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,.