Frequency response of sampled-data systems

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 Xϑ$= {x(t) ¦x(t) = ∑n∞ = −x xn exp (jϑt + jnωst), ∑n = −x ∥xn∥2 < ∞}, where ωs 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 Xϑ into Xϑ (≡l2) 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 L2 to L2 is given by maxϑ ∥Q(jϑ)∥/2//2, where 12ωs < ϑ ≤ 12ωs. 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.