# Quantitative aspects of the input-to-state stability property

L. Grüne:
**Quantitative aspects of the input-to-state stability property**

in:
*Optimal Control, Stabilization, and Nonsmooth Analysis (MCT03),*
*
M. de Queiroz, M. Malisoff, P.Wolenski
(eds.)
*

*Lecture Notes in Control and Information Sciences*

**301**, Springer Verlag, Heidelberg, 2004, 215 - 230

Smart-Link:
http://www.math.lsu.edu/~malisoff/LCMCT/

Keywords: input-to-state stability; input-to-state dynamic stability; nonlinear
stability theory; Lyapunov functions

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## Abstract:

From the Introduction:

Since its introduction by Sontag in 1989, the input-to-state stability (ISS) property has become one of the most influential concepts in nonlinear stability theory for perturbed systems.

The fact that this concept was used by many authors is mainly due to the intuitive simplicity of the concept, which captures the qualitative essence of robust asymptotic stability in a truly nonlinear manner. On the other hand, the use of comparison functions in its formulation immediately leads to the idea to explicitly use the quantitative information contained in the ISS inequality, i.e., the rate of convergence and the robustness gain, with one of the most prominent applications being the nonlinear small gain theorem by Jiang, Teel and Praly, for which the quantitative information contained in the robustness gain is crucial.

One of the most important features of the ISS property is that it can be characterized by a dissipation inequality using a so called ISS Lyapunov function. One of the central properties of the ISDS estimate is that it admits an ISDS Lyapunov function, which not only characterizes ISDS as a qualitative property but also represents the respective decay rate, the overshoot gain and the robustness gain.

Certainly, there are many applications where quantitative robust stability properties are of interest. A particular area of applications are numerical investigations, where one interprets a numerical approximation as a perturbation of the original system and vice versa. We describe an example from this application area as well as two control theoretic applications of the ISDS property, which also illustrate the difference to the ISS property.