A Simple Procedure for the Creation of Stability Charts in Delay Parameter Space for a Class of LTI Systems With Two Delays

The central task of the stability robustness with respect to delay uncertainties lies in creating the whole stability chart (delay map) in the delay parameter space. In this paper, we present a very simple frequency-sweeping procedure for creating stability chart in the delay parameter plane for a class of linear time-invariant (LTI) two-delay systems with a delay crossing talk. Actually, the procedure based on using Rekasius pseudo-delay substitution and discriminant of quadratic polynomial to characterize the crossing frequency set. The exact and exhaustive determination of crossing frequency intervals involves only finding all positive real roots of a real-coefficient polynomial. With the availability of the entire crossing frequency set, the famous cluster treatment of characteristic roots (CTCR) paradigm is applied to construct complete stability chart in the delay plane. A by-product of the proposed procedure is the revelation of the non-zero finite frequencies corresponding to infinite pseudo-delays. These frequencies divide the crossing frequency intervals into subintervals over each of which the frequency-sweeping technique provides continuous stability crossing curves in the domain of pseudo delays. For illustration and validation, two examples are provided.