Root locii for systems defined on Hilbert spaces

TL;DR

This paper extends the concept of root locus analysis from finite-dimensional systems to infinite-dimensional systems modeled by PDEs or delay equations, providing a rigorous definition and exploring their stability properties.

Contribution

It introduces a rigorous definition of the root locus for infinite-dimensional systems and analyzes their asymptotic behavior and pole-zero relationships.

Findings

Root locus is well-defined for a large class of infinite-dimensional systems.

Limit points of root locus branches are zeros, similar to finite-dimensional systems.

Pole-zero interlacing property extends to certain infinite-dimensional systems with self-adjoint or skew-adjoint generators.

Abstract

The root locus is an important tool for analysing the stability and time constants of linear finite-dimensional systems as a parameter, often the gain, is varied. However, many systems are modelled by partial differential equations or delay equations. These systems evolve on an infinite-dimensional space and their transfer functions are not rational. In this paper a rigorous definition of the root locus for infinite-dimensional systems is given and it is shown that the root locus is well-defined for a large class of infinite-dimensional systems. As for finite-dimensional systems, any limit point of a branch of the root locus is a zero. However, the asymptotic behaviour can be quite different from that for finite-dimensional systems. This point is illustrated with a number of examples. It is shown that the familiar pole-zero interlacing property for collocated systems with a Hermitian state matrix extends to infinite-dimensional systems with self-adjoint generator. This interlacing property is also shown to hold for collocated systems with a skew-adjoint generator.