Input-to-state dynamical stability and its Lyapunov function characterization

L Grune

doi:10.1109/tac.2002.802761

Abstract

We present a new variant of the input-to-state stability (ISS) property which is based on using a one-dimensional dynamical system for building the class /spl Kscr//spl Lscr/ function for the decay estimate and for describing the influence of the perturbation. We show the relation to the original ISS formulation and describe characterizations by means of suitable Lyapunov functions. As applications, we derive quantitative results on stability margins for nonlinear systems and a quantitative version of a small gain theorem for nonlinear systems.

Highlights

The input–to–state stability (ISS) property introduced by Sontag [13] has become one of the central properties in the study of stability of perturbed nonlinear systems
Instead, we use the value of the perturbation at each time instant as an initial value of a one–dimensional dynamical system, which leads to the concept of input–to–state dynamical stability (ISDS)
It turns out that ISDS is qualitatively equivalent to ISS and, in addition, that we can pass from ISS to ISDS with only slightly larger robustness gains

Summary

Introduction

The input–to–state stability (ISS) property introduced by Sontag [13] has become one of the central properties in the study of stability of perturbed nonlinear systems. It assumes that each trajectory φ of a perturbed system satisfies the inequality φ(t, x, u) ≤ {β( x , t), ρ( u ∞)}. Instead, we use the value of the perturbation at each time instant as an initial value of a one–dimensional dynamical system, which leads to the concept of input–to–state dynamical stability (ISDS) Proceeding this way, we are in particular able to “synchronize” the effects of past disturbances and large initial values by using the same dynamical system for both terms. We show two control theoretic applications of the ISDS property in Section 4, which illustrate the difference to the ISS property

Input–to–state dynamical stability

Lyapunov function characterization

Applications

Proofs