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+Cov[
V
(
k
+1)
,
HV
(
k
+1)+
W
(
k
+1)]
=
A
Cov[
X
(
k
)
,
X
(
k
)
X
(
k
)]
A
T
H
T
+Var[
V
(
k
+1)]
H
T
.
−
X
(
k
) with
X
(
k
)
X
(
k
), one gets
Yet, from the correlation of
−
X
(
k
)] = Var[
X
(
k
)
−
X
(
k
)] =
P
k
.
Cov[
X
(
k
)
,
X
(
k
)
−
Therefore,
HA X
(
k
)
Cov[
Y
(
k
+1)
HBu
(
k
)
,
X
(
k
+1)]
=(
AP
k
A
T
+
Q
)
H
T
=
P
k
+1
H
T
.
−
−
Finally, we obtain,
K
k
+1
=
P
k
+1
H
T
[
HP
k
+1
H
T
+
R
]
−
1
.
In order to iterate the algorithm, which is recursive, let us compute finally
the covariance matrix of the estimation error at time
k
+ 1. From its value,
X
(
k
+1)=
A
[
X
(
k
)
X
(
k
)] +
V
(
k
+1)
X
(
k
+1)
−
−
−
K
k
+1
[
Y
(
k
+1)
HA X
(
k
)
−
−
HBu
(
k
)]
X
(
k
+1)=(
I
X
(
k
)] +
V
(
k
+1)
X
(
k
+1)
−
−
K
k
+1
H
)
{
A
[
X
(
k
)
−
}
−
K
k
+1
W
(
k
+1)
,
the expression of the covariance matrix can be derived,
K
k
+1
H
)(
AP
k
A
T
+
Q
)(
I
K
k
+1
H
)
T
+
K
k
+1
RK
T
P
k
+1
=(
I
−
−
k
+1
.
4.7.2 The Delay Distribution Is Crucial for Recurrent Network
Dynamics
In this chapter, we provided examples of recurrent neural networks. Most of
them were of the input-output type, i.e., they were built from a feedforward
neural network whose outputs are fed back to the input with a unit time delay.
Other recurrent network models have been shown in this chapter (Hopfield
and Elman models) and in Chap. 2 (“gray box” modeling taking into account
algebraic and differential equations from prior knowledge for the network ar-
chitecture).
Let us emphasize that for a recurrent neural networks, delay distribution
has to be specified. If it is neglected, the network behavior is not properly de-
fined. To illustrate this, Fig. 4.21 shows a comparison of the delay specification
for a network without any closed circuit in the connection graph (feed-forward
network) and a recurrent network with circuits in the connection graph.
Pictures (a) and (b) show the graph of an elementary feedforward net-
work with four connections. In pictures (c) and (d), feedback was added. The
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