Modeling with Neural Networks: Principles and Model Design Methodology - Neural Networks: Methodology and Applications - page 162

• If the state variables are not measured, the ideal model should involve both

the state and the measured process output; therefore, it is in the form

x ( k )= ϕ ( x ( k − 1) , u ( k − 1) ,y p ( k − 1))

y ( k )= ψ ( x ( k )) .

2.7.2.6 Summary on the Structure, Training, and Operation of
Dynamic State-Space Models

Table 2.2 summarizes the noise assumptions and their consequences on the
training of state-space dynamic models.

Table 2.2. Consequences of noise assumptions on the training of dynamic state-
space models

Recommended

Assumption Training operation

State noise Directed One-step-ahead

(measured state) predictor

State noise Semidirected Simulator

(state not measured) (non optimal)

Output noise Semidirected Simulator

State noise and output Semidirected One-step-ahead

noise predictor

2.7.3 Nonadaptive Training of Dynamic Models in Canonical Form

The previous sections have shown how to choose the structure of the dynamic
model, as a function of the noise that is likely to be present in the process, so
that one can hope to approach the ideal model, i.e., the model that accounts
for the deterministic part of the process. We assume that appropriate sequence
of inputs and outputs are available: we consider nonadaptive (batch) training.
In all the following, we assume that the model whose training must be
performed is in canonical form, i.e., it is under the form

z ( k +1)= Φ ( z ( k ) , u ( k ))
y ( k +1)= Ψ ( z ( k ) , u ( k )) ,

where z ( k ) is the minimal set of ν variables, which allows the computation of
the model at time k + 1 knowing the state and the inputs of the model at time
k , and where the vector functions Φ and Ψ are feedforward neural networks.