Neural Identification of Controlled Dynamical Systems and Recurrent Networks - Neural Networks: Methodology and Applications - page 286

Fig. 4.21. The delays are crucial to update a neural network
connection weights are the same for cases (a) and (b) and for cases (c) and
(d). Those two pairs of networks are not simple replication because a delay
operator was added in the networks (b) and (d). We shall now investigate the
consequences of that feedback, assuming that the network inputs are constant.
In case (a), at time 1, the state of unit 3 depends on the initial states of
units 2 and 4, as the state of unit 4 is determined by the state of unit 1. At
time 2, the state of unit 3 is determined by the state of units 2 and 4, hence by
the states of units 1 and 2. In case (b), the state of unit 3 is only determined
at time 2. At that time, its state value is similar to the state value of unit 3
in case (a).
Remark. In fact, open-loop feedforward neural networks that are fed by
static inputs stabilize onto a final state, which depends only on the initial
input state, whatever the distribution delay. Therefore, it does not depend on
the update order; the units are assumed to be updated synchronously.
Moreover, the update order and the delay distribution are not taken into
account in a layered network with feedforward information propagation and
connections only from each layer to the following layers. In Fig. 4.14, even if
the network is operated in an open-loop fashion with a connection of the state
at time k towards the controller and the internal model, there is an ambiguity
for the update order. In that case, the relevant rule to update units is to
update synchronously the units of a same layer and sequentially along the
information propagation direction. Thus, the units of the first hidden layer of
the internal model have to wait, to be updated, for the controller to deliver
the control signal that is an input of the internal model. That rule is even
more important if the inputs are time-dependent.
In that case, one has to discriminate between the simulation time repre-
sentation (one time step for operating the whole simulation block including
the controller and the internal model) and the update steps of the different
layers of the whole simulation block that are nested in one time step of the
algorithm.