential equation (ODE) with a 1D heat equation. The nonlinear output of the ODE imposes a boundary condition on one side of the heat equation, while the measured output is on the other side. The observer design combines an infinitedimensional Kazantzis-Kravaris/Luenberger (KKL) observer for the ODE with a backstepping observer for the heat equation. This construction is the first extension of the KKL methodology to infinite-dimensional systems. We establish the convergence of the observer under a differential observability condition on the ODE. The effectiveness of the proposed approach is illustrated in numerical simulations.

A Backstepping-KKL observer for a cascade of a nonlinear ODE with a heat equation

TL;DR

This work addresses state estimation for a cascade system comprising a nonlinear ODE and a one-dimensional heat equation, with the ODE’s output shaping a boundary condition and a boundary measurement taken at the opposite end. The authors fuse an infinite-dimensional KKL observer for the ODE with a backstepping PDE observer by constructing a Backstepping-KKL transformation , where propagates along ODE trajectories and cancels the boundary term, while handles the PDE. They prove the existence of , establish injectivity of the full transformation on the relevant state space under a differential observability assumption, and show convergence of the observer to the true state provided a left-inverse of exists; the convergence analysis discusses exponential rates for the transformed error and the practical impact of the inverse on speed. Numerical simulations on two nonlinear examples illustrate effective state reconstruction and highlight trade-offs in choosing the PDE/ODE contraction parameter . The work provides a first step toward applying KKL observers to nonlinear infinite-dimensional interconnections and suggests extensions to broader classes of linear PDEs in cascade with nonlinear ODEs.

Abstract

We propose an observer design for a cascaded system composed of an arbitrary nonlinear ordinary differential equation (ODE) with a 1D heat equation. The nonlinear output of the ODE imposes a boundary condition on one side of the heat equation, while the measured output is on the other side. The observer design combines an infinitedimensional Kazantzis-Kravaris/Luenberger (KKL) observer for the ODE with a backstepping observer for the heat equation. This construction is the first extension of the KKL methodology to infinite-dimensional systems. We establish the convergence of the observer under a differential observability condition on the ODE. The effectiveness of the proposed approach is illustrated in numerical simulations.

Paper Structure

This paper contains 19 sections, 6 theorems, 81 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation:
  3. Problem statement and well-posedness
  4. Reminder of the KKL in finite dimension and conceptual similarities with the backstepping
  5. Observer design using a formal approach
  6. Existence of T0
  7. Injectivity of the Backstepping-KKL transformation and convergence of the observer
  8. Injectivity of T0
  9. Injectivity of T
  10. Observer convergence
  11. Numerical simulations
  12. Example 1: parameter estimation
  13. Example 2: harmonic oscillator with polynomial output
  14. Conclusion
  15. Proof of Lemma \ref{['pb stable heat bien posé']}
  16. ...and 4 more sections

Key Result

Lemma 5.1

For every $u \in C^{0}((-\infty, 0])$ such that $t \to e^{\gamma t}u(t)$ is bounded on $(-\infty, 0]$, the problem stable heat admits a unique solution $w_u \in C^1((-\infty, 0], C^2([0,1]))$ such that there exists $C>0$ such that for all $t\leqslant0$ Moreover, for all $t\leqslant0$, $w_u(t,\cdot)$ is an element of $\mathcal{A}$, where $\mathcal{A}$ is the space of analytical functions $v$ in $L^

Figures (4)

  • Figure 1: Representation of the cascaded system
  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (18)

  • Remark 2.3
  • Remark 4.1
  • Remark 4.2
  • Lemma 5.1
  • Definition 5.2
  • Proposition 5.3
  • proof
  • Theorem 6.1
  • proof
  • Lemma 6.2
  • ...and 8 more