Controller Discretisation: A Gap Metric Framework for Analysis and Synthesis

Although techniques for directly synthesising sampled-data (SD) compensators are available in the literature, feedback controller design is perhaps best understood in a purely continuous-time setting. As such, a feedback controller is often designed in the continuous-time domain and then discretised for digital implementation. It is important for the discretisation step involved to yield a SD approximation which captures the essential features of the original controller from the perspective of closed-loop behaviour. In this paper, a gap metric framework is developed for studying the controller discretisation problem for linear time-invariant (LTI) plants and controllers. Importantly, knowledge of a gap metric distance between an LTI controller and a SD approximation permits explicit characterisation of the possible difference in closed-loop performance, with any LTI plant for which the LTI controller is known to work well, accounting for inter-sample behaviour. The central result of the new framework gives rise to an algorithm for computing a gap metric measure of the distance between an LTI controller and a given discretisation, and a technique for synthesising a SD approximation which is optimal with respect to this metric.