Modeling with Neural Networks: Principles and Model Design Methodology - Neural Networks: Methodology and Applications - page 161

Fig. 2.42. Ideal model for a state-space representation with the “output noise”
assumption

Then, from an argument that is similar to those developed for input-output
models, the inputs of the ideal model should be, in addition to the control
inputs u , the state variables of the process. Two situations must be considered:

• Those variables are measured: then they can be considered as outputs, so

that the problem is amenable to the design of an input-output model: the
ideal model is a feedforward one, which can essentially be operated as a
one-step-ahead predictor.

• Those variables are not measured: then the ideal model cannot be con-

structed; in such a case, one should either use an input-output representa-
tion (although not completely general), or design a feedback model (al-
though non optimal).

Output Noise and State Noise (State-Space Representation)

Finally, we assume that the process can be appropriately described by the
equations

x ( k )= ϕ ( x ( k − 1) , u ( k − 1) , b 1 ( k − 1)) ,

y ( k )= ψ ( x ( k )) .

Here again, two cases must be considered:

• If the state variables are measured, they can be regarded as outputs, so that

the problem is amenable to the design of an input-output representation,
as described previously.