Stability and Control of Fractional Periodic Time-Delayed Systems

In this chapter, two new methods are proposed to study the stability of linear fractional periodic time-delayed (FPTD) systems. First, the explicit harmonic balance (EHB) method is proposed to find necessary and sufficient conditions for fold, flip, and secondary Hopf transition curves in linear FPTD systems, from which the stability boundaries are obtained as a subset. Transition curves of the fractional damped delayed Mathieu equation are obtained by using the EHB method. Next, an approximated monodromy operator in a Banach space is defined for FPTD systems, which gives the linear map between two solutions. The fractional Chebyshev collocation (FCC) method is proposed to approximate thismonodromy operator. The FCC method is outlined and illustrated with three practical problems including obtaining the parametric stability charts of the fractional Hayes equation and the fractional second-order system with delay, and designing an optimal linear periodic gain fractional delayed state feedback control for the damped delayed Mathieu equation.