Simultaneous rejection of signals below and above the nyquist frequency

This paper studies the disturbance rejection problem for sampled-data control systems, where disturbance signal occurs below and above the Nyquist frequency simultaneously. Two discrete-time controllers are designed via H
^∞
optimal control in two steps; at first a controller is designed to reject the low-frequency components, and then we construct the generalized plant including the first controller to design the second controller, which has the capability of rejecting the high-frequency components. In view of the well-known sampling theorem, one recognizes that any high-frequency components may be detected only as an alias in the low base band, and hence it is impossible to recover or detect such frequency components. The authors recently showed in [13] that this assumption depends crucially on the underlying analog model, and it is indeed possible to track or reject such frequency components by introducing multirate signal processing techniques. This paper aims to make this design technique applicable to the case in which the target frequencies lie both below and above the Nyquist frequency. Detailed analysis of multirate closed-loop systems are given. It is shown via examples that rejection of lower- and higher-frequency signals than the Nyquist frequency can be achieved.