Page 92 - Neural Network Modeling and Identification of Dynamical Systems

80 2. DYNAMIC NEURAL NETWORKS: STRUCTURES AND TRAINING METHODS

tem with state vectors and small-size controls 2.4.3 Indirect Approach to the
and with a moderate number of samples with Acquisition of Training Data Sets
respect to these variables are acceptable (the first for Dynamic Neural Networks
and second variants in (2.124)). Even a slight in- 2.4.3.1 General Characteristics of the
crease in the values of these parameters leads, as Indirect Approach to the Acquisition
can be seen from (2.125)), to unacceptable values of Training Data Sets
of the value of the training set. As noted in the previous section, the indirect
In real-world applied problems, where the approach is based on the application of a set of
possibilities of ANN modeling are particularly specially designed control signals to the dynam-
in demand, the result is even more impres- ical system, instead of direct sampling of the do-
main R X,U of feasible values of state and control
sive.
variables.
In particular, in the full model of the angular
With this approach, the actual motion of
motion of the aircraft (the corresponding ANN
the dynamical system (x(t),u(t)) is composed
model for this case is considered in Section 6.3) of a program (test maneuver) of the motion
of Chapter 6, we have 14 state variables and 3 (x (t),u (t)), generated by the control signal
∗
∗
control variables, hence the volume of the train- u (t),aswellasthe motion (˜x(t), ˜u(t)), gener-
∗
ing set for it in the direct approach to its forma- ated by the additional perturbing action ˜u(t), i.e.,
= 20 will be N =
tion and at N w = N q = M δ e ∗ ∗
18
20 17 = 2 · 10 , which, of course, is completely x(t) = x (t) +˜x(t), u(t) = u (t) +˜u(t). (2.126)
unacceptable.
Examples of test maneuvers include:
Thus, the direct approach to the formation
• a straight-line horizontal flight with a con-
of training sets for modeling dynamical systems
stant speed;
has a very small “niche,” in which its application
• a flight with a monotonically increasing angle
is possible – simple problems of low dimension-
of attack;
ality. An alternative indirect approach is more • a U-turn in the horizontal plane;
well-suited for complex high dimensional prob- • an ascending/descending spiral.
lems. This approach is based on the application
Possible variants of the test perturbing actions
of a set of specially designed control signals to
˜ u(t) are considered below.
a dynamical system of interest. This approach The type of test maneuver (x (t),u (t)) in
∗
∗
is discussed in more detail in the next section. (2.126) determines the obtained ranges for
The indirect approach has its advantages and changing the values of the state and control vari-
disadvantages. The indirect approach is the only ables; ˜u(t) is the variety of examples within these
viable option in situations where the training ranges.
What is the ideal form of a training set and
data acquisition is required to be performed in
how can it be obtained in practice using an indi-
real or even in advanced time. However, in cases
rect approach? We consider this issue in several
when there are no rigid time restrictions regard-
stages, starting with the simplest version of the
ing the acquisition and processing of training dynamical system and proceeding to more com-
data, the most appropriate approach is a mixed plex versions.
one, which is a combination of direct and indi- We first consider a simpler case of an uncon-
rect approaches. trolled dynamical system (Fig. 2.30).