Numerical simulation of dynamic object based on convolution operations
A recursive digital filter construction method is considered for simulation of inertial element as a typical component of complex dynamic object. New computational formulas are obtained. Their high accuracy as compared to traditional ones is shown. Decomposition of initial model of simulated object...
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Інститут кібернетики ім. В.М. Глушкова НАН України
2016
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| Цитувати: | Numerical simulation of dynamic object based on convolution operations / Jo Sterten // Математичне та комп'ютерне моделювання. Серія: Технічні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2016. — Вип. 13. — С. 160-165. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-1337392018-06-06T03:03:12Z Numerical simulation of dynamic object based on convolution operations Sterten, Jo A recursive digital filter construction method is considered for simulation of inertial element as a typical component of complex dynamic object. New computational formulas are obtained. Their high accuracy as compared to traditional ones is shown. Decomposition of initial model of simulated object by convolution operations with several typical exponential kernels is proposed instead of traditional operations of integration and differentiation. Розглянуто метод побудови рекурсивного цифрового фільтра для імітації ланки — типового елементу складного динамічного об’єкта. Отримано нові розрахункові формули. Показано їх високу точність порівняно з традиційними та доцільність декомпозиції вихідної моделі об’єкта, що імітується, за операціями згортки з декількома типовими експоненціальними ядрами замість традиційних операцій інтегрування та диференціювання. 2016 Article Numerical simulation of dynamic object based on convolution operations / Jo Sterten // Математичне та комп'ютерне моделювання. Серія: Технічні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2016. — Вип. 13. — С. 160-165. — Бібліогр.: 7 назв. — англ. 2308-5916 http://dspace.nbuv.gov.ua/handle/123456789/133739 681.325 en Математичне та комп'ютерне моделювання. Серія: Технічні науки Інститут кібернетики ім. В.М. Глушкова НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| description |
A recursive digital filter construction method is considered for simulation of inertial element as a typical component of complex dynamic object. New computational formulas are obtained. Their high accuracy as compared to traditional ones is shown. Decomposition of initial model of simulated object by convolution operations with several typical exponential kernels is proposed instead of traditional operations of integration and differentiation. |
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Article |
| author |
Sterten, Jo |
| spellingShingle |
Sterten, Jo Numerical simulation of dynamic object based on convolution operations Математичне та комп'ютерне моделювання. Серія: Технічні науки |
| author_facet |
Sterten, Jo |
| author_sort |
Sterten, Jo |
| title |
Numerical simulation of dynamic object based on convolution operations |
| title_short |
Numerical simulation of dynamic object based on convolution operations |
| title_full |
Numerical simulation of dynamic object based on convolution operations |
| title_fullStr |
Numerical simulation of dynamic object based on convolution operations |
| title_full_unstemmed |
Numerical simulation of dynamic object based on convolution operations |
| title_sort |
numerical simulation of dynamic object based on convolution operations |
| publisher |
Інститут кібернетики ім. В.М. Глушкова НАН України |
| publishDate |
2016 |
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http://dspace.nbuv.gov.ua/handle/123456789/133739 |
| citation_txt |
Numerical simulation of dynamic object based on convolution operations / Jo Sterten // Математичне та комп'ютерне моделювання. Серія: Технічні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2016. — Вип. 13. — С. 160-165. — Бібліогр.: 7 назв. — англ. |
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Математичне та комп'ютерне моделювання. Серія: Технічні науки |
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Математичне та комп’ютерне моделювання
160
Key words: informative parameters, steady-state visual evoked poten-
tials, orthogonal decomposition, discrete orthogonal polynomial, diagnos-
tics, visual system.
Отримано: 11.04.2016
UDK 681.325
Jo Sterten, Assistant Professor
Norwegian University of Science and Technology, Gjovik, Norway
NUMERICAL SIMULATION OF DYNAMIC
OBJECT BASED ON CONVOLUTION OPERATIONS
A recursive digital filter construction method is considered for
simulation of inertial element as a typical component of complex
dynamic object. New computational formulas are obtained. Their
high accuracy as compared to traditional ones is shown. Decompo-
sition of initial model of simulated object by convolution opera-
tions with several typical exponential kernels is proposed instead of
traditional operations of integration and differentiation.
Key words: mathematical model, transfer function, Volterra
integral operator, convolution operator, digital filter.
Introduction. A wide range of control system units can be described
with acceptable accuracy as a linear stationary dynamic object with
lumped parameters. There are several mathematical descriptions of such
an object. Common higher order linear differential equation is its tradi-
tional mathematical model.
Well-known software packages for simulation of continuous objects
(SPSCO), such as MATHLAB, SIMULINK, CC, PSPACE, MCAP,
SIGNAL, EURIKA, MATCAD, as well as domestic SPSCO, DISPAS, are
based on computational solution of differential equations. They have li-
mited accuracy and low tolerance to noise interference or rapidly changing
signals, that is especially evident while solving inverse problems of sys-
tems dynamics, in particular signal recovery problem [1].
In Laplace image space, traditional differential model of continuous
object is associated with rational fractional transfer function that can be
decomposed into partial fractions. Then, on the basis of the convolution
theorem and tables of Laplace transform basic functions one can pass to
the originals. As a result, we obtain an equivalent linear integral equation
based on convolution operator with a complex kernel in the form of super-
position of exponential and power functions.
Integrated mathematical model has a number of advantages as com-
pared to the differential one [1, 2]. In particular, when it is used, accuracy
of numerical simulation of the object under investigation is enhanced. The
© Jo Sterten, 2016
Серія: Технічні науки. Випуск 13
161
algorithms obtained are characterized by deeper parallelization. The last of
these properties is very important for simulation of an object in real time,
when the computation should be completed during the step of discretiza-
tion of input and output signals.
Digital filtration techniques. Application ofdigital filters seems to
be very promising for simulation of continuous objects.The simplest and
most common technique is to synthesize a transversal digital filter with
finite high-order impulse response (usually one or two hundred units).
This leads to emergence of technical difficulties. Advantage of this me-
thod consists in simplicity of organization of computer-aided calculation
of transversal filters [3]. Very promising perspective stems from develop-
ment of recursive digital filters [4] with simple technical implementation
(usually they have low orders — one, two or three units) and high level of
computational process parallelization.
Traditionally, an analog prototype is used for development of a re-
cursive digital filter. It means formally that with equivalent algebraic ma-
nipulations, operator of primary continuous mathematical model can be
reduced to a form suitable for realization in analog integrators, that are
replaced thereafter with digital ones.
Image F(p) of continuous original function f(t) according to defini-
tion of integral Laplace transform can be find as follows:
0
( ) exp( ) ( ) ,F p pt f t dt
where t is argument of the original function (as a rule, it is time); p is a
mapping function argument (frequency or spatial coordinate) [5].
It is convenient to use this transformation, as well as Fourier trans-
form, for spectral analysis of harmonic signals. Infinitely long in time sine
waves are transformed in images into infinitely narrow by spatial coordi-
nate spectral lines. Dynamic operations of differentiation or integration,
complicated in the originals, are reduced to simple multiplication opera-
tions in images. This method makes possible significant simplification of
analytical calculations of complex dynamical system response to an arbi-
trary continuous input signal. Sometimes performance of reverse trans-
formation for transition to the originals causes significant difficulties.
Image F(z) of the original discrete function f(nh) according to defini-
tion of Jury discrete transform (z-transform) is found as follows:
0
( ) ( ) ,n
n
F z f nh z
where n is argument of the original function (formally it is number h of
discretization step); z is argument of the image function [5].
Математичне та комп’ютерне моделювання
162
It is very convenient to use Jury transform when developing recurrence
formulas for numerical determination of values of the original function in
equidistant sampling nodes. It is enough to calculate value of the function in
some start nodes. Digital filters do work on this principle. They make it possi-
ble to simulate complex dynamic objects in real time, i.e. to calculate response
of such an object to an arbitrary input signal. Next output signal value is de-
termined from several input and output values of the object in previous discre-
tization time points. Simplicity of inverse transformation that leads to recur-
rence formulas is an important advantage of Jury transform.
From the formulas that determine Laplace transform and Jury trans-
form formally follows that z = exp(–ph), or p = (1/h)ln(z). Substitution of
continuous differentiation operator p with finite difference expression
leads to an infinite series. As a rule, truncation of the logarithm expansion
into Laurent series is used. Thus bilinear z-transform, that uses the first
two terms of expansion into Laurent series, is widely used: ln(z) =
=2 (z – 1) / (z + 1). Hence p = (2 / h) (z – 1) / (z + l). This method of recur-
sive digital filter synthesis is also called Tustin's method [3].
This is how traditional digital filter synthesis method for simulation of
continuous dynamic object is built [3]. Its main drawback consists in low ac-
curacy of calculation of the output signal of simulated dynamic object in dis-
cretization nods, caused by incorrect substitution of continuous operator with
discrete one. This is because the source model of dynamic object has been
defined as a rational fractional transfer function, i.e. as an initial decomposi-
tion of continuous operator of object on differentiation stage.
However, primary continuous operator of dynamic object can be
represented as a decomposition of rational fraction expression into partial
fractions. It corresponds in the originals to decomposition of continuous
operator into several convolution operations with several type kernels (ex-
ponential, exponential power and exponential trigonometric) [6]. Analog
model of dynamic object can be developed without integrators, solely on
inertial and oscillation elements [7].
It should be noted that integration is a subcase of convolution opera-
tion, when kernel is a singular constant function [2, 5].
Convolution method. Replacement of continuous convolution oper-
ator with finite difference expression can be performed more accurately.
Kernel of convolution operator is often an exponential function, that leads
to very low orders of difference equation of the desired digital filter. Let's
prove a simple but important theorem.
Theorem. Continuous operator 1 / (Tp + 1) in Laplace image space is
associated with discrete operator (l – g) z / (z – g), g = exp (–h / T)) in
z-image space.
Proof. Output signal Υ of inertial element is related with input signal
X by a simple equation in Laplace images
Серія: Технічні науки. Випуск 13
163
( ) ( ) ( ), ( ) 1 / 1 ,Y p V p X p V p Tp
where Τ is inertial element time constant. Product of two functions in im-
ages is associated in originals with convolution of two functions, that is
reduced to the problem of realization of Volterra integral operator:
0
( ) ( ) .
t
y t V t s x s ds
Convolution operation is formally defined for infinite integration
limits. However, since the argument of signal is time and not a spatial
coordinate, and kernel V(t) = 0 for t < 0, one can choose finite integration
limits. Signal start (t = 0) can be taken as a reference-starting point, upper
limit of integration will be variable here.
Let's find kernel function V(f) in originals according to Laplace
Transform Table. Let's reduce the V(p) function in images to tabular style:
( ) (1/ ) / ( (1/ )).V p T p T
Its parallel in the originals
( ) (1/ ) exp( / ).V t T t T
Consequently, passage of signal through inertial element in the origi-
nals is associated with operation of the signal function convolution with
kernel in the form of exponential function.
Let's discretize time t at increments of h, and move on to discrete
convolution:
0
( ) (( ) ) ( ) .
n
k
y nh n k h x kh h
Keep in mind that the value ds in continuous operator is associated
with increment of h in discrete operator. Thus, ν(nh) = (h / T) exp(–(nh/T)).
In z-images we obtain
( ) ( ) ( ), ( ) ( / ) / ( ), exp( / ).Y z V z x z V z h T z z g g h T
Let's move on to discrete originals through the shift theorem:
1
1
1
( )( ) ( / ) ( ),
( )(1 ) ( / ) ( ),
( ) ( ) ( / ) ( ),
( / ) , ( ), ( ).n n n n n
Y z z g h T zx z
Y z gz h T x z
Y z gz Y z h T x z
y gy h T x y y nh x x nh
Let's perform correction of digital filter static balance mode for the
case уn = уn–1 = хп with T 0. As a result, we obtain a working formula for
numerical simulation of inertial element:
1 (1 ) .n n ny gy g x
Next value of discrete output signal is calculated from the current
value of input signal and one of the previous values of the output signal.
Математичне та комп’ютерне моделювання
164
Coming back to z-images, we get V(z) = (1 – g) z / (z – g). Thus, from Lap-
lace images we can pass to z-images through the following substitution:
1/ ( 1) (1 ) / ( ), exp( / ).Tp g z z g g h T
The theorem is proven.
Accuracy of simulation of the primary continuous operator depends
solely on length of the filter order grid and according to [6] does not de-
pend on discretization increment (unlike Tustin's method). The validity of
this statement can be verified by analyzing disparity between calculations
of inertial element transient response obtained with digital filtering formu-
la and analytical formula y(t) = 1 – exp (–t / T).
If Tustin's method is used, significant dynamic error is observed in
numerical simulation of continuous object. Magnitude of this error de-
creases with decrease of discretization increment. Computational formula
structure is more complex [2].
Concepts of direct current static transmission factor for continuous
and discrete transfer functions are substantially different. In the first case,
this value is equal to one (constant), and in the second case it is close to
zero and has not physical, but mathematical meaning. In order to ensure
equal levels of input and output signals of dynamic object in equilibrium
state, this value can be chosen in such a way that its discrete transfer func-
tion shall be equal to one for z = 1. This condition ensures physical equali-
ty to one of transmission factor of continuous object discrete model.
Conclusions and generalization. If input signals used are smoother
than a jump, minor error in numerical simulation of inertial element output
signal can emerge as compared to analytical calculation. However, this
error can be eliminated with additional weak transversal filtering. In this
case, signal averaging in several adjacent points is actually performed in
accordance with high-order quadrature formulas.
Thus, convolution method can be considered the most efficient method
for synthesis of recursive digital filter to simulate linear stationary object with
lumped parameters. Structure of its recurrence computational formulas is
simpler than traditional one, and accuracy of numerical simulation of object is
higher, even with error down to zero with respect to analytical calculation.
Depending on the kind of roots (simple, multiple, complex conjugate) of
denominator polynomial of the primary rational fractional (in Laplace images)
transfer function of the channel we get several kinds of partial fractions. We
restricted ourselves to the case of one simple root, since all cases of complex
roots generally can be reduced to it. Thus, the case of multiple root corres-
ponds to a serial connection of several identical inertial elements [1], and the
case of complex conjugate roots corresponds to loop joint of two inertial ele-
ments [7]. Parameters of each elementary unit of dynamic objects can have
functional dependences on time or input signal amplitude, that allows for si-
mulation of nonstationary or nonlinear objects respectively.
Серія: Технічні науки. Випуск 13
165
Thus, it may be concluded that instead of common integration or dif-
ferentiation operations for development of dynamic models of continuous
objects it makes sense to use more complex operations in the form of con-
volutions with exponential power kernels. It improves accuracy of numeri-
cal simulation of continuous objects.
A recursive digital filter construction method is considered for simula-
tion of inertial element as a typical component of complex dynamic object.
New computational formulas are obtained. Their high accuracy as compared
to traditional ones is shown. Decomposition of initial model of simulated ob-
ject by convolution operations with several typical exponential kernels is pro-
posed instead of traditional operations of integration and differentiation.
References:
1. Methods and devices for interpretation of experimental dependencies in the study
and control of energy processes / A. F. Verlan, B. B. Abdusatarov,
A. A. Ignatchenko, N. A. Maksimovich. — Kyiv : Nauk. dumka, 1993. — 208 p.
2. Verlan A. F. Integral equations: methods, algorithms, programs / A. F. Verlan,
V. S. Sizikov. — Kyiv : Nauk. dumka, 1986. — 544 p.
3. Sibert U. M. Circuits, signals, systems. 4.1 / U. M. Sibert. — М. : Mir,
1988. — 336 p.
4. Verlan A. F. Development of recursive digital filter for recovery of continuous
signals / A. F. Verlan, N. A. Maksimovich // USiM (Control Systems and Ma-
chines). — 1999. — № 1. — P. 18–25.
5. Korn G. Mathematics handbook for scientists and engineers / G. Korn,
Т. Korn. — М. : Nauka, 1984. — 832 p.
6. Maksimovich N. A. Computational dynamic control system correctors. Au-
thor's extended abstract of PhD in Technical Sciences dissertation / N. A. Mak-
simovich. — Kyiv, 1991. — 16 p.
7. Maksimovich N. A. Integration-free models of oscillating element /
N. A. Maksimovich, V. A. Fedorchuk // Collection of research papers of Ka-
menets-Podolskiy State University. Series physics and mathematics. 3rd is-
sue. — 1997. — P. 58–64.
Розглянуто метод побудови рекурсивного цифрового фільтра для
імітації ланки — типового елементу складного динамічного об’єкта.
Отримано нові розрахункові формули. Показано їх високу точність
порівняно з традиційними та доцільність декомпозиції вихідної моде-
лі об’єкта, що імітується, за операціями згортки з декількома типови-
ми експоненціальними ядрами замість традиційних операцій інтегру-
вання та диференціювання.
Ключові слова: математична модель, передатна функція, інте-
гральний оператор Вольтерри, оператор згортки, цифровий фільтр.
Отримано: 26.04.2016
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/SUO <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>
/SVE <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>
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/UKR <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>
/RUS <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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
|