L<inf>1</inf> gain analysis of linear positive systems and its application

Abstract

In this paper, we focus on L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> gain analysis problems of linear time-invariant continuous-time positive systems. A positive system is characterized by the strong property that its output is always nonnegative for any nonnegative input. Because of this peculiar property, it is natural to evaluate the magnitude of positive systems by the L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> gain (i.e. the L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> induced norm) in terms of the input and output signals. In contrast with the standard L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> gain, in this paper, we are interested in L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> gains with weightings on the input and output signals. It turns out that the L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> gain with weightings plays an essential role in the stability analysis of interconnected positive systems. More precisely, as a main result of this paper, we show that an interconnected positive system is stable if and only if there exists a set of weighting vectors that renders the L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> gain of each positive subsystem less than unity. As such, using a terminology in the literature, the weighting vectors work as 'separators,' and thus we establish solid separator-based conditions for the stability of interconnected positive systems. We finally illustrate that these separator-based conditions are effective particularly when we deal with robust stability analysis of positive systems against both L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> gain bounded and parametric uncertainties.