Fast moving horizon estimation for a distributed parameter system

Partial differential equations (PDEs) pose a challenge for control engineers, both in terms of theory and computational requirements. PDEs are usually approximated by ordinary differential or partial difference equations via the finite difference method, resulting in a high-dimensional state-space system. The obtained system matrix is often symmetric, which allows this high-dimensional system to be decoupled into a set of single-dimensional systems using the state coordinate transformation defined by a singular value decomposition. Any linear constraints in the original control problem can also be simplified by replacement by an ellipsoidal constraint. This reformulated moving horizon estimation (MHE) problem can be solved in orders of magnitude lower computation time than the original MHE problem, by employing an analytical solution obtained by moving the ellipsoidal constraint to the objective function as a penalty weighted by a decreasing penalty parameter. The proposed MHE algorithm is demonstrated for a one-dimensional diffusion in which the concentration field is estimated using distributed sensors.