Algebraic Expression of Critical Imaginary Roots for Frequency-Sweeping Framework of Time-Delay Systems: A Characteristic-Set-Based Approach

• 成果类型: Article
• 署名作者: Li, Xu; Li, Xu-Guang; Zhang, Lu; Chen, Jun-Xiu
• 署名单位: Nanjing University of Posts & Telecommunications; Northeastern University - China; Northeastern University - China; Northeastern University - China; Dalian Polytechnic University
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2025.3647573
• 发表日期: 2026
• 关键词: Stability analysis; POLYNOMIALS; criteria

摘要

The frequency-sweeping approach is an important and well-known tool for stability analysis of time-delay systems with commensurate delays. However, when it is used to detect critical imaginary roots (CIRs), the accuracy is regarded to be highly related to the numerical grid for sweeping the frequency axis. To the best of the authors' knowledge, the characterization of CIRs for most reported approaches has not reached a completely algebraic level. In this article, we introduce a polynomial algebraic tool, the characteristic set, acting as a new perspective for characterizing CIRs. Based on the characteristic set algorithm, we decompose the zeros of the frequency-sweeping conditions that a CIR appears into a finite number of real quasi-components of triangular sets. The obtained triangular sets are an explicit algebraic expression of CIRs, which can be determined directly without any numerical criteria and conservatism. The proposed method not only overcomes the drawbacks of the frequency-sweeping approach but also provides a procedure for detecting CIRs in a completely algebraic way.