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may either converge to a local minimum or lead to undesirable nonlinear
computations [165,172].
2. The
equation-error method
, which minimizes the equation error. The LS
criterion gives a biased estimate. The TLS criterion gives a strong consistent
estimate (see Proposition 33).
With the second formulation, the IIR filter output is given by
N
−
1
M
−
1
y
(
t
)
=
s
i
(
t
)
d
(
t
−
i
)
+
c
i
(
t
)
u
(
t
−
i
)
(4.40)
i
=
1
i
=
0
with
{
u
(
t
)
}
and
{
d
(
t
)
}
the measured input and output sequences from the
unknown system. The error signal is defined as
e
(
t
)
=
y
(
t
)
−
d
(
t
)
(4.41)
The input vector for the TLS neurons is
]
T
a
(
t
)
=
[
d
(
t
−
)
...
,
d
(
t
−
N
+
)
,
u
(
t
)
...
,
u
(
t
−
M
+
)
1
,
1
,
1
(4.42)
y
(
t
)
and
d
(
t
)
are, respectively, the neuron output and the target. At convergence,
the weights yield the unknown parameters
s
i
(
t
)
and
c
i
(
t
)
. The neuron output error
e
(
t
)
yields the IIR error signal.
As a benchmark problem [63,64], consider that the unknown system parame-
ters are
[
s
1
,
s
2
,
c
0
,
c
1
,
c
2
]
T
=
[1
.
1993,
−
0
.
5156, 0
.
0705, 0
.
1410, 0
.
0705]
T
(4.43)
The unknown system is driven by a zero-mean uniformly distributed random
white signal with a variance of 1.
The initial conditions are null (it agrees with the TLS GAO assumption).
The index parameter
, introduced in Section 2.10, is used as a measure of the
accuracy. The learning rate
ρ
2
to 0
.
01, reached at null tangent at iteration 6000; then it is a constant equal to 0
.
01.
The TLS EXIN neuron is compared with the TLS GAO neuron and the recursive
least squares (RLS) algorithm [82] based on the equation-error formulation. In
the first simulations, the additive input and output white noises are distributed
α(
t
)
is a decreasing third-order polynomial from 0
.
uniformly over
−
κ/
2,
κ/
2
. Figures 4.2 to 4.6 show the dynamic behavior of
the weights for the two TLS neurons for
κ
=
0
.
4. They confirm the TLS EXIN
features cited above with respect to the behavior of TLS GAO. Figure 4.7 shows
the index parameter
ρ
. An improvement of more than 50 dB is obtained with
respect to TLS GAO. Table 4.1 (
σ
2
=
variance) summarires the results obtained
for TLS EXIN and shows the results obtained for TLS GAO and RLS, as given
in [63] and [64]. Despite a certain computational burden, TLS EXIN outperforms
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