External Positivity of Sampled-data Systems and Their <i>L<sub>q</sub></i>/<i>L</i><sub>∞</sub> Hankel Norm Analysis

Abstract

This paper extends the theory of positive systems to sampled-data systems. When considering positivity of LTI systems, where to take the initial time is unimportant. However, since sampled-data systems are periodically time-varying from the perspective of continuous-time, introducing an adequate definition for their positivity should be relevant to considering which time to take as the initial time. This paper begins by studying positivity of sampled-data systems first by considering the case where a sampling instant is regarded as the initial time, and secondly by further considering the case where an arbitrary intersample instant is regarded as the initial time. These arguments are combined to derive the necessary and sufficient conditions for a unified notion of (i.e., initial-time-independent) positivity of sampled-data systems. This paper then applies this concept to the analysis of the Lq/L∞ Hankel norm, known as a measure for evaluating the effects of disturbances in dynamical systems. It is shown that positive sampled-data systems lead to concise expressions for the (quasi) Lq/L∞ Hankel norms not only for q = ∞ but also for q = 1,2, where the latter are hard to handle for general sampled-data systems. Some relevant properties for these norms are also shown. The theoretical results derived in this paper are confirmed through numerical examples.