Page 94 - Neural Network Modeling and Identification of Dynamical Systems

82 2. DYNAMIC NEURAL NETWORKS: STRUCTURES AND TRAINING METHODS

∗
∗
x (t) − x(t) ε, x (t) ∈ X i ,x(t) ∈ X i ,t ∈ T.
i
i
(2.129)
∗ N R
The family of reference trajectories {x (t)}
i i=1
of the dynamical system must be such that
N R
)
= X, (2.130)
X i = X 1 ∪ X 2 ∪ ... ∪ X N R
i=1 FIGURE 2.31 Typical test excitation signals used in the
study of the dynamics of controllable systems. (A) Stepwise
where X is the family (collection) of all phase
excitation. (B) Impulse excitation. From [109], used with per-
trajectories (trajectories in the state space) poten- mission from Moscow Aviation Institute.
tially realized by the dynamical system in ques-
tion. This condition means that the family of influences in such a way as to obtain an informa-
∗
reference trajectories {x (t)} N R should together tive set of training data for a dynamical system
i i=1
represent all potentially possible variants of the are considered.
behavior of the dynamical system in question.
TYPICAL TEST EXCITATION SIGNALS FOR THE
This condition can be treated as a condition for
completeness of the ε-covering by support tra- IDENTIFICATION OF SYSTEMS
jectories of the domain of possible variants of the Elimination of uncertainties in the ANN
behavior of the dynamical system. model by refining (restoring) a number of el-
An optimal ε-covering problem for the do- ements included in it (for example, functions
main X of possible variants of the dynamical describing the aerodynamic characteristics of
system behavior can be stated, consisting in the aircraft) is a typical problem of identifying
minimizing the number N R of reference trajec- systems [44,93–99]. When solving identification
∗ N R problems for controllable dynamic systems, a
tories in the set {x (t)} , i.e.,
i i=1
number of typical test disturbances are used.
N ∗ Among them, the most common are the follow-
∗ R ∗ N R
{x (t)} = min{x (t)} , (2.131)
i i=1 i i=1
N R ing impacts [89,100–103]:
that allows to minimize the volume of the train- • stepwise excitation;
ing set while preserving its informativeness. • impulse excitation;
A desirable condition (but difficult to realize) • doublet (signal type 1–1);
is also the condition • triplet (signal type 2–1–1);
• quadruplet (signal type 3–2–1–1);
N R
* • random signal;
= ∅. (2.132)
• polyharmonic signal.
X i = X 1 ∩ X 2 ∩ ... ∩ X N R
i=1
Stepwise excitation (Fig. 2.31A) is a function
2.4.3.3 Formation of Test Excitation Signal
u(t) that changes at a certain moment in time t i
As already noted, the type of test maneuver from u = 0 to u = u , i.e.,
∗
in (2.126) determines the resulting ranges for +
changing the values of the state and control vari- 0, t < t i ;
u(t) = (2.133)
ables, while the kind of perturbation effect pro- u , t t i .
∗
vides a variety of examples within these ranges.
In the following sections, the questions of form- Let u = 1. Then (2.133) is the function of the
∗
ing (with a given test maneuver) test excitatory unit jump σ(t). With its use, you can define an-