Invariant zeros of SISO infinite-dimensional systems

This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, redistribution , reselling , loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. The zeros of a finite-dimensional system can be characterised in terms of the eigenvalues of an operator on the largest closed feedback-invariant subspace. This characterisation is also valid for infinite-dimensional systems, provided that a largest closed feedback-invariant subspace exists. We generalise this characterisation of the zeros to the case when the largest closed feedback-invariant subspace does not exist. We give an example which shows that the choice of domain of the operator on this invariant subspace is crucial to this characterisation. 1. Introduction The importance of the poles of a transfer function to system dynamics are well known. The zeros of the transfer function are also important to controller design e.g. Doyle, Francis, and Tannenbaum (1992) and Morris (2001). For example, the poles of a system controlled with a constant feedback gain move to the zeros of the open-loop system as the gain increases. Furthermore, regulation is only possible if the zeros of the system do not coincide with the poles of the signal to be tracked. Another example is sensitivity reduction – arbitrary reduction of sensitivity is only possible if all zeros lie in the open left-half-plane. There are a number of ways to define the zeros of a system; for systems with a finite-dimensional state space all these definitions are equivalent. However, systems with delays or partial differential equation models have state space representations with an infinite-dimensional state space. Since the zeros are often not accurately calculated by numerical approximations it is useful to obtain an understanding of their behaviour in the original infinite-dimensional context. Extensions from the finite-dimensional situation are complicated not only by the infinite-dimensional state space but also by the unboundedness of the generator A. There are results on the …