Best Achievable Tracking Performance in Sampled-Data Systems via LTI Controllers

In this paper, we study the problem of tracking a step reference signal using sampled-data control systems. We are interested in the tracking performance, defined as the integral square of the tracking error response between the system's output and the reference input. This performance is deemed the best achievable by a sampled-data controller with a linear time-invariant discrete-time compensator if it is the minimal attainable by all such controllers that stabilize the system. Our primary objective is to investigate the fundamental tracking performance limit in sampled-data systems, and to understand whether and how sampling and hold in a sampled-data system may impose intrinsic barriers to performance. We consider two tracking performance measures, with one defined with respect to the unit step signal, and another with respect to a delayed step signal and averaged over one sampling period. We derive an analytical closed-form expression in each case for the best achievable performance. The results show that a performance loss is generally incurred in a sampled-data system, in comparison to the tracking performance achievable by analog controllers. This loss of performance, as so demonstrated by the expressions, is attributed to the non-minimum phase behaviors as well as the intersample effects generated by samplers and hold devices. Thus, sampled-data controllers do result in an additional performance limit, which is seen as a necessary tradeoff for other advantages offered by this class of controllers.