Differential Transparency and Excitability Functions for Cooperative Systems

Abstract

In this article, we introduce novel concepts of $L_{1}$ differential transparency and $L_\infty$ differential excitability functions and analyze their properties in cooperative nonlinear systems. Specifically, we establish connections among the positive definiteness of the $L_{1}$ differential transparency (respectively, $L_\infty$ differential excitability) function, transparency (respectively, excitability), and stability of the associated variational systems, where transparency and excitability are weaker observability and reachability concepts compared to local observability and local accessibility, respectively. These $L_{1}$ differential transparency and $L_\infty$ differential excitability functions can be computed numerically by simulation of cooperative systems and their variational system. We also provide convex approaches to estimating their upper or lower bound. Furthermore, we define the $L_{1}$ incremental transparency and $L_\infty$ incremental excitability functions, which are studied in terms of the differential energy functions. Finally, we apply the proposed differential energy functions to model reduction of cooperative systems by truncation and singular perturbation.