Pole placement with minimised norm controllers

A convenient technique for placing some poles while preserving others is to use state feedback with unity-rank controllers. The paper discusses the problem of reducing the required control effort by considering the minimisation of a matrix functional whose value reflects the magnitudes of the elements of the feedback gain matrix. The investigation proceeds with vector-analysis methods, and it is shown that solutions can be obtained in closed form without recourse to numerical optimisation techniques. The algorithms are developed for use in a recursive scheme for placing several poles. A well established algorithm for constructing feedback controllers for pole placement preserves the acceptable modes by making use of the concept of unobservability. A proposed extension to this scheme is that of preserving modes by making use of the dual concept of uncontrollability, and it is shown that further substantial reductions in control effort can be made by the use of this generalisation.