Boundary estimation for a diffusion-reaction PDE driven by an unknown periodic input

In this paper, we design exponentially convergent observers for a class of parabolic partial differential equations (PDEs) driven by an unknown periodic input with only boundary sensing available. The problem is posed as a problem of designing an invertible coordinate transformation of the observer error system into an exponentially stable system. Observer gain (output injection function) is shown to satisfy a well-posed hyperbolic PDE. Moreover, the observer gains are obtained in closed form.