Image Processing Reference
In-Depth Information
where x
0
is shown in Equation 9.13, and is the value of the state vector when the
actuators are set equal to the nominal values, u
0
. X
d
(z) is the z-transform of the
desired target vector. Now, by rearranging Equation 9.37, we can write the expres-
sion for X(z) as follows:
X
(
z
) ¼ [
zI
(
I
BK
)]
1
BKX
d
(
z
) þ
z
[
zI
(
I
BK
)]
1
x
0
(
9
:
38
)
where matrix I is the identity matrix. Now the error vector can be written in z-domain
as follows:
E
(
z
) ¼
X
d
(
z
)
X
(
z
)
¼ [
I
{
zI
(
I
BK
)}
1
BK
]
X
d
(
z
)
z
[
zI
(
I
BK
)]
1
x
0
(
9
:
39
)
The reference vector containing desired target values can be written in z-domain as
follows:
z
z
X
d
(
z
) ¼
1
X
d
(
9
:
40
)
Using
final value property, Equation 3.70, we can show that the steady-state error is
driven to zero after multiple actuations.
e
ss
(
k
)j
closed
¼ lim
z
!
1
(
z
1
)
E
(
z
)
(
9
:
41
)
Substituting Equations 9.39 and 9.40 into Equation 9.41 and applying the limit as
z
!
1, the steady-state error will make its way to zero. This clearly shows that when
the system is stable (very important condition!) the output will settle to the desired
target with minimum error equal to or close to zero. This is true regardless of
the Jacobian
=
gain matrix and even the initial state, x
0
as long as the feedback loop
is stable.
In contrast, for the open-loop case, the following analysis shows that the steady-
state error is not independent of the initial state, x
0
, because the actual states will drift
with time (i.e., actual x(k) will be different from x
0
).
Applying z-transform on the open-loop state equation, we can write the state
vector in z-domain as follows:
B
z
V
(
z
) þ
x
0
z
z
X
(
z
) ¼
(
9
:
42
)
1
For the open-loop case the vector, V(z)
0.
Now the expression for the error vector between the desired inputs to the outputs
without feedback can be written as follows:
¼
E
(
z
) ¼
X
d
(
z
)
X
(
z
)
z
z
B
z
V
(
z
)
¼
X
d
x
0
ð
Þ
1
(
9
:
43
)
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