Limitations on optimal tracking performance of discrete time systems

In this paper, we investigate tracking properties of finite dimensional, linear, shift invariant feedback control systems. We use the energy of an error signal as a measure of tracking ability. Our main goal is to understand the fundamental limitations on tracking performance, which can arise due to plant nonminimum phase zeros and unstable poles, and which varies with input reference signals. We consider step, ramp, and sinusoidal signals, and for each we derive a closed form expression for the minimum tracking error attainable by any stabilizing controller. These results display an explicit dependence of the tracking error on nonminimum phase zeros, unstable poles, and in particular the coupling between the directions of the poles and zeros, and those of the input reference signal. An interesting outcome then is that not only zero and pole locations affect tracking performance, but their spatial properties also play a significant role.