General formula for stability testing of linear systems with fractional-delay characteristic equation

Abstract

AbstractIn some applications (especially in the filed of control theory) the characteristic equation of system contains fractional powers of the Laplace variable s possibly in combination with exponentials of fractional powers of s. The aim of this paper is to propose an easy-to-use and effective formula for bounded-input boundedoutput (BIBO) stability testing of a linear time-invariant system with fractional-delay characteristic equation in the general form of $$\Delta \left( s \right) = P_0 \left( s \right) + \sum\nolimits_{i = 1}^N {P_i \left( s \right)\exp ( - \zeta _i s^{\beta _i } ) = 0}$$, where P i(s) (i = 0,...,N) are the so-called fractional-order polynomials and ξ i and β i are positive real constants. The proposed formula determines the number of the roots of such a characteristic equation 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.