Cheap control tracking performance for non‐right‐invertible systems

There exists a well‐known fundamental limitation upon the achievable setpoint tracking performance of a non‐right‐invertible plant. This limitation manifests itself, for example, in the cost associated with the cheap control tracking problem. In this paper, we provide a new interpretation of this limitation. We show that the cheap control cost may be decomposed into the sum of two terms. The first of these depends upon certain non‐minimum phase zeroes that include the non‐minimum phase plant zeroes. The second term depends upon the extent to which the plant direction varies with frequency. To state these results, we first develop a co‐ordinate transformation that may be used to define the notion of frequency dependent plant direction and to display the relevant non‐minimum phase zeroes. We also show that the cheap control cost is connected to an integral relation that constrains the performance of any stable closed‐loop system (not necessarily under cheap control) for which the plant has a single control input and two performance outputs.