Backstepping observers for linear PDEs on higher-dimensional spatial domains

The extension of the backstepping-based state observer design is considered for systems governed by parabolic PDEs defined on a higher-dimensional cuboidal spatial domain with the measured output being restricted to a single boundary hyperplane. Thereby, a suitable coordinate and state transformation is employed, which allows for the application of the backstepping method for the determination of a Luenberger-type state observer with observer corrections entering the PDE and the boundary condition. This ensures that the observer error dynamics follows the behavior of a predefined exponentially stable target system. In view of a practical realization of the proposed observer approach, the idealized system output, previously assumed as measurable on the entire boundary hyperplane, is reconstructed from a limited set of optimally located measurements by means of a least squares approximation. The observer error convergence and the applicability of the proposed approach are confirmed analytically and are evaluated in numerical simulations.