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Shcherbakov S.V., Bychkov Y.A

Russia, Pskov, Pskov Polytechnical Institute at St.Petersburg State Technical University

Russia, St.Petersburg, St.Petersburg State Electrotechnical University


Analytical and numerical method of analysis and parametric synthesis of nonlinear nonautonomous deterministic models with lumped parameters of dynamic systems is presented. The method in conjunction with a consistent procedure of model formation, specialized computational algorithms and the software available makes up a unified problem-adaptable simulation complex.


Щербаков С.В., Бычков Ю.А.

Россия, Псков, Псковский политехнический институт

Санкт – Петербургского государственного технического университета

Россия, Санкт – Петербург, Санкт – Петербургский государственный электротехнический университет


Представлен аналитически-численный метод анализа и параметрического синтеза нелинейных неавтономных детерминированных с сосредоточенными параметрами моделей динамических систем. Совместно с согласованной процедурой формирования моделей, специализированными вычислительными алгоритмами и имеющимся программным обеспечением метод составляет единый проблемно-адаптируемый комплекс моделирования.


Advances in computing machines and nonlinear phenomena theory as well as in systems analysis and applied mathematics have resulted in strengthening the role of mathematical simulation in engineering projects and scientific researches. This versatile methodology has laid foundation for mathematization of scientific and technical progress. By the means applied and the results obtained mathematical simulation that combines both conventional investigation techniques and new methodology makes possible to obtain the necessary information about the character and peculiar features of the processes in dynamic systems relatively easily and quickly, make proper predictions and recommendations, thus ensuring the achievement of the goal to be sought.

One of the principle constituants of the complex dynamic systems simulation technology is the method of computation of formed models whose possibilities and peculiarities determine to a considerable extent reliability of the results obtained as well as computational complexity. Essential nonlinearity and nonstationarity of the models of complex dynamic systems, their objective stiffness and poor conditionality along with ambiguousness and instability of the character of changes of the coordinates dictate the whole spectrum of special and necessary requirements to the applied method of analysis and synthesis of such models. With consideration of the latest achievements in developing the qualitative theory of nonlinear phenomena and the main trends of improving the methods of calculus mathematics a one-step alternating-order method of analysis and parametric synthesis of nonlinear nonautonomous models of dynamic systems, so called analytical and numerical method, has been developed. The computational algorithm of the method imposes only two limitations on the description of intrinsic properties of the system being simulated requiring determinateness and lumpability of the parameters of the model being formed, with external effects being taken into consideration. The computational scheme of the analytical part of the method is based on the usage of generalized functions, generalized Laplace transformation and functional-power series. The numerical part of the method is realized in accordance with the principle of analytical continuation of the regular constituant of the desired solution of the equation of the model dynamics. The procedure of choosing the steps of the computation is that which ensures tackling all the problems by solving one inequality. These are the problems of finding out the unique existence of the desired solution of the inequality andny, whether it is the only desired solution , possibility to evaluate the upper level of the absolute local and global errors of the approximate solution computation; other problems are those of agreeing the step length and the speed of changing the regular constituant of the desired solution as well as observing the conditions of numerical stability. The order of the method is the function of the interaction of the intrinsic dynamic properties of the model and the preassigned level of the maximum absolute local error of its analytical and numerical computation. If necessary, the order of the method can be arbitrarily high, thus ensuring consistency and convergence of the method invariably. The main distinguishing feature and advantage of the method developed is that it offers possibility to obtain within optimal time period not only approximate solutions of the dynamic equations for the model being analyzed but also those with preassigned level of maximum absolute local or global errors in computing the ranges containing unknown exact values of the desired solutions of this equation. With consideration for inversion of causal relations the method can also be applied validly to solve the problem of the parametric synthesis within the frame of the formed model of the system.

To expand the possibilities of the method on the basis of its computational scheme problem-oriented algorithms of investigating poorly conditioned models with non-singled out linear part and discontinuous differentiable coordinates, for computing stiff models and models with complex functional nonlinearity, for analyzing upper values of coordinates of the singled-out type of nonautonomous and nonstationary models and spectral composition of the singled-out linear part of the dynamic equations of the models are formed.

To sum it up, it can be noted that analytical and numerical method as a basis for the combined approach towards complex dynamic systems simulation provides necessary schemes and computational algorithms while its functional possibilities make possible to carry out a comprehensive and exhaustive investigation of the intrinsic properties and peculiar features of the object under simulation. All the procedures and transformations used in the dynamic system simulation including formation of their models of various degree of complexity, purposeful exhaustive search, complementation and analytical and numerical computation of these models under the condition of optimization of the volume of computation per interval unit of Taylor series convergence for regular constituants of the desired solutions are well formalized and consistent in a proper way making in combination with the software developed a unified problem-adaptable simulation complex.

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