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

2025/10/06 by Adam Braun, Lucas Brivadis, Braun, Adam +3
Engineering · #Advanced Control Systems Optimization #Extremum Seeking Control Systems #FOS: Mathematics #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2510.04941

openalex publication_date 2025/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Backstepping for Partial Differential Equations:A Survey
• Recognition Models to Learn Dynamics from Partial Observations with Neural ODEs
• Luenberger observers for discrete-time nonlinear systems
• Output-Feedback Boundary Control of a Heat PDE Sandwiched Between Two ODEs
• Output feedback control of general linear heterodirectional hyperbolic PDE-ODE systems with spatially-varying coefficients
• Convergence Speed of Nonlinear Luenberger Observers
• Stabilization for a coupled PDE–ODE control system

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