Novel Properties of Spectrum for Stability Test of Linear Delay Differential Systems

Tiao-Yang Cai,Hua-Guang Zhang

doi:10.1109/access.2019.2936482

Abstract

This paper reveals novel properties of spectrum of linear delay differential systems, and concerns the stability of a general class of time delay systems. The stability issue can be well characterized by the distribution of spectrum of the associated characteristic equation. Benefit properties on finite and bounded are formulated. And the finiteness condition and distribution boundary are obtained in a practical way. Based on the argument principle, an analytical formula for computing the number of unstable zeros (with positive real part) is deduced, which is determined by definite integral. Moreover, a procedure put no restriction on system except for the delay engendering purely imaginary roots is proposed to investigate the stability of linear time delay systems. At last, typical examples are given to illustrate that the method carried out is reliable and efficient.

Highlights

In control theory, stability test and spectrum analysis of timedelay systems (TDS) have attracted considerable attention during last decades
The presence of delay in a feedback control system leads to a characteristic equation including exponential type transcendental terms, which are called exponential polynomials or quasi polynomials [1]
The problem comes, could we find a definite boundary of the integral region? It does exist in our work that the right hand side boundary of all the roots of exponential polynomials has been found

Summary

Introduction

Stability test and spectrum analysis of timedelay systems (TDS) have attracted considerable attention during last decades. The presence of delay in a feedback control system leads to a characteristic equation including exponential type transcendental terms, which are called exponential polynomials or quasi polynomials [1]. The exponential transcendentality brings infinitely many isolated roots, which makes the stability issue of TDS a challenging task. A vast bulk of fundamental and exploratory results was obtained and reported [2]–[4]. The stability assessment issues are performed in a fundamental way, such as, Pontrjagin’s criterion, D-partition technique, argument principle methods, integral criterion, Nyquist criterion and its equivalent edition Mikhailov criterion. Exploratory researches are conducted through direct method, bivariate polynomials, pseudo delay transformation, frequency sweeping tests and so on. Most of the achievements are committed to reduce the infinite dimension problem to finite one

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Results

Conclusion