Modeling with Neural Networks: Principles and Model Design Methodology - Neural Networks: Methodology and Applications - page 160

y p ( k )

Σ

ψ

x p 1 ( k )

x p 2 ( k )

ϕ

q -1

x p 2 ( k -1)

b ( k ) u ( k )

x p 1 ( k -1)

Fig. 2.41. State-space representation, output noise assumption

Output Noise Assumption (State-Space Representation)

In the previous sections, we discussed several noise assumptions, and derived
ideal models in each case, under the form of input-output representations. We
now discuss the same assumptions, but we seek models that are in state-space
representations, which are more general and parsimonious than input-output
representations.
We first make the output noise assumption, whereby the process can be
appropriately described by equations of the form

x ( k )= ϕ ( x ( k − 1) , u ( k − 1))

y ( k )= ψ ( x ( k )) + b ( k ) ,

as shown on Fig. 2.41 for a second-order model.
Because noise is present in the observation equation only, it has no in-
fluence on the dynamics of the model. From arguments similar to those we
developed for input-output representations, the ideal model is recurrent, as
shown on Fig. 2.42:

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

y ( k )= ψ NN ( x ( k )) ,

where ϕ NN is exactly function ϕ et ψ NN is exactly function ψ .

State Noise Assumption

We now assume that the process can be appropriately described by equations

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

y ( k )= ψ ( x ( k )) .