Real versus Complex Robustness Margin Continuity as a Smooth versus Holomorphic Singularity Problem

Abstract

It is well known that the real robustness margin can be discontinuous, while the complex robustness margin is always continuous relative to problem data. Using some concepts from set-valued analysis, continuity of μ can be viewed as structural stability of the neutral stability region. From this point of view, the crucial issue is whether 0+j0 is a critical value of the return difference map. This paper shows that the discrepancy between real and complex cases is due to the additional holomorphic property of the Nyquist return difference mapping of the complex μ-function. The critical points of the Nyquist map in the complex case are at most finite in number; in contrast, the critical points of the Nyquist map of the real smooth case form, generically, a curve. Furthermore and more importantly, in the complex case, even when 0+j0 is critical, the stability crossover is continuously deformed under the variation of “certain” parameters, while in the real case, the crossover could sustain a catastrophic change.