IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 20, NO. 2, MARCH 2012 465 Brief Papers Fractional Order Periodic Adaptive Learning Compensation for State-Dependent Periodic Disturbance Ying Luo, Yang Quan Chen, Hyo-Sung Ahn, and You Guo Pi Abstract—In this brief, a fractional order periodic adaptive learning compensation (FO-PALC) method is devised for the general state-dependent periodic disturbance minimization on the position and velocity servo platform. In the first trajectory period of the proposed FO-PALC scheme, a fractional order adaptive compensator is designed which can guarantee the boundedness of the system state, input and output signals. From the second repet- itive trajectory period and onward, one period previously stored information along the state axis is used in the current adaptation law. Asymptotical stability proof of the system with the proposed FO-PALC is presented. Experimental validation is demonstrated to show the benefits from using fractional calculus in periodic adaptive learning compensation for the state-dependent periodic disturbance. Index Terms—Fractional calculus, fractional order control, state-dependent periodic disturbance (SDPD), periodic adaptive learning compensation (PALC). I. INTRODUCTION F RACTIONAL calculus is a generalization of the tradi- tional integration and differentiation with integer order to the fractional (non-integer) order. The first reference about this conception may be associated with the correspondence be- tween Leibniz and L’Hospital in 1695 [1]. However, the non-in- teger order theory has not been widely applied into control en- gineering for hundreds of years, because of the unfamiliar idea and the realization limitation of the fractional order. In the last several decades, with the better theoretical understanding of the potential from fractional calculus, and the development of the electronic circuit and computer technology, it has been accepted Manuscript received April 22, 2010; revised September 29, 2010; accepted December 03, 2010. Manuscript received in final form February 15, 2011. Date of publication March 24, 2011; date of current version February 01, 2012. Rec- ommended by Associate Editor S. Saab. The work of Y. Luo was supported by the China Scholarship Council (CSC). Y. Luo is with the Department of Automation Science and Engineering, South China University of Technology, Guangzhou 510640, China and also with the Department of Electrical and Computer Engineering, Utah State University, UT 84322-4120 USA (e-mail: ying.luo@ieee.org). Y. Q. Chen is with the Department of Electrical and Computer Engineering, Utah State University, Logan, UT 84322-4120 USA (e-mail: yqchen@ieee.org). H.-S. Ahn is with the School of Mechatronics, Gwangju Institute of Science and Technology (GIST), Gwangju 500-712, Korea (e-mail: hyosung@gist.ac. kr). Y. G. Pi is with the Department of Automation Science and Engineering, South China University of Technology, Guangzhou 510640, China (e-mail: auygpi@scut.edu.cn). Digital Object Identifier 10.1109/TCST.2011.2117426 that the fractional calculus will be more and more attractive in various science and engineering areas [2]–[8]. In motion control area, the state-dependent periodic distur- bance (SDPD) exists in many control systems. For instance, the cogging effect in the permanent magnet (PM) motor is a typical SDPD [9]–[13], which degrades the servo control per- formance, especially in low-speed control systems. In [14], the friction force is demonstrated as a state-periodic parasitic effect and in [15], the friction and eccentricity in the wheeled mobile robots are shown as the SDPD. There also exists some state-pe- riodic external disturbances in rotary systems [16], [17]. Since the SDPD is widespread in practice, the suppression of this kind of disturbance has been paid more and more attention in re- cent years. Cogging effect minimization techniques have been proposed in some literatures [9], [10]. From [18] and [19], the SDPD can be well compensated by using the learning control method [20], [21]. Taking advantage of the state-dependent pe- riodicity, adaptive learning control is applied to compensate the cogging and Coulomb friction of permanent-magnet linear mo- tors [22], [18]. In the previous work [12], a periodic adaptive learning compensation (PALC) method is suggested for the per- manent magnet synchronous motor servo systems. In this brief, a fractional order periodic adaptive learning compensation (FO-PALC) method is devised for the SDPD minimization. In the first trajectory period of this proposed FO-PALC scheme, a fractional order adaptive compensator is designed with guaranteeing the boundedness of the system state, input, and output signals. From the second repetitive trajectory period and onward, the proposed adaptive controller uses one period previously stored information along the state axis to update the current adaptation law. Asymptotical stability proof of the system with the proposed FO-PALC is presented in frequency domain. The real-time experiments demonstrate the benefits from using fractional calculus in periodic adaptive learning compensation for the general SDPD. The major contributions of this brief include: 1) a fractional order periodic adaptive learning compensation method for the general state-dependent periodic disturbances; 2) asymptotical stability proof of the system with the proposed FO-PALC in fre- quency domain; 3) experimental verification of the FO-PALC for the state-dependent periodic disturbance in the real-time dy- namometer position servo system, and the advantages demon- stration of the FO-PALC by performing the experimental com- parisons with the traditional integer order PALC. The rest of this brief is organized as follows. In Section II, the general SDPD is introduced, and a FO-PALC is proposed for the 1063-6536/$26.00 © 2011 IEEE