Frequency response of sampled-data systems

Abstract

This paper develops a frequency-domain theory that provides a method to analyze and design sampled-data control systems, including their intersample behaviors. The key idea is to consider the signal space X ϑ $ ̂ = { x(t) ¦ x(t) = ∑ n ∞ = −x x n exp (jϑ t + j nω s t), ∑ n = −x ∥ x n ∥ 2 < ∞}, where ω s is the sampling angular frequency. It is shown that a stable sampled-data system equipped with a strictly-proper pre-filter before the sampler maps X ϑ into X ϑ (≡l 2) in the steady state. That mapping is denoted by Q (jϑ) and is referred to as an ‘FR operator’. It is proved that the norm of the sampled-data system as an operator from L 2 to L 2 is given by max ϑ ∥ Q (jϑ)∥ /2//2, where 1 2 ω s < ϑ ≤ 1 2 ω s . A set of equations relating outputs of a closed-loop system to its inputs in the frequency domain is derived, and their solution is given in an explicit form. Based on that solution, the sensitivity FR operator Y (jϑ) and the complementary sensitivity FR operator T (jϑ) are defined for feedback control systems, and it is shown that Y (jϑ) gives the improvement of the sensitivity of the transfer characteristics from the reference to the controlled output and also represents the ability of rejecting disturbances, and that T (jϑ) represents the degree of robust stability and, at the same time, gives the effect of detection noises. It is also shown that Y (jϑ) + T (jϑ) = I , and thus a frequency-domain paradigm for the design of sampled-data control systems, which is exactly parallel to the continuous-time case, is established.