Root locus analysis of a retarded quasipolynomial

Abstract

Time delay systems (TDS), called hereditary or anisochronic as well, can be frequently found in many engineering problems and they constitute a widespread family of industrial plants. Modelling, identification, stability analysis, stabilization, control, etc. of TDS are challenging and fascinating tasks in modern systems and control theory as well as in academic and industrial applications. One of their possible linear representations in the form of the Laplace transform yields the transfer function expressed as a fraction of quasipolynomials, instead of polynomials, with delay (exponential) terms in denominators. In this contribution, detailed root location analysis of a characteristic retarded quasipolynomial of degree one is presented, which gives rise to the spectrum of a retarded TDS. The presented analysis represents also a powerful tool for controller tuning in pole-placement control algorithms for delayed systems. A simulation example clarifies the results obtained vie proven propositions, lemmas and theorems.