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2.4 TRAINING SET ACQUISITION PROBLEM FOR DYNAMIC NEURAL NETWORKS 83

FIGURE 2.32 Typical test excitation signals used in the study of the dynamics of controllable systems. (A) Doublet (sig-
nal type 1–1). (B) Triplet (signal type 2–1–1). (C) Quadruplet (signal type 3–2–1–1). From [109], used with permission from
Moscow Aviation Institute.

FIGURE 2.33 Modified versions of the test excitation signals used in the study of the dynamics of controllable systems. (A)
Triplet (signal type 2–1–1). (B) Quadruplet (signal type 3–2–1–1). From [109], used with permission from Moscow Aviation
Institute.

other kind of test action – a rectangular pulse A triplet (signal of the type 2–1–1) is a combi-
(Fig. 2.31B): nation of a rectangular pulse of duration T = 2T r
and a complete rectangular oscillation with a pe-
u(t) = A(σ(t) − σ(t − T r )), (2.134) riod T = 2T r .
A quadruplet (a signal of the type 3–2–1–1–1)
is formed from a triplet by adding to its origin
where A is the pulse amplitude and T r = t f − t i
a rectangular pulse of width T = T r . In addition,
is the pulse duration.
we can also use triplet and quadruplet variants
On the basis of the rectangular pulse signal
in which each of the constituent parts of the sig-
(2.134), perturbing effects of oscillatory charac-
nal is a full-period oscillation (see Fig. 2.33). We
ter are determined, consisting of a series of rect- will designate them as signals of the type 2–1–1
angular oscillations with a definite relationship
and 3–2–1–1, respectively.
between their periods. Among the most com-
Another typical excitation signal is shown in
monly used influences of this kind are the dou- Fig. 2.34A. Its values are kept constant for all
blet (Fig. 2.32A), the triplet (Fig. 2.32B), and the time intervals [t i ,t i+1 ), i = 0,1,...,n − 1,and at
quadruplet (Fig. 2.32C). time instances t i it can be changed randomly. In
The doublet (also denoted as a signal of type more detail, a signal of this type will be consid-
1–1) is one complete rectangular wave with a pe- ered below by the example of solving the prob-
riod T = 2T r equal to twice the duration of the lem of the ANN simulation of the longitudinal
rectangular pulse. angular motion of an aircraft.