Input–output gain analysis for linear systems on cones

Abstract

This paper is concerned with the input–output gain analysis for dynamic systems that are defined on cones. For such monotone systems, we characterize their input–output gain induced by the so-called cone linear absolute norm, in terms of generalized inequalities. It is further shown that for a linear time-invariant system represented by a convolution operator that maps one proper cone to another in the L1 space, its operator norm is governed by the static gain of the system. As a byproduct, the spectral radius of a cone-preserving convolution operator turns out to be the same as that of the static gain matrix. Moreover, dual results based on the cone max norm are presented. Finally, the theoretical results are illustrated via linear systems over polyhedral cones or second-order cones, as well as a class of stochastic systems.