Direct stability-switching delays determination procedure with differential averaging

Abstract

In this paper, a direct computational method for the searching and determination of stability switching delays is introduced. The primary procedure is applicable to retarded linear time-invariant time-delay systems and it is based on the iterative (successive) estimation of the dominant pole of the infinite system spectrum by means of the Taylor’s series expansion in every node of the selected grid of discrete delay values. Whenever a crossing of the stability border is detected, the switching pole loci and the corresponding set of switching delays are further enhanced. To perform it, a linear Regula Falsi interpolation has been used in the original version. Here, two versions of the use of root tendency property are applied and compared. Root tendency expresses the change in the pole position with respect to the infinitesimal change in delays; that is, the complex valued gradient. Once a finite set of stability switching delays’ values is determined, these delays can be joined so that infinitely many switching delays are obtained. In this paper, the linear and the quadratic interpolations are compared in addition. The whole procedure is simply implementable by using standard software tools and it does not require special ones; neither a deep mathematical knowledge is required, which is favorable for the practice. A numerical example performed in MATLAB/Simulink environment demonstrates the accuracy of the algorithm and its substrategies compared with a well-established method for the delay-dependent stability analysis. Some beneficial and worthwhile ideas of how to cope with neutral delay systems are given and supported by an example as well.