IDENTIFICATION OF FRACTIONAL ORDER SYSTEMS WITH NONZERO INITIAL CONDITIONS AND CORRUPTED BY NONZERO-MEAN NOISES
Аннотация
Most methods in the literature for fractional order systems identification ignore the initial condition and assume the measurement noise is zero mean, which may lead to incorrect estimation. In this paper, the problem of accurate parameter estimation of fractional order system with nonzero initial conditions and corrupted by nonzeromean Gaussian noises is investigated. The initial conditions along with the mean of noise are treated as extra parameters of the system. The parameter, the differential orders and the extra parameters are simultaneously estimated via minimizing the error between the output of actual fractional order system and that of the identified system. In order to reduce the computation complexity of fractional derivatives of input and output signals, the operation matrix of block pulse function is adopted to convert the fractional order system to an algebraic one. Experimental results demonstrates the effectiveness of the proposed method
Литература
1.Alaimo, G., Piccolo, V., Cutolo, A., Deseri, L., Fraldi, M., Zingales, M.: A fractional order theory of poroelasticity. Mech. Res. Commun. 100, 103395 (2019).
2.Babolian, E., Masouri, Z.: Direct method to solve volterra integral equation of the first kind using operational matrix with block-pulse functions. J. Comput. Appl. Math. 220(1), 51 – 57 (2008).
3.Balcı, E., I˙lhan O¨ ztu¨rk, Kartal, S.: Dynamical behaviour of fractional order tumor model with caputo and conformable fractional derivative. Chaos, Solitons & Fractals 123, 4351 (2019).
4.Belkhatir, Z., Laleg-Kirati, T.M.: Parameters and fractional differentiation orders esti- mation for linear continuous-time non-commensurate fractional order systems. Systems and Control Letters 115, 26 – 33 (2018)
5. Bin, D., Yiheng, W., Liang, S., Wang, Y.: Estimation of exact initial states of fractional order systems. Nonlinear Dyn. 86(3), 2061–2070 (2016)
6.C.A.Monje, Y.Chen, B.M.Vinagre, V.Xue, V.Feliu-Batrlle: Fractional-order Systems and Controls: Fundamentals and Applications. London: Springer (2010)
7.Cois, O., Oustaloup, A., Battaglia, E., Battaglia, J.L.: Non integer model from modal de- composition for time domain system identification. IFAC Proceedings Volumes 33(15), 989 – 994 (2000). 12th IFAC Symposium on System Identification (SYSID 2000), Santa Barbara, CA, USA, 21-23 June 2000
8.Cois, O., Oustaloup, A., Poinot, T., Battaglia, J.L.: Fractional state variable filter for system identification by fractional model (2001).
9.Dai, Y., Wei, Y., Hu, Y., Wang, Y.: Modulating function-based identification for frac- tional order systems. Neurocomputing 173, 1959 – 1966 (2016).
10.Gao, Z.: Modulating function-based system identification for a fractional-order system with a time delay involving measurement noise using least-squares method. Int. J. Syst. Sci. 48(7), 1460–1471 (2017).
11. Gao, Z., Lin, X., Zheng, Y.: System identification with measurement noise compensation based on polynomial modulating function for fractional-order systems with a known time-delay. ISA Trans. 79, 62 – 72 (2018).
12. Hu, Y., Fan, Y., Wei, Y., Wang, Y., Liang, Q.: Subspace-based continuous-time identi- fication of fractional order systems from non-uniformly sampled data. Int. J. Syst. Sci. 47(1), 122–134 (2016).
13. Kilicman, A., Zhour, Z.A.A.A.: Kronecker operational matrices for fractional calculus and some applications. Appl. Math. Comput. 187(1), 250–265 (2007)
14. Kothari, K., Mehta, U., Vanualailai, J.: A novel approach of fractional-order time delay system modeling based on haar wavelet. ISA Trans. 80, 371 – 380 (2018).
15.Lay, L.L.: Identification frequentielle et temporelle par modele non entier (ph. d. thesis). Ph.D. thesis (1998)
16. Leibniz, G.: Letter from hanover, germany, september 30, 1695, GA L’Hospital. JLeib- nizen Mathematische Schriften 2(301-302), 1849 (1849) 17. Li, Y., Meng, X., Zheng, B., Ding, Y.: Parameter identification of fractional order linear system based on haar wavelet operational matrix. ISA Trans. 59, 79 – 84 (2015).
18. Liao, Z., Peng, C., Wang, Y.: Subspace identification for commensurate fractional order systems using instrumental variables. In: Proceedings of the 30th Chinese Control Conference, pp. 1636–1640. IEEE (2011)
19. Lin, J.: Modelisation et identification de system dordre non entier (ph. d. thesis). Ph.D. thesis (2001)
20. Liu, D., Laleg-Kirati, T., Gibaru, O., Perruquetti, W.: Identification of fractional order systems using modulating functions method. In: 2013 American Control Conference, pp. 1679–1684 (2013).
21. Liu, D.Y., Laleg-Kirati, T.M.: Robust fractional order differentiators using generalized modulating functions method. Signal Process. 107, 395 – 406 (2015). Special Issue on ad hoc microphone arrays and wireless acoustic sensor networks Special Issue on Fractional Signal Processing and Applications
22. Malti, R., Victor, S., Oustaloup, A., Garnier, H.: An optimal instrumental variable method for continuous-time fractional model identification. IFAC Proceedings Volumes 41(2), 14379–14384 (2008)
23. Mashayekhi, S., Razzaghi, M.: Numerical solution of nonlinear fractional integro- differential equations by hybrid functions. Eng. Anal. Boundary Elem. 56, 81–89 (2015)
24. Podlubny, I., Dorcak, L., Kostial, I.: On fractional derivatives, fractional-order dynamic systems and PIλ Dµ -controllers. In: Decision and Control, 1997., Proceedings of the IEEE Conference on, pp. 4985–4990 vol.5 (1998)
25.Ren, H.P., Wang, X., Fan, J.T., Kaynak, O.: Fractional order sliding mode control of a pneumatic position servo system. J. Franklin Inst. 356(12), 6160 – 6174 (2019).
26. Silva, C.J., Torres, D.F.: Stability of a fractional hiv/aids model. Math. Comput. Simul 164, 180 – 190 (2019). The 7th International Conference on Approximation Meth- ods and Numerical Modelling in Environment and Natural Resources, held in Oujda, Morocco, May 17-20, 2017.
27. Tang, Y., Li, N., Liu, M., Lu, Y., Wang, W.: Identification of fractional-order systems with time delays using block pulse functions. Mech. Syst. Sig. Process. 91, 382 – 394 (2017).
28. Tang, Y., Liu, H., Wang, W., Lian, Q., Guan, X.: Parameter identification of fractional order systems using block pulse functions. Signal Process. 107, 272–281 (2015)
29. Thomassin, M., Malti, R.: Subspace method for continuous-time fractional system iden- tification. IFAC Proceedings Volumes 42(10), 880–885 (2009)
30.V.E.Tarasov: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Beijing:Higher Education Press (2011)
31. Victor, S., Malti, R., Garnier, H., Oustaloup, A.: Parameter and differentiation order estimation in fractional models. Automatica 49(4), 926 – 935 (2013).
32. Zarei, M.: A reflection on the fractional order nuclear reactor dynamics. Appl. Math. Modell. 73, 349 – 364 (2019).
33. Zhuting, Z., Zeng, L., Shu, L., Cheng, P., Yong, W.: Subspace-based identification for fractional order time delay systems. In: Proceedings of the 31st Chinese Control Con- ference, pp. 2019–2023. IEEE (2012)
2.Babolian, E., Masouri, Z.: Direct method to solve volterra integral equation of the first kind using operational matrix with block-pulse functions. J. Comput. Appl. Math. 220(1), 51 – 57 (2008).
3.Balcı, E., I˙lhan O¨ ztu¨rk, Kartal, S.: Dynamical behaviour of fractional order tumor model with caputo and conformable fractional derivative. Chaos, Solitons & Fractals 123, 4351 (2019).
4.Belkhatir, Z., Laleg-Kirati, T.M.: Parameters and fractional differentiation orders esti- mation for linear continuous-time non-commensurate fractional order systems. Systems and Control Letters 115, 26 – 33 (2018)
5. Bin, D., Yiheng, W., Liang, S., Wang, Y.: Estimation of exact initial states of fractional order systems. Nonlinear Dyn. 86(3), 2061–2070 (2016)
6.C.A.Monje, Y.Chen, B.M.Vinagre, V.Xue, V.Feliu-Batrlle: Fractional-order Systems and Controls: Fundamentals and Applications. London: Springer (2010)
7.Cois, O., Oustaloup, A., Battaglia, E., Battaglia, J.L.: Non integer model from modal de- composition for time domain system identification. IFAC Proceedings Volumes 33(15), 989 – 994 (2000). 12th IFAC Symposium on System Identification (SYSID 2000), Santa Barbara, CA, USA, 21-23 June 2000
8.Cois, O., Oustaloup, A., Poinot, T., Battaglia, J.L.: Fractional state variable filter for system identification by fractional model (2001).
9.Dai, Y., Wei, Y., Hu, Y., Wang, Y.: Modulating function-based identification for frac- tional order systems. Neurocomputing 173, 1959 – 1966 (2016).
10.Gao, Z.: Modulating function-based system identification for a fractional-order system with a time delay involving measurement noise using least-squares method. Int. J. Syst. Sci. 48(7), 1460–1471 (2017).
11. Gao, Z., Lin, X., Zheng, Y.: System identification with measurement noise compensation based on polynomial modulating function for fractional-order systems with a known time-delay. ISA Trans. 79, 62 – 72 (2018).
12. Hu, Y., Fan, Y., Wei, Y., Wang, Y., Liang, Q.: Subspace-based continuous-time identi- fication of fractional order systems from non-uniformly sampled data. Int. J. Syst. Sci. 47(1), 122–134 (2016).
13. Kilicman, A., Zhour, Z.A.A.A.: Kronecker operational matrices for fractional calculus and some applications. Appl. Math. Comput. 187(1), 250–265 (2007)
14. Kothari, K., Mehta, U., Vanualailai, J.: A novel approach of fractional-order time delay system modeling based on haar wavelet. ISA Trans. 80, 371 – 380 (2018).
15.Lay, L.L.: Identification frequentielle et temporelle par modele non entier (ph. d. thesis). Ph.D. thesis (1998)
16. Leibniz, G.: Letter from hanover, germany, september 30, 1695, GA L’Hospital. JLeib- nizen Mathematische Schriften 2(301-302), 1849 (1849) 17. Li, Y., Meng, X., Zheng, B., Ding, Y.: Parameter identification of fractional order linear system based on haar wavelet operational matrix. ISA Trans. 59, 79 – 84 (2015).
18. Liao, Z., Peng, C., Wang, Y.: Subspace identification for commensurate fractional order systems using instrumental variables. In: Proceedings of the 30th Chinese Control Conference, pp. 1636–1640. IEEE (2011)
19. Lin, J.: Modelisation et identification de system dordre non entier (ph. d. thesis). Ph.D. thesis (2001)
20. Liu, D., Laleg-Kirati, T., Gibaru, O., Perruquetti, W.: Identification of fractional order systems using modulating functions method. In: 2013 American Control Conference, pp. 1679–1684 (2013).
21. Liu, D.Y., Laleg-Kirati, T.M.: Robust fractional order differentiators using generalized modulating functions method. Signal Process. 107, 395 – 406 (2015). Special Issue on ad hoc microphone arrays and wireless acoustic sensor networks Special Issue on Fractional Signal Processing and Applications
22. Malti, R., Victor, S., Oustaloup, A., Garnier, H.: An optimal instrumental variable method for continuous-time fractional model identification. IFAC Proceedings Volumes 41(2), 14379–14384 (2008)
23. Mashayekhi, S., Razzaghi, M.: Numerical solution of nonlinear fractional integro- differential equations by hybrid functions. Eng. Anal. Boundary Elem. 56, 81–89 (2015)
24. Podlubny, I., Dorcak, L., Kostial, I.: On fractional derivatives, fractional-order dynamic systems and PIλ Dµ -controllers. In: Decision and Control, 1997., Proceedings of the IEEE Conference on, pp. 4985–4990 vol.5 (1998)
25.Ren, H.P., Wang, X., Fan, J.T., Kaynak, O.: Fractional order sliding mode control of a pneumatic position servo system. J. Franklin Inst. 356(12), 6160 – 6174 (2019).
26. Silva, C.J., Torres, D.F.: Stability of a fractional hiv/aids model. Math. Comput. Simul 164, 180 – 190 (2019). The 7th International Conference on Approximation Meth- ods and Numerical Modelling in Environment and Natural Resources, held in Oujda, Morocco, May 17-20, 2017.
27. Tang, Y., Li, N., Liu, M., Lu, Y., Wang, W.: Identification of fractional-order systems with time delays using block pulse functions. Mech. Syst. Sig. Process. 91, 382 – 394 (2017).
28. Tang, Y., Liu, H., Wang, W., Lian, Q., Guan, X.: Parameter identification of fractional order systems using block pulse functions. Signal Process. 107, 272–281 (2015)
29. Thomassin, M., Malti, R.: Subspace method for continuous-time fractional system iden- tification. IFAC Proceedings Volumes 42(10), 880–885 (2009)
30.V.E.Tarasov: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Beijing:Higher Education Press (2011)
31. Victor, S., Malti, R., Garnier, H., Oustaloup, A.: Parameter and differentiation order estimation in fractional models. Automatica 49(4), 926 – 935 (2013).
32. Zarei, M.: A reflection on the fractional order nuclear reactor dynamics. Appl. Math. Modell. 73, 349 – 364 (2019).
33. Zhuting, Z., Zeng, L., Shu, L., Cheng, P., Yong, W.: Subspace-based identification for fractional order time delay systems. In: Proceedings of the 31st Chinese Control Con- ference, pp. 2019–2023. IEEE (2012)
Опубликован
2020-10-30
Как цитировать
Shuen , W., L. Yao, и T. Yinggan. 2020. «IDENTIFICATION OF FRACTIONAL ORDER SYSTEMS WITH NONZERO INITIAL CONDITIONS AND CORRUPTED BY NONZERO-MEAN NOISES ». EurasianUnionScientists 6 (9(78), 33-42. https://archive.euroasia-science.ru/index.php/Euroasia/article/view/82.
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