The toolkit for nonparametric identification nonlinear dynamical systems based on Volterra models in frequency domain
The software-hardware tools used for nonlinear dynamical systems nonparametric identification based on Volterra models in frequency domain are presented. The polyharmonic test impacts are selected as the test ones. The proposed methodology and the toolkit are used for building the communication chan...
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Інститут кібернетики ім. В.М. Глушкова НАН України
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irk-123456789-864492015-09-18T03:02:07Z The toolkit for nonparametric identification nonlinear dynamical systems based on Volterra models in frequency domain Pavlenko, V.D. Speranskyy, V.A. The software-hardware tools used for nonlinear dynamical systems nonparametric identification based on Volterra models in frequency domain are presented. The polyharmonic test impacts are selected as the test ones. The proposed methodology and the toolkit are used for building the communication channel model. Представлено програмно-апаратні засоби, що використовуються для непараметричної ідентифікації нелінійних динамічних систем на основі моделей Вольтерра в частотній області. В якості тестових впливів обрано полігармонічні сигнали. Запропонована методологія та інструментарій використовуються для побудови моделі каналу зв’язку. 2014 Article The toolkit for nonparametric identification nonlinear dynamical systems based on Volterra models in frequency domain / V.D. Pavlenko, V.A. Speranskyy // Математичне та комп'ютерне моделювання. Серія: Технічні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2014. — Вип. 11. — С. 107-116. — Бібліогр.: 10 назв. — англ. 2308-5916 http://dspace.nbuv.gov.ua/handle/123456789/86449 681.5.015:[52+87] en Математичне та комп'ютерне моделювання. Серія: Технічні науки Інститут кібернетики ім. В.М. Глушкова НАН України |
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The software-hardware tools used for nonlinear dynamical systems nonparametric identification based on Volterra models in frequency domain are presented. The polyharmonic test impacts are selected as the test ones. The proposed methodology and the toolkit are used for building the communication channel model. |
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Article |
| author |
Pavlenko, V.D. Speranskyy, V.A. |
| spellingShingle |
Pavlenko, V.D. Speranskyy, V.A. The toolkit for nonparametric identification nonlinear dynamical systems based on Volterra models in frequency domain Математичне та комп'ютерне моделювання. Серія: Технічні науки |
| author_facet |
Pavlenko, V.D. Speranskyy, V.A. |
| author_sort |
Pavlenko, V.D. |
| title |
The toolkit for nonparametric identification nonlinear dynamical systems based on Volterra models in frequency domain |
| title_short |
The toolkit for nonparametric identification nonlinear dynamical systems based on Volterra models in frequency domain |
| title_full |
The toolkit for nonparametric identification nonlinear dynamical systems based on Volterra models in frequency domain |
| title_fullStr |
The toolkit for nonparametric identification nonlinear dynamical systems based on Volterra models in frequency domain |
| title_full_unstemmed |
The toolkit for nonparametric identification nonlinear dynamical systems based on Volterra models in frequency domain |
| title_sort |
toolkit for nonparametric identification nonlinear dynamical systems based on volterra models in frequency domain |
| publisher |
Інститут кібернетики ім. В.М. Глушкова НАН України |
| publishDate |
2014 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/86449 |
| citation_txt |
The toolkit for nonparametric identification nonlinear dynamical systems based on Volterra models in frequency domain / V.D. Pavlenko, V.A. Speranskyy // Математичне та комп'ютерне моделювання. Серія: Технічні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2014. — Вип. 11. — С. 107-116. — Бібліогр.: 10 назв. — англ. |
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Математичне та комп'ютерне моделювання. Серія: Технічні науки |
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Серія: Технічні науки. Випуск 11
107
10. Qian Y. N. Lattice BGK models for Navier-Stokes equation / Y. N. Qian,
D. D’Humieres, P. Lallemand // Europhysics Letters. —1992. — Vol. 17,
№ 6. — P.479–484.
11. Cercignani C. The Boltzmann Equation and Its Applications / C. Cercigna-
ni. — New York etc. : Springer-Verlag, 1988. — 455 p.
12. Inamuro T. A non-slip boundary condition for lattice Boltzmann simulations /
T. Inamuro, M. Yoshina, F. Ogino // Physics of Fluids. — 1995. — Vol. 7,
№ 12. — P. 2928–2930.
Actuality of study of the peristaltic processes that ensure the movement of
fluids in the human body is substantiated. Whereas the traditional approach to
mathematical modeling of real peristaltic processes causes a number of prob-
lems associated with increasing of the computation complexity and violating of
the conservation laws, the paper presents the application of mathematical mod-
eling technology based on lattice Boltzmann equation. Theoretical Foundations
of lattice Boltzmann model and especially its use in two-dimensional lattice is
described. Much attention is paid to the peculiarities of formation of boundary
conditions at the macroscopic level and the level of the lattice sites. Results of
simulations of the peristaltic process in the digestive tract are given.
Key words: lattice Boltzmann equation, mathematical modeling, peri-
staltic process, BGK-model.
Отримано: 15.09.2014
UDC 681.5.015:[52+87]
V. D. Pavlenko, D. Sc, Professor,
V. A. Speranskyy, Ph. D.
Odessa national polytechnic university, Odessa
THE TOOLKIT FOR NONPARAMETRIC IDENTIFICATION
NONLINEAR DYNAMICAL SYSTEMS BASED ON VOLTERRA
MODELS IN FREQUENCY DOMAIN
The software-hardware tools used for nonlinear dynamical sys-
tems nonparametric identification based on Volterra models in fre-
quency domain are presented. The polyharmonic test impacts are
selected as the test ones. The proposed methodology and the toolkit
are used for building the communication channel model.
Key words: nonlinear dynamical systems, Volterra models, fre-
quency domain, nonparametric identification, polyharmonic signals,
multidimensional frequency characteristics, identification toolkit.
Introduction. Increasing complexity of the technical systems and ob-
jects being studied and planned necessitates the development of mathematical
models. Such models have to take into account nonlinear and dynamic proper-
ties of mentioned systems and objects. Also the development of efficient com-
puter implementation of tools for constructing such models is needed.
© V. D. Pavlenko, V. A. Speranskyy, 2014
Математичне та комп’ютерне моделювання
108
One of the most important classes of such systems is the modern
communication channels. In real conditions with the presence of high val-
ues of the crest factor such systems are nonlinear dynamical systems.
The quality of data transfer in communication channels depends on
the media and the characteristics of the devices used for data transmission.
This is very important in such areas: in sonar information channels for
resource development problems, environmental monitoring of shelve wa-
ters, for remote inspection of the environment, taking into account the
complicated boundary conditions and geometry of spatial inhomogeneities
of the aquatic environment; to improve the accuracy of measurements in
remote sensing of the Earth surface and subsurface, where the nonlinearity
is introduced by atmosphere layers and weather phenomenon and prevent
the passage of the test signals; in special-purpose communications systems
to provide high reliability steganographic information transfer; in systems
of indirect control and diagnostics of different physical nature objects.
Communication channels due to their complexity and lack of knowl-
edge can be regarded as a «black box» system. The nonparametric dy-
namic models based on integral power Volterra series are usually used for
mathematical modeling of such systems.
The models in form of integral Volterra series [1–4] are widely used
to identify nonlinear dynamical systems [5–6]. Herewith the nonlinear and
dynamical properties of the system are fully characterized by sequence of
multidimensional weighting functions — Volterra kernels.
The aim of the work is the toolkit developing used for constructing
the nonparametric models of nonlinear dynamical systems in a form of
Volterra series in frequency domain using polyharmonic test signals. The
efficiency of the developed toolkit had to be tested in practical issue for
building the nonlinear dynamical model of the communication channel.
Experimental researches methodology. Identification problem for
modelling nonlinear dynamical system in a form of Volterra series consists in
n-dimensional weighting functions determination wn(1,…,n) for time domain
or it’s Fourier transforms Wn(j1,…,jn) — n-dimensional transfer functions
for frequency domain. This based on data of experimental researches of the
«input-output» system. Identification of nonlinear system in frequency domain
coming to determination of absolute value |Wn(j1,…,jn)| and phase
argWn(j1,…,jn) of multidimensional transfer function at given frequencies —
multidimensional amplitude–frequency characteristics (AFC) and phase-
frequency characteristics (PFC) respectively. They are defined by formulas:
2 2
1 1 1( , , ) Re( ( , , )) Im( ( , , )) n n n n n nW j j W j j W j j (1)
1
1
1
Im[ ( , , )]
arg ( , , ) arctg
Re[ ( , , )]
n n
n n
n n
W j j
W j j
W j j
, (2)
Серія: Технічні науки. Випуск 11
109
where Re and Im are real and imaginary parts of a complex function of n
variables respectively.
So the nonlinear system identification procedure consists in extract-
ing the partial components yn[x(t)] and determination of multidimensional
Volterra kernels or frequency characteristics: AFC and PFC [7].
The test polyharmonic effects for identification in the frequency do-
main representing by signals of such type:
1
cos
n
k k k
k
x t A t
, (3)
where n — the order of transfer function being estimated; Ak, k and k —
amplitude, frequency and a phase of k-th harmonics respectively. In research,
it is supposed every amplitude of Ak to be equal, and phases k equal to zero.
The identification algorithm is based on nonlinear dynamical model con-
structing in a form of Volterra series and consists in selecting the form of the
test signals. The identification methodology is implemented with approxima-
tion and interpolation methods [8–10]. The structured scheme of the computa-
tional process of the identification procedure is shown in the fig. 1.
Begin
–
+
+
–
Experiment
Processing of responses of the system
Calculating the spectra of a signal yn(t)
Filtering the harmonics with frequency
ω1+ + ωn
ω < ω final
Ω < Ω final
End
Setting amplitudes
Arrays with responses of the system being identifiedto
the polyharmonic test signals with the different
amplitudes
Arrays with corresponding partial components yn(t)
Arrays and
Arrays of AFC and PFC at the frequencies ω1+…+ ωn
Setting frequencies
Setting coefficients
Arrays with test signals frequencies
Arrays with test signals amplitudes
Arrays with test signals coefficients
ω – current frequency of the experiment
Ω – shift-parameter for the frequency for the
multidimensional AFC building
1Re[ ( , , )] n nW j j 1Im[ ( , , )] n nW j j
Fig. 1. Structured scheme of the computational process
of the identification procedure
The hardware platform of the experimental researches using developed
toolkit consists of IBM–PC compatible computer with two soundcards Creative
Audigy 4 (signal to noise ratio less than 89 dB and distortions not higher than
0,003%). This allows characterizing the final results as reliable ones. Onboard
soundcards (motherboard built-in) has much worst characteristics and higher
unevenness of its AFC. Thus it cannot be used in experimental researches.
Математичне та комп’ютерне моделювання
110
Maximum allowed amplitude in the described experiment with use of
sound card was A = 0,25 V (defined experimentally). The range of fre-
quencies was defined by the sound card pass band (20…20000 Hz), and
frequencies of the test signals has been chosen from this range, taking into
account restrictions specified in [5]. Such parameters were chosen for the
experiment: start frequency fs = 125 Hz; final frequency fe = 3125 Hz; a
frequency change step F = 125 Hz; to define AFC of the second order
determination, an offset on frequency F1 = f2 – f1 was increasingly growing
from 201 to 3401 Hz with step 100 Hz.
The toolkit is organized from two software parts and written in С++ and
Matlab languages. The first part is assigned for test signals generation with
minimal impact of the operation system of used PC. The experimental results
processing is implemented in the second part. The 13 modules were developed
during the software part implementation. The structured scheme of the soft-
ware toolkit for nonlinear dynamical systems identification is shown in fig. 2.
Fig. 2. The structured scheme of the software toolkit
for nonlinear dynamical systems identification
An initialization of the identification process parameters of the
nonlinear system being identified is performed in the main module
(main_form.m). The list of those parameters:
1) start frequency f1 of harmonic signals for the experiment;
2) quantity of the experimental steps (defines the quantity of the subdi-
agonal sections of the AFC and the final frequency of the experiment);
3) polyharmonic test frequency step;
4) quantity of the experiments repeats that allows to average results re-
ceived for current nonlinear object;
Серія: Технічні науки. Випуск 11
111
5) quantity of the signal samples being sent to the nonlinear dynamical
system input (this number have to be divisible by 2 to obtain correct
work oа the Fast Fourier Transform);
6) sampling frequency of the sound signal being sent to the nonlinear
dynamical system input;
7) kernel order for the Volterra model;
8) approximation order / experiments quantity for the approximation /
interpolation method of the coefficients calculation and experiments
providing;
9) shift between the frequencies f2 and f1 for the polyharmonic test signals
(for nonlinear models);
10) shift between the frequencies f3 and f2 for the polyharmonic test signals
(for nonlinear models);
11) amplitudes array for the mono- or polyharmonic test signals (depends
on model order);
12) corresponding coefficients array calculated using amplitudes of the test
signals;
13) time array for sound test signal forming;
14) response array (consists of the values of tested nonlinear dynamical
system responses to the test harmonic signal);
15) AFC array (consists AFC points values of tested nonlinear dynamical
system).
The implemented in module main_gui.fig visual interface contains all
visual components of the windows application used to control the parame-
ters of the experiment (fig. 3 и fig. 4).
Fig. 3. Part of the main window of the software toolkit used for standard
characteristics constructing (subdiagonal sections and 3D surfaces)
Visual elements located at the main form allow controlling the identi-
fication process by changing the values of variable parameters. Also you
Математичне та комп’ютерне моделювання
112
can manage the visualization of received results in a form of two-
dimensional and three-dimensional plots of the identified system AFC.
Fig. 4. The main form of the software toolkit used for identification
of nonlinear dynamical systems
The coefficients for identification during the forming the test signals
are depending on its amplitudes. The calculation of such coefficients is
performed by the matrix method of SLAE decision and it’s performed in
calc_coef module. One of the identification modules of the chosen order
(ident_1, ident_2, ident_3) begins working after starting the identification
process. In common way:
the test harmonic signals with selected amplitudes (depending on
method) are generated;
generated signals data arrays are sent to the input of the nonlinear system;
the signals received after nonlinear system has the visual form shown
in fig. 5 and they have to be preprocessed (multiplication the signals
and output coefficients);
the total (sum) response of all test signals is calculated;
the points of AFC array for the responses of the system being identi-
fied are calculated;
the data received during identification at the current frequency step is
saved to the file.
The identification method is organized using the approximation [8]
and interpolation method [10]. The identification of the nonlinear system
of the 1st order is presented as textual algorithm:
Step 1. The main module (main_form.m) sending the parameters of the
identification to the module ident_1.m;
Step 2. All supporting local variables and arrays are initialized;
Step 3. The loop for experiment repetitions with selected iterations
Серія: Технічні науки. Випуск 11
113
Step 3.1. The loop for changing the test frequency from start value with
selected step for the selected quantity of steps (specified in pa-
rameters of the identification).
Step 3.1.1. The f1 frequency increment at current experiment step.
Step 3.1.2. The test harmonic signal generating.
Step 3.1.3. Searching for the f1 frequency position in signal spectra.
Step 3.1.4. The loop for each test signal amplitude in the experiment.
Step 3.1.5. The signal with specified amplitude is forming.
Step 3.1.6. The formed signal is sending to the output of the
soundcard.
Step 3.1.7. The signal is receiving from the input of the soundcard.
Step 3.1.8. The multiplication of the received signal data (re-
sponse) with specified amplitudes by corresponding
coefficients.
Step 3.1.9. The total response calculation by summation of re-
ceived signals groups.
Step 3.1.10. If the selected method is interpolation then performing
division of the total response by additional method co-
efficient.
Step 3.1.11. Calculating the signal spectra using Fast Fourier
Transform.
Step 3.1.12. Calculating the value of informational harmonics with
position detected at the Step 3.1.3.
Step 3.1.13. Calculating the AFC and amplitude correction.
Step 3.1.14. Saving the experimental data in files as arrays with
identification results for the frequency f1.
Fig. 5. The responses (4 signals) received from channel with noises for the 2nd
order model: a=–1 (1); a=1 (2); a=–0,644 (3); a=0,644 (4)
To save the data of the identification results for subsequent system
model in frequency domain building the m-files are used. The names of
Математичне та комп’ютерне моделювання
114
those files are fully showing the parameters of the experiment. The files
are saved at the Results subfolder of the program folder.
The format of file name with identification data of the test model al-
lowing cataloging results looks as Vn(N)_Na_metb_nc_mad_mke__w1_f-
g-h_dw1_i_w2_j.mat and consists of such fields: n — model (Volterra
kernel) order, N — approximation order, a — discretization order, b —
method (1 — interpolation, 0 — approximation), c — noise level (in %
relatively to test signal level), d — scaling the test signals amplitudes rela-
tively to initial (in %), e — scaling the test signals coefficients relatively to
initial (in %), f — start frequency f1 value, g — step of the frequency f1
changing, h — final value of the frequency f1, i — shifting between the
frequencies f2 and f1, j — the value of the frequency f3.
The format of file name with identification data of the real communica-
tion channel allowing cataloging results looks as Final_xxxx_yyyy_zzzz_Vk-
N_mm-dd-yy_HH-MM-SS.mat and consists of such fields: xxxx — start
frequency of the current experiment (section), yyy — shift between the fre-
quencies f2 and f1, zzz — the value of the frequency f3, n — model (Volterra
kernel) order, N — approximation order, mm-dd-yy_HH-MM-SS — current
date and time in selected format.
On the results of the identification data contained in the files, it is possible
to draw two-dimensional plots (module sections_vis.m) — subdiagonal AFC
sections of the nonlinear dynamical system. A 3-dimensional plots (modules
surf_vis.m, plotter.m) — surfaces are built of the subdiagonal AFC sections of
the nonlinear dynamical system by changing shifting between the frequencies f2
and f1 and changing frequency f3 for the 2nd and 3rd order models respectively.
The automatic wavelet filtration of the graphic data is performed during
the plots building. The chosen wavelet is 2nd level Coiflet. It allows to smooth
output data of the real communication channel characteristics while minimal
ERMSE growing. The examples of the received models for the 1st (subdiago-
nal sections) and 2nd (surface built of set of subdiagonal sections) orders with
wavelet smoothing are shown in fig. 6а and fig. 6b respectively.
а) б)
Fig. 6. a) Subdiagonal sections of AFCs of the second order after wavelet
«Coiflet» 2nd level noise-suppression at different frequencies f1:201(1), 401(2),
601(3), 801(4), 1001(5), 1401(6) Hz; b) Surface built of AFCs of the second order
after wavelet «Coiflet» 3rd level noise-suppression
Серія: Технічні науки. Випуск 11
115
Conclusions. The hardware-software toolkit was made as a result of the
development and performed researches. This toolkit allows identifying and
constructing the models of the system with unknown structure using the
Volterra series models and polyharmonic signals in frequency domain. Using
the developed toolkit for obtaining characteristics of the nonlinear systems in
future will allow correcting its characteristics. Current toolkit was applied for
constructing the nonparametric models of the communication channel. Results
of identification of the linear and nonlinear communication channels models
were presented in [8–10]. There showed significant nonlinearities of the iden-
tified systems. Thus it is necessary to take into consideration the characteris-
tics of the system to obtain its high efficient and reliable operating modes.
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Представлено програмно-апаратні засоби, що використовуються для
непараметричної ідентифікації нелінійних динамічних систем на основі
моделей Вольтерра в частотній області. В якості тестових впливів обрано
полігармонічні сигнали. Запропонована методологія та інструментарій
використовуються для побудови моделі каналу зв’язку.
Ключові слова: нелінійні динамічні системи, моделі Вольтерра,
частотна область, непараметрична ідентифікація, полігармонічні
сигнали, багатовимірні частотні характеристики, інструментальні
засоби ідентифікації.
Отримано: 29.07.2014
УДК 621.396.982.2
В. В. Палагін, д-р техн. наук, професор,
О. В. Івченко, аспірант
Черкаський державний технологічний університет, м. Черкаси
ПРОГРАМНІ ЗАСОБИ КОМП’ЮТЕРНОГО
МОДЕЛЮВАННЯ ОЦІНЮВАННЯ ПАРАМЕТРІВ
НЕГАУСОВИХ КОРЕЛЬОВАНИХ ВИПАДКОВИХ ПРОЦЕСІВ
У роботі наведені результати оцінювання параметрів нега-
усових випадкових процесів на основі застосування адаптова-
ного методу максимізації полінома та моментно-кумулянтного
опису випадкових величин. Отримані результати моделювання
і ефективності запропонованого методу в порівнянні з відоми-
ми підходами.
Ключові слова: негаусові корельовані випадкові процеси,
оцінювання параметрів, метод максимізації полінома, диспер-
сія оцінки.
Вступ. Проблемі статистичного аналізу багатомірних випадко-
вих величин присвячено багато робіт [1–3], де в основному робиться
припущення про їх нормальний розподіл. На практиці не завжди ви-
конується умова нормалізації багатомірних випадкових величин [4],
тому виникає необхідність розширення математичного апарату з об-
робки даних при негаусових завадах. Одним з підходів для вирішення
даної проблеми є застосування методу максимізації поліному [5–6],
© В. В. Палагін, О. В. Івченко, 2014
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>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AllowImageBreaks true
/AllowTableBreaks true
/ExpandPage false
/HonorBaseURL true
/HonorRolloverEffect false
/IgnoreHTMLPageBreaks false
/IncludeHeaderFooter false
/MarginOffset [
0
0
0
0
]
/MetadataAuthor ()
/MetadataKeywords ()
/MetadataSubject ()
/MetadataTitle ()
/MetricPageSize [
0
0
]
/MetricUnit /inch
/MobileCompatible 0
/Namespace [
(Adobe)
(GoLive)
(8.0)
]
/OpenZoomToHTMLFontSize false
/PageOrientation /Portrait
/RemoveBackground false
/ShrinkContent true
/TreatColorsAs /MainMonitorColors
/UseEmbeddedProfiles false
/UseHTMLTitleAsMetadata true
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/BleedOffset [
0
0
0
0
]
/ConvertColors /ConvertToRGB
/DestinationProfileName (sRGB IEC61966-2.1)
/DestinationProfileSelector /UseName
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements true
/GenerateStructure false
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles true
/MarksOffset 6
/MarksWeight 0.250000
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /DocumentCMYK
/PageMarksFile /RomanDefault
/PreserveEditing true
/UntaggedCMYKHandling /UseDocumentProfile
/UntaggedRGBHandling /LeaveUntagged
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [600 600]
/PageSize [419.528 595.276]
>> setpagedevice
|