Hr(s) has unity gain for both low and high frequencies, whereas the gain in a frequency band about ω = αs/2ζ is reduced; we have Hr(j(αs/2ζ)) = 1/(1 + (1/4ζ2)).
The 2DOF PI controller in [2] uses a first-order lead–lag prefilter. Consequently, the ideal closed-loop system Gc(s) = αs/(s + αs) cannot be exactly obtained, but a good approximation can.