Scientific Journal

Applied Aspects of Information Technology

DETERMINISTIC IDENTIFICATION METHODS FOR NONLINEAR DYNAMICAL SYSTEMS BASED ON THE VOLTERRA MODEL
Abstract:

The paper solves an important scientific and practical problem, which is to improve the accuracy and computational stability of the methods of deterministic identification of nonlinear dynamic systems in the form of Volterra model based on experimental data of observations ”input-output” taking. On the base of theoretical and experimental studies created effective instrumental algorithmic and software tools for estimating Volterra kernels in the time domain Into account measurement errors. Results of the further development of methods of deterministic identification of nonlinear dynamic systems based on Volterra models using irregular pulse sequences show. The methods are based on the use of the Tikhonov regularization procedure. The amplitude of test impulses is used as a regularization parameter. In the identification, procedure applies wavelet filtering for smooth the estimates of the Volterra kernels apply. This gives increase the accuracy and noise immunity of identification methods. The approximation method of identification of the nonlinear dynamic systems based on Volterra models is improved. Method is consists in the choice of amplitudes of test signals and of coefficients scaling of the partial components of responses a nonlinear system in procedure of processing of signals-responses. The improvement is reduced to minimizing the methodological error in the allocation of partial components from the response of the identification object and allows obtaining more accurate estimates of Volterra nuclei. To improve the computational stability of the developed identification algorithms and for noise reduction in the obtained estimates of multidimensional Volterra kernels the wavelet filtration is used. This allows obtaining smoothed solutions and decreases error of the identification by 1,5-2,5 times. A new robust method of deterministic identification of nonlinear dynamic systems based on Volterra models in the time domain is developed. In contrast to the interpolation method, where finite difference formulas with a predetermined number of experimental studies of the object of identification are used for numerical differentiation. It is proposed to solve the corresponding Volterra integral equations of the first kind, for the numerical implementation of which an unlimited number of experiments can be used. This makes it possible to increase the accuracy of the calculation of derivatives, and consequently, the accuracy of identification. Software tools on the system Matlab platform have been developed to implement the developed computational algorithms for deterministic identification of nonlinear dynamic systems in the form of Volterra kernels.

Authors:
Keywords
DOI
10.15276/aait.01.2018.1
References

1. Ivahnenko, A. G., & Yurachkovskiy, Yu. P. (1987). Modelirovanie slozhnyh sistem na osnove jeksperimental'nyh dannyh [Modeling of complex systems based on experimental data], Moscow, Russian Federation, Radio and Svyaz', 120 p. (in Russian).

2. (2002). Modelirovanie dinamicheskih sistem: aspekty monitoringa i obrabotki signalov [Modeling of dynamic systems: Aspects of monitoring and Signal processing], Edited by V. V. Vasilyev. Kyiv Institute of Modeling Problems in Power Engineering, named G. E. Pukhov, National Academy Science of Ukraine, 344 p. (in Russian).

3. Patton, R. J., Fantuzzi C., & Simani, S. (2003), “Model-Based Fault Diagnosis in Dynamic Systems Using Identification Techniques”. New York, Springer-Verlag, 368 p.

4. (2004). Korbicz, J., Kościelny, J. M., Kowalczuk, Z., Cholewa, W. (Eds.). “Fault Diagnosis: Models, Artificial Intelligence, Applications”. Berlin, Springer, 920 p.

5. (2010). Korbicz, J., & Kościelny, J. M. (Eds.), “Modeling, Diagnostics and Process Control: Implementation in the DiaSter System”. Berlin, Springer, 384 p.

6. Levy, P. (1951). Probnyj betonnyj analiz [Problèmes concrets d'analyse fonctionnelle]. Translated from French by G. E. Shilov, Moscow, Russian Federation, Nauka Publ, 551 p. (in Russian).

7. Pupkov, K. A., Kapalin, V. I., & Yushchenko, A. S. (1976). Funkcional'nye rjady v teorii nelinejnyh system [Functional Series in the Theory of Nonlinear Systems]. Moscow, Russian Federation, Nauka Publ., 448 p. (in Russian).

8. Volterra Vito. (1982). Teorija funkcionalov i integral'nyh i integro-differencial'nyh uravnenij. [Theory of Functionals and of Integral and Integro-differential Equations]. Translated from English by M. K. Kerimov. Moscow, Russian Federation, Nauka Publ., 304 p. (in Russian).

9. Besler, I. O., & Daugavet, I. K. (1990). O priblizhenii nelinejnyh operatorov polinomami Vol'terra. [On the approximation of nonlinear operators by Volterra polynomials]. Leningrad, Russian Federation, Mathematical Society Bulletin, Vol.1, pp. 53-64 (in Russian).

10. Suvorov, S. G. (2005). Priblizhenie nelinejnyh operatorov rjadami Vol'terra v mnogomernom sluchae [Approximation of nonlinear operators by Volterra series in the multidimensional case]. Ukrainian Mathematical Bulletin, Vol. 2, No. 3, pp. 418-441 (in Russian).

11. Apartsyn, A. S., Solodusha, S. V., & Spiryaev, V. A. (2013). “Modeling of nonlinear dynamic systems with Volterra polynomials: elements of theory and applications”. International Journal of Energy Optimization and Engineering. Vol. 2, No. 4, pp. 16-43.

12. Venikov, V. A., & Suhanov, O. A. (1982). Kiberneticheskie modeli jelektricheskih sistem: uchebnik akademii. [Cybernetic models of electrical systems: the academy workbook]. Moscow, Russian Federation, Energoizdat Publ., 328 p. (in Russian).

13. Danilov, L. V., Mathanov, L. N., & Filippov, V. S. (1990). Teorija nelinejnyh jelektricheskih cepej. [Theory of Nonlinear Electrical Circuits]. Leningrad, Russian Federation, Energoatomizdat Publ., 256 p. (in Russian).

14. Popkov, Ju. S., Kiselev, O. N., Petrov N. P., & Shmul'jan, B. L. (1976). Identifikacija i optimizacija nelinejnyh stohasticheskih sistem. [Identification and optimization of nonlinear stochastic systems]. Moscow, Russian Federation, Energiya Publ., 440 p. (in Russian).

15. Pupkov, K. A., & Egupov, N. D. (2004). Metody klassicheskoj i sovremennoj teorii avtomaticheskogo upravlenija. Statisticheskaja dinamika i identifikacija sistem avtomaticheskogo upravlenija]: [Methods of classical and modern automatic control theory. Statistical dynamics and identification of automatic control systems]: Textbook for Universities, Vol. 2, 2nd ed., Moscow, Russian Federation, Bauman Moscow, State Technical University (MSTU), 638 p. (in Russian).

16. Doyle, F. J., Pearson, R. K., & Ogunnaike, D. A., (2001). “Identification and Control Using Volterra Models”. Published Springer Technology & Industrial Arts, 314 p.

17. Giannakis, G. B., & Serpedin, E. (2001), A bibliography on nonlinear system identification and its applications in signal processing, communications and biomedical engineering”. Signal Processing EURASIP. Vol. 81, No. 3, pp. 533-580.

18. Cheng, C. M., Peng, Z. K., Zhang, W. M., & Meng G. (2016). “Volterra-series-based nonlinear system modeling and its engineering applications: A state-of-the-art review”, Mechanical Systems and Signal Processing. November, pp. 1-25. http://dx.doi.org/10.1016/j.ymssp.2016.10.029. https://www.researchgate.net/publication/309724868

19. Apartsin, A. S., & Solodusha, S. V. (1999). O matematicheskom modelirovanii nelinejnyh dinamicheskih sistem rjadami Vol'terra. [On mathematical modeling of nonlinear dynamic systems by Volterra series]. Electronic Modeling International Scientific-Theoretical Journal, No. 2, pp. 3-12 (in Russian).

20. Apartsin, A. S. (2001). Ob uluchshenii tochnosti modelirovanija nelinejnyh dinamicheskih sistem polinomami Vol'terra. [On improving the accuracy of modeling nonlinear dynamical systems by Volterra polynomials]. Electronic Modeling International Scientific-Theoretical Journal, No. 6. pp. 3-12 (in Russian).

21. Pavlenko, V. D. (2009). Metod kompensacii dlja identifikacii nelinejnyh dinamicheskih sistem v vide jader Vol'terra [Compensation method for identifying nonlinear dynamic systems in the form of Volterra kernels]. Odessa National Polytechnic University Bulletin, Iss. 2 (32), pp. 121-129 (in Russian).

22. Pavlenko, S. V. (2010). Primenenie vejvlet-fil'tracii v procedure identifikacii nelinejnyh sistem na osnove modelej Vol'terra. [Application of wavelet filtering in the procedure for identifying nonlinear systems based on Volterra models]. Eastern-European Journal of Enterprise Technologies, Kharkiv, Ukraine, No. 6/4 (48), pp. 65-70 (in Russian).

23. Pavlenko, V., Pavlenko, S., & Speranskyy, V. (2014). “Identification of Systems using Volterra Model in Time and Frequency Domain”. In book: Advanced Data Acquisition and Intelligent Data Processing. V. Haasz & K. Madani (Eds.). Chapter 10. River Publishers, pp. 233-270. ISBN 978-87-93102-73-6.

24. Pavlenko, S. V., & Polozhaenko, S. A. (2013). Optimizacija vychislitel'nyh algoritmov dlja metoda approksimacii dlja identifikacii nelinejnyh sistem v vide modelej Vol'terra. [Optimization of computational algorithms for the approximation method for the identification of nonlinear systems in the form of Volterra models]. Informatics and Mathematical Methods in Modeling. Odessa, Ukraine, ONPU, Vol. 3, No. 2, pp. 102-112 (in Russian).

25. Pavlenko, V. D., & Pavlenko, S. V. (2010). Issledovanie oshibok metoda approksimacii dlja identifikacii nelinejnyh dinamicheskih ob#ektov v vide jader Vol'terra. [Investigation of the errors of the approximation method for the identification of nonlinear dynamic objects in the form of Volterra kernels]. Scientific and Technical Journal Electrotechnic and Computer Systems, Iss. 01 (77), pp. 102-108 (in Russian).

26. Pavlenko, V. D. (2006). “Estimation of the Volterra Kernels of a Nonlinear System Using Impulse Response Data”. Signal/Image Processing and Pattern Recognition: Proceedings the Eighth All-Ukrainian International Conference UkrOBRAZ’2006, August 28-31, Kyiv, Ukraine, pp. 191-194.

27. Pavlenko, V., Massri, M., & Ilyin, V. (2008). Computing of the Volterra Kernels of a Nonlinear System Using Impulse Response Data”. Proceedings of 9th International Middle Eastern Simulation Multiconference MESM'2008, August 26-28, Philadelphia University, Amman, Jordan, pp. 131-138.

28. Pavlenko, V. D. (2010). Identifikacija nelinejnyh dinamicheskih sistem v vide jader Vol'terra na osnove dannyh izmerenija impul'snoj harakteristiki. [Identification of nonlinear dynamic systems in the form of Volterra kernels based on impulse response measurement data]. Electronic Modeling International Scientific–Theoretical Journal, Vol. 32, No. 3, pp. 3-18 (in Russian).

29. Marmarelis, P., & Marmarelis, V. (1981). Analiz fiziologicheskih sistem. [Analysis of physiological systems]. White noise method, Moscow, Russian Federation, Mir Publ., 480 p. (in Russian).

30. Tikhonov, A. N., & Arsenin, V. Y. (1983). Metody reshenija nekorrektnyh zadach [Methods for solving ill-posed problems], Nauka Publ, Moscow, Russian Federation, 288 p. (in Russian).

31. (1983), Tikhonov, A. N., Goncharsky, A. V., Stepanov, V.V., & oth. Reguljarizujushhie algoritmy i apriornaja informacija] [Regularizing algorithms and a priori information], Nauka Publ, Moscow, Russian Federation, 200 p. (in Russian).

32. Chui, C. K. (1992). “An Introduction to Wavelets”. Publisher Academic Press, 265 p. DOI: 10.2307/2153134.

33. Daubechies, I. (1992). “Ten Lectures on Wavelets”. Vol. 61 of CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA.

34. Smolentsev, N. K. (2005). Osnovy teorii vejvletov. Vejvlety v MATLAB], [Fundamentals of the theory of wavelets. Wavelets in MATLAB], Moscow, Russian Federation, DMK-Press Publ., 304 p. (in Russian).

35. Godunov, S. K., & Ryabenkiy, V. S. (1973). Raznostnye shemy [Difference Schemes], Moscow, Russian Federation, Nauka Publ., 400 p. (in Russian).

36. Pavlenko, S. V., & Pavlenko, V. D. (2015). Reguljarizacija procedury identifikacii nelinejnyh sistem v vide modelej Vol'terra. [Regularization of the procedure for identifying nonlinear systems in the form of Volterra models]. (Digital resource). Papers of X International Conference “System Identification and Control Problems”, SICPRO‘15. Moscow, V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, pp. 230-238, ISBN 978-5-91450-162-1 (in Russian).

37. Pavlenko, S. V. (2012). Robastnaja ocenka jader Vol'terra nelinejnyh sistem po izmerenijam impul'snogo otklika. [Robust estimation of Volterra nuclei of nonlinear systems from impulse response measurements]. Computer science and Mathematical Methods in Modeling. Odessa, Ukraine, ONPU, Vol. 2, No. 4, pp. 380-387 (in Russian).

38. Pavlenko, S. V. (2013). Robastnyj metod identifikacii nelinejnyh dinamicheskih sistem na osnove modelej Vol'terra. [Robust method of identification of nonlinear dynamic systems based on Volterra models]. Electrical and computer systems. Kyiv, Ukraine, TehnIka Publ., Iss. 09 (85), pp. 84-88 (in Russian).

39. Pavlenko, V. D., Pavlenko, S. V., & Romanov, D. Yu. (2016). Issledovanie tochnosti i vychislitel'noj ustojchivosti reguljarizovannogo metoda identifikacii nelinejnyh sistem [Investigation of the accuracy and computational stability of a regularized method for the identification of nonlinear systems]. Herald of the National Technical University “KhPI”, Ukraine, Subject issue: Informatics and Modelling, Kharkov, Ukraine, NTU “KhPI”, No. 44 (1216), pp. 88-99. DOI:10.20998/2411-0558.2016.44.06 (in Russian).

40. Pavlenko, V. D., and Lomovoy, V. I. (2018). Postroenie approksimacionnoj modeli nelinejnoj dinamicheskoj sistemy v vide polinoma Vol'terra. [Construction of an approximation model of a nonlinear dynamical system in the form of a Volterra polynomial]. Scientific notes of the Taurida National University named after V.I. Vernadsky, series: Engineering, Vol. 29(68), No. 6, pp. 200-205 (in Ukrainian).

41. Pavlenko, V. D. (2008). Informacionnye tehnologii dlja kosvennogo monitoringa i diagnostiki dinamicheskih ob#ektov na osnove modelej Vol'terra. [Information technology for indirect monitoring and diagnostics of dynamic objects based on Volterra models]. Odessa National Polytechnic University Bulletin, Iss. 2 (30), pp. 194-199 (in Russian).

42. Pavlenko, V. D., Fomin A. A., Pavlenko, S. V., & Ilyin, V. M. (2008). Metod diagnostiki nepreryvnyh sistem na osnove modelej jadra Vol'terra. [Method for diagnosing continuous systems based on Volterra kernel models]. Modeling and controlling the state of the ecological-economic systems of the region: a collection of papers. International Scientific and Educational Center of Information Technologies and Systems, NAS of Ukraine, Kyiv, Ukraine, Iss. 4, pp. 180-191 (in Russian).

43. Pavlenko, V., & Fomin, A. (2008), Methods for Black-Box Diagnostics Using Volterra Kernels”. Proceedings 2-nd International Conference on Inductive Modeling, ICIM-2008, September 15-19, Kyiv, Ukraine, рр. 104-107.

44. Pavlenko, V., & Fomin, O. (2008). ”Method for Modeling and Fault Simulation using Volterra kernels”. Proc. 6-th IEEE East-West Design & Test Symposium (EWDTS’08), Lviv, Ukraine, October 9-12, pp. 204-207.

45. Pavlenko, V., Fomin, O., & Ilyin, V., (2009). Technology for Data Acquisition in Diagnosis Processes by Means of the Identification Using Models Volterra”. Proc. of the 5-th IEEE International Workshop on Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications IDAACS'2009, Rende (Cosenza), Italy, September 21-23, pp. 327-332.

46. Pavlenko, V. D., & Fomin, O. O. (2015). “Intelligent Information Technology Building Systems Diagnostics Using Nuclear Moments Volterra”. Herald of the National Technical University “KhPI”. Subject Issue: Information Science and Modeling, Kharkov, Ukraine, NTU “KhPI”, No. 33 (1142), pp. 106-119.

47. Pavlenko, V., & Speranskyy, V. (2014). “Identification of Communication Channels for Remote Sensing Systems Using Volterra Model in Frequency Domain”. In book: Advanced Geoscience Remote Sensing. Edited by Maged Marghany. Chapter 8. Publisher InTech, Rijeka, Croatia, pp. 175-203. ISBN 978-953-51-1581-6

http://dx.doi.org/10.5772 /58354.

48. Pavlenko, V. D., Pavlenko, S. V., & Lomovoy, V. I. (2018). Vychislitel'nye instrumenty dlja postroenija modelej Vol'terra nelinejnyh dinamicheskih sistem v chastotnoj oblasti [Computational Tools for Building Volterra Models of Nonlinear Dynamic Systems in the Frequency Domain]. Herald of the National Technical University “KhPI”. Subject issue: Informatics and Modelling, Kharkov, Ukraine, NTU “KhPI”, No. 42 (1318), pp. 115-130. DOI: 10.20998/2411-0558.2017.50.07 (in Russian).

49. Pavlenko, V. D., & Speranskyy, V. A. (2014), “The Toolkit for Nonparametric Identification Nonlinear Dynamical Systems Based on Volterra Models in Frequency Domain”. Mathematical and Computer Modelling. Series: Technical sciences [V. M. Glushkov Institute of Cybernetics of NAS of Ukraine & Kamianets-Podilsky National Ivan Ohienko University], Issue 11, pp. 107-116, ISSN 2308-5916.

50. Pavlenko, V. D., & Speranskyy, V. O. (2013). “Analysis of Identification Accuracy of Nonlinear System Based on Volterra Model in Frequency Domain”. American Journal of Modeling and Optimization, 1(2), pp. 11-18. DOI:10.12691/ajmo-1-2-2.

51. Pavlenko, V., Fomin, A., Pavlenko, S., & Grigorenko, Y. (2013). “Identification Accuracy of Nonlinear System based on Volterra Model in Frequency Domain”. AASRI Procedia, Vol. 4, pp.297-305. http://dx.doi.org/10.1016/j.aasri.2013.10.044.

52. Pavlenko V., Pavlenko S., & Speranskyy, V. (2013), “Interpolation Method of Nonlinear Dynamical Systems Identification Based on Volterra Model in Frequency Domain”. Proceedings of the 7th IEEE International Conference on Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications (IDAACS’2013), 15-17 September, Berlin, Germany, Vol. 1, pp. 173-178.

53. Pavlenko, V., & Speranskyy, V. (2013). “Interpolation Method Modification for Nonlinear Objects Identification Using Volterra Model in Frequency Domain”. Proceedings 23-th International Crimean Conference Microwave and Telecommunication Technology, CriMiCo-2013, 8-13 September, Sevastopol, Vol. 1, pp. 427-429.

54. Pavlenko, V., Speranskyy, V., Ilyin, V., & Lomovoy, V. (2012). “Modified Approximation Method for Identification of Nonlinear Systems Using Volterra Models in Frequency Domain”. Applied Mathematics in Electrical and Computer Engineering. Proc. of the American Conf. on Applied Mathematics (AMERICAN-MATH'12) & Proc. of the 6-th WSEAS Intern. Conf. on Circuits, Systems, Signal and Telecommunications (CSST'12) & Proc.of the 6th WSEAS Intern. Conf. on Computer Engineering and Applications (CEA'12), Harvard, Cambridge, USA January 25-27. Published by WSEAS Press, pp. 423-428.

55. Pavlenko, V., & Speranskyy, V. (2015). “The methodology of experimental researches and the software tools for Volterra model construction of infocommunication system”. Problems of Infocommunications Science and Technology (PIC S&T), IEEE Second International Scientific-Practical Conference, Kharkiv National University of Radio Electronics, Kharkiv, Ukraine, pp. 141-144. DOI: 10.1109/INFOCOMMST.2015.7357296.

56. Pavlenko, V. D., & Speranskyy, V. O., (2012). “Simulation of Telecommunication Channel Using Volterra Model in Frequency Domain”. 10-th IEEE East-West Design & Test Symposium EWDTS’, Kharkov, Ukraine, September 14-17, pp. 401-404.

57. Pavlenko, V. D., Speranskyy, V. O., and Lomovoy, V. I. (2011). “The Test Method for Identification of Radiofrequency Wireless Communication Channels Using Volterra Model”. Proceedings of the 9-th IEEE East-West Design & Test Symposium (EWDTS’2011), Sevastopol, Ukraine, September 9-12. Published Kharkov, KNURE, pp. 331-334.

58. Pavlenko, V. D., & Speranskyy, V. O. (2011), “Communication Channel Identification in Frequency Domain Based on the Volterra Model”. Recent Advances in Computers, Nonlinear Objects Identification Using Volterra Model in Frequency Domain”. Proceedings 23-th International Crimean Conference Microwave and Telecommunication Technology, CriMiCo-2013, 8-13 September, Sevastopol, Vol. 1, pp. 427-429.Communications, Applied Social Science and Mathematics. Proceedings of the International Conference on Computers, Digital Communications and Computing (ICDCC'11), Barcelona, Spain, September 15-17. Published by WSEAS Press, pp. 218-222.

59. Pavlenko, V. D., Speranskyy, V. O, & Lomovoy, V. I. (2011), “Modelling of Radio-Frequency Communication Channels Using Volterra Model”. Proc. of the 6-th IEEE International Conference on Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications (IDAACS’2011), 15-17 September, Prague, Czech Republic, pp. 574-579.

60. Pavlenko, V. D., Speranskiy, V. A., & Lomovoy, V. I. (2011), “Radio Frequency Test Method for Wireless Communications using Volterra Model”. 21-st International Crimean Conference Microwave and Telecommunication Technology, CriMiCo-2011, 12-16 September, Sevastopol, Vol. 1, pp. 370-371.

61. Pavlenko, V. D., Speranskiy V. A., Lomovoy, V. I., & Ilyin, V.M. (2011). “Radio frequency test method for wireless communications using Volterra model”. Proceedings of the 11-th conference on dynamical systems theory and applications (DSTA'2011), December 5-8. Łódź, Poland. Editors: J. Awrejcewicz, M. Kaźmierczak, P. Olejnik, & J. Mrozowski. Publisher Łódź, Poland, pp. 446-452.

62. Pavlenko, V., Salata, D., Dombrovskyi M., & Maksymenko Yu. (2017). “Estimation of the Multidimensional Transient Functions Oculo-Motor System of Human”. Mathematical Methods and Computational Techniques in Science and Engineering: AIP Conf. Proc. MMCTSE, Cambridge, UK, 24-26 February. Vol. 1872. Melville, New York. 020014-1-020014-8; DOI: 10.1063/1.4996671. Published by AIP Publishing. 978-0-7354-1552-2, pp.110-117.

63. Pavlenko, V. D., Salata D. V., & Chaikovskyi, H. P. (2017). “Identification of an Oculo-Motor System Human Based on Volterra Kernels”. International Journal of Biology and Biomedical Engineering, Vol. 11, pp 121-126.

64. Pavlenko, V., Salata, D., & Maksymenko, Yu. (2017), “Nonlinear Dynamic Model of an Oculo-Motor System Human based on Volterra Kernels”, WSEAS Transactions on Systems, Vol. 16, pp. 234-241.

65. Pavlenko, V., Ivanov, I., & Kravchenko, E. (2017), “Estimation of the Multidimensional Dynamical Characteristic Eye-Motor System”. Proceedings of the 9-th IEEE International Conference on Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications (IDAACS’2017), 21-23 September, Bucharest, Romania, Vol.2, pp. 645-650.

66. Pavlenko, V. D., Salata, D. V. (2017). “Identification Eye-Motor System with Using Volterra Model,System analysis and information technology”. Proceeding of the 19-th International conference SAIT 2017, Kyiv, Ukraine, May 22-25. Published by ESC “IASA” NTUU “Igor Sikorsky Kyiv Polytechnic Institute”, pp. 28-31. http://sait.kpi.ua/books/sait2017.ebook.pdf

67. Pavlenko, V., & Salata, D. (2017). “Constructing Nonlinear Dynamic Model of Oculo-Motor System based on Experimental Studies of Input-Output”. 4-th International Conference “Computational intelligence (Results, Problems and Perspectives)” (ComInt 2017), Taras Shevchenko National University of Kyiv, Ukraine, May 16-18, 2017, pp. 184-185.

68. Pavlenko, V. D., Kravchenko, E. I., & Salata D. V. (2017). “Volterra Model Building of Oculo-Motor System Based on Experimental Input-Output Data”. Intellectual decision-making systems and problems of computational intelligence (ISDMCI’2017). Proceedings XII International Scientific Conference, May 22-26, Zaliznyi Port, Ukraine, pp. 16-18.

69. Pavlenko, V. D., Fomin, O. O., Fedorova, A. N., & Dombrovskyi, M. M. (2016). “Identification of Human Eye-Motor System Base on Volterra Model”. Herald of the National Technical University “KhPI”. Subject issue: Informatics and Modelling. Kharkov, NTU “KhPI”, No. 21 (1193), pp. 74-85.

70. Pavlenko, S. V., & Pavlenko, V. D. (2016). “Building an Information Model of the Photosynthetic Reaction Center in the Form of Volterra Kernels”. Proceedings of the Eleventh International Conference “Analytical and numerical methods for modeling natural science and social problems”, ANM-2016, Penza, Russia, Russian Federation, December 6-9. Ed. prof. I. V. Boykov. Penza State University, Publishing House, pp. 100-104. http://dep_vipm.pnzgu.ru/files/dep_vipm.pnzgu.ru/konference/achmm2016.pdf.

 

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