Page 131 - Neural Network Modeling and Identification of Dynamical Systems

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

where each of the neurocontrollers i , i ∈ 3.4.2.3 Optimization of an Ensemble of
{1,...,N},isusedonits owndomain D i ⊂ , Neural Controllers for a Multimode
whichwecallthe area of specialization of the neu- Dynamical System
rocontroller i : For the ENC, it is very important to optimize
it. The solution of this problem should ensure
N
the minimization of the number of neurocon-
D i ⊂ , D i = , D i D j = ∅,
trollers in the ensemble for a given external set,
i=1 i =j
that is, for a given region of MDS operation
∀i,j ∈{1,...,N}. modes. If, for some reason, besides the exter-
nal set of MDS, the number of neural controllers
We determine the rule of transition from one
in the ENC is also specified, then its optimiza-
neurocontroller to another using the distribution
tion allows choosing the values of the neurocon-
function E(λ), whose argument is the external
troller parameters so as to minimize the error
vector λ ∈ , and its integer values are the num-
generated by the ENC. The key problem here,
bers of the specialization areas and, respectively,
as in any optimization problem, is the formation
neurocontrollers operating them, i.e.,
of an optimality criterion for the system under
consideration.
E(λ) = i, i ∈{1,...,N},
(3.43)
D i ={λ ∈ | E(λ) = i}. FORMATION OF AN OPTIMALITY CRITERION
FOR AN ENSEMBLE OF NEURAL CONTROLLERS
The distribution function E(λ) is realized ac- FOR A MULTIMODE DYNAMICAL SYSTEM
cording to (3.43)bythe 0 element of the neu- One of the most important points in solving
rocontroller ensemble. ENC optimization problems is the formation of
It should be emphasized that the ENC (3.42) the optimality criterion F( , ,J,E(λ)), taking
is a set of mutually agreed neurocontrollers. All into account all the above mentioned features of
these neurocontrollers have the same current the MDS and ENC. Based on the results obtained
value of the external vector λ ∈ . in [88], it is easy to show that such a criterion
In (3.42) there are two types of neurocon- can be constructed knowing the way to calcu-
trollers. Neurocontrollers of the first type form late the efficiency of the considered system at
aset { 1 ,..., N }, members of which imple- the current λ point of the external set for a
ment the corresponding control laws. The neu- fixed set { i },i = 1,...,N, of neural controllers
rocontroller of the second type ( 0 )isakindof in the ENC as well as for the fixed distribution
1
“conductor” of the ENC 1 ,..., N . This neu- function E(λ). In addition, we need to know the
rocontroller for each current λ ∈ according to functional, which takes the form (3.38), (3.40)or
(3.43) produces the number i, 1 i N,thatis, (3.39), (3.41). A function describing the efficiency
indicates which of the neurocontrollers i ,i ∈ of the ENC under these assumptions,
{1,...,N}, have to control at a given λ ∈ .
Thus, the ENC is a mutually agreed set of f = f(λ, ,J,E(λ)), ∀λ ∈ , (3.44)
neurocontrollers, in which all neurocontrollers
get the current value of the external vector λ ∈ we call criterial function of the ENC. Since (3.44)
as an input. Further, the neurocontroller 0 by depends actually only on λ ∈ , and all other ar-
the current λ in accordance with (3.43) generates guments can be treated as parameters frozen in
the number i, 1 i N, indicating one of the
neurocontrollers i , which should control for a 1 That is, for the given quantity N of neurocontrollers i in
given λ ∈ . the ENC and the values of the parameters W and V.