Phase-reduction-theory-based treatment of extended delayed feedback control algorithm in the presence of a small time delay mismatch

The delayed feedback control (DFC) methods are noninvasive, which means that the control signal vanishes if the delay time is adjusted to be equal to the period of a target unstable periodic orbit (UPO). If the delay time differs slightly from the UPO period, a nonvanishing periodic control signal is observed. We derive an analytical expression for this period for a general class of multiple-input multiple-output systems controlled by an extended DFC algorithm. Our approach is based on the phase-reduction theory adapted to systems with time delay. The analytical results are supported by numerical simulations of the controlled Rossler system.