Chapter 1 - On the numerical determination of stability regions in the delay space via dominant pole estimation

Abstract

This contribution intends to provide the reader with the presentation of a numerical gridding iterative algorithm to determine all stability regions within the prescribed area of the delay space. In every single grid node, the iterative estimation of the rightmost pole is computed based on the polynomial approximation of the characteristic quasipolynomial, by utilizing the knowledge of the dominant pole estimation in the nearest grid node. The polynomial approximation is made via the Taylor series-based expansion in the vicinity of the closest dominant poles estimation, and by using the bilinear transformation followed with prewarping for a discrete-time approximation. Exponential terms are subjected to a quadratic extrapolation method to get commensurate delays. Two-step Newton’s iteration method with averaging is used to detect imaginary axis crossings. Neutral delay case is concisely discussed as well. Two numerical examples demonstrate the accuracy and efficiency of the algorithm. Possible future directions of this research and algorithm modifications are proposed and discussed in brief as well.