We can rewrite the above equation in matrix form as:
R y = Φθ a ,
then we have
θ a = Φ T Φ)
−1 Φ T R y ,
δ Φ c R y .
Hence the amplitude of each sine and cosine wave can be obtained. It should
be pointed out here that,
1. if one is interested to find the amplitudes of a few selected harmonics,
then the corresponding columns of the Φ matrix in the above equation
can be retained omitting the remaining columns, and
2. R y (k)intheaboveequationistheRROmeasuredfromtheclosed-loop
and is not the realtime PES signal which contains the NRRO signal.
Let us assume that the transfer functions of the feedback controller and the
plant are G c and G p , respectively, then the counter signal to be injected at a
of Figure 3.40 to cancel RRO of frequency if 0 should be
a fi = A i e jφ i | 1+G c G p
| f=if 0 = A i
B i e j(φ i −θ i ) ,
where B i and θ i are the magnitude and phase of the transfer function between
a and y at frequency if 0 . The above process is essentially means inverting
the RRO signal with respect to a suitable transfer function to have a counter
signal for RRO cancelation.
TheBodeplotfroma to y for an example system is shown in Figure 3.23.
The bode plot is evaluated using sweept sine excitation of the model and then
measuring the output signal's amplitude ratio and phase delay with respect to
the input signal at each frequency.
Using the equations given above, we find for the signal shown in Figure 3.44,
a i =[2.1608, 0.5680, 1.1550,−0.4569, ···] × 10 −3 ,
b i =[−2.0524,−0.0731,−0.5640, ···] × 10 −3 .
Then we can calculate using equation 3.90thecontrolsignalrequiredtocom-
pensate for these RRO components. The spectrum of the resultant PES with
all the RRO components taken out is shown in Figure 3.46. Comparing Fig-
ure 3.46 with Figure 3.45, there is a 33% reduction in the standard deviation
of PES when all components of RRO are cancelled.
Implementation of this method requires measurement of the RRO signal,
whichisthenusedtocalculatethecontrol input that can corrects the phase
and amplitude of RRO at certain frequencies via plant inversion. This can as