Document Type: Research Paper


Department of Electrical Engineering, Shahid Bahonar University, Kerman, Iran,


In this paper, a novel problem of observer-based adaptive fuzzy fractional control for fractional order dynamic systems with commensurate orders is investigated; the control scheme is constructed by using the backstepping and adaptive technique. Dynamic surface control method is used to avoid the problem of “explosion of complexity” which is caused by backstepping design process. Fuzzy logic systems are used to approximate the unknown nonlinear functions. A fractional order Lyapunov function is defined at each stage and the negativity of an overall Lyapunov function is ensured by proper selection of the control law. It is proven that the proposed controller guarantees the boundedness property for all the signals and also the tracking error can converge to a small neighborhood of the origin. Simulation examples are given to demonstrate the effectiveness and robustness of the proposed controllers.


[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,
[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{
[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),
[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,
[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),
[10] S. Ladaci and A. Charef, On fractional adaptive control, Nonlinear Dyn., 43(4) (2006),
[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),
[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,
[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,
[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),
[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,
[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),
[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),
[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.