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