# Synthesis of Adequate Mathematical Description as Solution of Special Inverse Problems

### Abstract

The problem of mathematical simulation of dynamic system characteristics behavior andÂ their adequacy to real experimental data, which correspond to these characteristics, is considered inÂ this paper. The specified problem is still poorly investigated and hardly adapted to formalization. TheÂ requirements of related to the adequate mathematical simulation of dynamic system are considered forÂ the case when mathematical description is represented by system of the ordinary differential equations.Â The conditions are obtained which allow to reduce a problem of the adequate mathematical descriptionÂ to the solution of the several integral equations of the first type. The methods of obtaining of the steadyÂ solutions are suggested. The domains of application of the obtained solutions are specified. For a case,Â when the differential equations of dynamic system are given with errors in coefficients, several variantsÂ of synthesis of the adequate mathematical descriptions depending on final goals of this description useÂ are suggested. The examples of the adequate descriptions of concrete dynamic systems are given.

### References

M. Alexik. Modelling and Identification of Eye-Hand Dynamics. Simulation Practice and

Theory, 8, 25-38, 2000.

J. Awrejcewicz and V. Krysko. Introduction to Asymptotic Methods. Taylor and Francis,

F. Breitenecker, F. Judex, N. Popper, K. Breitenecker, A. Mathe and S. Wassertheurer.

Laura and Petrarca - True Emotions vs. Modelled Emotions. 6-th Vienna Conference on

mathematical Modelling, Vienna, full Papers CD Volume, Vienna Univ. of Technology,

ISBM 978-3-901608-35-3, 46-69, 2009.

Ju. Gelfandbein and L. Kolosov. Retrospective Identification of Perturbations and Interferences.

moscow, Science, 1972.

V. Gubarev. Method of Iterative Identification of Many-Dimensional Systems with Inexact

Data. Part 1. Theoretical basises. Problems of Control and Information, Kiev, Ukraine, 2,

-26, 2008.

O. Gukov. Algorithms of Iterative Identification of Many-Dimensional Systems. XV International

Conference on Authomatical Control "Authomatics -2008", Odessa: INI,

Ukraine, 774-777, 2008.

H. Hirahara. Engine Modeling and Control System Design Considering Twist of a Crank

Shaft. 6-th Vienna Conference on mathematical Modelling, Vienna - Full Papers CD

Volume, Vienna Univ. of Technology, ISBM 978-3-901608-35-3, 173-179, 2009.

S. Ikeda, S. Migamoto, Y. Sawaragi. Regularization Method for Identification of Distributed

Systems. IY a Symposium IFAC, Identification and Evaluation of Parameters of

Systems, Tbilisi, USSR, Preprint. moscow, 3, 153-162, 1976.

N. Krasovskij. Theory of Motion Control. Science, Moscow, 1968.

V. C. Krass and B. P. Chuprinin. Shape Mathematics in Economics. Mathematical Methods

and Models, Finance and Statistics, Moscow, 2007.

Y. Liu, D. Soffker. Robust Control Approach for Input-Output Linearizable Nonlinear

Systems with Modeling Errors Based on High-Gain PI-Observer. 6-th Vienna Conference

on mathematical Modelling, Vienna, full Papers CD Volume, Vienna Univ. of Technology,

ISBM 978-3-901608-35-3, 193-199, 2009.

Yu. Menshikov. Identification of Moment of Technological Resistance on Rolling Mill.

Journal of Differential Equations and Their Applications in Physics, Dnepropetrovsk University,

Ukraine, 1, 22-28, 1976.

Yu. Menshikov. The Models of External Actions for Mathematical Simulation. System

Analysis and Mathematical Simulation, 14(2), 139-147, 2004.

Yu. Menshikov. About Adequacy of Mathematical Modeling Results. International Conference

"Simulation-2008", Kiev, Ukraine, 119-124, 2008.

Yu. Menshikov. Robotics, Automation and Control. Chapter 7. The Identification of Models

of External Loads, In-Teh is Groatian Branch of I-Tech and Publishing KG, Vienna,

Austria, 2008.

REFERENCES 271

Yu. Menshikov. Algorithms of Construction of Adequate Mathematical Description of

Dynamic System. 6-th Vienna Conference on mathematical Modelling, Vienna - Full

Papers CD Volume, Vienna Univ. of Technology, ISBM 978-3-901608-35-3, 2482-2485,

Yu. Menshikov. Inverse Problems in Non-classical Statements. International Journal of

Pure and Applied Mathematics, 67(1), 79-96, 2011.

V. Perminov. Mathematical Modeling of Large Forest Fires Initiation. 6-th Vienna Conference

on Mathematical Modelling, Vienna, full Papers CD Volume, Vienna Univ. of

Technology, ISBM 978-3-901608-35-3, 1165-1172, 2009.

W. Porter. Modern Foundations of Systems Engineering. The Macmillan Company, New

York, Collier-Macmillan Limited, London, 1970.

I. Sarmar and A. Malik. Modeling, Analysis and Simulation of a Pan Tilt Platform Based

on Linear and Nonlinear Systems. IEEE/ASME MESA, China, 147-152, 2008.

R. Shannon. Systems Simulation - The Art and Science. Prentice-Hall, Inc., Englewood

Cliffs, New Jersey, 1975.

V. Stepashko. Method of Critical Dispersions as Analytical Apparatus of Theory of Inductive

Modeling. Problems of Control and Information, Kiev, Ukraine, 2, 27-32, 2008.

J. Tillack, S. Noack, K. Noh, A. Elsheikh and W. Wiechert. A Software Framework for

Modeling and Simulation of Dynamic Metabolic and Isotopic Systems. 6-th Vienna Conference

on Mathematical Modelling, Vienna, full Papers CD Volume, Vienna Univ. of

Technology, ISBM 978-3-901608-35-3, 769-778, 2009.

A.Tikhonov and V.Arsenin. Methods of Solution of Incorrectly Problems. Science,

Moscow, 1979.

S. Vilenkin. Application of Regularization for Evaluation of Input Signal under Realization

of Output Signal. Automation and Telemechanics, 21, 52-55, 1968.

*European Journal of Mathematical Sciences*,

*2*(3), 256-271. Retrieved from https://ejmathsci.org/index.php/ejmathsci/article/view/151