Hardware Reference
In-Depth Information
To enhance the stability of AFC algorithm as well as to avoid amplification
of other RRO components, a suitable O
−1
(z) should satisfy: R(z) < 1atother
harmonics and O(z) O
−1
(z)mustbestable.AsC
r
(e
jω
i
T
) is pure imaginary,
the objective is to find a suitable O
−1
(z) that shapes the plant such that
O(z) O
−1
(z) is close to SPR in a wide frequency range so that R(e
jωT
) < 1.
If the equivalent plant O(z) is not a zero phase system, pole/zero cancelation
and phase cancelation can be used to design O
−1
(z)toshapetheplantsuch
that O(z) O
−1
(z) is close to a zero-phase system [224].
Without loss of generality, let the equivalent plant O(z)beexpressedas
D(z)
=
b
0
z
m
+ b
1
z
m−1
+
···
+ b
m
O(z)=
N(z)
.
(3.101)
z
n
+ a
1
z
n−1
+ ···+ a
n
Because the equivalent plant is the closed-loop system under nominal con-
troller, all the roots of D(z) are inside the unit circle in the Z-plane, and the
roots of N(z) can be either inside, on or outside the unit circle.
If the equivalent plant O(z) is a minimum phase system, i.e., all the roots
of N(z) are inside the unit circle, then all the poles and zeros of the plant are
cancelable. Therefor we can choose:
−1
(z)=
D(z)
O
z
d
N(z)
,
(3.102)
where d = n−m. After that all the poles and zeros of O(z) are canceled by
O
−1
(z).
If the plant is a non-minimum phase system, and suppose that there is no
zero on the unit circle, then the numerator polynomial N(z) can be factored
into two parts such that,
N(z)=N
s
(z)N
u
(z),
(3.103)
where N
s
(z) includes stable zeros which are cancelable, and N
u
(z)includes
zeros which are not inside the unit circle. Then
O
−1
(z) can be designed as
−1
(z)=
D(z)N
u
∗
(z)
z
d+2u
N
s
(z)
,
O
(3.104)
where u is the order of N
u
∗
(z). N
u
∗
(z) can be designed according to Butter-
worth transforms. If N
u
(z) is represented as,
N
u
(z)=d
0
+ d
1
z + ···+ d
u
z
u
,
(3.105)
then N
u
∗
(z) can be designed as,
N
u
∗
(z)=d
u
+ d
u−1
z + ···+ d
0
z
u
.
(3.106)
Note that
N
u
∗
(z)
z
u
is the complex conjugate of N
u
(z)whenz =e
jωT
s
. Therefore,
N
u
(z)
N
u
∗
(z)
z
u
is positive real.
