It is very interesting to talk to and see all the people that some time during their work have "tried it out" just to see if it would work, e.g. have had a masters project or graduate student working on the subject.
From a theoretical point of view, the controllers are applications of the Internal Model Principle, and as such could be thought of as well explored already. Thus, a dead research area. However, there are still properties, at least by me, not fully understood yet. Also, applications always tend to have the nasty habit of violating the assumptions we made in our analysis.
A few years ago I wrote my PhD thesis "On Repetitive Control", which started out from an application point of view; but gradually got in more theory. The applications, for which these controllers naturally come in, which I also focused on, normally involves some kind of rotation. Either you try to control the rotation directly, or some noise or vibrations arising from it. It is considered as "implementation know how" to synchronize sampling with this rotation, and sample at fixed angular positions instead of in time if possible; to make the periodic signal independent of rotational speed.
In the first application I worked with, a peristaltic pump used in dialysis, the normal assumption of rotation being fairly constant (as been used for e.g. CDs, harddisks ...) is not valid. In fact, this turned out to be a fundamental limitation on performance. As many other people, I started out with the internal model comprising a delay chain storing one period of the signal. (Which some people define as what should be used within the subject area of Repetitive Control.) It turned out that this scheme, even with really sophisticated filtering within the loop of the delay chain (known as F- or Q-filtering), the controller would not stabilize for the full working range. After further analysis, like the Robustness below in the Int. Journal of Control, I abandoned this model, for the original approach of including the sinusoids needed to reject the disturbance (a lower order, or reduced order, compared with the delay internal model). Even if this controller is slightly messier than the delay model, it turn out to have two good advantages: 1) enhanced robustness margins, and 2) a lower order controller giving a shorter transient.
With the reduced order model, or the delay model without filtering (which implies bad robustness), included in the design, we know that the modelled frequencies will be perfectly cancelled asymptotically. This result is of little use if it takes ages to reach this state. During my postdoctoral year I have focused on basic trade-offs between the transient performance, and robustness in terms of the gap metric. To put it short, the faster the transient response the smaller the robustness margin, for the interesting regions of the controller design. Richard Lee, who I have been working with during this year, has e.g. shown that a bound on the obtainable robustness may directly be calculated from the open loop plant gain for the modelled frequencies; see either the paper "Some Remarks on Repetitive Control" or, more elaborated with examples, our Technical Report below.
Anyone who wants to discuss the subject further, or has a challenging application, please feel free to contact me.
G. Hillerström and Richard C.H. Lee, "Trade-offs in Repetitive control," Technical Report CUED/F-INFENG/TR 209, University of Cambridge, June 1997. Abstract and PostScript file.
G. Hillerström and J. Sternby, "Robustness properties of repetitive controllers," In the International Journal of Control , 65(6), pp. 939-961, Dec. 1996.
G. Hillerström and Kirthi Walgama, "Repetitive Control Theory and Applications - A Survey", In proc of the 13th IFAC World Congress, Vol. D, pp. 1--6, San Francisco, July 1996.
G. Hillerström and J. Sternby, "Application of repetitive control to a peristaltic pump," In the ASME Journal of Dynamic Systems Measurement and Control , 116(4), pp. 786--789, Dec. 1994.