Frequency-response of Sampled-data Systems I: Open-loop Consideration

Abstract

The first part of the theory which provides a frequency-domain paradigm for the design of sampled-data control systems is developed. The key idea is to cousider the signal space χφ=▵{∑xmexpπ(jφt+jmwst):∑||xm||2<∞,m=0,±1,ċ} where -ws/2 < φ ≤ ws/2, ws = 2π/Γ is the sampling augular frequency, and τ is the sampling period. In this paper stable open-loop sampled-data systems equipped with strictly-proper pre-filters before samplers are considered. It is shown that such a sampled-data system maps χφintoχφ(≡l2) and that this mapping, denoted by Q(jφ), is bounded. It is also shown that the, norm of the sampled-data system as all operator from L2 to L2 is given by maxϕQ(jϕ)l2/l2.