ℌ2 and ℌ∞ norm computations for LTI systems with generalized frequency variables

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 ℌ₂ and ℌ_∞ norm computations based on the generalized frequency variable. Specifically, we first derive a way of ℌ₂ 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_∞ norm bound for a simple feedback case, and a graphical test and numerical methods for computing the ℌ_∞ norm are derived. The last part is devoted to the loop shaping type ℌ_∞ norm, where a graphical test for the the ℌ_∞ norm condition is provided.