Page 88 - Neural Network Modeling and Identification of Dynamical Systems

76 2. DYNAMIC NEURAL NETWORKS: STRUCTURES AND TRAINING METHODS

(e) (i) (j)
out given the initial state x under the influ- We define on these ranges a grid { , } as
0
(e)
ence of some control u (t) on the time interval follows:
[t 0 ,t f ]. Comparing the process {x (e) (t k )} with the (i) (s i ) min
: x = x + s i x i ,
process {x(t k )}, obtained for the same initial con- i i
(e) (rusinde) i = 1,...,n; s i = 0,1,...,N i ,
ditions x and control u (t), in fact, that
0 (2.116)
is, for some fixed value of the parameters w,it (j) : u (p j ) = u min + p j u j ,
j j
would be possible in some way to determine the
j = 1,...,m; p i = 0,1,...,M j .
distance between the required and actually im-
plemented processes, and then try to minimize In expressions (2.116), we have
it by varying the values of the parameters w.
This kind of “straightforward” approach, how- x i max − x min
i
x i = ,i = 1,...,n,
ever, leads to a sharp increase in the amount of N i
computation at the stage of training of the ANN max min
u j − u j
and, in particular, at the stage of the formation u j = ,j = 1,...,m.
of the corresponding training set. M j
There is, however, the possibility of drasti- Here we denote the following: N i is the number
cally reducing these volumes of calculations if of segments divided by the range of values for
we take advantage of the fact that the state into the state variable x i ,i = 1,...,n; M j is the num-
which the system goes (2.111), (2.112)for the ber of segments divided by the range of values
time t = t i+1 − t i , depends only on its state for the control variable u j ,j = 1,...,m.
x(t i ) at time t i , and also on the value u(t i ) of the The nodes of this grid are tuples of length
control action at the same instant of time. This (n + m) of the form x (s i ) ,u (p j ) , where the com-
j
i
circumstance gives grounds to replace the mul- ponents x (s i ) , i = 1,...,n, are taken from the
tistep process {x (e) (t k )}, k = 0,1,...,N t ,asetof i (p j )
(i)
corresponding , and the components u ,
N t one-step processes, each of which consists in j
(2.111), (2.112) of one step in time of length t j = 1,...,m,from (j) in (2.116). If the domain
from some initial point x(t k ). R XU is a subset of the Cartesian product X × U,
In order to obtain a set of initial points x t (t 0 ), then this fact can be taken into account by ex-
u t (t 0 ), which completely characterizes the be- cluding the “extra” tuples from the grid (2.116).
havior of the system (2.111), (2.112)onthe whole In [90] an example of the solution of the ANN
modeling problem was considered, in which the
range of admissible values R XU ⊆ X × U, x ∈ X,
training set was formed according to the method
u ∈ U, we construct the corresponding grid.
presented above. The source model of motion in
Let the state variables x i ,i = 1,...,n, in equa-
this example is a system of equations of the fol-
tion (2.111) take values from the ranges defined
lowing form:
for each of them, i.e.,
m(V z − qV x ) = Z,
˙
(2.117)
min max
x x i x ,i = 1,...,n. (2.114) I y ˙q = M,
i i
where Z is the aerodynamic normal force; M
Similar inequalities hold for control variables
is the aerodynamic pitching moment; q is the
u j ,j = 1,...,m,in(2.111), i.e.,
pitch angular velocity; m is the aircraft mass; I y
is the pitch moment of inertia; V x and V z are the
u min u j u max ,j = 1,...,m. (2.115) longitudinal and normal velocities, respectively.
j
j