Neural Identification of Controlled Dynamical Systems and Recurrent Networks - Neural Networks: Methodology and Applications - page 270

empirical knowledge on perturbations and system noise. Thus, the tracking
abilities of the algorithm may be tuned. We will apply that method to neural
control in the next chapter.
4.5 Recurrent Neural Networks
4.5.1 Neural Simulator of an Open-Loop Controlled Dynamical
System
In the section that was dedicated to neural identification of a controlled dy-
namical system, a feedforward neural network was designed as a one-step-
ahead prediction model. We presented on Fig. 4.11 the diagram of the learn-
ing process of an input-output model according to the NARX hypothesis.
We showed in Chap. 2, in the section entitled “black box modeling” and in
the section “state noise hypothesis, input-output representation”, that that
approach is relevant when a state noise is present: the output of the model
at time k may be reconstructed from the past values of the process output
and the past values of the control signal. After completion of training, the
network output may be plugged into the state input, with a delay operator in
the feedback loop. That generates a recurrent neural network. Actually, the
network graph exhibits a closed circuit. One may use that recurrent network,
which models the function ψ RN , to predict the process output within a finite
horizon.
Figure 4.14 shows an input-output recurrent neural network: the network
state input consists in past values of the output. If the network parameters
have been estimated using open-loop training as in Fig. 4.11, and if the net-
work is used to predict the process output beyond one time step, then it was
shown in Chap. 2 that such an approach is not optimal: the prediction is
corrupted by the iteration of the state noise. Conversely, it was shown theo-
retically, and illustrated through examples, that if the noise is a measurement
noise (output noise), if training was performed by a semidirected algorithm,
and if, during learning, the model outputs were used as input states, then the
accuracy of the prediction is optimal.
In that context, we assumed that the control signal u ( k )didnotdependon
the state (actually the output of the network). It was therefore an open-loop
controlled dynamical system. We shall now model a closed-loop controlled
dynamical system using a combination of neural networks.
4.5.2 Neural Simulator of a Closed Loop Controlled Dynamical
System
Just as a feedforward neural network whose inputs were the (state+control)
signals, and whose output was a state output, was used to model a controlled