Environmental Engineering Reference
In-Depth Information
If
R
ϕϕ
is nonsingular (a condition achieved through persistent excitation [44]), an
explicit solution
θ
R
−
ϕϕ
R
ϕ
y
=
(4.58)
is obtained for the optimal vector θof LIP models, such as ARX and NARX models
(see (4.34) in Example 1, and (4.46) in Example 2). In Section 4.4 the same result
is obtained (see (4.110)) in a general inner product space by projecting the primary
measurement
y
(
t
)
onto the space spanned by the base
B
m
= {
ϕ
1
,
ϕ
2
,...,
ϕ
m
}
to
which
y
belongs, and specializing the results to the case of a random variable
inner product space.
But, again, the problem is that the optimal model vector is given in terms of ex-
pected values, and in the large majority of cases the probability functions needed to
compute these expected values are not available. Therefore, they must be estimated
using time averages.
(
t
|
t
−
1
)
t
∑
1
T
R
ϕϕ
ϕ
T
ϕϕ
T
=
ϕ
(
i
)
(
i
) ≈
E
{
},
(4.59)
i
=
t
−
T
+
1
t
∑
1
T
R
ϕ
y
=
ϕ
(
i
)
y
(
i
) ≈
E
{
ϕ
y
},
(4.60)
i
=
t
−
T
+
1
θ
R
−
ϕϕ
R
ϕ
y
=
.
(4.61)
In Section 4.3.2 it can be seen that
R
ϕϕ
is a random matrix and
R
ϕ
y
is a random
vector, so θ is a random vector having an expected value and a covariance matrix.
Hence the estimator θ for a given set of measurements is just one of all possible
outcomes that could have resulted. It should be that realizations of this estimator are
distributed around θ and that variances of the parameter estimations are small. Re-
quired properties for these estimators through time averages then are unbiasedness
and ergodicity or consistency, so that variances decrease to zero as length
T
of time
window
θ converges in the mean to
θ. Ljung
[
t
−
T
+
1
,
t
]
increases to infinity and
[44] shows that estimator θ satisfies these conditions.
If the model is NLIP,
e.g.
, a NARMAX model (see Examples 3 and 4), the opti-
mal parameter vector must be obtained using algorithmic methods [48] to directly
minimize the time average (4.3),
e.g.
, using the System Identification Toolbox of
MATLAB
®
[49], or algorithms used in neural network optimal parameter (weight)
determination [12, 14].
From (4.135) in Section 4.3
θ
X
T
X
)
−
1
X
T
Y
=(
,
(4.62)
where
Search WWH ::
Custom Search