Page 98 - Neural Network Modeling and Identification of Dynamical Systems

86 2. DYNAMIC NEURAL NETWORKS: STRUCTURES AND TRAINING METHODS

eral, the individual harmonic components (oscil- GENERATION PROCEDURE FOR
lations), being summed, can give at t(i) a value POLYHARMONIC EXCITATION SIGNALS
of the amplitude of the sum signal u j (i) at which The procedure for forming a polyharmonic
the conditions of proximity of the disturbed mo- input for a given set of controls consists of the
tion to the reference one are violated. following steps.
In (2.136), ϕ k is the phase shift that must be
1. Set the value of the time interval T ,during
selected for each of the harmonic components in
which a disturbing effect will be applied to
such a way as to provide a small value of the
7
peak factor (amplitude factor) PF(u j ),which is the input of the control object. The value of T
determines the smallest value of the resolving
defined by the relation
power in frequency f = 1/T , as well as the
(u max − u min ) minimum frequency limit f min 2/T .
j j
PF(u j ) = . (2.137) 2. Set the frequency range [f min ,f max ],from
T
2 (u u j )/N which the frequencies of disturbing effects
j
for the dynamical system under considera-
or tion will be selected. It corresponds to the
frequency range of the expected reactions of
(u max − u min )
j j ||u j || ∞ this system to the applied effects. These ef-
PF(u j ) = = , (2.138)
2rms(u j ) ||u j || 2 fects cover the interval [f min ,f max ] uniformly,
with step f . The total number of used fre-
where the last equality is satisfied only in the quencies is
case when u j oscillates symmetrically with re-
spect to zero. In the relations (2.137)and (2.138), / f max − f min 0
M = + 1,
f
min max
u = min[u j (i)], u = max[u j (i)].
j j
i i where · is the integer part of the real num-
ber.
For an individual sinusoidal component in
3. Divide the set of indices K ={1,2,...,M}
(2.135), if the value of the peak factor equals
√ into approximately equal in number of ele-
PF = 2, then the value of the peak factor re- ments subsets I j ⊂ K, each of which deter-
lated to such a component RPF(u j ) (relative mines the set of frequencies for the corre-
8
peak factor, relative amplitude factor) is de- sponding jth body management. This sepa-
fined as ration should be performed in such a way
that the frequencies for different controls al-
max
min
(u − u ) PF(u j )
j j ternate. For example, for two controls, the set
RPF(u j ) = √ = √ . (2.139)
2 2rms(u j ) 2 K ={1,2,...,12} is divided according to this
rule into subsets I 1 ={1,3,...,11} and I 2 =
Minimizing the exponent (2.139) by selecting the {2,4,...,12}, and for three controls into sub-
appropriate phase shift values ϕ k for all k al- sets I 1 ={1,4,7,10}, I 2 ={2,5,8,11},and I 3 =
lows to prevent the occurrence of the situation {3,6,9,12}. This approach ensures the pro-
mentioned above, with the deviation of the dis- duction of small peak factor values for indi-
turbed motion from the reference to an invalid vidual input signals and also allows uniform
value. coverage of the frequency range [f min ,f max ]
for each of these signals. If necessary, this
7 PF – Peak Factor. kind of uniformity can be avoided, for exam-
8 RPF – Relative Peak Factor. ple, if certain frequencies are to be empha-