Interconnecting a System Having a Single Input-to-State Gain With a System Having a Region-Dependent Input-to-State Gain

TL;DR

This paper analyzes the interconnection of systems with different input-to-state gains, establishing conditions for local and global stability using the Small Gain Theorem, and demonstrating the results with an example.

Contribution

It introduces a method to analyze interconnected systems with region-dependent ISS gains, extending stability analysis techniques.

Findings

Local and global stability conditions derived

Interconnection stability depends on gain composition

Example demonstrates practical application

Abstract

For an ISS system, by analyzing local and non-local properties, it is obtained different input-to-state gains. The interconnection of a system having two input-to-state gains with a system having a single ISS gain is analyzed. By employing the Small Gain Theorem for the local (resp. non-local) gains composition, it is concluded about the local (resp. global) stability of the origin (resp. of a compact set). Additionally, if the region of local stability of the origin strictly includes the region attraction of the compact set, then it is shown that the origin is globally asymptotically stable. An example illustrates the approach.