Boundary Analysis of Control Systems using Schwarz Lemma

Abstract

In this paper, Schwarz lemma at the boundary is considered for analysis of transfer functions used in control systems. Two theorems are presented with their proofs by performing boundary analysis of the derivative of positive real functions evaluated at the origin. Considering that the transfer function,H(s) , is an analytic function defined on the right half of the s- plane, inequalities for |H(0)| are obtained by assuming that   is also analytic at the boundary point s=0 on the imaginary axis with H(0)=0. Finally, the sharpness of these inequalities is proved. As result of the sharpness analysis, different extremal functions corresponding to different transfer functions are obtained. The related block diagrams and root-locus curves are also presented for considered transfer functions. According to the root-locus diagrams, marginally stable transfer functions are obtained as the natural results of the theorems proposed in the study.