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.