Asymptotic properties of root locus for distributed parameter systems

The authors examine various examples of boundary feedback control in the context of a root locus theory for distributed parameter systems. Using root locus principles they design proportional-error and proportional-rate feedback laws for some explicit examples which are shown to achieve closed-loop stability. They note that while it is possible to develop in abstract a geometric state space interpretation of root locus methods, in the context of singular perturbations, it turns out that even for some very simple examples the corresponding root-locus plots have somewhat disturbing discontinuities, which do not exist in the lumped case and seem not to be explained in an abstract state-space setting. This phenomenon is illustrated in a classical example due to F. Rellich (1969)