Adaptive Repetitive Control Using a Modified Filtered-x Lms Algorithm

In this paper we develop a modified filtered-x least mean squares (MFX-LMS) method to synthesis an adaptive repetitive controller for rejecting periodic disturbances at selective frequencies. We show how a MFX-LMS algorithm can be utilized when the reference signal is deterministic and periodic. A new adaptive step size is proposed with the motivation to improve the convergence rate of the MFX-LMS algorithm and fade the steady state excess error caused by the variation of estimated parameters in a stochastic environment. A novel secondary path modeling scheme is proposed to compensate for the modeling mismatches online. We further discuss the application of this adaptive controller in hard disk drives that use Bit Patterned Media Recording. Finally we present the results of comprehensive realistic numerical simulations and experimental implementations of the algorithms on a hard disk drive servo mechanism that is subjected to periodic disturbances known as repeatable runout. INTRODUCTION Repetitive control is used to cancel disturbances or execute commands which are periodic in time. This type of control methodology has been widely used for the applications in which a trajectory should be followed repeatedly or attenuation of a periodic disturbance is desired. For instance, a robot link that is performing a periodic motion in the work space is a dynamic system that is subject to periodic gravity force and torque. Hard disk drive servo system, satellite attitude control, helicopter forward flight, orbital stabilization of under-actuated systems, and rotating machinery are other applications of repetitive control. The concept of repetitive control was first introduced in 1980’s and early works include [1, 2]. This type of controllers are typically classified into either internal model based or external model based compensators [3]. The key distinction between these two structures is that in an internal model based controller a periodic signal generators is embedded [4], while the latter structure views the cancellation signal as being injected from outside of the plant/controller feedback loop [5]. In this paper we pursue employing modified filtered-x least 1 Copyright c © 2014 by ASME mean squares (MFX-LMS) algorithm – which can be categorized as an external model based method – to develop an adaptive repetitive controller. Least mean squares (LMS) as a stochastic gradient descent method was first developed by Widrow and Hoff [6] in 1960’s and began to flourish due to its simplicity and stable behavior when implemented with finite-precision arithmetic [7]. However, this algorithm is generally unstable when its output enters to a dynamic system at a point which is distinct from the injection point of reference signal. The reason behind this instability behavior is that the error signal is not correctly “aligned” in time with the reference signal [8]. The concept of using a secondary path in the adaptation algorithm to solve this problem was later introduced independently by Burgess [9], and Widrow [10]. This technique is called filtered-x LMS (FXLMS) and it appears to be very robust to the mismatches between the model used in the secondary path and the real system [11]. However, the convergence rate of FX-LMS algorithm when the secondary path exists is slower than the simple LMS in the absence of secondary path. Bjarnason [12] and Kim [13] proposed a modification in the update equation of FX-LMS to adapt the algorithm to the standard LMS method when the secondary path model is identical to the real system. This algorithm is known as MFX-LMS and has been extensively used in active noise cancellation. Motivated by the fact that MFX-LMS algorithm has benefit of being robust as FX-LMS, and fast as LMS algorithm, we pursue this method to synthesis an adaptive repetitive controller for rejecting periodic disturbances at desired frequencies. We show how a MFX-LMS algorithm can be utilized when the reference signal is deterministic and periodic. A new variable step size is proposed with the motivation to improve the convergence rate of the MFX-LMS algorithm and fade the steady state excess error caused by the variation of estimated parameters in a stochastic environment. We also propose an online secondary path modeling scheme to compensate for the mismatch between the plant model embedded in the adaptive controller and the real plant dynamics. As mentioned earlier, one application of repetitive control is disturbance rejection in hard disk drives (HDD). The servo control mechanism of hard disk drives are usually subject to periodic disturbances that are due to the imperfection in both fabrication and assembly processes. A great deal of research effort has been focused on bit patterned media recording (BPMR) – a method in which data is stored in an array of single-domain magnetic particles – in recent years since it is recognized as a potential breakthrough method to increase the areal density of hard disk drives significantly [14, 15]. Later in the paper, we will discuss the application of the proposed repetitive control in BPMR and finally we present the results of comprehensive numerical simulations and experimental implementation of the algorithms on a hard disk drive that is subjected to periodic disturbances known as repeatable runout. e _ v w