Controller discretization: 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 discretized for digital implementation. It is important for the discretization step involved to yield an SD approximation which captures the essential features of the original controller from the perspective of closed-loop behavior. In this note, a gap metric framework is developed for studying the controller discretization problem for linear time-invariant (LTI) plants and controllers. Importantly, knowledge of a gap metric distance between an LTI controller and an SD approximation permits explicit characterization of the possible difference in closed-loop performance, with any LTI plant for which the LTI controller is known to work well, accounting for intersample behavior. 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 discretization, and a technique for synthesising a SD approximation which is optimal with respect to this metric.