Stability analysis of time-delay systems in the parametric space

TL;DR

This paper introduces a frequency domain method based on Rouche's theorem for stability analysis of various linear time-delay systems, capable of identifying stable parameter regions even with complex delays.

Contribution

The paper presents a novel, unified approach for stability analysis of diverse time-delay systems using frequency domain techniques and parametric region identification.

Findings

Method effectively identifies stability regions in parametric space.

Applicable to systems with incommensurate and distributed delays.

Demonstrated success through illustrative examples.

Abstract

This paper presents a novel method for stability analysis of a wide class of linear, time-delay systems (TDS), including retarded non-neutral ones, as well as those incorporating incommensurate and distributed delays. The proposed method is based on frequency domain analysis and the application of Rouche's theorem. Given a parametrized TDS, and some parametric point for which the number of unstable poles is known, the proposed method is capable of identifying the maximum surrounding region in the parametric space for which the number of unstable poles remains invariant. First, a procedure for investigating stability along a line is developed. Then, the results are extended by the application of Holder's inequality to investigating stability within a region. Contrary to existing approaches, the proposed method is uniformly applicable to parameters of different types (delays, distributed delay limits, time constants, etc.). Efficacy of the proposed method is demonstrated using illustrative examples.