Ring Systems

Abstract

The topology of a ring system consists of a closed chain of identical subsystems that interact in a repeated pattern. It is a common, simple pattern found in natural and man-made systems. Mathematically, ring systems can be referred to as circulant systems, because the matrices in a state space model of a ring have a circulant, or more generally block circulant, form. Circulant systems are probably the most prominent class [69] we have identified within the family of patterned systems.

Their control has been studied by previous researchers largely using the diagonalization approach first presented by Brockett and Willems [6]. The class has not been thoroughly examined from a control perspective in the literature; rather certain individual problems, mostly concerning optimization, have been tackled.

This chapter begins with a formal definition of circulant matrices and a presentation of their key properties. The special case of symmetric circulants is examined, as well as a variation on circulant matrices called factor circulants. Block circulants fall into the category of block patterned systems, which are not covered by the theories presented in this book. However, a special sub-class of block circulants systems is hierarchies of circulants, which is a patterned system class within our framework, and it is presented herein. Finally, we apply some of our theoretical results on patterned systems to a selection of simple, physical examples of ring systems.