Boundary observer design for coupled reaction-diffusion systems with spatially-varying reaction

Following a recent solution to the problem of boundary stabilization of linear coupled reaction-diffusion systems by means of the backstepping method, we present an observer for a coupled pair of reaction-diffusion partial differential equations (PDEs) with boundary measurements. We show that, as in the case of stabilization, the backstepping kernel PDEs are essentially equivalent to the PDEs governing the kernels for stabilization of first-order hyperbolic coupled PDEs. A numerical example is provided for the computation of the kernels. The problem we solve is motivated by diffusion phenomena in lithium-ion batteries with electrodes that comprise multiple active materials. The particular choice of boundary conditions also comes from this application.