Page 93 - Neural Network Modeling and Identification of Dynamical Systems

2.4 TRAINING SET ACQUISITION PROBLEM FOR DYNAMIC NEURAL NETWORKS 81
or trajectory in the state space) of the dynamical
system.
The behavior of an (uncontrolled) dynami-
cal system is determined, as already noted, by
n
its initial state {x i (t 0 )} i=1 and “the nature of the
FIGURE 2.30 Uncontrolled dynamical system. (A) With- dynamical system,” i.e., the way in which the
out external influences. (B) With external influences.
variables x i are related to each other in the evo-
lution law (the law of functioning) of the dy-
Suppose that there is some dynamical system, namical system F(x,t). This evolution law de-
that is, a system whose state varies with time. termines the state of the dynamical system at
This dynamical system is uncontrollable, its be- time (t + t), if these states are known at pre-
havior is affected only by the initial conditions vious time instants.
and, possibly, by some external influences (the
impact of the environment in which and in in- 2.4.3.2 Formation of a Set of Test
teraction with which the dynamical system real- Maneuvers
izes its behavior). An example of such a dynam- It was noted above that the selected program
ical system can be an artillery shell, the flight motion (reference trajectory) as part of the test
path of which is affected by the initial conditions maneuver determines the range of values of the
of shooting. The impact of the medium in this state variables in which the training data will
case is determined by the gravitational field in be obtained. It is required to choose such a set
which the projectile moves, and also by the at- of reference trajectories that covers the whole
mosphere.
range of changes in the values of the state vari-
The state of the dynamical system in ques-
ables of the dynamical system. The required
tion at a particular moment in time t ∈ T =
number of trajectories in such a collection is de-
[t 0 ,t f ] is characterized by a set of values x =
termined from the condition of ε-proximity of
(x 1 ,...,x n ). The composition of this set of quan-
the phase trajectories of the dynamical system,
tities, as noted above, is determined by the ques-
tions that are asked about the dynamical system i.e.
in question.
The state of the dynamical system in ques- x i (t) − x j (t) ε, x i (t),x j (t) ∈ X, t ∈ T.
tion at a particular moment in time t ∈ T =
[t 0 ,t f ] is characterized by a set of values x = (2.127)
(x 1 ,...,x n ). The composition of this set of quan-
tities, as noted above, is determined by the ques-
We define a family of reference trajectories of
tions that are asked about the considered dy- the dynamical system,
namical system.
At the initial instant of time t ∈ T ,the state
∗
∗
0 {x (t)} N R ,x (t) ∈ X, t ∈ T. (2.128)
of the dynamical system takes on the value x = i i=1 i
0
0
0
x(t 0 ) = (x ,...,x ), where x = x(t 0 ) ∈ X.
1 n
∗
Since the variables {x i } n describe exactly We assume that the reference trajectory x (t),
i
i=1
some dynamical system, they, according to the i = 1,...,N R ,isan ε-representative of the family
definition of the dynamical system, vary with X i ⊂ X of the phase trajectories of the dynamical
time, that is, the dynamical system is character- system in the domain X i ⊂ X if for each of the
n
ized by a set of variables {x i (t)} , t ∈ T .This phase trajectories x(t) ∈ X i the following condi-
i=1
set will be called the behavior (phase trajectory tion is satisfied: