Abstract
This paper presents an optimal sliding mode output tracking control scheme for a class of fractional-order uncertain systems. Firstly, an augmented fractional-order system, composed of the original system and the external system, is constructed to transform the optimal output tracking issue into the design problem of linear quadratic regulator. The optimal tracking control problem for the nominal augmented fractional-order system is then studied. Secondly, the fractional-integral sliding mode controller is introduced to robustify the augmented fractional-order system, which satisfy the matching conditions. As a result, the original system output can track the external system output trajectory effectively even the uncertainties exist. Finally, the developed design techniques are applied to the tracking control of fractional-order permanent magnet synchronous motor. The simulation results demonstrate the validity of this approach.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Explore related subjects
Discover the latest articles and news from researchers in related subjects, suggested using machine learning.References
Podlubny, I.: Fractional Differential Equations. Academie Press, New York (1999)
Hilfer, R.: Application of Fractional Calculus in Physics. World Science Publishing, Singapore (2000)
Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D.: Fractional-Order Systems and Controls: Fundamentals and Applications. Springer, New York (2010)
Li, Y., Chen, Y.Q., Ahn, H.S.: Fractional-order iterative learning control for fractional-order linear systems. Asian J. Control 13, 54–63 (2011)
Luo, Y., Chen, Y.Q.: Fractional order proportional derivative controller for a class of fractional order systems. Automatica 45, 2446–2450 (2009)
Krishna, B.T.: Studies on fractional order differentiators and integrators: a survey. Signal Process. 91, 386–426 (2011)
Gao, Z., Yan, M., Wei, J.: Robust stabilizing regions of fractional-order PD controllers of time-delay fractional-order systems. J. Process Control 1, 37–47 (2014)
Zhang, Y.M., Liu, G.R., Liu, B.: Finite-time cascaded tracking control approach for mobile robots. Inf. Sci. 284, 31–43 (2014)
Xiao, L.F., Zhu, Y.: Sliding mode output feedback control based on tracking error observer with disturbance estimator. ISA Trans. 53, 1061–1072 (2014)
Lan, Y.H., Li, H.B., Liu, X.: Convergence analysis of high-order P-type iterative learning control for fractional order nonlinear systems. Nat. Sci. J. Xiangtan Univ. 37, 1–9 (2015)
Lewis, F.L., Syrmos, V.L.: Optimal Control. Wiley, New York (1995)
Agrawal, O.P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38, 323–337 (2004)
Li, Y., Chen, Y.Q.: Fractional order linear quadratic regulator. In: Proceedings of Mechtronic and Embedded Systems and Applications, Beijing, China (2008)
Razminia, A., Majd, V.J., Dizaji, A.F.: An extended formulation of calculus of variations for incommensurate fractional derivatives with fractional performance index. Nonlinear Dyn. 69, 1263–1284 (2012)
Agrawal, O.P.: A quadratic numerical scheme for fractional optimal control problems. ASME J. Dyn. Syst. Meas. Control 130, 1–6 (2008)
Özdemir, N., Agrawal, O.P., Iskender, B.B., Karadeniz, D.: Fractional optimal control of a 2-dimensional distributed system. Nonlinear Dyn. 55, 251–260 (2009)
Ding, Y., Wang, Z., Ye, H.: Optimal control of a fractional-order HIV-immune system with memory. IEEE Trans. Control Syst. Technol. 20, 763–769 (2012)
Wang, J.R., Zhou, Y., Wei, W.: Optimal feedback control for semilinear fractional evolution equations in Banach spaces. Syst. Control Lett. 61, 472–476 (2012)
Biswas, R.K., Sen, S.: Fractional optimal control problems with specified final time. J. Comput. Nonlinear Dyn. 6, 1–6 (2011)
Sierociuk, D., Vinagre, B.M.: Infinite horizon state-feedback LQR controller for fractional systems. In: 49th IEEE Conference on Decision and Control, Hilton Atlanta, Atlanta 6674-6680 (2010)
Callier, F.M., Desoer, C.A.: Linear Systems Theory. Springer, New York (1991)
John, M.D., Wang, W.L.: Output feedback sliding mode control in the presence of unknown disturbances. Syst. Control Lett. 58, 188–193 (2009)
Eduardo, V.L., Peixoto, A.J., Oliveira, T.R., Liu, H.: Global exact tracking for uncertain MIMO linear systems by output feedback sliding mode control. J. Frankl. Inst. 351, 2015–2032 (2010)
Li, C.L., Yu, S.M., Luo, X.S.: Fractional-order permanent magnet synchronous motor and its adaptive chaotic control. China Phys. B 21, 100506–6 (2012)
Lan, Y.H., Zhou, Y.: Non-fragile observer-based robust control for a class of fractional-order nonlinear systems. Syst. Control Lett. 62, 1143–1150 (2013)
Acknowledgments
This work was supported by National Natural Science Foundation of People’s Republic of China (61573298, 61363073), Hunan Provincial Natural Science Foundation of China (14JJ2073) and Scientific Research Fund of Hunan Provincial Education Department (15B238).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lan, YH., Wang, LL., Chen, CX. et al. Optimal Sliding Mode Robust Control for Fractional-Order Systems with Application to Permanent Magnet Synchronous Motor Tracking Control. J Optim Theory Appl 174, 197–209 (2017). https://doi.org/10.1007/s10957-015-0827-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-015-0827-4