Environmental Engineering Reference
In-Depth Information
Figure 4.4
Example of general black box or gray box model structure (4.8), where
y
is the model
output, ξ is a vector of selected model inputs, possibly including delayed versions of
y
,and
parameters θ
k
define a linear combination of functions ϕ
k
=
ϕ
k
(
t
,
μ
)
cance or be terms of expansions, as in truncated wavelet, Fourier, and Taylor series
expansions.
There is a general expression for a one step ahead predictor
y
(
t
|
t
−
1
)
for an
assumed discrete time plant model of the form
z
−
1
z
−
1
(
)=
(
)
(
)+
(
)
(
),
y
t
G
u
t
H
w
t
(4.17)
z
−
1
(
)
(
)
(
)
where
w
t
is a white noise random sequence uncorrelated with
u
t
,and
G
z
−
1
are fractions of polynomials in the delay operator
z
−
1
(
)
and
H
. This predictor
[1, 44] is given by
z
−
1
z
−
1
z
−
1
H
(
)
y
(
t
|
t
−
1
)=
G
(
)
u
(
t
)+(
H
(
)−
1
)
y
(
t
).
(4.18)
In the ARMAX case
z
−
1
z
−
1
z
−
1
A
(
)
y
(
t
)=
B
(
)
u
(
t
)+
C
(
)
w
(
t
),
(4.19)
where
A
,
B
,and
C
are polynomials in the delay operator
z
−
1
[1, 44],
z
−
1
z
−
1
B
(
)
C
(
)
y
(
t
)=
u
(
t
)+
w
(
t
),
(4.20)
A
(
z
−
1
)
A
(
z
−
1
)
so
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