Methods of Complex Dynamic Systems' Models' Equivalent Conversion
Proposed and considered the formal description of equivalent conversions which can be applied to obtain the various models of different kinds of complex dynamic systems (including electrical systems, power installations, etc.), as well as for transition from one representation to another. The set of...
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irk-123456789-1814672021-11-18T01:26:42Z Methods of Complex Dynamic Systems' Models' Equivalent Conversion Verlan, A. Sterten, Jo Proposed and considered the formal description of equivalent conversions which can be applied to obtain the various models of different kinds of complex dynamic systems (including electrical systems, power installations, etc.), as well as for transition from one representation to another. The set of basic operations which realize elementary conversions of models is described. The methods and algorithms for conversion of differential equations into integral or integro-differential are considered Запропоновано та розглянуто формальний опис еквівалентних перетворень, які можна застосувати для отримання різних моделей складних динамічних систем (включаючи електричні системи, енергоустановки тощо), а також для переходу від одного представлення до іншого. Описано набір основних операцій, які реалізують перетворення моделей. Розглянуто методи та алгоритми перетворення диференціальних рівнянь в інтегральні та інтегро-диференціальні. 2020 Article Methods of Complex Dynamic Systems' Models' Equivalent Conversion / A. Verlan, Jo Sterten // Математичне та комп'ютерне моделювання. Серія: Технічні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2019. — Вип. 20. — С. 16-25. — Бібліогр.: 6 назв. — англ. 2308-5916 DOI: https://doi.org/10.32626/2308-5916.2019-20.16-25 http://dspace.nbuv.gov.ua/handle/123456789/181467 519.626, 519.6,004.942:519.876.2 en Математичне та комп'ютерне моделювання. Серія: Технічні науки Інститут кібернетики ім. В.М. Глушкова НАН України |
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Proposed and considered the formal description of equivalent conversions which can be applied to obtain the various models of different kinds of complex dynamic systems (including electrical systems, power installations, etc.), as well as for transition from one representation to another. The set of basic operations which realize elementary conversions of models is described. The methods and algorithms for conversion of differential equations into integral or integro-differential are considered |
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Verlan, A. Sterten, Jo |
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Verlan, A. Sterten, Jo Methods of Complex Dynamic Systems' Models' Equivalent Conversion Математичне та комп'ютерне моделювання. Серія: Технічні науки |
| author_facet |
Verlan, A. Sterten, Jo |
| author_sort |
Verlan, A. |
| title |
Methods of Complex Dynamic Systems' Models' Equivalent Conversion |
| title_short |
Methods of Complex Dynamic Systems' Models' Equivalent Conversion |
| title_full |
Methods of Complex Dynamic Systems' Models' Equivalent Conversion |
| title_fullStr |
Methods of Complex Dynamic Systems' Models' Equivalent Conversion |
| title_full_unstemmed |
Methods of Complex Dynamic Systems' Models' Equivalent Conversion |
| title_sort |
methods of complex dynamic systems' models' equivalent conversion |
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Інститут кібернетики ім. В.М. Глушкова НАН України |
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| citation_txt |
Methods of Complex Dynamic Systems' Models' Equivalent Conversion / A. Verlan, Jo Sterten // Математичне та комп'ютерне моделювання. Серія: Технічні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2019. — Вип. 20. — С. 16-25. — Бібліогр.: 6 назв. — англ. |
| series |
Математичне та комп'ютерне моделювання. Серія: Технічні науки |
| work_keys_str_mv |
AT verlana methodsofcomplexdynamicsystemsmodelsequivalentconversion AT stertenjo methodsofcomplexdynamicsystemsmodelsequivalentconversion |
| first_indexed |
2025-07-15T22:41:02Z |
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1837754505666494464 |
| fulltext |
Математичне та комп’ютерне моделювання
16
UDC 519.626, 519.6,004.942:519.876.2
DOI: 10.32626/2308-5916.2019-20.16-25
A. Verlan, PhD,
Jo Sterten, PhD
Norwegian University of Science and Technology, Gjovik, Norway
METHODS OF COMPLEX DYNAMIC SYSTEMS’
MODELS’ EQUIVALENT CONVERSION
Proposed and considered the formal description of equivalent
conversions which can be applied to obtain the various models of
different kinds of complex dynamic systems (including electrical
systems, power installations, etc.), as well as for transition from
one representation to another. The set of basic operations which re-
alize elementary conversions of models is described. The methods
and algorithms for conversion of differential equations into integral
or integro-differential are considered.
Key words: Dynamic systems, model conversion, integral
equations.
Introduction. For investigation of dynamic systems their modeling
on the base of differential equations is used in most cases. Appropriate
models and methods of their solution are well known and widely applied
in practice. At the same time, it is not always evident what kind of model
is better to use for a particular system. Selection of the adequate and at the
same time enough simple model is actually an art from many points of
view. Besides differential equations, there are many other means for de-
scription of dynamic systems. Those are, primarily, integral and integro-
differential equations. For many problems integral equations are preferable
than differential ones. Thus, it is obviously important to create and devel-
op mathematical methods and computer tools which would allow to con-
vert one model description to another.
Formalization of equivalent conversions’ description. Let us consider
the formalized description of equivalent conversions which can be applied
to obtain the various models of the researched dynamic system and for
transition from one representation to another.
Let we have an operator model defined by the equation
1(u) = 2(u). The basic operations realizing elementary conversions of
the model are:
the additive conversion
1(u) = 2(u) 1(u)+3(u) = 2(u)+3(u);
the multiplicative conversion
1(u) = 2(u) 1(u) 3(u) = 2(u) 3(u);
© А. Verlan, Jo Sterten, 2019
Серія: Технічні науки. Випуск 20
17
the additive splitting
1(u) = 2(u) 11(u) + 12(u) = 2(u);
the multiplicative splitting
1(u) = 2(u) 11(u) 12(u) = 2(u);
the partial additive inversion
1(u) = 2(u) u = –1
11(u) (2(u) – 12(u));
the partial multiplicative inversion
1(u) = 2(u) 12(u) = –1
11(u) (2(u)).
Combining these basic operations, we can obtain more complex con-
versions of mathematical models.
Let's consider some realizations of algorithms of equivalent convert-
ing differential equations to integral or integro-differential ones [1, 2, 4].
In general case, it should be noted that precise reverse transition from in-
tegral to differential form of mathematical model is not always possible.
The integral form of mathematical models representation is more universal
than the differential one. It allows to describe much more physical objects,
both with lumped and distributed parameters.
Method of analytical inversion with operator splitting. Let a model of
the object is given in a form of the ordinary differential equation (ODE)
1
[ ] , 0 , 0, 1,
n
n i in
i i
i
D y y t a y t f t y C i n
(1)
or, in the functional form,
[ ] .D y f (2)
To obtain a series of equivalent integral dynamic models [3, 5, 6],
i. e. relations containing integral operators, rather general method based on
different versions of splitting the initial differential operator can be ap-
plied. Indeed, splitting the operator D into two operators, i. e. putting
D = D1 + D2, we obtain the following differential equation
1[ ] ,D y (3)
where (t) = f(t) – D2[y]. Choosing such form of decomposition which
admits analytical solution (3) is available, can allow us to get the equation
1
1 ,y D
(4)
The operator
1
1D
which is inverse to D1 is in integral operator,
therefore (4) is the integral or integro-differential equation.
The considered method of equivalent conversion can be applied both
to linear and nonlinear problems. If for example the nonlinear differential
equation Dn[y] = f with a given nonlinear operator Dn, then for its decom-
position it is reasonable to separate its linear part, i. e. to use the represen-
tation Dn = D1 + D2n, where D1 is a linear operator. Then the initial equa-
Математичне та комп’ютерне моделювання
18
tion is reduced to the form (4), which is generally the nonlinear integro-
differential equation.
Let's consider this method in more details on the example of the
equation (1), which can be rewritten as
1 1
.
m n
n n i n i
i i
i i m
y t a y t f t a y t
(5)
After the substitution of variables
1
, ,..., ,
n m n m m n
u t y t u t y t u t y t
(6)
we obtain the m-th order equation
1
1
,
m
m i
i
i
u t a u t t
(7)
where
1
.
n
n i
i
i m
t f t a y t
(8)
Converting the equation (7) to an equivalent the 1st order ODEs sys-
tem and building its solution, e. g. using fundamental solutions of this sys-
tem, we obtain the equation with exponential kernel:
0
0
, , ,
t
At At
u t e u e a u (9)
where
u(t) = [u'(t), u''(t), …, u(m)(t)], u0(t)=[u'(0), u''(0), …, u(m)(0)],
(a, u, t)=[0, 0, …, (t)],
and we get the following m-th order matrix A:
1 2 1
0 1 0 0
0 0 1 0
.
m m m
A
a a a a
The unknown variables in the equations (1) and (9) are connected by
the relation
1
0 0 0
... .
1 !
n mt t t
n m
t s
y t u s ds u s ds
n m
(10)
The transition from one form of a model to another is carried by
modifying value of m from 1 to n.
The method of sequential integration. If we put m = n in the regarded
method, decomposition of the operator D is reduced to solution of the ini-
Серія: Технічні науки. Випуск 20
19
tial equation with respect to the higher derivative. In this case the solution
of the equation (3) is carried out by n sequential integrations. As the result
we get the following integral equation:
0
,
t
y t K t s y s ds F t (11)
where
1
1
1 1 1
0
1 10
1 12
1 2
1
,
1 !
...
1 ! ! !
...
1 ! 1 !
in
i
i
nt i in n
i i
i i
i nn
i n
i
t s
K t s q
i
t s t t
F t f s ds C C q
n i i
t t
C q C
i n
(12)
The method of higher derivative. This method is usually considered
in the literature. It represents a special case of the splitting method based
on the substitution
1
1
0
, , ...
t
n n
u t y t u s ds c y t
It allows to obtain the equivalent integral equation with respect to the
higher derivative of the initial equation (1):
1
10
1 1 1 2 2
1
1 1 0
,
1 !
...
... .
1 !
kt n
n n
k
k
n n n
n
n n
t s
y t a y s ds t
k
t f t C a C t C a
t
C C t C a
n
(14)
The analytical methods of equivalent transition from the ordinary dif-
ferential equations to integral ones can be effectively implemented using
the packages oriented to analytical conversions (Mathematica, Maple etc.).
The structure of the algorithm which allows to carry out the most general
method of analytical inversion with operator splitting is shown in the Fig. 1.
This method at m = n is reduced to the sequential integration method,
and at m = 0 — to the higher derivative method. In general case it allows
us to obtain an integro-differential equation, and in the two last special
cases we obtain pure integral equations.
It is useful to develop intelligent program environments for selecting
appropriate models oriented to simulation and modeling of dynamic sys-
tems. Some approaches to solving this problem were discussed in [3].
Математичне та комп’ютерне моделювання
20
Fig. 1. Algorithm of analytical inversion with operator splitting
Comparing the sequential integration method and the higher deriva-
tive method, we have to note that only in the first respect to the required
function u(t) = y(t) is obtained. In the rest cases (at m < n) we obtain the
equation with respect to derivative of required function, and obtained solu-
tion should be integrated m – n times.
Continuing a comparison of the methods of transition from differen-
tial to integral form of mathematical models, we have to note that for the
sequential integration method in algorithm in the Fig. 1 the item 6, and in
the higher derivative method the item 4 are omitted. Anyway the problem
of numerical integration remains. In the sequential integration method we
have to integrate the right hand side of the differential equation, and in the
higher derivative method — the obtained solution.
For an investigation of dynamic models with approximate initial data
(for example, obtained by measurements) the sequential integration meth-
Серія: Технічні науки. Випуск 20
21
od is preferable. In this case approximate initial data are integrated on the
first step of algorithm. In this case the influence of errors in initial data
(especially if errors look like a white noise) can be considerably reduced.
Conversion of nonlinear models. Let’s consider a possibility to ob-
tain the equivalent integral equations for nonlinear object in the case when
one of derivative is included into the initial differential equation under the
sign of continuous nonlinear function, i. e. when the nonlinear differential
equation has the form
0
,
n
i m
i
i
a t y t F y t f t
(15)
with the initial conditions y(i)(t0) = Ci , and continuous variable coefficients
ai (t), i = 0, …, n – 1.
We suppose that an = 1, am = 0 and m n.
An integral equation with respect to the m-th derivative can be ob-
tained integrating the equation (15) n – m times.
Let's consider at first the case of m = 0. Integrating the equation (15)
n + 1 times, we reduce it to the following form that does not contain de-
rivatives:
0 0 0
1 , .
!
nt t t t
n n
t s t t
t s
M s d y s ds F y s ds s ds
n
(16)
Here
11
1 1
, 1 ,
1 !
i jn n
i jj
n n i n i
i j
s
M s C a s
i j
0
1
,
1 !
nt
n n
t
t s
t f s ds P t
n
11
0
1
1 0 1
1 ,
1 !
i jn k k i
i i i
n j k jk
k i j
t t
P t a C C
i j
!
, .
! !
i
i m
k nk i
t s
d n
a a t C
m n mdt
In a special case when the higher derivative is included into the equa-
tion under the sign of continuous nonlinear function F, i. e. when the non-
linear differential equation has the form
1
0
,
n
n i
i
i
F y t a t y t f t
(17)
with the initial conditions y(i)(t0) = Ci and continuous variable coefficients
ai(t), i = 0,…, n – 1, the equivalent integral equation can be written as
Математичне та комп’ютерне моделювання
22
0
, ,
t
n n n n
t
F Z t K t s Z s ds t (18)
where
11
0
1
11 1
0
1
0 1
, ,
1 !
,
.
1 !
n in
n i
i
n
n
n jn n
n i i j
i j
t t
K t s a t
n i
Z t y t
t t
t y t a t C
n j
The equation (18) is obtained by substituting the derivatives y(i)(t),
expressed in terms of y(n)(t), into (17):
0
1
0
1
1
.
1 ! !
n i jn it n i
i
n i j
jt
t tt s
y t Z s ds C
n i n i j
In the case m 0 the equation (15) can be represented as
1
1 0
.
n m
i m i
i i
i m i
a t y t F y t a t y t f t
Taking into account (18) it can be written in the following form:
0
1
1
1
0
, .
tn m
i
m i m m m m m
i t
a t Z t F Z t K t s Z s ds t
(19)
Sequentially integrating (19) and considering (16), we obtain the
nonlinear integral equation
0 0
0
1
1
1
1 ,
1 !
, ,
n mt t t
n m m m
t s t
t
n m
m m
t
t s
M s d Z s ds F Z s ds
n m
I K t s Z s ds t
where
0 0
1
1 ,
1 !
n mt t
m
t t
t s
t s ds Pn n t dt
n m
In–m–1[Km(t, s)] means the application of the operator
, ,
t
m m
s
I K t s K s d
n – m – 1 times. The operator I arises when the integration limits are
changed:
Серія: Технічні науки. Випуск 20
23
0 0 0
0 0
, ,
, , .
t s t t
m m m m
t t t
t t t
m m m m
t s t
ds K s Z d Z d K s ds
Z s ds K s d Z s I K t s ds
Thus, the integral equation equivalent to (15) has the following form
0 0
1
, ,
1 !
n mt t
m m
t t
t s
F Z s ds Q t s Z s ds t
n m
(20)
where
1
1, 1 , , , 0, .
t
n m
n m m
s
Q t s M s d I K t s m n
It follows from the formulas (16), (18), (20), that the left-hand side of
the integral equation, equivalent to the given nonlinear differential equa-
tion, in which one of derivatives is included nonlinearly, consists of two
components. One of these components is the nonlinear with respect to the
function y(m)(t), and the second one is an application of integral operator
with the kernel Q(t – s) to y(m)(t).
If a model is represented by a system of differential equations, we
have more wide opportunities to transform it to integral or integro-
differential form then in the case of a single equation. Every equation in
the system can be transformed in different way, equations can be com-
bined etc. We can also reduce the model dimension decreasing the number
of governing equations. This can be done more flexible than if we use the
differential approach: the higher derivatives demanding special treatment
at numerical solution do not appear, and resulting integral equations are
solved in usual way.
To illustrate these opportunities, we consider the following example.
The simple quarter car model of automotive suspension (Fig. 2) is de-
scribed by following system of ODEs:
1 2
1 2
,
,
b b s b w s b
w w s b w s t w r t w r w
m x C x x x m g
m x C x x x C x x x x m g
(21)
where mb and mw are masses of the body and the wheel, xb, xw are their
displacements, xs2 is a displacement in the second section used as the lam-
inated spring model, Ci, i are stiffnesses and viscosities in the models of
the spring and the tire, Ffr is the friction force in the spring model.
Instead of the last equation in the system (21), on the stage when the
second element of spring model is deformed we can use a differential
equation with respect to the spring tension Fs = Cs1(xb – xw – xs2):
Математичне та комп’ютерне моделювання
24
2 2
1
1
,
s f s
s s s
s
C x F sgn x
F F C x
n
(22)
where
1 2
s
s s
n
C C
is a characteristic relaxation time [1], and
b wx x x is the spring deformation.
Fig. 2. Automotive suspension
Solving this equation analytically, we obtain the following relation
which includes integral operators with the relaxation kernel
t s
ne
:
0
0 0
0 1 0
1
1 2 .
t t
n
s s s
t s t st t
frs n n
s s
s st t
F t F C x e
FC
C x t e x s ds sgn x s e x s ds
(23)
The substitution of (23) into the dynamic equations (the first two ones in
the system (21) leads to a system of integro-differential equations. Integrating
them twice, we obtain two Volterra integral equations of the 2nd kind [4].
Conclusions. Using the proposed approach, we can obtain integral
equations for each of the variables xb, xw and xs2 or Fs by the sequential
integration method. Also, we can obtain the simple system of integral
equations (but with larger number of equations) by transition from (21) to
Серія: Технічні науки. Випуск 20
25
the 1st order ODEs system and integrating it. One integral equation with
respect to a single governing function (e. g. the car's body displacement or
acceleration) can be obtained instead of two dynamic equations.
Thus, the obtained equivalent conversions’ formalized description along
with the proposed conversion algorithms provide the possibility to create vari-
ous models of different kind of complex dynamic systems and make conven-
ient and effective conversion of the models from one representation to another,
i.e. from differential equations to integral or integro-differential.
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МЕТОДИ ЕКВІВАЛЕНТНОГО ПЕРЕТВОРЕННЯ
МОДЕЛЕЙ СКЛАДНИХ ДИНАМІЧНИХ СИСТЕМ
Запропоновано та розглянуто формальний опис еквівалентних пе-
ретворень, які можна застосувати для отримання різних моделей
складних динамічних систем (включаючи електричні системи, енер-
гоустановки тощо), а також для переходу від одного представлення до
іншого. Описано набір основних операцій, які реалізують перетво-
рення моделей. Розглянуто методи та алгоритми перетворення дифе-
ренціальних рівнянь в інтегральні та інтегро-диференціальні.
Ключові слова: динамічні системи, перетворення моделі, інтег-
ральні рівняння.
Отримано: 21.08.2019
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