Definitions of Internal/External Positivity of Sampled-Data Systems and the Necessary and Sufficient Conditions

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 systems from the perspective of the continuous-time input and output signals, it is important to consider which time should be viewed as the initial time. Therefore, this paper studies positivity of sampled-data systems first by considering the case where a sampling instant is viewed as the initial time, and secondly by further considering the case where an arbitrary intersample instant is viewed as the initial time. The two different situations lead to quite relevant issues of what to take as the initial conditions and what to assume on the initial conditions in appropriately defining the internal/external positivity of sampled-data systems. These arguments are combined to derive the necessary and sufficient conditions for ultimate notions of (i.e., initial-time-independent) internal/external positivity of sampled-data systems.