The l_q/l_p Hankel norms of discrete-time positive systems across switching

In this study, we focus on the (Formula presented.) Hankel norms of linear time-invariant (LTI) discrete-time positive systems across a single switching. The (Formula presented.) Hankel norms are defined as the induced norms from vector-valued (Formula presented.) past inputs to vector-valued (Formula presented.) future outputs across a system switching and a state transition at the time instant zero. A closed-form characterization of the (Formula presented.) Hankel norm in this switching setting for general LTI systems can readily be derived as the natural extension of the standard (Formula presented.) Hankel norm. Thanks to the strong positivity property, we show that we can successfully characterize the (Formula presented.) Hankel norms for the positive system switching case even in some combinations of p, q being (Formula presented.). In particular, some of them are given in the form of linear programming (LP) and semidefinite programming (SDP). These LP- and SDP-based characterizations are particularly useful for the analysis of the (Formula presented.) Hankel norms where the systems of interest are affected by parametric uncertainties.