Graph topology and gap topology for unstable plants

Abstract

This paper provides a refonnulation of the graph topology and the gap topology in a very general setting in the frequency domain. Many essential properties and their comparison are clearly presented in the refonnulation. It is shown that the gap topology is suitable for the general systems rather than square systems with unit feedback, which is the situation studied in [2,3,9]. It is also revealed that, whenever an unstable plant can be stabilized by a feedback, it is a closed operator, mapping input space to output space. Hence the gap topology can always be applied whenever the unstable plants can be stabilized. The graph topology and the gap topology are suitable for different unstable subsets, and have many similar characteristics. If one confines them to the same subset, they will be identical Finally, the definitions of the graph metric and the gap metric are discussed. Keywords: graph topology, gap topology, unstable plants, feedback system