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

Abstract

This paper is an attempt to extend the theory of positive systems to sampled-data systems. Even though where to take the initial time is unimportant in positivity of linear time-invariant systems, this is not the case for sampled-data systems because of their periodically time-varying nature from the perspective of the continuous-time input and output signals. Hence, we first study positivity of sampled-data systems by considering the case where a sampling instant is viewed as the initial time. We then further consider the case where an arbitrary intersample instant is viewed as the initial time, and we finally give the necessary and sufficient conditions for ultimate notions of (i.e., initial-time-independent) internal/external positivity of sampled-data systems. The arguments of this paper are then illustrated through numerical examples.

Keywords

• External positivity
• Hankel norm analysis
• induced norm analysis
• internal positivity
• periodic systems
• sampled-data systems