| The waterbed effect is a fundamental limitation to the performance of feedback control systems. It says that if the frequency response of a feedback control system has the property that it can significantly attenuate the effects of disturbances in one frequency range, then it must amplify disturbances in some other frequency range. Learning and repetitive control often aim to eliminate all periodic disturbances of a fixed period, i.e. for a fundamental frequency and all harmonics. This suggests that one must pay for this elimination by amplifying errors that occur between these evenly spaced frequencies. In this thesis, learning and repetitive control are studied using linear phase lead as well as low pass filtering for a frequency cutoff of the learning for purposes of stabilization. It is shown that learning control has the ability to bypass the waterbed effect, and the range of mechanisms for doing so are described. Technically, in finite time problems such as learning control one is never in steady state response. Hence, the waterbed conclusions apply to that part of the trajectory beyond a settling time of the system. In the repetitive control problem, time progresses indefinitely, allowing the system to converge to steady state response. It is shown that for real time repetitive control, one is subject to the waterbed effect. On the other hand, it is possible to bypass this effect by making batch updates of the repetitive control signal, at least for periodic inputs and disturbances of the period being addressed. Sufficiently slow updates will result in the situation that applies to learning control. Methods are developed to predict the disturbance to error steady state frequency response characteristics for batch update repetitive control. |
