Modeling with Neural Networks: Principles and Model Design Methodology - Neural Networks: Methodology and Applications - page 172

z ( k +1)= φ ( z ( k )) , u ( k ))
y ( k +1)= ψ ( z ( k ) , u ( k )) ,
where z ( k ) is the minimal set, made of ν variables, that allows the derivation
of the state of the model at time k + 1, given the state of the model and
its inputs at time k , and where functions φ and ψ can be implemented as
feedforward neural networks.
The order of the canonical form is ν . It is convenient, but not mandatory, to
design the predictor as a single neural network, whose inputs are the control
inputs and the state variables at time k , and whose outputs are the state
variables at time k + 1 (Fig. 2.48).
Fig. 2.48. Canonical form of a recurrent network
A general technique, which allows a fully automatic derivation of the
canonical form of any recurrent network, is described in [Dreyfus et al. 1998].
An illustrative example is given below.
2.7.5.2 An Example of Derivation of a Canonical Form
The analysis of a process has led to the following model:
x 1 = φ 1 ( x 1 ,x 2 ,x 3 ,u )
x 2 = φ 2 ( x 1 ,x 3 )
x 3 = φ 3 ( x 1 , x 2 )
y = x 3 .
Its discrete-time equivalent, derived with the explicit Euler discretization
method, is given by