The ${L}_{{q}}/{L}_{{p}}$ Hankel Norms of Positive Systems

Abstract

This letter is concerned with the $L_{q}/L_{p}$ Hankel norms of linear time-invariant positive systems, where the Hankel norms are defined as the induced norms from vector-valued $L_{p}$ -past inputs to vector valued $L_{q}$ -future outputs. The $L_{q}/L_{p}$ Hankel norms are studied in detail for general (nonpositive) LTI systems, and closed-form formulas have been derived for their characterization. However, some of them unavoidably include the absolute integral of impulse responses and/or implicit functions in their characterization and hence there remain difficulties in practical computation. In this letter we show that such difficulties are circumvented for positive systems, where positivity drastically facilitates the characterization of the past input that attains the Hankel norm. We also provide linear-programming- and semidefinite-programming-based characterizations of the $L_{q}/L_{p}$ Hankel norms of positive systems, which are useful in the case where we analyze the $L_{q}/L_{p}$ Hankel norms of positive systems affected by parametric uncertainties.