Environmental Engineering Reference
In-Depth Information
Since the one step ahead predictions depend on the model parameters, the explicit
notation
ϕ
T
y
(
t
|
t
−
1
)=
(
t
,
θ
)
θ
(4.32)
has been used, since (4.32) is nonlinear in the parameter vector θ .
In the case of the ARX class of models, from (4.23), or letting
c
1
=
c
2
=
c
3
=
0
in (4.29),
y
(
t
|
t
−
1
)=
b
1
u
(
t
−
1
)−
a
1
y
(
t
−
1
)−
a
2
y
(
t
−
2
)−
a
3
y
(
t
−
3
),
(4.33)
so
T
ϕ
(
t
)=[−
y
(
t
−
1
) −
y
(
t
−
2
) −
y
(
t
−
3
)
u
(
t
−
1
)]
(4.34)
does not depend on θ .Then
ϕ
T
y
(
t
|
t
−
1
)=
(
t
)
θ
(4.35)
is linear in the parameter vector
T
θ
=[
a
1
a
2
a
3
b
1
]
.
(4.36)
Example 2.
Let the assumed plant model plant model be of the NARX class
b
2
u
2
y
(
t
)=−
a
1
y
(
t
−
1
)−
a
2
y
(
t
−
2
)−
a
3
y
(
t
−
3
)+
b
1
u
(
t
−
1
)+
(
t
−
1
)
u
(
t
−
1
)
+
b
3
−
u
)
+
b
4
l
(
t
)+
w
(
t
).
(4.37)
3
(
t
−
1
Here
l
is a function having a real value 1 for all discrete times
t
that must be
considered as an additional input to account for the constant term
b
4
.
Let
(
t
)
u
2
(
−
)=
(
−
)
(
−
)=
(
−
)
u
1
t
1
u
t
1
u
2
t
1
t
1
(
−
)
u
t
1
u
3
(
t
−
1
)=
u
u
4
(
t
−
1
)=
l
(
t
).
(4.38)
−
(
−
)
3
t
1
Replacing (4.38) in (4.37) gives
y
(
t
)=−
a
1
y
(
t
−
1
)−
a
2
y
(
t
−
2
)−
a
3
y
(
t
−
3
)+
b
1
u
1
(
t
−
1
)+
b
2
u
2
(
t
−
1
)
+
b
3
u
3
(
t
−
1
)+
b
4
u
4
(
t
−
1
)+
w
(
t
),
(4.39)
i.e.
, a single output ARX class model with four inputs.
(
−
)+
(
−
)+
(
−
)+
(
−
)
(
)
b
1
u
1
t
1
b
2
u
2
t
1
b
3
u
3
t
1
b
4
u
4
t
1
w
t
y
(
t
)=
+
)
,
(4.40)
A
(
z
−
1
)
A
(
z
−
1
z
−
1
a
1
z
−
1
a
2
z
−
2
a
3
z
−
3
.
where
A
(
)=
1
+
+
+
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