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sinusoidal test signal. Statistical estimation methods largely rely on
normal
operating records
and are robust against the noise.
Consider the multivariate time series in which supposedly a single output
y
depends on some lagged values
i
= 0, 1, 2, …, of the input time series in
x
,
ti
the following way:
ycxcx cx
...
[
t
0
t
1
t
1
2
t
2
t
where [(
t
) is the output noise component of
y
. After introducing the unit delay
operator
D
this becomes
2
yccDcD
[
,
[
...
]
t
0
1
2
t
which finally results in
y Dx
[
[]
,
t
t
t
where
[]
cD
represents the transfer function model of the system. It is supposed that
the input series and the noise component [(
t
) are mutually independent.
In systems engineering, the ARMA(
n
,
m
) model
yt
()
ayt
(
1) ...
a yt n
(
)
et
()
cet
() ...
c et m
(
)
1
n
1
m
plays a key role in model building. The compact form of the above model is
1 1
()() ()( ,
A zyt
Czet
where the polynomials
A
(
z
-1
) and
C
(
z
-1
) are the respective operators,
i.e.
the
polynomials in
z
-1
. The corresponding transfer function of the system is
1
yt
()
C z
(
)
.
1
et
()
A z
(
)
2.6.2 State-space Models
In systems theory, the widely preferred class of models are the
state-space models
.
The models are made of two sets of equations. One set represents the state-space
model of the system
x
Ax
Bu
w
,
t
1
t
t
t
t
t
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