Page 43 - Neural Network Modeling and Identification of Dynamical Systems

1.3 A GENERAL APPROACH TO DYNAMICAL SYSTEM MODELING 31

(u(t),ξ,ζ) − (u,ξ,ζ) (˜u,ξ,ζ) − (˜u,ξ,ζ)
= max | (u(t),ξ,ζ) − (u(t),ξ,ζ)|. (1.31) 1
N T
t 0 t t n 2
= [ (˜u j ,ξ,ζ) − (˜u j ,ξ,ζ)] ; (1.36)
N T
The second, more common, approach is to es- j=0
timate the values of the difference between (·) it can also be represented in the form
and (·) as the norm of the form
(˜u,ξ,ζ) − (˜u,ξ,ζ)
(u,ξ,ζ) − (u,ξ,ζ)

t n
N T
2
= [ (u(t),ξ,ζ) − (u(t),ξ,ζ)] dt. (1.32) 1 2
= [ (˜u j ,ξ,ζ) − (˜u j ,ξ,ζ)] .
t 0 N T
j=0
The number of experiments available for gen- (1.37)
eration of the data set (1.29) is finite. There-
fore, instead of (1.32), one of the possible finite- Now we can formulate the problem of the
dimensional versions of the given expression model synthesis for the dynamical system S.
should be used, for example, the standard de- It is required to construct a model (·),which
viation of the form will reproduce the mapping (·), implemented
by the system S, with the required level of ac-
(u,ξ,ζ) − (u,ξ,ζ)
curacy. This means that the magnitude of the
N P simulation error (1.36)or(1.37) on the test set
1 2
= [ (u i ,ξ,ζ) − (u i ,ξ,ζ)] (1.33) (1.35) should not exceed the specified maximum
N P
i=0 allowed value ε in (1.30) for such model (·).
or The model synthesis procedure should be based
on the data (1.29) used to adjust (learning) the
(u,ξ,ζ) − (u,ξ,ζ) model, and data (1.35) used to test the model. In
addition, we may involve available knowledge
N P
1 about the simulated system S.
2
= [ (u i ,ξ,ζ) − (u i ,ξ,ζ)] .
N P It is assumed that in order to solve this prob-
i=0 lem, we need to select the optimal (in some
(1.34)
∗
sense) model (·) from some finite or infinite
Testing the mapping (·) in order to evaluate family (set) of options j (·), j = 1,2,....Inthis
its generalization properties is performed using regard, the following two questions arise:
a set of ordered pairs similar to (1.29), • What is the given family of variants (F) =
{ j (·)}, j = 1,2,...?
{ ˜u j , ˜y j }, ˜u ∈ U, ˜y ∈ Y, i = 1,...,N T ; (1.35)
• How can we choose (·) from the family
∗
(F) , so that it satisfies the condition (1.30).
it is necessary that the condition u i = ˜u i is ful-
filled, ∀i ∈{1,...,N P }, ∀j ∈{1,...,N T },thatis, Abovementioned family of options for the
all pairs in sets model should meet the following two require-
ments, which, in general case, may contradict
N P N T
{ u i ,y i } , { ˜u j , ˜y i }
i=1 j=1 each other:
must be distinct. • a family of models { j (·)},j = 1,2,...,should
The error on the test set (1.35)iscomputedin be as rich as possible in order to “have a lot of
the same way as for the training set (1.29), i.e., options to choose from”;