Page 133 - Neural Network Modeling and Identification of Dynamical Systems

122 3. NEURAL NETWORK BLACK BOX APPROACH TO THE MODELING AND CONTROL OF DYNAMICAL SYSTEMS

These properties are attached to the construc- of estimating the effectiveness of a system on an
tion function to ensure the independence of the external set are reduced to two classes: guar-
criterion F( ) from the order of combining the anteed estimates, when the optimality criterion
elements on the set . F( ) takes the form
The function (η,ν) gives a way to calculate
the efficiency F( αβ ) of the ENC for the MDS F( ) = max [ρ(λ)f (λ,k)], (3.50)
λ∈ ,
with the external set k=const

αβ = α β , and integral, for which F( ) is an expression of
the form
by known efficiencies f( α ) and f( β ), where
α ⊂ , β ⊂ , i.e., F( ) = [ρ(λ)f (λ,k)], (3.51)
λ∈ ,
k=const

F( α β ) = (f ( α ),f ( β )).
where ρ(λ) is the degree of relative significance
In the same way we can define on an exterior set for the element λ ∈ .
Taking into account the above, the criterial

αβγ = ( α β ) γ , γ ⊂ , function f(λ,k) can be treated as a degree of
nonoptimality of ENC for the MDS that operates
the efficiency F( αβγ ) for the system on it, i.e., on the λ ∈ mode. Then we can say that for a
criterion of the form (3.50), the problem of min-

F( αβγ ) = F(( α β ) γ ) imizing the maximum degree of nonoptimality
of ENC for the MDS with an external set

= (f ( γ ), ( α β )). will be solved; we have
∗
∗
Repeating this operation further, we get F = F G ( ) = min max [ρ(λ)f (λ,k)]. (3.52)
G
λ∈ ,
k=const
F( 1 ( 2 ( 3 ...( N−1 N )...)...)
With the integral criterion, the ENC optimiza-
= (f ( 1 ), (f ( 2 ), (f ( 3 ),...))). tion problem reduces to minimizing the integral
degree of nonoptimality of the ENC for an
Bearing in mind that
MDS acting on the external set , i.e.,

1 ( 2 ( 3 ...( N−1 N )...)) = ,
F = F I ( ) = min [ρ(λ)f (λ,k)]dλ.
∗
∗
I
we get
λ∈ ,
k=const
F( ) = (f ( 1 ), (f ( 2 ), (f ( 3 ),...))). (3.53)
(3.49) Applied to the integral criterion (3.53), the
rule from (3.49) actually defines the weight func-
The method of constructing the rule in tion ρ(λ), which specifies the relative impor-
(3.49), which allows us to know f(λ,k), ∀λ ∈ tance of the elements λ of the external set .
, ∀k ∈ K, and find the value of the optimality Accordingly, the integral criterion (3.51), (3.53)
criterion F( ) of a multimode system of general can be varied when it is formed within wide lim-
form on the entire external set , is described its, corresponding to the specifics of the applied
in [88]. It is shown here that all possible types task.