Environmental Engineering Reference
In-Depth Information
z
−
1
z
−
1
B
(
)
C
(
)
z
−
1
z
−
1
G
(
)=
H
(
)=
)
.
(4.21)
(
z
−
1
)
(
z
−
1
A
A
From (4.18) and (4.21) the one step ahead predictor for the ARMAX case is
given by
z
−
1
z
−
1
z
−
1
z
−
1
C
(
)
y
(
t
|
t
−
1
)=
B
(
)
u
(
t
)+(
C
(
)−
A
(
))
y
(
t
).
(4.22)
For convergence, asymptotic stability of (4.22) is required, for which a necessary
condition is that the roots of
C
z
−
1
(
)
have absolute values that are less than one.
z
−
1
(
)=
For the ARX model class
C
1 in (4.22), so
z
−
1
z
−
1
(
|
−
)=
(
)
(
)+(
−
(
))
(
).
y
t
t
1
B
u
t
1
A
y
t
(4.23)
In the case of soft sensors there are usually several inputs
u
1
,
u
2
, ...
u
r
. Let then
T
z
−
1
z
−
1
z
−
1
z
−
1
u
(
t
)=[
u
1
(
t
)
u
2
(
t
) ...
u
r
(
t
)]
G
(
)=[
G
1
(
)
G
2
(
) ...
G
r
(
)]
(4.24)
z
−
1
z
−
1
z
−
1
z
−
1
G
(
)
u
(
t
)=
G
1
(
)
u
1
(
t
)+
G
2
(
)
u
2
(
t
)+...
G
r
(
)
u
r
(
t
).
(4.25)
Following the same procedure used by Ljung [44] to derive the one step ahead
prediction for one input, given by (4.18), it may easily be shown that in this multiple
input case the one step ahead prediction has the same form,
i.e.
,
z
−
1
z
−
1
z
−
1
H
(
)
y
(
t
|
t
−
1
)=
G
(
)
u
(
t
)+(
H
(
)−
1
))
y
(
t
),
(4.26)
z
−
1
where vectors
u
(
t
)
and
G
(
)
are defined by (4.24).
Example 1.
Let the assumed plant model be the ARMAX model
y
(
t
)+
a
1
y
(
t
−
1
)+
a
2
y
(
t
−
2
)+
a
3
y
(
t
−
3
)=
b
1
u
(
t
−
1
)+
w
(
t
)+
c
1
w
(
t
−
1
)
+
c
2
w
(
t
−
2
),
(4.27)
where
w
is a white noise sequence uncorrelated with
u
. Therefore the noise in (4.27)
is colored (non white). Then
z
−
1
a
1
z
−
1
a
2
z
−
2
a
3
z
−
3
z
−
1
b
1
z
−
1
z
−
1
c
1
z
−
1
c
2
z
−
2
(4.28)
A
(
)=
1
+
+
+
,
B
(
)=
,
C
(
)=
1
+
+
and from (4.22),
y
(
t
|
t
−
1
)=−
c
1
y
(
t
−
1
|
t
−
2
)−
c
2
y
(
t
−
2
|
t
−
3
)+
b
1
u
(
t
−
1
)
−(
a
1
−
c
1
)
y
(
t
−
1
)−(
a
2
−
c
2
)
y
(
t
−
2
)−
a
3
y
(
t
−
3
).
(4.29)
Let
T
θ
=[
c
1
c
2
(
a
1
−
c
1
)(
a
2
−
c
2
)
a
3
b
1
]
.
(4.30)
and
T
ϕ
(
t
)=[−
y
(
t
−
1
|
t
−
2
) −
y
(
t
−
2
|
t
−
3
) −
y
(
t
−
1
) −
y
(
t
−
2
) −
y
(
t
−
3
)
u
(
t
−
.
(4.31)
1
)]
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