Performance limitation of a linear system in tracking sinusoidal signals

This paper studies the performance limitation of a feedback system with a given linear time-invariant (LTI) plant in tracking a sinusoidal signal. It continues and goes beyond some recent studies in the same topic in which it is assumed that the controller can access all the past and future values of the reference signal. In this paper, we consider the more realistic (and more difficult) situation where the controller only accesses the current and past values of the reference. An explicit formula of the best attainable performance is obtained for a SISO system which depends on the nonminimum phase zeros of the plant and the frequency of the reference sinusoid. Compared to the previously studied case when the future of the reference is available, this formula clearly shows the extra effort one has to pay to predict the future values of the reference. A partial result for a MIMO system is also given.