Adaptive observer design of coupled ODE-heat PDE systems subject to uncertain parameters, sensor delays and non-linearities

Abstract This paper investigates adaptive observer design for a class of ordinary differential equation–partial differential equation (ODE–PDE) cascaded systems featuring time-delay characteristics, unknown parameters and inherent non-linear terms. The framework consists of a primary ODE subsystem, whose output is observed through a non-linear time-delay PDE subsystem modelled by heat reaction-diffusion equations. Relying on measurable outputs, an adaptive observer is constructed to achieve synchronous online exponential convergent estimation of all states and unknown parameters for both ODE and PDE subsystems. Decoupling transformation and Lyapunov–Krasovskii functional (LKF) are utilized for observer construction and stability analysis. The transformation fully decouples state and parameter estimation, and LKF suppresses negative impacts from time delays and non-linearities to secure closed-loop stability. Strict mathematical proofs establish the observer’s existence, uniqueness and global stability, and the exponential convergence of estimation errors is confirmed. Numerical simulations fully validate the effectiveness and distinct superiority of the presented approach.