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.