BIBO stability analysis of fractional order time delay systems

Abstract

In this paper, an analytical method is introduced for determining the stability map of a general class of fractional order systems with single and multiple commensurate time delays, against delay uncertainties. The complexity arises due to the exponential type transcendental terms and fractional order in their characteristic equation. It is shown that this method analytically reveals all possible stability windows exclusively in the parametric space of the time delay. Using the approach presented in this study, first, without using any approximation or substitution, the transcendental characteristic equation is converted to an algebraic one with some specific crossing points. The resulting algebraic equation also enables us to easily determine the delay dependency of the system stability and the sensitivities of crossing roots with respect to time delay. Then, an expression in terms of system parameters and imaginary root of the characteristic equation is derived for computing the delay margin. As an added strength, it does not require the delay-free system under consideration to be stable. The number of unstable roots in each interval is calculated with the definition of root tendency on the boundary of each interval. Finally, the concept of stability is expressed as a function of delay. Five illustrative examples are presented to confirm the proposed method results.