Robust output-feedback stabilisation of parabolic PDEs via stubborn observers with dynamic saturation

In this work, we study the output-feedback stabilisation of a class of unstable parabolic partial differential equations (PDEs) with boundary measurements corrupted by large-amplitude short-duration disturbances. We develop a novel control architecture that integrates a stubborn observer with dynamic saturation and a backstepping-based boundary controller. The observer employs a nonlinear injection term with an adaptively adjusted saturation level, providing robustness against measurement outliers while maintaining exponential convergence in disturbance-free conditions. The boundary controller is designed via the backstepping transformation technique, utilising the state estimates from the observer. We establish the well-posedness of the closed-loop system and prove its input-to-state stability with respect to measurement disturbances. A comprehensive performance analysis demonstrates the superiority of the proposed approach over linear observer-based designs, particularly under such disturbances. Numerical simulations on an unstable reaction-diffusion equation validate the theoretical results and illustrate the practical efficacy of the method.