Image Processing Reference
In-Depth Information
that discussion on arbitrary pole placement is only possible if the controllability
condition is satis
ed.
In recapitutation, an LTI continuous-time system can be written as
x
¼
Ax
þ
Bu
y
¼
Cx
þ
Du
(
:
)
9
28
or in the discrete-time
x
(
k
þ
) ¼
Ax
(
k
) þ
Bu
(
k
)
y
(
k
) ¼
Cx
(
k
) þ
Du
(
k
)
1
(
9
:
29
)
Taking the Laplace transform of both sides of Equation 9.28, we get
sX
(
s
)
x
(
) ¼
AX
(
s
) þ
BU
(
s
)
Y
(
s
) ¼
CX
(
s
) þ
DU
(
s
)
0
)
(
9
:
30
)
X
(
s
) ¼ (
sI
A
)
1
X
(
) þ (
sI
A
)
1
BU
(
s
)
Y
(
s
) ¼
CX
(
s
) þ
DU
(
s
)
¼
C
(
sI
A
)
1
x
(
0
) þ (
C
(
sI
A
)
1
B
þ
D
)
U
(
s
)
0
where X(s), Y(s), and U(s), denote the Laplace transforms of the signals x(t), y(t), u(t),
respectively, and where the inverse (sI
A)
1
exists for all real values of s except at
the eigenvalues of matrix A.
Similarly, applying one-sided z-transform on Equation 9.29, we get
zX
(
z
)
x
(
) ¼
AX
(
z
) þ
BU
(
z
)
Y
(
z
) ¼
CX
(
z
) þ
DU
(
z
)
0
)
(
9
:
31
)
X
(
z
) ¼ (
zI
A
)
1
x
(
) þ (
zI
A
)
1
BU
(
z
)
Y
(
z
) ¼
CX
(
z
) þ
DU
(
z
)
¼
C
(
zI
A
)
1
x
(
0
) þ (
C
(
zI
A
)
1
B
þ
D
)
U
(
z
)
0
where X(z), Y(z), and U(z) denote the one-sided z-transforms of the signals x(k),
y(k), u(k), respectively, and where the inverse (zI
A)
1
exists for all values of
z except at the eigenvalues of matrix A. This shows that the two (matrix) transfer
functions have exactly the same form, only s is replaced by z. Therefore, a pole-
placement algorithm to design state feedback for a continuous-time system can also
be used for the discrete case, since the transfer function has the exact same form in
both cases.
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