Numerical construction of LISS Lyapunov functions under a small-gain condition

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 operator. 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 over an earlier algorithm is demonstrated. Furthermore, an example is given which shows how to analyze a given perturbed interconnected system.