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,.
