PROBLEMS OF DIFFERENTIAL AND TOPOLOGICAL DIAGNOSTICS. PART 3. THE CHECKING PROBLEM



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Abstract

Proposed work is the third in the cycle, therefore, we explain such notions as checking sphere, checking ellipsoid and checking tubes. The checking problem is stated and the algorithms for solving it are formulated. The criterion for a malfunction in a controlled system whose motion is described by ordinary differential equations is taken to be the attainment of a checking surface by the checking vector. We first propose the methods for solving the checking problems in which the checking surfaces are chosen in the form of a checking sphere, checking ellipsoid or checking tube. Then we consider the general techniques for constructing the checking surface by using the statistical testing method. We also give the extended statement of the checking problem. And we also prepare the material for the consideration of the problem of diagnostics.

About the authors

M. V. Shamolin

Lomonosov Moscow State University, 1, Leninskie Gory, Moscow, 119192, Russian Federation.

Author for correspondence.
Email: morenov.sv@ssau.ru
ORCID iD: 0000-0002-9534-0213

Doctor of Physical and Mathematical Sciences, professor, leading researcher of the Institute of Mechanics, academic of the Russian Academy of Natural Sciences

Russian Federation

References

  1. Shamolin M.V. Zadachi differentsial’noi i topologicheskoi diagnostiki. Chast’ 1. Uravneniya dvizheniya i klassifikatsiya neispravnostei . Vestnik Samarskogo universiteta. Estestvennonauchnaya seriya , 2019, vol. 25, no. 1, pp. 32–43. DOI: https://doi.org/10.18287/2541-7525-2019-25-1-32-43 .
  2. Shamolin M.V. Zadachi differentsial’noi i topologicheskoi diagnostiki. Chast’ 2. Zadacha differentsial’noi diagnostiki . Vestnik Samarskogo universiteta. Estestvennonauchnaya seriya , 2019, Vol. 25, no. 3, pp. 22–31. DOI: https://doi.org/10.18287/2541-7525-2019-25-3-22-32. .
  3. Borisenok I.T., Shamolin M.V. Reshenie zadachi differentsial’noi diagnostiki . Fundament. i prikl. matem. , 1999, vol. 5, issue 3, pp. 775–790. Available at: http://www.mathnet.ru/links/595e3ba8d2482ea55b741cb75c91b4ca/fpm401.pdf .
  4. Shamolin M.V. Nekotorye zadachi differentsial’noi i topologicheskoi diagnostiki. Izdanie 2-e, pererab. i dopoln. . Moscow: Ekzamen, 2007. Available at: http://eqworld.ipmnet.ru/ru/library/books/Shamolin2007-2ru.pdf .
  5. Shamolin M.V. Foundations of Differential and Topological Diagnostics. Journal of Mathematical Sciences, 2003, vol. 114, no. 1, pp. 976–1024. DOI: https://doi.org/10.1023/A:1021807110899 .
  6. Parkhomenko P.P., Sagomonian E.S. Osnovy tekhnicheskoi diagnostiki . Moscow: Energiya, 1981. Available at: https://www.studmed.ru/parhomenko-pp-red-osnovy-tehnicheskoy-diagnostiki-kniga-1-modeli-obektov-metody-ialgoritmy-diagnoza_5853e5d7550.html .
  7. Mironovskiy L.A. Funktsional’noe diagnostirovanie dinamicheskikh sistem . Avtomatika i telemekhanika , 1980, no. 8, pp. 96–121. Available at: http://www.mathnet.ru/links/e98eaea228af4a5515d22fb76185e5a8/at7158.pdf .
  8. Okunev Yu.M., Parusnikov N.A. Strukturnye i algoritmicheskie aspekty modelirovaniya dlya zadach upravleniya . Moscow: Izd-vo MGU, 1983. .
  9. Chikin M.G. Sistemy s fazovymi ogranicheniyami . Avtomatika i telemekhanika , 1987, no. 10, pp. 38–46. Available at: http://www.mathnet.ru/links/ea42bf4ed24a9fb9a3608386e29f5a24/at4566.pdf .
  10. Zhukov V.P. O dostatochnykh i neobkhodimykh usloviyakh asimptoticheskoi ustoichivosti nelineinykh dinamicheskikh sistem . Avtomatika i telemekhanika , 1994, no. 55, no. 3, pp. 321–330. Available at: http://www.mathnet.ru/links/b6907f0c0ee94a4f45d426d179367e37/at3855.pdf. .
  11. Zhukov V.P. O dostatochnykh i neobkhodimykh usloviyakh grubosti nelineinykh dinamicheskikh sistem v smysle sokhraneniya kharaktera ustoichivosti . Avtomatika i telemekhanika , 2008, no. 69, pp. 27–35. DOI: https://doi.org/10.1134/S0005117908010037 .
  12. Zhukov V.P. O reduktsii zadachi issledovaniya nelineinykh dinamicheskikh sistem na ustoichivost’ vtorym metodom Lyapunova . Avtomatika i telemekhanika , 2005, no. 66, pp. 1916–1928. DOI: https://doi.org/10.1007/s10513-005-0224-9 .
  13. Borisenok I.T., Shamolin M.V. Reshenie zadachi differentsial’noi diagnostiki metodom statisticheskikh ispytanii . Vestnik MGU. Ser. 1. Matematika. Mekhanika , 2001, no. 1, pp. 29–31. Available at: http://www.mathnet.ru/links/5fa90ac847725484c2bde066a8cc64f7/vmumm1441.pdf .
  14. Beck A., Teboulle M. Mirror Descent and Nonlinear Projected Subgradient Methods for Convex Optimization. Operations Research Letters, 2003, vol. 31, no. 3, pp. 167–175. DOI: https://doi.org/10.1016/S0167-6377(02)00231-6 .
  15. Ben-Tal A., Margalit T., Nemirovski A. The Ordered Subsets Mirror Descent Optimization Method with Applications to Tomography. SIAM J. Optimization, 2001, vol. 12, no. 1, pp. 79–108. URL: https://pdfs.semanticscholar.org/e19f/7697c83e692d7a459b09c229d0faef3b31ea.pdf?_ga=2.188452072.1367780915.1591514604-1525477732.1586505106 .
  16. Su W., Boyd S., Candes E. A Differential Equation for Modeling Nesterov’s Accelerated Gradient Method: Theory and Insights. J. Mach. Learn. Res., 2016, no. 17(153), pp. 1–43. Available at: https://arxiv.org/pdf/1503.01243.pdf. .
  17. Shamolin M.V. Diagnostika girostabilizirovannoi platformy, vklyuchennoi v sistemu upravleniya dvizheniem letatel’nogo apparata . Elektronnoe modelirovanie , 2011, vol. 33, no. 3, pp. 121–126. .
  18. Shamolin M.V. Diagnostika dvizheniya letatel’nogo apparata v rezhime planiruyushchego spuska . Elektronnoe modelirovanie , 2010, vol. 32, no. 5, pp. 31–44. .
  19. Fleming W.H. Optimal Control for Partially Observable Diffusions. SIAM Journal on Control and Optimization, 1968, vol. 6, no. 2, pp. 194–214. DOI: https://doi.org/10.1137/0320021 .
  20. Choi D.H., Kim S.H., Sung D.K. Energy-efficient Maneuvering and Communication of a Single UAV-based Relay. IEEE Transactions on Aerospace and Electronic Systems, 2014, vol. 50, no. 3, pp. 2119–2326. doi: 10.1109/TAES.2013.130074. .
  21. Ho D.-T., Grotli E.I., Sujit P.B., Johansen T.A., Sousa J.B. Optimization of Wireless Sensor Network and UAV Data Acquisition. Journal of Intelligent and Robotic Systems, April 2015, vol. 78, issue 1, pp. 159–179. DOI: https://doi.org/ 10.1007/sl0846-015-0175-5 .
  22. Ceci C., Gerardi A., Tardelli P. Existence of Optimal Controls for Partially Observed Jump Processes. Acta Applicandae Mathematica, 2002, vol. 74, no. 2, pp. 155–175. doi: 10.1023/A:1020669212384. .
  23. Rieder U., Winter J. Optimal Control of Markovian Jump Processes with Partial Information and Applications to a Parallel Queueing Model. Mathematical Methods of Operational Research, 2009, vol. 70, pp. 567–596. doi: 10.1007/s00186-009-0284-7. .
  24. Chiang M., Tan C.W., Hande P., Lan T. Power control in wireless cellular networks. Foundations and Trends in Networking, 2008, vol. 2, no. 4, pp. 381–533. doi: 10.1561/1300000009 .
  25. Altman E., Avrachenkov K., Menache I., Miller G., Prabhu B.J., Shwartz A. Power control in wireless cellular networks. IEEE Transactions Autom. Contr., 2009, Vol. 54, No. 10, pp. 2328–2340. .
  26. Ober R.J. Balanced parameterization of classes of linear systems. SIAM Journal on Control and Optimization, 1991, vol. 29, no. 6, pp. 1251–1287. doi: 10.1137/0329065 .
  27. Ober R.J., McFarlane D. Balanced Canonical Forms for Minimal Systems: A normalized Coprime Factor Approach. Linear Algebra and its Applications, 1989, vol. 122-124, pp. 23–64. doi: 10.1016/0024-3795(89)90646-0 .
  28. Antoulas A.C., Sorensen D.C., Zhou Y. On the Decay Rate of Hankel Singular Values and Related Issues. Syst. Contr. Lett., 2002, vol. 46, pp. 323–342. doi: 10.1016/S0167-6911(02)00147-0. .
  29. Wilson D.A. The Hankel Operator and its Induced Norms. International Journal on Control, 1985, vol. 42, pp. 65–70. doi: 10.1080/00207178508933346 .
  30. Anderson B.D. O., Jury E.I., Mansour M. Schwarz Matrix Properties for Continuous and Discrete Time Systems. International Journal on Control, 1976, vol. 3, pp. 1–16. doi: 10.1080/00207177608922133. .
  31. Peeters R., Hanzon B., Olivi M. Canonical Lossless State-Space Systems: Staircase Forms and the Schur Algorithm. Linear Algebra and its Applications, 2007, vol. 425, no. 2-3, pp. 404–433. doi: 10.1016/j.laa.2006.09.029. .
  32. Tang X., Wang S. A low hardware overhead self-diagnosis technique using reed-solomon codes for self-repairing chips. IEEE Transactions on Computers, 2010, vol. 59, no. 10, pp. 1309–1319. doi: 10.1109/TC.2009.182. .

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