Learning Control and Related Problems in Infinite-Dimensional Systems

The basic features of a special type of learning control scheme, currently known as repetitive control are reviewed. It is seen that this control scheme also induces varied interesting theoretical problems— particularly those related to infinite-dimensional systems. They include such problems as the internal model principle, minimal representation of transfer functions, fractional representations, stability characterizations, correspondence of internal and external stability, etc. This article intends to give a comprehensive overview of the repetitive control scheme as well as the discussion of these related theoretical problems for infinite-dimensional systems. 1 What is Learning Control? The word learning almost always intrigues us. It reflects our very fundamental desire of having an intelligent artificial object that can perform complicated tasks instead of ourselves. Control itself was in fact a product of such a desire. From the very beginning, control was destined to have some intelligent power to be able to automatically adjust itself to the operating environment. It is not hard to imagine that the concept of feedback (in connection with the sensitivity reduction against disturbances) had a striking conceptual appeal to those working in the field. The secret here is that the control system can adjust itself (or, more appealingly, “learn to react”) to the fluctuation, disturbances, etc. that could be an obstacle to performing its task. ∗Supported in part by the Tateishi Science and Technology Foundation.