Hardware Reference
In-Depth Information
In state space form, the above can be written as
A s
A N− s B s
A N− s B s
···
B s
x[k + N]
y[k]
y[k +1]
···
y[k + N −1]
C s
D s
0
···
0
C s A s
C s B s
D s
···
0
=
,
.
.
.
.
···
C s A N−1
C s A N− s B s C s A N− s B s
···
D s
s
¸
A B
C D
=
.
(3.127)
After multi-rate discretization, the system becomes an n-dimensional plant
with N inputs and N outputs, with D being a square matrix of full rank. Thus
we can get the system's inverse state-space model O −1 (z)={A, B, C, D} −1
directly:
¸
¸
−1
A−BD −1 C BD −1
−D −1 C
A B
C D
=
.
(3.128)
D −1
Recall that in Section 3.5.3 we discussed that by adding the plant inverse in
the AFC scheme can work in a wider frequency range. We shall explain the
advantage of using the multi-rate inverse scheme with the help of an example.
Example
Let us consider a first order SISO system described by a transfer function
O(s)= s 10
s +23 .
(3.129)
Its single rate discrete form with sampling frequency 15 kHz is,
O 1 (z)= 0.9989z 0.9996
z −0.9985
.
(3.130)
When discretized using dual rate sampling, its state space representation be-
comes,
¸
0.9985
0.0002664
0.0002666
A B
C D
O 2 (z)=
=
−4.125
1
0
.
(3.131)
−4.122
−0.0011
1
Thedualrateinversemodelis,
¸
1.001
0.0002667
0.0002666
−1
A B
C D
O −1
2
(z)=
=
4.125
1
0
.
(3.132)
−4.126
0.0011
1
Because there is one unstable zero of the plant, the phase of the shaped
plant O(z) O −1 (z)isthesameas z when using O −1 (z) is obtained using the
pole/zero and phase cancelation scheme,.
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