ADAPTIVE BACKSTEPPING CONTROL OF UNCERTAIN FRACTIONAL ORDER SYSTEMS BY FUZZY APPROXIMATION APPROACH

[1] N. Aguila-Camacho, M. A. Duarte-Mermoud and J. A. Gallegos, Lyapunov functions for
fractional order systems, Commun. Nonlinear Sci. Numer. Simul., 19(9) (2014), 2951{2957.
[2] D. Baleanu, J. A. T. Machado and A. C. J. Luo, Fractional Dynamics and Control, Springer,
2011.
[3] B. Chen, X. Liu, K. Liu and C. Lin, Direct adaptive fuzzy control of nonlinear strict-feedback
systems, Automatica, 45(6) (2009), 1530{1535.
[4] W. Chen, L. Jiao and Z. Du, Output-feedback adaptive dynamic surface control of stochastic
nonlinear systems using neural network, IET Control Theory Appl., 4(12) (2010), 3012{
3021.
[5] D. Ding, D. L. Qi, Y. M. Meng and L. Xu, Adaptive MittagLeffler stabilization of commen-
surate fractional-order nonlinear systems, in: The 53rd IEEE Conference on Decision and
Control, Los Angeles, USA, December 2014, 6920-6926.
[6] D. Ding, D. L. Qi and J. Peng, Asymptotic pseudo-state stabilization of commensurate frac-
tional order nonlinear systems with additive disturbance, Nonlinear Dyn., 81(1-2) (2015),
667{677.
[7] M. . Efe, Fractional order systems in industrial automation a survey, IEEE Trans. Ind.
Inform., 7(4) (2011), 582{591.
[8] R. Hilfer, Applications of Fractional Calculus in Physics, Hackensack, NJ: World Scientific,
2001.
[9] Z. Jiang, I. Mareels, D. Hill and J. Huang, A unifying framework for global regulation via
nonlinear output feedback: From ISS to iISS, IEEE Trans. Autom. Control, 49(4) (2004),
549{562.
[10] S. Ladaci and A. Charef, On fractional adaptive control, Nonlinear Dyn., 43(4) (2006),
365{378.
[11] H. Lee and M. Tomizuka, Robust adaptive control using a universal approximation for SISO
nonlinear systems, IEEE Trans. Fuzzy Syst., 8(1) (2001), 95{106.
[12] Y. M. Li, T. S. Li and S. C. Tong, Adaptive fuzzy backstepping dynamic surface control
of uncertain nonlinear systems based on filter observer, Int. J. Fuzzy Syst., 14(2) (2012),
320{329.
[13] Y. Li, Y. Chen, and I. Podlubny, Mittag-Leffler stability of fractional order nonlinear dynamic
systems, Automatica, 45(8) (2009), 19651969.
[14] K. S. Miller and S. G. Samko, Completely monotonic functions, Int. Transf. Spec. F., 12(4)
(2001), 389{402.

[15] K. Miroslav, K. Ioannis and V. K. Petar, Nonlinear and Adaptive Control Design, Wiley,
1995.
[16] C. A. Monje, Y. Q. Chen, B. M. Vinagre, D. Y. Xue and F. Vicente, Fractional-Order
Systems and Controls: Fundamentals and Applications, Springer, London, 2010.
[17] N. Nikdel, M. A. Badamchizadeh, V. Azimirad and M. A. Nazari, Fractional-order adap-
tive backstepping control of robotic manipulators in the presence of model uncertainties and
external disturbances, IEEE Trans. Ind. Electron., 63(10) (2016), 6249{6256.
[18] K. Oldham and J. Spanier, The Fractional Calculus: Theory and Application of Differenti-
ation and Integration to Arbitrary Order, New York: Academic, 1974.
[19] I. Pan and S. Das, Intelligent Fractional Order Systems and Control, Springer, Berlin, 2013.
[20] I. Petras, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation, Springer,
2011.
[21] H. N. Pishkenari, N. Jalili, S. H. Mahboobi, A. Alasty and A. Meghdari, Robust adaptive
backstepping control of uncertain Lorenz system, Chaos. Interdisciplinary J. Nonlinear Sci.,
20(2) (2010), 023105.
[22] I. Podlubny, Fractional Differential Equations, San Diego, CA, USA: Academic Press, 1998.
[23] I. Podlubny, Fractional-order systems and controllers, IEEE Trans. Autom. Control, 44(1)
(1999), 208{214.
[24] J. B. Qiu, G. Feng and H. J. Gao, Fuzzy-model-based piecewise H1 static output feedback
controller design for networked nonlinear systems, IEEE Trans. Fuzzy Syst., 18(5) (2010),
919{934.
[25] J. B. Qiu, G. Feng and H. J. Gao, Observer-based piecewise affine output feedback controller
synthesis of continuous-time TS fuzzy affine dynamic systems using quantized measurements,
IEEE Trans. Fuzzy Syst., 20(6) (2010), 1046{1062.
[26] J. B. Qiu, G. Feng and J. Yang, A new design of delay-dependent robust H1 filtering for
discrete-time TS fuzzy systems with time-varying delay, IEEE Trans. Fuzzy Syst., 17(5)
(2009), 1044{1058.
[27] M. R. Rahmani Mehdiabadi, E. Rouhani, S. K. Mousavi Mashhadi and A. A. Jalali. Adap-
tive Fractional-order Control for Synchronization of Two Coupled Neurons in the External
Electrical Stimulation, Neurosci., 5(2) (2014), 144{155.
[28] J. Sabatier, O.P. Agrawal and J. A. T. Machado, Advances in Fractional Calculusl, Springer,
2007.
[29] J. Sabatier, A. Oustaloup, A. G. Iturricha and P. Lanusse, CRONE control: principles and
extension to time-variant plants with asymptotically constant coefficients, Nonlinear Dyn.,
29(1) (2002), 363{385.
[30] D. Sheng, Y. Wei, S. Cheng and J. Shuai, Adaptive backstepping control for fractional order
systems with input saturation, J. Franklin inst., 354(5) (2017), 2245{2268.
[31] H. Shi, A novel scheme for the design of backstepping control for a class of nonlinear systems,
Appl. Math. Model., 35(4) (2011), 1893{1903.
[32] D. Swaroop, J. Hedrick, P. Yip and J. Gerdes,Dynamic surface control for a class of nonlinear
systems, IEEE Trans. Autom. Control, 45(10) (2000), 1893{1899.
[33] S. C. Tong, X. L. He and H. G. Zhang, A combined backstepping and small-gain approach
to robust adaptive fuzzy output feedback control, IEEE Trans. Fuzzy Syst., 17(5) (2009),
1059{1069.
[34] S. C. Tong and Y. M. Li, Adaptive backstepping output feedback control for SISO nonlinear
system using fuzzy neural networks, Int. J. Aut. Comp., 6(2) (2009), 145{153.
[35] S. C. Tong and Y. M. Li, Observer-based adaptive fuzzy backstepping control of uncertain
nonlinear pure-feedback systems, Sci. China Inf. Sci., 57(1) (2014), 1{14.
[36] S. C. Tong and Y. M. Li, Observer-based fuzzy adaptive control for strict feedback nonlinear
systems, Fuzzy Sets Syst., 160(12) (2009), 1749{1764.
[37] S. C. Tong, C. Liu and Y. M. Li, Fuzzy-adaptive decentralized output feedback control for
large-scale nonlinear systems with dynamical uncertainties, IEEE Trans. Fuzzy Syst., 18(5)
(2010), 845{861.
[38] V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers, Springer, 2013.

[39] N. Ullah, W. Khan and S. Wang, High performance direct torque control of electrical aero-
dynamics load simulator using fractional calculus, Acta Polytech. Hung., 11(10) (2014),
59{78.
[40] N. Ullah, S. Wang and M. I. Khattak, Fractional Order Fuzzy Backstepping Torque Control
of Electrical Load Simulator, Prz. Elektrotech., 89(5) (2013), 237{240.
[41] D. Wang and J. Huang, Neural network-based adaptive dynamic surface control for a class
of uncertain nonlinear systems in strict-feedback form, IEEE Trans. Neural Netw., 16(1)
(2005), 195{202.
[42] Z. Wang, X. Huang and H. Shen, Control of an uncertain fractional order economic system
via adaptive sliding mode, Neurocomputing, 83 (2012), 83{88.
[43] Y. H. Wei, Y. Q. Chen, S. Liang and Y. Wang, A novel algorithm on adaptive backstepping
control of fractional order systems, Neurocomputing, 127 (2015),395{402.
[44] Y. Wu and H. Lv,Adaptive neural network backstepping control for a class of uncertain
fractional-order chaotic systems with unknown backlash-like hysteresis, AIP Adv. 6(8)
(2016), 85{121.
[45] Y. S. Yang, G. Feng and J. S. Ren, A combined backstepping and small gain approach to
robust adaptive fuzzy control for strict-feedback nonlinear systems, IEEE Trans. Syst. Man,
Cybern. A, Syst. Humans, 34(3) (2004), 406{420.
[46] P. Yip and J. Hedrick, Adaptive dynamic surface control: A simplified algorithm for adaptive
backstepping control of nonlinear systems, Int. J. Control 71(5) (1998), 959{979.
[47] Q. Zhou, P. Shi, S. Xu and H. Li, Adaptive output feedback control for nonlinear time-delay
systems by fuzzy approximation approach, IEEE Trans. Fuzzy Syst., 21(2) (2013), 301{313.
[48] Z. Zhang, S. Xu, and H. Shen, Reduced-order observer-based output-feedback tracking control
of nonlinear systems with state delay and disturbance, Int.l J. Robust Nonlin. Control, 20(15)
(2010), 1723{1738.
[49] Z. Zhang, S. Xu and B. Wang, Adaptive actuator failure compensation with unknown control
gain signs, IET Control Theory Appl., 5(16) (2011), 1859{1867.