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Research Papers

# Stabilization of a Fractional-Order Nonlinear Brushless Direct Current Motor

[+] Author and Article Information
Sunhua Huang

Department of Electrical Engineering,
Northwest A&F University,
Yangling 712100, Shaanxi, China

Bin Wang

Department of Electrical Engineering,
Northwest A&F University,
Yangling 712100, Shaanxi, China
e-mail: binwang@nwsuaf.edu.cn

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received April 18, 2016; final manuscript received October 7, 2016; published online January 20, 2017. Assoc. Editor: Gabor Stepan.

J. Comput. Nonlinear Dynam 12(4), 041005 (Jan 20, 2017) (6 pages) Paper No: CND-16-1198; doi: 10.1115/1.4034997 History: Received April 18, 2016; Revised October 07, 2016

## Abstract

This paper describes the stabilization of a fractional-order nonlinear brushless DC motor (BLDCM) with the Caputo derivative. Based on the Laplace transform, a Mittag-Leffler function, Jordan decomposition, and Grönwall's inequality, sufficient conditions are proposed that ensure the local stabilization of a BLDCM as fractional-order $α$: $0<α≤1$ is proposed. Then, numerical simulations are presented to show the feasibility and validity of the designed method. The proposed scheme is simpler and easier to implement than previous schemes.

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## References

Wen, X. J. , Wu, Z. M. , and Lu, J. G. , 2008, “ Stability Analysis of a Class of Nonlinear Fractional-Order Systems,” IEEE Trans. Circuits Syst. II-Express Briefs, 55(11), pp. 1178–1182.
Wang, J. R. , and Zhou, Y. , 2012, “ Mittag-Leffler–Ulam Stabilities of Fractional Evolution Equations,” Appl. Math. Lett., 25(4), pp. 723–728.
Bhrawy, A. H. , and Alofi, A. S. , 2013, “ The Operational Matrix of Fractional Integration for Shifted Chebyshev Polynomials,” Appl. Math. Lett., 26(1), pp. 25–31.
Huang, S. H. , Zhang, R. F. , and Chen, D. Y. , 2016, “ Stability of Nonlinear Fractional-Order Time Varying Systems,” ASME J. Comput. Nonlinear Dyn., 11(3), p. 031007.
Qian, D. L. , Li, C. P. , Agarwal, R. P. , and Wong, P. J. Y. , 2010, “ Stability Analysis of Fractional Differential System With Riemann–Liouville Derivative,” Math. Comput. Modell., 52(5–6), pp. 862–874.
Podlubny, I. , 1999, “ Fractional-Order Systems and PI-Lambda-D-Mu-Controllers,” IEEE Trans. Autom. Control, 44(1), pp. 208–214.
Li, H. , and Haldane, F. D. M. , 2008, “ Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States,” Phys. Rev. Lett., 101(1), p. 010504. [PubMed]
Bhrawy, A. H. , and Al-Shomrani, M. M. , 2012, “ A Shifted Legendre Spectral Method for Fractional-Order Multi-Point Boundary Value Problems,” Adv. Differ. Equations, 2012, p. 8.
Bhrawy, A. H. , Baleanu, D. , and Assas, L. , 2014, “ Efficient Generalized Laguerre-Spectral Methods for Solving Multi-Term Fractional Differential Equations on the Half Line,” J. Vib. Control, 20(7), pp. 973–985.
Chen, D. Y. , Zhang, R. F. , Liu, X. Z. , and Ma, X. Y. , 2014, “ Fractional Order Lyapunov Stability Theorem and Its Applications in Synchronization of Complex Dynamical Networks,” Commun. Nonlinear Sci. Numer. Simul., 19(12), pp. 4105–4121.
Yu, J. M. , Hu, H. , Zhou, S. B. , and Lin, X. R. , 2013, “ Generalized Mittag-Leffler Stability of Multi-Variables Fractional Order Nonlinear Systems,” Automatica, 49(6), pp. 1798–1803.
Xu, Y. , Li, Y. G. , and Liu, D. , 2014, “ Response of Fractional Oscillators With Viscoelastic Term Under Random Excitation,” ASME J. Comput. Nonlinear Dyn., 9(3), p. 031015.
Sun, H. H. , Abdelwahad, A. A. , and Onaral, B. , 1984, “ Linear Approximation of Transfer Function With a Pole of Fractional Order,” IEEE Trans. Autom. Control, 29(5), pp. 441–444.
Bhrawy, A. H. , 2016, “ A Jacobi Spectral Collocation Method for Solving Multi-Dimensional Nonlinear Fractional Sub-Diffusion Equations,” Numer. Algorithms, 73(1), pp. 91–113.
Bhrawy, A. H. , 2016, “ A New Spectral Algorithm for Time-Space Fractional Partial Differential Equations With Subdiffusion and Superdiffusion,” Proc. Rom. Acad., Ser. A: Math. Phys. Tech. Sci. Inf. Sci., 17(1), pp. 39–47.
Xin, B. G. , and Zhang, J. Y. , 2015, “ Finite-Time Stabilizing a Fractional-Order Chaotic Financial System With Market Confidence,” Nonlinear Dyn., 79(2), pp. 1399–1409.
Lazopoulos, K. A. , 2006, “ Non-Local Continuum Mechanics and Fractional Calculus,” Mech. Res. Commun., 33(6), pp. 753–757.
Kusnezov, D. , Bulgac, A. , and Dang, G. D. , 1999, “ Quantum Levy Processes and Fractional Kinetics,” Phys. Rev. Lett., 82(6), pp. 1136–1139.
Dumitru, B. , Richard, L. M. , Sachin, B. , and Varsha, D. G. , 2015, “ Chaos in the Fractional Order Nonlinear Bloch Equation With Delay,” Commun. Nonlinear Sci. Numer. Simul., 25(1–3), pp. 41–49.
Kiani, B. A. , Fallahi, K. , Pariz, N. , and Leung, H. , 2009, “ A Chaotic Secure Communication Scheme Using Fractional Chaotic Systems Based on an Extended Fractional Kalman Filter,” Commun. Nonlinear Sci. Numer. Simul., 14(3), pp. 863–879.
Muthukumar, P. , Balasubramaniam, P. , and Ratnavelu, K. , 2014, “ Synchronization of a Novel Fractional Order Stretch-Twist-Fold (STF) Flow Chaotic System and Its Application to a New Authenticated Encryption Scheme (AES),” Nonlinear Dyn., 77(4), pp. 1547–1559.
Aghababa, M. , 2012, “ Finite-Time Chaos Control and Synchronization of Fractional-Order Chaotic (Hyperchaotic) Systems Using Fractional Nonsingular Terminal Sliding Mode Technique,” Nonlinear Dyn., 69(1–2), pp. 247–261.
Wang, B. , Xue, J. Y. , Wu, F. J. , and Zhum, D. L. , 2016, “ Stabilization Conditions for Fuzzy Control of Uncertain Fractional Order Nonlinear Systems With Random Disturbances,” IET Control Theory Appl., 10(6), pp. 637–647.
Balasubramaniam, P. , and Tamilalagan, P. , 2015, “ Approximate Controllability of a Class of Fractional Neutral Stochastic Integro-Differential Inclusions With Infinite Delay by Using Mainardi's Function,” Appl. Math. Comput., 256, pp. 232–246.
Bhrawy, A. H. , and Zaky, M. A. , 2016, “ Shifted Fractional-Order Jacobi Orthogonal Functions: Application to a System of Fractional Differential Equations,” Appl. Math. Modell., 40(2), pp. 832–845.
Hemati, N. , and Leu, M. C. , 1992, “ A Complete Model Characterization of Brushless DC Motors,” IEEE Trans. Ind. Appl., 28(1), pp. 172–180.
Hemati, N. , 1994, “ Strange Attractors in Brushless DC Motors,” IEEE Trans. Circuits Syst. I, 41(1), pp. 40–45.
Ge, Z. M. , Chang, C. M. , and Chen, Y. S. , 2006, “ Anti-Control of Chaos Single Time Scale Brushless DC Motors and Chaos Synchronization of Different Order System,” Chaos, Solitons Fractals, 27(5), pp. 1298–1315.
Wei, D. Q. , Wan, L. , Luo, X. S. , Zeng, S. Y. , and Zhang, B. , 2014, “ Global Exponential Stabilization for Chaotic Brushless DC Motors With a Single Input,” Nonlinear Dyn., 77(1–2), pp. 209–212.
Zhou, P. , Bai, R. J. , and Zheng, J. M. , 2015, “ Stabilization of a Fractional-Order Chaotic Brushless DC Motor Via a Single Input,” Nonlinear Dyn., 82(1–2), pp. 519–525.
Ge, Z. M. , and Chang, C. M. , 2004, “ Chaos Synchronization and Parameters Identification of Single Time Scale Brushless DC Motors,” Chaos, Solitons Fractals, 20(4), pp. 883–903.
Liu, G. , Cui, C. J. , Wang, K. , Han, B. C. , and Zheng, S. Q. , 2016, “ Sensorless Control for High-Speed Brushless DC Motor Based on the Line-to-Line Back EMF,” IEEE Trans. Power Electron., 31(7), pp. 4669–4683.
Wang, W. , Jin, R. J. , and Jiang, J. P. , 2007, “ Optimal State Feedback Control of Brushless Direct-Current Motor Drive Systems Based on Lyapunov Stability Criterion,” J. Zhejiang Univ., Sci., A, 8(12), pp. 1889–1893.
Zhang, F. R. , and Li, C. P. , 2011, “ Stability Analysis of Fractional Differential Systems With Order Lying in (1, 2),” Adv. Differ. Equations, 2011, p. 213485.
Zhou, P. , Cai, H. , and Yang, C. D. , 2016, “ Stabilization of the Unstable Equilibrium Points of the Fractional-Order BLDCM Chaotic System in the Sense of Lyapunov by a Single-State Variable,” Nonlinear Dyn., 84(4), pp. 2357–2361.
Zheng, W. J. , Luo, Y. , Chen, Y. Q. , and Pi, Y. G. , 2016, “ Fractional-Order Modeling of Permanent Magnet Synchronous Motor Speed Servo System,” J. Vib. Control, 22(9), pp. 2255–2280.
Xu, B. B. , Chen, D. Y. , Zhang, H. , and Wang, F. F. , 2015, “ Modeling and Stability Analysis of a Fractional-Order Francis Hydro-Turbine Governing System,” Chaos, Solitons Fractals, 75, pp. 50–61.
Bao, H. B. , Park, J. H. , and Cao, J. D. , 2016, “ Synchronization of Fractional-Order Complex-Valued Neural Networks With Time Delay,” Neural Networks, 81, pp. 16–28. [PubMed]
Hei, X. D. , and Wu, R. C. , 2016, “ Finite-Time Stability of Impulsive Fractional-Order Systems With Time-Delay,” Appl. Math. Modell., 40(7–8), pp. 4285–4290.

## Figures

Fig. 1

Chaotic vibration in the fractional-order BLDCM (19) with α=0.97 : (a) xd-xq-xa, (b) xd-t, (c) xq-t, and (d) xa-t

Fig. 2

The controlled fractional-order BLDCM system (41) with α=0.97

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