On an approximation with prescribed zeros of SISO abstract boundary control systems

Finite-dimensional approximations of partial differential equations are used not only for simulation, but also for controller design. Modal truncation and numerical approximation are common practical methods for approximating distributed parameter systems. The modal approximation preserves the exact, low-order poles of the original system. However, the zeros of modal approximations may differ significantly from those of the original distributed parameter system. In particular, right half-plane zeros, which are not present in the original infinite-dimensional model, may appear in modal truncations. In this paper we consider a boundary control system and propose a moment matching based approximation which preserves a prescribed set of zeros. To illustrate the advantages of the method, we consider its application to the heat equation with Neumann boundary control at the right end (HENBCR). Although the modal approximation provides good error bounds for the HENBCR, it contains non-minimum phase zeros which lead to erroneous predictions. The moment matching approach sketched in this paper yields an approximation of the HENBCR with minimum phase zeros only. We consider that the numerical example is very interesting and convincing for the reader. Due to space limitations, further theoretical analysis will be addressed in the full paper.