Inherent design limitations for linear sampled-data feedback systems

In this paper, we develop a theory of design limitations for sampled-data feedback systems wherein we consider the response of the analog system output. To do this, we use the fact that the steady state response of a hybrid feedback system to a sinusoidal input consists of a fundamental component at the frequency of the input together with infinitely many harmonics at frequencies spaced integer multiples of the sampling frequency away from the fundamental. This fact allows us to define fundamental sensitivity and complementary sensitivity functions that relate the fundamental component of the response to the input signal. These sensitivity and complementary sensitivity functions must satisfy integral relations analogous to the Bode and Poisson integrals for purely analog systems. The relations show, for example, that design limitations due to nonminimum phase zeros of the analog plant constrain the response of the sampled-data feedback system regardless of whether the discretized system is minimum phase and independently of the choice of hold function.