ℌ<inf>2</inf> and ℌ<inf>∞</inf> norm computations for LTI systems with generalized frequency variables

Abstract

A class of large-scale systems with decentralized information structures such as multi-agent dynamical systems can be represented by a linear system with a generalized frequency variable. In this paper, we propose efficient ℌ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> and ℌ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> norm computations based on the generalized frequency variable. Specifically, we first derive a way of ℌ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> norm computation from state-space realizations of subsystems. We then discuss a region on the complex plane specified by the generalized frequency variable for achieving the ℌ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> norm bound for a simple feedback case, and a graphical test and numerical methods for computing the ℌ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> norm are derived. The last part is devoted to the loop shaping type ℌ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> norm, where a graphical test for the the ℌ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> norm condition is provided.