Modeling with Neural Networks: Principles and Model Design Methodology - Neural Networks: Methodology and Applications - page 164

Fig. 2.43. Copy k of the feedforward neural network of the canonical form, for
semidirected training

2.7.3.2 Nonadaptive (Batch) Training of Recurrent Input-Output
Models: Semidirected Training

Under the output noise assumption, or the output noise and state noise as-
sumption, the ideal model is a recurrent model, the inputs of which are

• the control inputs and the outputs of the model at the n previous time

steps (under the assumption of output noise alone),

• the control inputs, the outputs of the model and the modeling errors on a

suitable horizon p (under the NARMAX assumption).

Output Noise Assumption

Because the model is recurrent, its training, from a sequence of length N ,
requires unfolding the network into a large feedforward neural network, made
of N identical copies of the feedforward part of the canonical form. The input
of copy k (shown on Fig. 2.43) is

m +1)] T , • the control input vector u ( k )=[ u ( k ) ,...,u ( k −

• the vector of outputs at time k and at the previous n time steps [ y ( k ) ,...,

y ( k n +1)] T .

−

The output vector of copy k is the vector of the outputs at time k +1 and at
the previous n time steps [ r ( k +1) ,...,r ( k

n +2)] T . Therefore, the network
actually computes r ( k + 1) only, the other components of the output vector
being derived from the input vector by a unit delay. The output vector of copy
k is part of the input vector of the next copy, corresponding to time k +1.

−