Dealing with periodic disturbances in controls of mechanical systems☆
Introduction
Reference inputs and disturbances are often periodic in mechanical control applications. Periodic disturbances may be due to rotational components such as motors, and in such cases the period may be assumed to be known or measurable. On the other hand, if the periodic disturbance comes from vibrations external to the servo control loop, the period may not be known in advance. Repetitive control was originally formulated by Inoue, Nakano, and Iwai (1981) to deal with repetitive disturbances with a known period, and was developed by many researchers (Hara, Yamamoto, Omata, & Nakano, 1988; Hu & Tomizuka, 1993; Tomizuka, Tsao, & Chew, 1989; Yamada, Riadh, & Funahashi, 1999 among others). Successful applications of repetitive control include machining (Tsao & Tomizuka, 1994) and computer hard disk drives (Chew & Tomizuka, 1990). Early work on repetitive control was based on the key assumption that the period of repetitive disturbance is precisely known. While this assumption holds in many applications, more recent research efforts have been directed at dealing with cases where the period is not known in advance or is time varying (Hu, 1992; Tsao & Nemani, 1992; Tsao, Qian, & Nemani, 2000).
Repetitive control attempts to compensate for all repetitive frequency components, the fundamental frequency component as well as all higher order harmonics. In many cases, this is not necessary. For example, if a control system is perturbed by a single sinusoidal disturbance, it suffices if the compensator is designed for the single frequency component. Peak filters popular in hard disk drive (HDD) controls represent an example (Kim, Kang, & Tomizuka, 2005) of this methodology. Such approaches may be easier to extend to adaptive cases including cases for repetitive signals with unknown periods (Landau, Constantinescu, & Rey, 2005).
Closely related to repetitive control is iterative learning control. Iterative learning control was motivated by robots that must perform a repetitive operation over a finite time interval (Uchiyama, 1978). Early work on the subject most frequently cited is the betterment approach by Arimoto, Kawamura, and Miyazaki (1984), but there have been other independent developments of similar ideas at about the same time as Arimoto (Longman, 2000). Iterative learning control and repetitive control have formed a substantial research community.
The objective of this paper is to provide the fundamental design and implementation issues of repetitive control as related to the original discrete time repetitive controller (Tomizuka et al., 1989), and to introduce selected recent research activities on compensations for periodic disturbances. Application examples are drawn from mechanical systems.
Section snippets
Basics of repetitive control
Consider a discrete time system described bywhere u, y and d are, respectively, the input, output and disturbance, z−1 represents one time step delay, and d is the pure delay steps. Note that the input–output transfer function is
Assume that the system is asymptotically stable, i.e. the poles of the transfer function are all inside the unit circle. B(z−1) is written as
Adaptive repetitive control
As stated in the previous section, repetitive controllers have a built-in high gain nature, and the trade-off between the robustness by q-filter and performance is a design issue. In a recent work, Mishra and Tomizuka (2004) studied the use of an online indirect adaptive pole placement technique to implement an adaptive-repetitive controller. The scheme involves identification of the plant parameters and adaptation of the plant parameters, A and B, in the controller (6). A stability proof has
Concluding remarks
This paper presented the basics of repetitive control and selected recent research. As stated in the Introduction, repetitive control and iterative learning control are closely related to each other. One fundamental difference is the way the memory contents are updated. In repetitive control, one memory element is updated at every sampling instance while in iterative learning control, all memory elements are updated simultaneously every cycle. Repetitive and learning controls provide learning
Masayoshi Tomizuka was born in Tokyo, Japan in 1946. He received his B.S. and M.S. degrees in Mechanical Engineering from Keio University, Tokyo, Japan and his Ph.D. degree in Mechanical Engineering from the Massachusetts Institute of Technology in February 1974. In 1974, he joined the faculty of the Department of Mechanical Engineering at the University of California at Berkeley, where he currently holds the Cheryl and John Neerhout, Jr., Distinguished Professorship Chair. At UC Berkeley, he
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Masayoshi Tomizuka was born in Tokyo, Japan in 1946. He received his B.S. and M.S. degrees in Mechanical Engineering from Keio University, Tokyo, Japan and his Ph.D. degree in Mechanical Engineering from the Massachusetts Institute of Technology in February 1974. In 1974, he joined the faculty of the Department of Mechanical Engineering at the University of California at Berkeley, where he currently holds the Cheryl and John Neerhout, Jr., Distinguished Professorship Chair. At UC Berkeley, he teaches courses in dynamic systems and controls. His current research interests are optimal and adaptive control, digital control, signal processing, motion control, and control problems related to robotics, machining, manufacturing, information storage devices and vehicles. He served as Program Director of the Dynamic Systems and Control Program of the Civil and Mechanical Systems Division of NSF (2002–2004). He served as Technical Editor of the ASME Journal of Dynamic Systems, Measurement and Control, J-DSMC (1988–1993), Editor-in-Chief of the IEEE/ASME Transactions on Mechatronics (1997–1999), an Associate Editor of the Journal of the International Federation of Automatic Control, Automatica and the European Journal of Control. He was General Chairman of the 1995 American Control Conference, and served as President of the American Automatic Control Council (1998–1999). He is a Fellow of the ASME, the Institute of Electric and Electronics Engineers (IEEE) and the Society of Manufacturing Engineers. He is the recipient of the Best J-DSMC Best Paper Award (1995), the DSCD Outstanding Investigator Award (1996), the Charles Russ Richards Memorial Award (ASME, 1997), the Rufus Oldenburger Medal (ASME, 2002) and the John R. Ragazzini Award (2006).
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An earlier version of the paper was presented at the 3rd IFAC Workshop on Periodic Control Systems, PSYCO-07, St. Petersburg, Russia, August 29–31, 2007.