Canonical Forms for a Class of Distributed Parameter Control Systems

A class of linear distributed parameter control systems with scalar control is considered in which the uncontrolled system is governed by a holomorphic semigroup. A canonical form is constructed for these systems, and is used to help construct feedback controls solving an eigenvalue specification problem. Explicit formulas for the feedback element and the closed-loop eigenvectors are obtained in terms of the original eigenvectors and the basis which is biorthogonal to the eigenvectors. This will be applied to an eigenvalue specification problem for a structurally damped beam. These feedback controls are given by infinite series, which cannot be computed in practice, so the canonical form is used to analyze the effect of truncated series on the distributed system. The canonical form is a functional equation with an associated canonical state space equation that is equivalent to it. The development in this paper is primarily done in the frequency domain, and hinges on an infinite-dimensional generalization of the characteristic polynomial, which is a meromorphic function.