An adaptive observer for a class of parabolic PDEs based on a convex optimization approach for backstepping PDE design

This article addresses the observer design problem for simultaneous state and parameter estimation for a class of one-dimensional linear parabolic PDEs. The design is based on the backstepping PDE methodology, including a modified integral transformation to compensate for the parameter uncertainty. The solution of the resulting Kernel-PDE is recast as a convex optimization problem via a Sum-of-Squares formulation which is solved by polynomial optimization techniques (semidefinite programming). This allows computing - in a fast and direct way - the state and parameter observer gains at every sampling time. In addition to an observer based on the Volterra transformation, an observer design method with two boundary measures via a Fredholm-type transformation is presented. The uniqueness and invertibility of this transformation are proved for polynomial kernels in the space of continuous functions. The effectiveness of this approach is illustrated by numerical simulations.