Environmental Engineering Reference
In-Depth Information
z
−
1
z
−
1
z
−
1
y
(
t
+
j
)=
F
j
(
))
y
(
t
)+
E
j
(
)
B
(
)
Δ
u
(
t
−
d
+
j
−
1
)
(5.46)
z
−
1
z
−
1
z
−
1
+
E
j
(
)
H
(
)
Δ
p
(
t
−
d
+
j
−
1
)+
E
j
(
)
w
(
t
+
j
)
z
−
1
As the last term associated with noise
E
j
(
)
w
(
t
+
j
)
only has terms in the future,
the best prediction is
z
−
1
z
−
1
z
−
1
y
(
t
+
j
|
t
)=
F
j
(
))
y
(
t
)+
G
j
(
)
Δ
u
(
t
−
d
+
j
−
1
)+
L
j
(
)
Δ
p
(
t
−
d
+
j
),
(5.47)
−
1
z
−
1
z
−
1
z
−
1
z
−
1
z
−
1
z
−
1
where
G
j
.
Thus, the set of
N
-ahead predictions starting from
d
steps in the future is:
(
)=
E
j
(
)
B
(
)
and
L
j
(
)=
E
j
(
)
H
(
)
z
−
1
z
−
1
z
−
1
y
(
t
+
d
+
1
|
t
)=
F
d
+
1
(
))
y
(
t
)+
G
d
+
1
(
)
Δ
u
(
t
)+
L
d
+
1
(
)
Δ
p
(
t
)
z
−
1
z
−
1
z
−
1
y
(
t
+
d
+
2
|
t
)=
F
d
+
2
(
))
y
(
t
)+
G
d
+
2
(
)
Δ
u
(
t
+
1
)+
L
d
+
1
(
)
Δ
p
(
t
+
1
)
.
z
−
1
z
−
1
z
−
1
).
(5.48)
The coefficients of
F
j
,
G
j
and
L
j
can be efficiently calculated by recursive methods
[16] or by matrix calculations [17].
These predictions can be expressed in matrix notation as follows:
y
(
t
+
d
+
N
|
t
)=
F
d
+
N
(
))
y
(
t
)+
G
d
+
N
(
)
Δ
u
(
t
+
N
)+
L
d
+
N
(
)
Δ
p
(
t
+
N
G
z
−
1
z
−
1
z
−
1
=
+
(
)
(
)+
(
)
(
−
)+
(
)
,
y
Gu
F
y
t
Δ
u
t
1
L
p
(5.49)
where
T
y
=[
y
(
t
+
d
|
t
),...,
y
(
t
+
d
+
N
|
t
)]
T
u
=[
Δ
u
(
t
),...,
Δ
u
(
t
+
N
−
1
)]
(5.50)
T
p
=[
Δ
p
(
t
),...,
Δ
p
(
t
+
N
−
1
)]
⎡
⎤
g
0
0
...
0
⎣
⎦
g
1
g
0
...
0
G
=
(5.51)
.
.
.
.
.
.
g
N
−
1
g
N
−
2
...
g
0
⎡
⎣
⎤
⎦
z
−
1
(
G
d
+
1
(
)−
)
g
0
z
z
−
1
g
1
z
−
1
z
2
(
G
d
+
2
(
)−
g
0
−
)
G
z
−
1
(
)=
(5.52)
.
z
−
1
g
1
z
−
1
g
N
−
1
z
N
−
1
z
N
(
G
d
+
N
(
)−
g
0
−
−...−
)
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
T
z
−
1
z
−
1
L
d
+
1
(
)
F
d
+
1
(
)
z
−
1
z
−
1
L
d
+
2
(
)
F
d
+
2
(
)
z
−
1
z
−
1
L
(
)=
,
F
(
)=
(5.53)
.
L
d
+
N
.
z
−
1
z
−
1
(
)
F
d
+
N
(
)
=
,...,
−
Parameters
g
i
,
i
0
N
1arethecoefficients of
G
N
−
1
and can be obtained by
solving (5.43) for
E
j
and
F
j
.
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