Necessity of a sampled-data approximation result

A computationally tractable, sufficient condition was given in Cantoni and Vinnicombe (2000) for a periodic sampled-data (SD) controller to: (i) lie within a specified pointwise gap metric distance, closely related to the ν-gap metric, from a given LTI controller C; and (ii) stabilise any LTI plant that achieves a certain level of closed-loop performance with C. In this paper, it is shown that the condition is also necessary. As such, the result gives rise to an algorithm for computing the distance between an LTI controller and a given discretisation, and an optimal technique for SD approximation in the gap metric. An example is presented to demonstrate the use of these tools