Structural-Schematic Approach in the Modelling of Dynamic Systems

Abstract

The structural-schematic approach is offered for the stage of construction of model of mathematical modeling process of the dynamic systems. The phase-space (state-space) method is utilized. The methods of construction of schemes and equations are illustrated for more simple cases, and ideology of construction of mathematical models of the multi-circuit systems is demonstrated. Differential equation systems, operator equations, phase space, control.

1. Introduction

2023 Copyright for this paper by its authors. CEUR

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2. Formulation of the problem

In this work, we will mainly consider the methods ofconstruction the mathematical models of dynamic control systems with single input and single output (SISO) using structural diagrams in state variables, the dynamics of which can be described by an ordinary differential equation of the form. d n yt  dt n  an1 d n1 yt  dt n1    a0 yt   bm d mut  dt m  bm1 d m1ut  dt m1    b0ut  ( 1 ) under initial conditions

y0  y0 , y1 0  y01 ,, yn1 0  y0n1; m  n or by a system of ordinary differential equations (ODE).

In the last part of our work, we will briefly show that the methods that are applicable for systems such as SISO also work successfully in the case of multi-circuit systems with multiple inputs and outputs (MIMO). 2.1.

The method of combining derivatives.

Let us first consider a simpler case when the initial conditions of equation ( 1 ) are zero. At the same time, using Laplace transformation [20], equation ( 1 ) in operator form will be rewritten as

A pY  p  B pU  p, where

Let's present equation ( 2 ) in the form We replace this equation with the following two:

A p  p n   ai pi ;

B p  bi pi .

Y  p B p  U  p A p

Y  p B p  X  p, U  p A p  X  p, where X  p - auxiliary variable Laplace-image of xt .

In the differential form, the last two equations will take the form: bm d n xt  dt n d m xt  dtm  an1  bm1 d n1 xt 

dt n d m1xt  dtn    a0 xt   ut    b0 xt   yt 

Thus, instead of solving equation ( 1 ), you can solve equations ( 3 ), ( 4 ). Let's build a structural diagram of the solution of equation ( 3 ), for which we will solve it with respect to the higher derivative d n xt  dtn  an1 d n1xt  dtn

d n2 xt   an1 dtn2  a0 xt   ut 

Suppose that the n -th derivative of the function xt  is known, then, applying it to a chain of serially connected integrators, at the output of each subsequent integrator we will have derivatives of the function xt  of lower orders. If we multiply each of these derivatives by the corresponding coefficient ai and form their negative sum together with the function ut , then we get the right-hand side of equation ( 5 ), i.e., the n -derivative of the function xt , which was previously assumed to be known. The function yt , which is a solution of equation ( 1 ), is obtained by a linear combination of derivatives of xt , taken with coefficients b1 in accordance with equation ( 4 ). The structural diagram of the solution of equations ( 3 ), ( 4 ) is shown in Fig. 1

If the initial conditions of equation ( 1 ) are non-zero, they must be converted into the initial conditions of equation ( 3 ). Assuming that equations ( 3 ), ( 4 ) are equivalent to ( 1 ) and with non-zero initial conditions, we rewrite these equations in operator form: ( 2 ) ( 3 ) ( 4 ) ( 5 ) where Y0  p, a, X 0  p, a, X 0  p, b - polynomials of the initial conditions.

Substituting U  p  and Y  p from the last two equations ( 6 ) into the first, we get

A pB pX  p  X 0  p, b Y0  p, a  B pA pX  p  X 0  p, a ( 6 ) ( 7 ) ( 8 ) By equating the coefficients with the same degrees of variable p of this equality, we find the initial conditions of the integrators of the structural scheme of Fig. 1.

The application of this method for constructing state equations does not require transformation of equation ( 1 ). It is possible to make a scheme in state variables directly according to the form of equation ( 1 ), because its coefficients are the coefficients of the structural scheme in state variables. Recalculation of the initial conditions is carried out according to equation ( 7 ). 2.2.

The method of successive integration.

Let's rewrite, using Laplace-transform [20], equation ( 1 ) in operator form with zero initial conditions in the form from where we have

n1 m pnY  p   ai piY  p   bi piU  p

i0 i0 n1 1

Y  p  i0 pn1 biU  p  aiY  p; bi  0 if i  m

Let's make a chain of n serially connected integrators. We will take the signal at the output of the rightmost integrator as yt  . The fulfillment of equality ( 8 ) is ensured if the difference b0ut   ao yt  is applied to the input of the leftmost integrator, the difference b1ut   a1 yt  and the output of the previous integrator are applied to the input of the next one, etc. The structural diagram of the solution of equation ( 8 ) will then take the form shown in Fig. 2.

If we again choose as state variables xt  the outputs of the integrators of the structural diagram of xn1t   xnm1t   am t x1t   bmut ; xnt   a0 x1t   b0ut ; yt   x1t .

Y  p  dU p   c1  c2   p  1   p  2   

cn U  p.  p  n 

The coefficient d will be different from zero only when m  n . The structural scheme of the state variables of equation ( 12 ) is shown in Fig. 3.

With non-zero initial conditions of equation ( 1 ), the initial conditions of the integrators of the structural diagram are determined by the ratio [18] i xi 0   ank y0ik 

k0

As for the method of combining derivatives, the coefficients of equation ( 1 ) are simultaneously the coefficients of the structural scheme in the state variables. 2.3. Method of decomposition of the transfer function into elementary fractions.

A rather promising approach for the synthesis of systems using the state space method is the construction of a structural scheme in state variables by decomposing the transfer function into elementary fractions. Its idea is as follows. Equation ( 1 ) is presented in the form of a transfer function Let's decompose this transfer function into elementary fractions

Y  p U  p

b pm  bm1 pm1    b0  m

pn  an1 pn1    a0 Y  p U  p  d 

c1  c2  p  1   p  2   

cn  p  n  hence ( 9 ) ( 10 ) ( 11 ) ( 12 )

The state equations have the form ( 13 ) ( 14 )

3. Equation of state of multi-circuit systems.

The advantages of the state space method are most evident when building models of complex multidimensional and multi-circuit systems.

It is appropriate to use the following steps of constructing state equations for modeling complex systems. Usually, a functional scheme of the system is given, the analysis of which must be performed. From the functional scheme by determining the dynamics equations (transfer functions) of the elements of this system, you can go to the algorithmic scheme. Such a scheme makes it possible to preserve the mutual relations of the elements of the functional scheme of the system and at the same time obtain a mathematical description of its dynamics. (This step is demonstrated in Example 2). Then, for all elements of the system, structural schemes are drawn up in state variables, which are combined with each other in the same way as the elements of the system under study. This method of constructing state equations has an important advantage. It consists in the fact that some state variables will correspond to the real variables of the system, which makes it possible to trace their behavior over time after solving the state equations. Let us consider the methodology for determining the equations of state for a multidimensional system of the MIMO type using a specific example.

Example 1. Figure 4 shows the algorithmic diagram of a multi-circuit system (MIMO). It has three inputs u1, u2, u3 and two outputs y1, y2. The presence of many internal connections between its elements makes it difficult to analyze this system using classical methods of automatic control theory. Let's consider a method for obtaining equations of state for this system. Figure 5 shows a circuit-scheme in state variables, and each element of the algorithmic circuit-scheme has its own scheme in state variables and they are connected to each other in the same way as the elements of the algorithmic circuit-scheme.

The equations connecting the state variables ( 1 − 5) with each other, as well as the input ( 1 − 3) and output ( 1, 2), are obtained directly from the diagram in the state variables: x5  2x3  3x4  4x5  u3

Finally, we want to show one simple example, of which confirms the practical value of the results of the presented article.

Example 2. Let's consider a simplified diagram of the operation of a radar station (Fig. 6). Namely, we are interested in modeling the tracking system, which receives as input the angle of extinction of the observed target, and its precise processing by the radar. In Fig. 7 shows a functional scheme of a power tracking system, containing as a measuring-converting element an SD-SP synchro pair, a phase discriminator FD, a magnetic amplifier MU, an electric machine power amplifier EMU, a DC motor DW, a gearbox Red, a working mechanism RM, a tach generator TG in the circuit local feedback. The algorithmic scheme of this system is shown in Fig. 8. The corresponding scheme in state variables is shown in Fig. 9. For clarity of the circuit in state variables, individual elements are constructed using different methods. (See for detail [18]). System of equations for the scheme in Fig. 9 looks like x7  KTG x1 

TTG

1 TTG

x7   x1

K DW x T 2

5 x7  K P

The all obtaining results are constructive from the point of view of performing computational experiments. All presented analytics are currently implemented without problems in a fairly wide range of IT, for example, one of the most common is the SIMULINK toolkit of the MATLAB package [1416]. In the future, they can be extended to the case of multi-circuit systems, which is described in terms of functional-differential equations (FDE), taking into account the factor of argument deviation.

4. Conclusion

The following results were achieved in this work: - Using a fairly simple approach, we showed three methods for constructing systems of equations of state in phase space for the case when the type of input and output of the system (SISO) is known, but its internal structure is not known. - We have demonstrated the possibility of using state space methods in the case of multi-circuit systems MIMO, that allow a certain decomposition into SISO systems.

5. Acknowledgements

This work is conducted under the Agreement on scientific cooperation between Faculty of Computer Science and Cybernetics Taras Shevchenko National University of Kyiv, Ukraine and the Faculty of Electrical Engineering and Communication, Brno University of Technology, Brno, Czech Republic.

Andriy Shatyrko and Denys Khusainov were supported by the project #BF015-04 at Taras Shevchenko National University of Kyiv.

Josef Diblik has been supported by the project of specific university research at Brno University of Technology FEKT-S-23-8179 (Faculty of Electrical Engineering and Communication).

6. References

References

[1] Fedotkin U. М., Burley U. Yu ., Ryumshin N. А . Mathematical modeling of technological processes. Reactor theory, recirculation and boundary layer - К ., Теchnika , 2003 . - 424p . (in Ukraine)

[2] Emmanuel Okechukwu Chukwumuanya , Kizito Chidozie Onyeneho. Modeling of Control Systems for Industrial Operations // Journal of Research in Mechanical Engineering Volume 8 , Issue 5 , - 2022 . - pp. 50 - 59 ISSN(Online): 2321 - 8185 .

[3] Ellis Cumberbatch , Alistair Fitt. Mathematical Modeling. Case Studies from Industry . Cambridge University Press., 2001 . - 291p .

[4] Tahmineh Azizi , Bacim Alali, Gabriel Kerr. Mathematical Modeling: With Applications in Physics, Biology, Chemistry, and Engineering, India. United Kingdom. BP International, 2021 . - 127p .

[5] Sandip Banerjee . Mathematical Modeling Models, Analysis and Applications. 2nd Edition . Chapman and Hall/CRC., 2021 . - 433p .

[6] Leah Edelstein Keshet . Mathematical Models in Biology. SIAM , Philadelphia. 2005 . - 615p.

[7] Novoseltcev V.N. Mathematical modeling in biology: systems capable of living and dying // Automation and Remote Control, Iss.6 , 2006 . - pp. 3 - 26 .

[8] Oleksii Byckov , Evgen Gurko, Denys Khusainov, Andriy Shatyrko, Bedrich Puza and Veronika Novotna . Construction of a Mathematical Model of the Heart Muscle // CEUR Workshop Proceedings “IT&I- 2021 ”, 2022 , Vol. 3132 , р. 272 - 280 .

[9] Liashenko O.I. Mathematical modeling of the dynamics of an open economy . - Rivne, Volyn oberig, 2005 . - 360p . (in Ukraine).

[10] Bahrushin V.E. Mathematical modeling. Textbook . - Zaporizhzhya, 2004 .- 140p . (in Ukraine).

[11] Khusainov D. Ya ., Kharchenko I.I. , Shatyrko А .V. Modeling of dynamic systems : Textbook. - Kyiv: Кyiv university, 2011 .- 136p . (in Ukraine).

[12] Krack Yu .V., Shatyrko A.V. Control theory for computer scientists . Textbook . - Kyiv: Кyiv university, 2015 . - 175с . (in Ukraine).

[13] Krasnoschekov P. , Petrov А . Principles of model construction. - Phasis . 2000 . - 411p.

[14] Devendra K. Chatutvedi . Modeling and Simulation of Systems using MATLAB and Simulink . CRC Press., 2010 . - 734p .

[15] Dingyü Xue , Yangquan Chen . Modeling, Analysis and Design of Control Systems in Matlab and Simulink . World Scientific Pub. Co Inc., 2014 . - 558p .

[16] Sulaymon L. Eshkabilov Practical MATLAB Modeling with Simulink . Apress , Springer, 2020 . GitHub - Apress/practical-matlab-modeling-w-simulink: Source Code for 'Practical MATLAB Modeling with Simulink' by Sulaymon L . Eshkabilov

[17] D. Basmadjian , R.Farnood The Art of Modeling in Science and Engineering with Mathematica 2 - nd ed. // Chapman&Hall, CRC , New York, 2007 . - 509p .

[18] Kozirev V.D. The application of digital computers in the study of automatic radio-electronic tracking systems . - Kyiv, KVIRTU PVO , 1976 . - 183p . (in Ukraine).

[19] Ingo Steinbach , Hesham Salama. Lectures on Phase Field . Springer, 2023 . - 122p .

[20] Joel L. Schiff . The Laplace Transform: Theory and Applications . Springer, 2012 . - 235 p.