Pergamon PII:S0967-0661(96)00040-8 ControlEng. Practice, Vol. 4, No. 4, pp. 563-569, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0967-0661/96 $15.00 + 0.00 ZERO PLACEMENT FOR DESIGNING DISCRETE-TIME REPETITIVE CONTROLLERS W.C. Messner* and C.J. Kempf** *Department of Mechanical Engineering, CarnegieMellon University, Pittsburgh, Pennsylvania, USA **NSK Ltd., PrecisionMachinery and Parts Technology Center, 78 Toriba-machi, Maebashi, Gunma371, Japan (Received July 1995; in final form January 1996) Abstract: The analysis and design of two discrete-time repetitive control methods using root locus techniques are presented. These methods are theoretically capable of perfect cancellation of all harmonics of a periodic disturbance up to the Nyquist frequency while remaining robust to a class of unmodeled dynamics. They are designed to be stable in the presence of an unmod- eled pole on the real axis inside the unit circle. The robustness results are achieved by adding zeros to the dynamics of the compensators. Keywords:Root locus, repetitive control, digital control 1. INTRODUCTION "Repetitive control" refers to the collection of tech- niques used to compensate for periodic disturbances or periodic reference signals where the period is known. Many mechanical systems are subject to such disturbances or reference signals, most notably rotat- ing machinery (such as computer disk drives, lathes, rotating magnetic bearings) or those which repeatedly execute the same trajectory (such as milling machines, X-Y stages, robots in manufacturing). The term "betterment control" is also used, although usu- ally in the context of systems operated in an intermit- tent fashion. A variety of methods have been developed both in continuous time (Arimoto et al., 1985; Bodson et al., 1992) and in discrete time (Chew and Tomizuka, 1990; Hu and Tomizuka, 1991; Sidman,1988; Messner, 1992; Tomizuka and Kempf, 1990). The methods have two approaches. One is to attempt perfect cancellation of a few harmonics of the fundamental mode of the periodic disturbance. The other approach is to achieve some attenuation of all harmonics of the fundamental mode. Perfect cancella- tion of all of the harmonics of the fundamental mode is usually not attempted for robustness reasons. The methods developed in this paper are based on the "pro- totype" repetitive controller with Q-filter (Chew and Tomizuka, 1990), which is a discrete-time controller and which implements the latter strategy. However, these methods attempt perfect cancellation of all of the harmonics of the disturbance. Prototype repetitive control was developed for cancel- ing a periodic disturbance of period N by using an N- step delay chain with unity positive feedback gain to ~ I U m delay s Fig. 1. Tapped delay chain signal generator. 563