Page 99 - Neural Network Modeling and Identification of Dynamical Systems

2.4 TRAINING SET ACQUISITION PROBLEM FOR DYNAMIC NEURAL NETWORKS 87
sized, or, if necessary, some frequency com- noted that to obtain a constant time shift of
ponents should be eliminated (in particular, all components u j their phase shifts will be
for fear of causing undesired reaction of the different in magnitude, since each of the com-
control object). In the paper [106]itwas estab- ponents has its own frequency different from
lished empirically that if the sets of I j indices the frequency of the other components. Since
are formed in such a way that they contain all components of the signal u j are harmon-
numbers greater than 1, multiples of 2 or 3 ics of the same fundamental frequency for the
(for example, k = 2,4,6 or k = 5,10,15,20), period of oscillations T , if the phase angles ϕ k
then the phase shift for them can be opti- of all components are changed so that the ini-
mized in such a way that the relative peak tial value of the input signal was zero, then
factor for the corresponding input action will its value at the final moment of time will also
be very close to 1, and in some cases it will be zero. In this case, the energy spectrum, or-
be even less than 1. For the distribution of thogonality, and relative peak factor of the in-
indices over subsets I j , the following condi- put signals remain unchanged.
tions must be satisfied: 7. Go back to step 5 and repeat the appropri-
ate actions until either the relative peak fac-
) *
I j = K, K ={1,2,...,M}, I j = ∅. tor reaches the prescribed value, or the limit
number of iterations of the process is reached.
j j
For example, the target value of the relative
Each index k ∈ K must be used exactly once. peak factor can be set as 1.01, the maximum
Compliance with this condition ensures mu- number of iterations 50.
tual orthogonality of the input actions both
There are a number of methods that allow to
in the time domain and in the frequency do- optimize the frequency spectrum of input (test)
main. signals when solving the problem of estimating
4. Generate, according to (2.136), the input ac-
the parameters of a dynamic system. However,
tion u j for each of the controls used, and then
all these methods require a significant amount of
calculate the initial phase angle values ϕ k ac-
computation, as well as a certain level of knowl-
cording to the Schröder method, assuming
edge about the dynamical being investigated,
the uniformity of the power spectrum. usually tied to a certain nominal state of the sys-
5. Find the phase angle values ϕ k for each of the tem. With respect to the situation considered in
input actions u j which minimize the relative this chapter, such methods are useless because
peak factor for them. the task is to identify the dynamics of the system
6. For each of the input actions u j ,perform a in real time for various modes of its functioning
one-dimensional search procedure to find a that vary widely. In addition, the solution of the
constant time offset value such that the cor- task of reconfiguring the control system in the
responding input signal starts at a zero value event of failures and damage of the dynamical
of its amplitude. This operation is equiva- system requires the solution of the problem of
lent to shifting the graph of the input signal identification with significant and unpredictable
along the time axis so that the point of inter- changes in the dynamics of the system. Under
section of this graph with the abscissa axis such conditions, the laborious calculation of the
(i.e., with the time axis) coincides with the input effect optimized for the frequency spec-
origin. The phase shift corresponding to such trum does not make sense, and in some cases it
a displacement is added to the values of ϕ k is impossible, since it does not fit into real time.
of all sinusoidal components (harmonics) of Instead, the frequency spectrum of all generated
the considered input actions u j .Itshouldbe input influences is selected in such a way that it