Numerical Construction of LISS Lyapunov Functions under a Small Gain Condition

TL;DR

This paper introduces a homotopy algorithm for numerically constructing LISS Lyapunov functions by finding decay points of a monotone gain operator, aiding stability analysis of large interconnected systems.

Contribution

It presents a novel homotopy algorithm that efficiently computes decay points of monotone operators, improving upon previous methods for stability analysis.

Findings

The algorithm successfully computes decay points in large-scale systems.

It demonstrates improved efficiency over earlier algorithms.

An example illustrates its application to perturbed systems.

Abstract

In the stability analysis of large-scale interconnected systems it is frequently desirable to be able to determine a decay point of the gain operator, i.e., a point whose image under the monotone operator is strictly smaller than the point itself. The set of such decay points plays a crucial role in checking, in a semi-global fashion, the local input-to-state stability of an interconnected system and in the numerical construction of a LISS Lyapunov function. We provide a homotopy algorithm that computes a decay point of a monotone op- erator. For this purpose we use a fixed point algorithm and provide a function whose fixed points correspond to decay points of the monotone operator. The advantage to an earlier algorithm is demonstrated. Furthermore an example is given which shows how to analyze a given perturbed interconnected system.