General formula for stability testing of fractional-delay systems

TL;DR

This paper introduces a straightforward and effective formula for stability testing of fractional-delay systems, utilizing Rouche's theorem to determine the number of unstable roots in the characteristic equation.

Contribution

The paper proposes a novel, easy-to-apply formula for stability analysis of fractional-delay systems with general characteristic equations, enhancing existing methods.

Findings

The formula accurately predicts the number of unstable roots.

Numerical simulations confirm the efficiency of the proposed method.

The approach simplifies stability testing for complex fractional-delay systems.

Abstract

An easy-to-use and effective formula for stability testing of a system with fractional-delay characteristic equation in the general form of $\Delta(s)=P_0(s)+\sum_{i=1}^N P_i(s)\exp(-\zeta_i s^{\beta_i}) =0$, where $P_i(s)$ ($i=0,..., N$) are the so-called fractional-order polynomials and $\zeta_i$ and $\beta_i$ are positive real constants, is proposed in this paper. The proposed formula determines the number of unstable roots of the characteristic equation (i.e., those located in the right half-plane of the first Riemann sheet) by applying Rouche's theorem. Numerical simulations are also presented to confirm the efficiency of the proposed formula.