Small gain theorems for large scale systems and construction of ISS Lyapunov functions

TL;DR

This paper develops small gain theorems for interconnected nonlinear systems within the ISS framework, providing a constructive method to build Lyapunov functions for large networks under a small gain condition.

Contribution

It introduces a constructive approach to create ISS Lyapunov functions for large-scale systems using a gain matrix and small gain assumptions.

Findings

Constructive method for ISS Lyapunov functions for interconnected systems

General formulation covering summation, maximization, and separation cases

Applicable to large networks with mutual dependencies

Abstract

We consider interconnections of n nonlinear subsystems in the input-to-state stability (ISS) framework. For each subsystem an ISS Lyapunov function is given that treats the other subsystems as independent inputs. A gain matrix is used to encode the mutual dependencies of the systems in the network. Under a small gain assumption on the monotone operator induced by the gain matrix, a locally Lipschitz continuous ISS Lyapunov function is obtained constructively for the entire network by appropriately scaling the individual Lyapunov functions for the subsystems. The results are obtained in a general formulation of ISS, the cases of summation, maximization and separation with respect to external gains are obtained as corollaries.