Limitations on maximal tracking accuracy, Part I: tracking step signals

This paper studies optimal tracking performance issues pertaining to finite dimensional, linear, time invariant feedback control systems. The problem under consideration amounts to determining the minimal tracking error between the output and input signals of a system, attainable by all possible stabilizing compensators. An integral square error criterion is used as a measure for the tracking error, and explicit expressions are derived for this measure with respect to step signals. It is shown that plant nonminimum phase zeros have a negative effect upon a system's ability in reducing the tracking error, and that in a multivariable system this effect results in a way depending upon not only the zero locations, but also the zero directions. It is also shown that plant non-minimum phase zeros and unstable poles can together play a particularly detrimental role to tracking performance, especially when the zeros and poles are nearby and their directions are closely aligned. These results lead to new insights into the optimal tracking problem, and more generally, insights into certain fundamental issues concerning limitations on performance achievable via feedback control.