When O −1 is used, the phase of the shaped plant is 1/z d+u and is near zero
phase in a wider low frequency range compared with the case without O −1 .
Such a modification can improve the robustness and convergence rate of the
AFC scheme. Additionally, such a modification might lower the sensitivity
transfer function hump from D to y and prevent significant amplification
of other RRO harmonics or NRRO signal when canceling the selected RRO
We note here that by using (3.104), C r with a suitable gain g i might be
able to generate a notch at the desired frequency and at the same time have
attenuation at frequencies other than those of C r 's center frequency .
Nevertheless, such a system is still governed by the Bode Integral theorem, and
hence there will be amplification of RRO and NRRO at some other frequencies.
RRO Compensation via Periodic Signal Generator using Delay Terms
In the above designs, compensating each RRO frequency requires a second
order controller. If we want to compensate for more frequencies of RRO,
the order of the compensator increases to twice the number of frequencies
to the compensated for. Adopting the same “plug-in” structure as shown in
Figure 3.49, periodic signal generators (PSGs) with a simple delay term in a
feedback loop can be used to generate the internal model for disturbance 
Figure 3.49: “Plug-in” repetitive compensation using periodic signal generator-
continuous time case .
Hara et al has proved in  that exponential stabilization is not achiev-
able for such repetitive control systems with strictly proper transfer functions.
However, when a low-pass filter is used in conjuction with the delay section,
the internal model is able to generate the signal with certain cancelation of the
disturbances  . Figure 3.50 shows an example of a scheme with PSG
based control. Discrete time version of the such a scheme can be found in 
and the references therein.
In the case of digital control, we can assume that the plant model shown