Numerical Method for Stability Testing of Fractional Exponential Delay Systems

A numerical method for stability testing of fractional exponential systems including delays is presented in this contribution. We propose the numerical test of stability for a very general class of systems with a transfer function, which includes polynomials and exponentials of fractional powers of the Laplace variable s combined with delay terms. Such a system is unstable if any root of its characteristic equation, which usually involves transcendental functions, is localized on the right half-plane of the s-domain. Due to the infinite size of the right half-plane, the bilinear transformation is employed to map it onto the unit disc on the complex plane. Then, the global roots and poles finding algorithm based on phase analysis is executed on the unit disc. That is, the roots of the characteristic equation for the considered system are detected with the use of an efficient algorithm based on domain meshing and Cauchy’s argument principle. In order to demonstrate the efficiency and applicability of the proposed numerical method, we executed stability tests for fractional exponential delay systems, which are considered benchmarking cases in other publications. It occurs that, each time, the proposed method correctly evaluates the system stability. However, unlike other methods, it is a very general technique that allows evaluation of almost any system, which does not require any preprocessing of the characteristic equation to execute the stability test.