Robustness of Boundary Observers for Radial Diffusion Equations to Parameter Uncertainty

Boundary observers for radial diffusion equations can be derived to achieve exponential convergence of the estimation error system provided that coefficients are known; which can be either constant or possibly spatially and time varying. When the coefficients depend on the state, their values are not longer known and this might prevent the estimation error to converge to zero. Here, we address the state estimation problem for a radial diffusion equation in which the diffusion coefficient depends on the spatial average of the state value; using an observer with a constant diffusion coefficient. The error introduced to the observer, in this particular situation, can be quantified from an input-to-state stability (ISS) analysis. This study is motivated mainly by the problem of state estimation from electrochemical models of lithium-ion batteries, namely the Single Particle Model (SPM). In this application, the variation in the diffusion coefficient as a function of the spatial average of the states is of several orders of magnitude. We consider this result an additional effort in the broader goal of designing estimation algorithms from electrochemical models of lithium-ion batteries without relying in the discretization of the PDEs in these models.