Page 97 - Neural Network Modeling and Identification of Dynamical Systems

2.4 TRAINING SET ACQUISITION PROBLEM FOR DYNAMIC NEURAL NETWORKS 85
is highly desirable to be able to apply a test exci- harmonic polynomial
tation signal to all these organs at the same time
, -
to reduce the total time required for data collec- 2πkt
u j = A k sin + ϕ k ,
tion. T (2.135)
k∈I k
Schröder’s work [108] showed the promise of
using a polyharmonic excitation signal for these I k ⊂ K, K ={1,2,...,M},
purposes, which is a set of sinusoids shifted in as a finite linear combination of the fundamental
phase relative to each other. Such a signal makes harmonic A 1 sin(ωt + ϕ 1 ) and higher-order har-
it possible to obtain an excitation signal with a monics A 2 sin(2ωt + ϕ 2 ), A 3 sin(3ωt + ϕ 2 ),andso
rich frequency content and a small peak factor on.
value (amplitude coefficient). Such a signal is re- The input effect for each of the m controls (for
ferred to as a Schröder sweep.
example, the steering surfaces of the aircraft) is
The peak factor is the ratio of the maximum
formed as the sum of the harmonic signals (si-
amplitude of the input signal to the energy of
nusoids), each of which has its own phase shift
the input signal. Inputs with small peak factor
ϕ k . The input signal u j corresponding to the jth
values are effective in that they provide a good
control body has the following form:
frequency content of the dynamical system re-
sponse without large amplitudes of the output , 2πkt -
signal (reaction) of the dynamical system in the u j = A k cos + ϕ k ,j = 1,...,m,
T
time domain. k∈I k
The paper [107] proposes the development of I k ⊂ K, K ={1,2,...,M},
an approach to the formation of a sweep signal (2.136)
by Schröder, which makes it possible to obtain
such a signal for the case of several controls used where M is the total number of harmonically re-
simultaneously, with optimization of the peak lated frequencies; T is the time interval during
factor values for them. This development is ori- which a test excitation signal acts on the dy-
ented to work in real time. namical system; A k is the amplitude of the kth
The excitation signals generated in [107]are sinusoidal component. The expression (2.136)is
mutually orthogonal in both the time and fre- writtenindiscretetimefor N samples
quency domain; they are interpreted as pertur-
bations that are additional to the values of the u j ={u j (0),u j (1),...,u j (i),...,u j (N − 1)},
corresponding control inputs required for the re-
alization of the given behavior of the dynamical where u j (i) = u j (t(i)).
Each of the m inputs (disturbance effects) is
system.
formed from sinusoids with frequencies
To generate test excitation signals, only a pri-
ori information is needed in the form of approxi-
2πk
mate estimates of the frequency band inherent in ω k = , k ∈ I k , I k ⊂ K, K ={1,2,...,M},
T
the dynamical system in question, as well as the
relative effectiveness of the controls for correctly where ω M = 2πM/T is the upper boundary
scaling the amplitudes of the input signals. value of the frequency band of the exciting input
signals (influences). The interval [ω 1 ,ω M ] speci-
GENERATION OF A SET OF POLYHARMONIC fies the frequency range in which the dynamics
EXCITATION SIGNALS of the aircraft under study is expected to lie.
The mathematical model of the input pertur- If the phase angles ϕ k in (2.136)arechosen
bation signal u j affecting the jth control is the randomly in the interval (−π,π], then in gen-