Page 134 - Neural Network Modeling and Identification of Dynamical Systems

3.4 ANN-BASED CONTROL OF DYNAMICAL SYSTEMS 123
Given the above formulations, the concept 3. General optimization problem for the ENC,
of mutual consistency of the neurocontrollers
∗
∗
i ,i ∈{1,...,N}, that are part of the ENC , F( , ,J,E (λ)) = min F( , ,J,E(λ)).
E(λ, ),
mentioned at the end of Section 3.4.2.2, can now N=var
be clarified. (3.56)
Namely, the mutual consistency of the neuro-
In the problem of the optimal distribution
controllers in the ENC (3.42) is as follows:
(3.54) there is a region of the λ operation
• parameters of each of the neurocontrollers modes of the dynamical system (the external set
i ,i ∈{1,...,N}, are selected taking into of the system) and N given by some neurocon-
account all other neurocontrollers j ,j = trollers i ,i = 1,...,N. It is required to assign
i, i,j ∈{1,...,N}, based on the requirements to each neurocontroller i the domain of its spe-
imposed by the optimality criterion of the cialization D i ⊂ ,
MDS (3.50)or(3.51)for theENC as a whole;
D i = D( i ) ={λ ∈ | E(λ) = i},i ∈{1,...,N},
• it is guaranteed that each of the modes (tasks
to be solved) λ ∈ will be worked out by
N
the neurocontroller, the most effective of the
available within the ENC (3.42), that is, such D i = , D j D k = ∅, ∀j,k ∈{1,...,N},
neurocontroller i ,i ∈{1,...,N},for which i=1 j =k
the value of the criterion function (degree of where the use of this neurocontroller i is
nonoptimality for neurocontroller i ) f i (λ,k), preferable to the use comparing all other neuro-
defined by expression (3.48), is the least for controllers j ,j = i, i,j ∈{1,...,N}.The divi-
the given λ ∈ and k ∈ K. sion of the domain into the D i ⊂ specializa-
tion domains is given by the distribution func-
OPTIMIZATION TASKS FOR AN ENSEMBLE OF tion E(λ) defined on the set and takes integer
NEURAL CONTROLLERS WITH A CONSERVA- values 1,2,...,N. The function E(λ) assigns to
TIVE APPROACH each λ ∈ the number of the neurocontroller
With regard to optimization of the ENC, the corresponding to the given mode, such that its
following main tasks can be formulated: criterial function (3.44) will be for this λ ∈ the
smallest in comparison with the criterial func-
1. The problem of optimal distribution for the tions of the remaining neurocontrollers that are
ENC,
part of the ENC.
The problem of the optimal choice of parame-
∗
F( , ,J,E (λ)) = min F( , ,J,E(λ)). ters (3.55) for neurocontrollers i ,I = 1,...,N,
E(λ),
N=const, included in the ENC has the optimal distri-
k=const bution problem (3.54) as a subproblem. It con-
(3.54)
sists in the selection of parameters W (i) and V (i)
of neurocontrollers i ,i = 1,...,N,included
2. The problem of the optimal choice of the pa-
in the ENC , in such a way as to minimize
rameters for neurocontrollers included in the
the value of the ENC optimality criterion (3.50),
ENC,
(3.52)or(3.51), (3.53), depending on the type of
the corresponding application task. We assume
∗
∗
F( , ,J,E (λ)) = min F( , ,J,E(λ)).
E(λ, ), that the number of neurocontrollers N in the
N=const ENC is fixed from any considerations external
(3.55) to the problem to be solved.