General formula for stability testing of fractional-delay systems

An easy-to-use and effective formula for stability testing of a system with fractional-delay characteristic equation in the general form of $\mathrm{\Delta }(s)=P_0(s)+{\sum }_{i=1}^N{P}_i(s)\exp (-{\zeta }_i{s}^{{\beta }_i})=0$, where $P_i(s)$ ($i=0,\mathrm{.}\mathrm{.}\mathrm{.},N$) 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.