Page 132 - Neural Network Modeling and Identification of Dynamical Systems

3.4 ANN-BASED CONTROL OF DYNAMICAL SYSTEMS 121
∗
one way or another, to shorten the expressions If k ∈ K is the point of absolute minimum of
in this section, we write, instead of (3.44), the functional J for λ ∈ , then the following in-

equality will be satisfied:
f = f(λ), (3.45)
∗
∗
J(λ,k ) J(λ,k ), ∀λ ∈ . (3.47)
and instead of F( , ,J,E(λ)), F = F( ).
Thus, the problem of finding the value of the Based on the condition (3.47), we write the ex-
optimality criterion for the ENC is divided into pression for the criterial function f(λ,k) in the
two subtasks: first, we need to be able to calcu- form
late f(λ), ∀λ ∈ , taking into account the above
∗
assumptions; second, it is necessary to deter- f(λ,k) = J(λ,k) − J(λ,k ), (3.48)
mine (generate) the rule that would allow us
to find F( ), that is, the ENC optimality crite- where J(λ,k) is the value of the functional J
rion on the whole external set ,by f(λ), i.e., for arbitrary admissible λ ∈ and k ∈ K,and
∗
J(λ,k ) is the minimal value of the functional J
F( ) = [f(λ)]. (3.46) for the parameter vector of the neurocontroller,
optimal for the given λ ∈ .
Let us first consider the question of construct- Finding the value J(λ,k) does not cause any
ing a criterial function f(λ). We will assume difficulties and can be performed by calculating
that the ENC is defined if for each of the neu- it together with (3.31) by one of the numerical
rocontrollers { i },i = 1,...,N, we know the set methods for solving the Cauchy problem for a
of synaptic weight matrices W (i) ={W (i) }, i = 1, system of ODEs of the first order. The calcula-
j
∗
...,N, j = 1,...,p (i) + 1, where p (i) is the num- tion of the value J(λ,k ), that is, minimum of
the functional J(λ,k) under the parameter vec-
ber of hidden layers in the ANN used in the
∗
tor k ∈ K, which is optimal for a given current
neurocontroller i , as well as the value of the
vector v (i) ∈ V of additional adjusting parame- λ ∈ , is significantly more difficult, because in
this case we need to solve the problem of synthe-
ters of this neurocontroller. Suppose that we also
s
have an external set ⊂ R for the MDS. If we sis of the optimal control law for a single-mode
dynamical system. This problem in the case un-
also freeze the “point” λ ∈ , then we get the

der consideration relates to the training of the
usual problem of synthesizing the control law
for a single-mode system. Solving this problem, ANN used in the correcting module of the cor-
responding neurocontroller.
∗
we find k ∈ K, that is, the optimal value of the
We now consider the problem of determin-
regulator parameters vector (3.33)–(3.35) under ing the rule (construction function) , which al-
the condition λ ∈ . In this case, the functional J lows us to find the value of the optimality cri-

takes the value
terion F( ) for the given ENC ,ifweknow
f(λ,k), ∀λ ∈ , ∀k ∈ K.
∗
∗
J (λ) = J(λ,k ) = minJ(λ,k).

k∈K The construction function is assumed to be
symmetric, i.e.,
If we apply the neurocontroller with the pa-
rameter vector k ∈ K, which is optimal for the (η,ν) = (ν,η),
∗
point λ ∈ ,inthe point λ ∈ , then the func-

tional J will be and associative, i.e.,
∗
∗
∗
J = J(λ,k ). (η, (ν,ξ)) = (ν, (η,ξ)) = (ξ, (η,ν)).