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 .
Without loss of generality, let the equivalent plant O(z)beexpressedas
D(z) = b 0 z m + b 1 z m−1 + ··· + b m
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)
z d N(z) ,
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),
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 d+2u N s (z) ,
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 ,
then N u ∗
(z) can be designed as,
N u ∗
(z)=d u + d u−1 z + ···+ d 0 z u .
Note that N u ∗ (z)
is the complex conjugate of N u (z)whenz =e jωT s . Therefore,
N u (z) N u ∗ (z)
is positive real.