H∞H∞ design with fractional-order PDμPDμ controllers

This article focuses on H∞H∞ performance design for fractional-order systems with time-delay, using fractional-order proportional-derivative (PDμ)(PDμ) controllers. First, the stabilizing parameters region in proportional-derivative plane of PDμPDμ controller, for a fixed derivative-order, is determined in terms of a graphical stability criterion applicable to fractional-order time-delay systems. Then, in the stabilizing region, the pairs of proportional and derivative gains of PDμPDμ controller are calculated for a range of frequencies, which satisfy the H∞H∞-norm constraint of complementary sensitivity function and define the H∞H∞ boundary curve. Finally, by changing the derivative-order of PDμPDμ controller, we observe the relationship between the H∞H∞ curve and the derivative-order. Examples are followed to illustrate the design procedure.