Hardware Reference
In-Depth Information
150
100
50
0
−50
10 1
10 2
10 3
10 4
−100
−150
−200
−250
−300
−350
10 1
10 2
10 3
10 4
Frequency (Hz)
Figure 3.50: Frequency response of a control system using PSG. Solid line:
with controller C(s)andwithouttheC rp , Dashed-line, with controller C(s)
and without the C rp .
in Figure 3.49 is expressed as
−1 )= z −k B(z −1 )
A(z −1 )
G p (z
,
(3.107)
where k is the number of delays in the plant. Following [107], the controller
with a periodic signal generator is given by:
z −N+k q(z −1 )B u (z −1 )
(1−q(z −1 )z −N )B s (z −1 )b ,
−1 )=K r
G c (z
(3.108)
where K r is the repetitive control gain, N is the number of discrete-time sam-
ples of the periodical disturbance per revolution, B u (z −1 ) is the non-minimum
phase zeros (non-cancelable part of the numerator), B s (z −1 ) is the minimum
phase zeros (cancelable part of the numerator),
−1 )= z +2+z −1
4
q(z
,
(3.109)
and b =[B u (1)] 2 . It is an inverse model of the plant, modified for unstable
zeros, and the remainder of the controller places poles on the unit circle at the
harmonics of the fundamental frequency. The low-pass filter q(z −1 )brings
the poles inside the unit circle and sacrifices high-frequency regulation, in
order to improve robustness to the unmodeled dynamics and to guarantee
stability [34], [168].
The formulation of the RRO and a few methods to cancel the effects of RRO
on the performance of the servomechanism are explained in this section. Based
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