State and disturbance estimator for unstable reaction-advection-diffusion PDE with boundary disturbance

We propose an asymptotically convergent state and disturbance estimator for a class of reaction-advection-diffusion PDEs where the disturbance is anti-collocated with control input, using measurements at both boundaries. Two auxiliary systems are designed for disturbance estimation, adopting the output error injection method, where the disturbance estimator is determined by the plant model structure and measured signals. A sufficient condition on the reaction coefficient is derived for which the disturbance estimator achieves asymptotic convergence to the true value. All states in the disturbance estimator are proved to be bounded using Lyapunov stability theory. We further propose a state estimator using the backstepping technique, by injecting the disturbance estimation signal into the state observer.