On pole-zero placement by unit-rank modification

The dynamic behaviour of a general linear system is characterised by the poles (resonances) and the zeros (anti-resonances). This paper deals with the general class of dynamical systems that can be described by a second order ordinary differential equation. For spatially discretised models the poles are given by the roots of the characteristic polynomial of the system matrix whilst the zeros correspond to the roots of the polynomials related to the elements of the adjugate of the system matrix. The question of how the poles and zeros can be assigned by a structural modification is important in many areas of application, for example in control, vibration absorption and model updating. In this paper we derive the basic equation of pole-zero placement by unit-rank modification. For simple real matrix pencils we simplify the basic equation which enables an analytical solution of the transformed vector of the modification. These results are related to the original unit-rank modification by a system of linear equations. Finally we will give insight into solution methods of the original problem, and suggest a method which is based on the incorporation of additional model structure information.