Fuzzy Methods of Hemispherical Resonator Gyroscope Simulation

The study deals with a fuzzy simulation describing a hemispherical resonator gyroscope. We used the following methods to solve resulting fuzzy differential equations with fuzzy initial and boundary conditions: fuzzy Laplace transform, fuzzy operator and fuzzy Galerkin. In the last case, when we use a trigonometric basis to approximate the solution, there appears a fully fuzzy system of linear algebraic equations that may be solved through the fuzzy inverse method. We present our simulation results

References

[1] Matveev V.A., Lunin B.S., Basarab M.A. Navigatsionnye sistemy na volnovykh tverdotelnykh giroskopakh [Navigation systems based on hemispherical resonator gyro]. Moscow, Fizmatlit Publ., 2008. 240 p.

[2] Lunin B.S., Matveev V.A., Basarab M.A. Volnovoy tverdotelnyy giroskop. Teoriya i tekhnologiya [Hemispherical resonator gyro. Theory and technology]. Moscow, Radiotekhnika Publ., 2014. 174 p.

[3] Basarab M.A., Lunin B.S., Matveev V.A., et al. Miniature gyroscope based on elastic waves in solids for small spacecraft. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Priborostr. [Herald of the Bauman Moscow State Tech. Univ., Instrum. Eng.], 2014, no. 4, pp. 80–96 (in Russ.).

[4] Basarab M.A., Kravchenko V.F., Matveev V.A. Metody modelirovaniya i tsifrovoy obrabotki signalov v giroskopii [Modelling and digital processing methods of signals in gyroscopy]. Moscow, Fizmatlit Publ., 2008. 248 p.

[5] Allahviranloo T., Kermani M.A. Numerical methods for fuzzy linear partial differential equations under new definition for derivative. Iran Journal of Fuzzy Systems, 2010, vol. 7, no. 3, pp. 33–50.

[6] Corveleyn S. The numerical solution of elliptic partial differential equations with fuzzy coefficients. PhD thesis. University of Leuven, 2014. 184 p.

[7] Pownuk A. Numerical solutions of fuzzy partial differential equations and its applications in computational mechanic. M. Nikravesh, L.A. Zadh, V. Korotkikh, eds. In: Fuzzy partial equations and relational equations. Springer, 2004, pp. 308–347.

[8] Arara A., Benchohra M., Nthouyas S.R., Ouahab A. Fuzzy solutions for hyperbolic partial differential equations. International Journal of Applied Mathematical Sciences, 1998, vol. 2, no. 2, pp. 241–248.

[9] Buckley J.J., Feuring T. Introduction of fuzzy partial and differential equations. Fuzzy Sets and Systems, 1999, vol. 105, iss. 2, pp. 241–248. DOI: 10.1016/S0165-0114(98)00323-6

[10] Vorobiev D., Seikkala T. Forwards the theory of fuzzy differential equations. Fuzzy Sets and Systems, 2002, vol. 125, iss. 2, pp. 231–237. DOI: 10.1016/S0165-0114(00)00131-7

[11] Buckley J.J., Feuring T. Fuzzy differential equations. Fuzzy Sets and Systems, 2000, vol. 110, iss. 1, pp. 43–54. DOI: 10.1016/S0165-0114(98)00141-9

[12] Kaleva O. The Cauchy problem for fuzzy differential equations. Fuzzy Sets and Systems, 1990, vol. 35, iss. 3, pp. 389–396. DOI: 10.1016/0165-0114(90)90010-4

[13] Mochalov I.A., Khrisat M.S., Shikhab Eddin M.Ya. Fuzzy differential equations in control. Part I. Informatsionnye tekhnologii [Information Technologies], 2015, vol. 21, no. 3, pp. 171–178 (in Russ.).

[14] Mochalov I.A., Khrisat M.S., Shikhab Eddin M.Ya. Fuzzy differential equations in control. Part II. Informatsionnye tekhnologii [Information Technologies], 2015, vol. 21, no. 4, pp. 243–250 (in Russ.).

[15] Khalilpour K., Allahviranloo T. An initial — value method for two-point fuzzy boundary value problems. Word Applied Science Journal, 2001, vol. 13, no. 10, pp. 2142–2155.

[16] Ormand A., Gouyandeh Z. Solving two-point fuzzy boundary value problems using variational iteration method. CACSA, 2013, vol. 2013, pp. 1–13. DOI: 10.5899/2013/cacsa-00006

[17] Liu H.K. Comparison result of two-point fuzzy boundary value problems. Word Academy of Science, Engineering and Technology, 2011, vol. 51, pp. 697–703.

[18] Mochalov I.A., Khrisat M.S. Estimation parameter model using fuzzy random data. Informatsionnye tekhnologii [Information Technologies], 2014, no. 2, pp. 14–22 (in Russ.).

[19] Friedman M., Ming M., Kandel A. Fuzzy linear systems. Fuzzy Sets and Systems, 1998, vol. 96, iss. 2, pp. 201–209. DOI: 10.1016/S0165-0114(96)00270-9

[20] Dehghan M., Hashemi B. Iterative solution of fuzzy linear systems. Applied Mathematics and Computation, 2006, vol. 175, iss. 1, pp. 645–674. DOI: 10.1016/j.amc.2005.07.033

[21] Allahviranloo T., Salahshour S., Homayoun-nejad M., Baleanu B. General solutions of fuzzy linear systems. Abstract and Applied Analysis, 2013, vol. 2013, art. 593274. DOI: 10.1155/2013/593274 Available at: https://www.hindawi.com/journals/aaa/2013/593274

[22] Murnganandam S., Razak A.K. Matrix inversion method for solving fully fuzzy linear systems with triangular fuzzy numbers. International Journal of Computer Application, 2013, vol. 65, no. 4, pp. 9–11.

[23] Martinson L.K., Malov Yu.I. Differentsialnye uravneniya matematicheskoy fiziki [Differential equations of mathematical physics]. Moscow, Bauman MSTU Publ., 2006. 368 p.

[24] El Jaoui E., Melliani S., Chadli S. Solving second order fuzzy differential equations by the fuzzy Laplace transform method. Advances in Difference Equations, 2015, vol. 66. DOI: 10.1186/s13662-015-0414-x

[25] Allahviranloo T., Abbasbandy S., Salahshour S., Hakimzadehet A. A new method for solving linear differential equations. Computing, 2011, vol. 92, iss. 2, pp. 181–197. DOI: 10.1007/s00607-010-0136-6

[26] Demenkov N.P., Mochalov I.A. Fuzzy system dynamics of automatic optimization. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Priborostr. [Herald of the Bauman Moscow State Tech. Univ., Instrum. Eng.], 2016, no. 1, pp. 59–74 (in Russ.). DOI: 10.18698/0236-3933-2016-1-59-74