Root Locus for SISO Infinite-dimensional systems

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 we provide a rigorous definition of the root locus and show that it 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. We also show that the familar pole-zero interlacing property for collocated systems generated by a self-adjoint operator extends to infinite-dimensional systems.