2. Home
  3. ARCHIVE
  4. 2018
  5. VOL 40, NO 2 (2018)
  6. Архив номеров
  7. 2014 год
  8. 36-5

Electronic Modeling

VOL 36, NO 5 (2014)

CONTENTS

Mathematical Modeling and Computation Methods

  ARISTOV V.V. Integro-Algorithmic Method for Computation the Matrix Logarithm with Arbitrary Accuracy 3-16
  IVANOV I.L. Regulation of Power Systems under Impulse Perturbations 17-26 
  DOMBROVSKYI V.V., SMAKOVSKYI D.S., SMAKOVSKA A.N. Modeling of Heat Transfer Processes with the Areas of Significant Gradients of Solution with the Help of Nested Adaptive Grids 27-36 

Informational Technologies

  MELIKOV A.Z., FATTAKHOVA M.I., VELIDJANOVA G.M. Method to Calculate the Parameters of Integral Cellular Communication Networks with Isolated Partition of Channels 37-48 

Computational Processes and Systems

  KALINOVSKY Ya.A., TURENKO A.S., BOYARINOVA Yu.Ye., KHITSKO Ya.V. Research of Computing Operations in Hypercomplex Number System of Antiquaternions 49-66 

Application of Modelling Methods and Facilities

  YEVDOKIMOV V.F., PETRUSHENKO E.I. Integral Model of Three-Dimensional Distribution of Eddy Flows in Continuous Casting of Round Cross-Section under Electromagnetic Stirring in Vertical MCC I 67-80 
  KRYZHANIVSKYY V.B. Optimization of Placement of Discrete Sources of Physical Field Described by Mixed Boundary Problem 81-94 
  CHEMERYS V.T., BORODIY I.A., MARINCHENKO A.Ye. Simulation of the Pulsed Electric Current Passage across the Contact Surfaces of Multilayer Electrical Conductivity 95-106 
  DOLGIN V.P., DOLGIN I.V. Quadrant-Criterion of Stability of Dynamic Systems 107-114 

Short Notes

  MAMEDOV R.K., IMANOVA U.G. Increasing the Reliability of Decision-making in Pattern Recognition 115-121 

INTEGRO-ALGORITHMIC METHOD FOR COMPUTATION THE MATRIX LOGARITHM WITH ARBITRARY ACCURACY

V.V. Aristov

ABSTRACT

The integro-algorithmic method of approximation and iteration correction for high-accuracy computation of matrix logarithms is proposed. The method is based on the use of linear multistep formulas of numerical integration of the difference type as well as the Obreshkov difference-differential formulas with allowance for higher derivatives. Due to iterations in this case there is not a necessity to choose a high-fidelity primary approximating formula and a basic criterion is a receipt of high-rate of convergence. In addition, the offered method summarizes the known algorithms of taking logs based on the Pade formulas of iteration corrections and increases their order and accuracy. The proposed relations and program solutions permit determining necessary parameters for organizing the processes of matrix logarithms calculations with arbitrary preset high accuracy.

KEYWORDS

matrix logarithms, numerical integration, integro-algorithmic method, multistep formulas of integration, transfer function, equivalent transformations.

REFERENCES

1. Gantmacher F.R. The Theory of Matrices.—Moscow: Nauka, 1988.—552 p. (in Russian).
2. Culver W.J. On the existence and uniqueness of the real logarithm of a matrix // Proc. of the American Mathematical Society.— 1966. — Vol. 17, № 5. — P. 1146—1151.
3. Al-Mohy A., Higham N. Improved inverse scaling and squaring algorithms for the matrix logarithm // SIAM J. Sci. Comput. — 2012. — Vol. 34, № 4. — P. C153—C169.
4. Cheng S.H., Higham N.J., Kenney C.S., Laub A.J. Approximating the logarithm of a matrix to specified accuracy // SIAM J. Matrix Anal. Appl.—2001.—Vol. 22.—P. 1112—1125.
5. Kenney C., Laub A. Condition estimates for matrix functions // Ibid.—1989.—Vol. 10.—P. 707— 730.
6. Kenney C., Laub A. A Schur-Frechet algorithm for computing the logarithm and exponential of a matrix // SIAM J. Matrix Anal. Appl. — 1998. — Vol. 19, № 3. — P. 640—663.
7. Higham N.J. Functions of matrices. Theory and computation // Society for Industrial and Applied Mathematics. SIAM-2008. Philadelphia, 2008.—425 p.
8. Aristov V.V. Functional macrooperation: basics of iterative algorithms. — Kiev: Nauk. Dumka, 1992.— 280 p. (in Russian).
9. Aristov V.V. Model multistep difference-differential methods integration with regard to the influence of the starting section // Electronic Modeling.—2013.—Vol. 35,№6.—P. 3—26 (in Russian).
10. Aristov V.V. Integro-algorithmic computations.—Kiev: Nauk. Dumka, 1980.—192 p. (in Russian).
11. Aristov V.V. Mathematical models of iterative relations generalized CORDIC-algorithms // Electronic Modeling.— 2011.— Vol. 33, № 1. — P. 3—29 (in Russian).
12. Higham N.J. Evaluating Pade approximants of the matrix logarithm // SIAM J. Matrix Anal. Appl. — Vol. 22, № 4. — P. 1126—1135.

Full text: PDF (in Russian)

REGULATION OF POWER SYSTEMS UNDER IMPULSIVE PERTURBATIONS

I.L. Ivanov

ABSTRACT

Estimations of the region of asymptotic stability in the space of control parameters are found for a power systemmodelwith pulse effects and delayed control by applying Lyapunov discontinuousmatrix functions and Razumikhin conditions. The piecewise exponential and piecewise linear functions were used as Lyapunov functions. Numerical estimations of stability regions in the space of control parameters were obtained by using each of these two types of Lyapunov functions. A comparison of the effectiveness of these two types of functions has been performed. The mechanisms of losing stability by the power system while varying its parameters are described.

KEYWORDS

power system, Lyapunov method, delay, pulse effects, asymptotic stability.

REFERENCES

1. Chiang H.D., Chu C.C., Cauley G. Direct stability analysis of electric power systems using energy functions: Theory, applications, and perspective // Proc. of the IEEE.—1995.— Vol. 13. — P. 1497—1529.
2. Martynyuk A.A., Ivanov I.L. On the connective stability of three-machine power system under impulsive perturbations // ReportsNASof Ukraine.—2013.—№7.—P. 64—71 (inRussian).
3. Ivanov I.L. Stability of a model of a power system with delay and pulse effects // Analytical Mechanics and Its Applications: Proc. Institute of Mathematics of NAS of Ukraine.—2012.— Vol. 9, № 1. — P. 114—127 (in Russian).
4. Jia H. et al. Power system small signal stability region with time delay // Int. J. Electrical Power & Energy Systems. — 2008.— Vol. 30, № 1. — P. 16—22.
5. Ayasun S., Nwanpka C.O. Probability of small-signal stability of power systems in the presence of communication delays // Int. Conf. on Electrical and Electronics Engineering. ELECO 2009. Bursa, Turkey, 2009 5—8 Nov. // IEEE, 2009. — P. I-70—I-74.
6. Slyn'ko V.I. Stability conditions for linear impulsive systems with delay // International Applied Mechanics. —2005.— Vol. 41, № 6. —P. 130—138 (in Russian).
7. Demidovich B.P. Lectures on the Mathematical Theory of Stability. — Moscow: Nauka, 1967.— 472 p. (in Russian).

Full text: PDF (in Russian)

MODELING OF HEAT TRANSFER PROCESSES WITH THE AREAS OF SIGNIFICANT GRADIENTS OF SOLUTION WITH THE HELP OF NESTED ADAPTIVE GRIDS

V.V. Dombrovskyi, D.S. Smakovskyi, A.N. Smakovska

ABSTRACT

This paper is devoted to the method of constructing the adaptive nested grid in the areas of rapid change of the unknown function for solving two-dimensional unsteady heat equation. Partial differential equations are widely used in modeling of various natural phenomena. The effectiveness of solving a particular problem largely depends on the number of grid points and their location. The purpose of the paper is to develop the method of constructing the grid which dynamically adapts to the zones of significant gradients. The construction of nested grids is based on the local error estimate obtained from the difference of solutions with various spatial and temporal grid steps. A search algorithm of zones with significant gradients of the unknown function is proposed. The applied software tool can be used for computer modeling of problems described by the heat equation. The proposed method of constructing the adaptive nested grid permits reducing the use of machine resources required for solving the problem.

KEYWORDS

partial differential equations, finite difference method, finite difference grid, nested grid.

REFERENCES

1. Liseykin V.D. Survey of adaptive structural grid generation technology // Computational Mathematics and Mathematical Physics.—1996.—Vol. 36,№1.—P. 3—41 (in Russian).
2. Lukyanenko S.O. Adaptive Computational Methods for Modeling Objects with Distributed Parameters. – Kyiv: Politechnica, 2004. — 236 p. (in Ukrainian).
3. Spall M.A., Holland W.R. A nested primitive equation model for oceanic applications // J. Physical Oceanography. — 1996. — № 21. — P. 205—220.
4. Martin P.J. Description of the Navy Coastal Ocean Model Version 1.0 // Naval Research Laboratory Technical Report. — 2000. — 45 р. — NRIVFR/7322-00-9962.
5. Zavatarelli M., Pinardi N. The Adriatic Sea modeling system: a nested approach // Annales Geophysicae. — 2003. — № 21. — P. 345—364.
6. Belotserkovskaya M.S., Oparin A.M. Nested grids using for filtration process modelling // Mathematical Models.— 2004. — Vol. 16, № 12. — P. 3—10 (in Russian).
7. Smakovskyi D.S. Adaptive interpolation based on Ferguson curves for constructing the grid functions // Electronic Modeling. — 2010. — Vol. 32, № 5. — P. 11—18 (in Russian).

Full text: PDF (in Russian)

METHOD TO CALCULATE THE PARAMETERS OF INTEGRAL CELLULAR COMMUNICATION NETWORKS WITH ISOLATED PARTITION OF CHANNELS

A.Z.Melikov, M.I.Fattakhova, G.M.Velidjanova

ABSTRACT

A scheme for isolated partition of channels in the integral cellular network where voice and data calls are processed is proposed. In accordance to the proposed scheme the voice calls require a free channel in their own zone and, if all channels in this zone are occupied, a free channel in another zone might be assigned only for handover voice calls. There is a threshold parameter for the number of handover voice calls in the zone for data calls. To determine the rule for access of data calls a cutoff scheme for new data calls is used. It is shown that the adequate model of this system is two-dimensional finite Markov chain. The state of space of the system is defined and a system of equilibrium equations is developed. It is proved that this system of equations has analytical solution in multiplicative form. An algorithm to calculate quality of service metrics of the proposed partition scheme based on given multiplicative solution is created. Results of numerical experiments are presented.

KEYWORDS

integral cellular communication networks, channels partition, algorithms to calculate the quality of service metrics.

REFERENCES

1. Schneps-Schneppe M., Iversen V.B. Call admission control in cellular networks // Mobile Networks. Ed. J.H. Ortiz. — InTech., 2012. — P. 111—136.
2. Melikov A.Z., Ponomarenko L.A., Kim C.S. Numerical investigation of a multi-threshold access strategy in multiservice cellular networks // Cybernetics and System Analysis.—2009.— Vol. 45, № 5. — P. 13—24 (in Russian).
3. Oh Y.J., Kim C.S., Melikov A.Z., Fattakhova M.I. Numerical analysis of multi-parameter strategy of access in multiservice wireless cellular networks // Automation and Remote Control. — 2010.— Vol. 71, № 12. — P. 70—85 (in Russian).
4. Melikov A.Z., Fattakhova M.I., Babayev A.T. Investigation of cellular communication networks with private channels for servicing of handover calls // Automatic Control and Computer Sciences. — 2005.— Vol. 39, № 3. — P. 71—81 (in Russian).
5. Ponomarenko L., Kim C.S., Melikov A. Performance Analysis and Optimization of Multitraffic on Communication Networks. — Heidelberg: Springer, 2010.
6. Feng W., KowadaM. Performance analysis of wireless mobile networks with queueing priority and guard channels // International Transactions in Operational Research.—2008.—Vol. 15. — P. 481—508.
7. Kelly F. P. Reversibility and Stochastic Networks. — N Y : John Wiley & Sons, 1979.
8. Carvalho G.H.S., Martins V.S., Frances C.R.L., et al. Performance analysis of multi-service wireless network: An approach integrating CAC, scheduling, and buffer management // Computers & Electrical Engineering.— 2008. — Vol. 34. — P. 346—356.

Full text: PDF (in Russian)