Extending the admissible control-loop delays for the inverted pendulum by fractional-order proportional-derivative controller

Stabilization of the inverted pendulum by fractional-order proportional-derivative (PD) feedback with two delays is investigated. This feedback law is obtained as a combination of PD feedback with two delays and fractional-order PD feedback with a single delay. Different types of stabilizability boundaries and the corresponding geometric and multiplicity conditions are determined using the D-subdivision method. The stabilizable region is depicted in the plane of the delay parameters for given fractional derivative orders. Several special cases and the concept of delay detuning are also discussed. It is shown that the admissible delay can be slightly increased compared to the integer-order PD feedback by introducing a fractional-order feedback term.