1) la(x)1 s boo 18a(x)j8xl S ba < 00 ex:> Fdisturbance == L Ai sin(WiX + 'Pi), i=lwhere Ai is the amplitude, Wi is the state-dependent periodic disturbance frequency, and 'Pi is the phase angle. Note that, this general form can well represent state-dependent periodic disturbance in real world, for example, the state-dependent friction in [2], the position-dependent cogging force in permanent magnetic linear motor [8] or permanent magnetic synchronous motor [9] [10], the eccentricity in the wheeled mobile robots [7] and the experiment apparatus [1], and so on.For simplicity, in the sequel, we denote the above statedependent periodic disturbance as a(x). From physical limit consideration, it is reasonable to believe that a(x) is bounded, that is, errors of more than one previous periods are utilized in the adaptive updatingllearning law. Experimental results reported in [10] confirm that UO-PALe does achieve better compensation performance than the first-order PALC scheme.In the present work, we propose a new high-order periodic adaptive learning compensation (HO-PALC) method for statedependent periodic disturbance where the stored information of more than one previous periods is used and, the information includes composite tracking error as well as the estimate of the periodic disturbance. This is called dual HO-PALC (DHO-PALe) scheme which we show offers potential to achieve faster learning convergence. In particular, when the reference signal is also periodically changing, the proposed DHO-PALC can achieve much better convergence performance in terms of both convergence speed and final error bound. Asymptotical stability proof of the proposed DHO-PALC is presented. Extensive lab experimental results are presented to illustrate the effectiveness of the proposed DHO-PALC scheme over the first-order periodic adaptive learning compensation (FO-PALC).The major contributions of this paper include 1) A new dualhigh-order periodic adaptive learning compensation method for state-dependent periodic disturbance and the proof of the asymptotical stability of the system with the DHO-PALC; 2) Experimental study of the DBO-PALC for state-dependent periodic disturbance on a dynamometer position control system; 3) Experimental demonstration of the advantages of the DHO-PALC over the FO-PALC scheme.
II. THE GENERAL FORM OF STATE-DEPENDENTPERIODIC DISTURBANCE This paper is mainly concerned with the general state-dependent periodic disturbance similar to [1], [2], [3], [8], [9], [10], [14]. The disturbance could be any type of nonlinear periodic function depending on a state variable x which usually represents the linear displacement or rotational angle. In Fourier series, the general state-dependent periodic disturbance can be expressed by Abstraet-State-periodic disturbances an' frequently found in motiOb control systems. Eump1eS intlUde tOIling in permanent maanetic linear motor, ectentricity in rotary machines aDd etc. This paper considers general form or state-dependent periodic disturbance and proposes a new blgb-order perIod...