sontag2008input

Input to State Stability: Basic Concepts and Results

Authors: Eduardo D. Sontag · Year: 2008 · Venue: in Nonlinear and Optimal Control Theory (Lecture Notes in Mathematics), pp. 163–220, Springer · Raw: md · pdf

Summary

An expository survey of input-to-state stability (ISS) and its relatives, written as a graduate lecture course (C.I.M.E. Summer School, Cetraro 2004). It develops — informally but precisely — the paradigm of quantifying robustness to disturbances by bounding the state as a $\mathcal{K}\mathcal{L}$ function of the initial condition plus a $\mathcal{K}$ function of the input magnitude. In this corpus it is the canonical provenance for the ISS layer: the source that resolves plain-text “Sontag ISS” mentions to a citable bibkey.

Key Claims

• ISS as the unifying robustness notion. ISS packages “bounded input ⇒ bounded state”, “converging input ⇒ converging state” (CIBS/CICS), and asymptotic gain into one definition, equivalent to the existence of an ISS-Lyapunov function.
• Toolkit for decomposition. The framework is built for reasoning about interconnections — cascades and feedback — via the ISS/gain properties of the components; this is the mechanism panteley2001growth invokes for cascaded systems.
• Related notions. integral-ISS, input-to-output stability, and the nonlinear small-gain theorem are surveyed as members of the same family.

Method

Survey/tutorial: definitions, equivalent characterizations (dissipation and gain forms), and pointers to the primary literature; no new theorems — oriented toward giving the reader a working toolkit.

Regime note. Pure systems theory — no manipulator or spacecraft model — so the free-flying vs free-floating distinction does not arise; a tool text.

Relevance to thesis

The free-flying manipulator’s closed-loop tracking-error dynamics are non-autonomous and driven by residual/interconnection terms; ISS is the property that certifies bounded, converging error under bounded disturbance, and underlies the cascade arguments (R4) this project builds on. This page supplies the resolving bibkey so those derivations cite Sontag rather than a plain-text mention. See the topic page input to state stability.

Connections

Topics: input to state stability · cascaded systems · Lyapunov stability Sources: panteley2001growth · panteley1998global · khalil2002nonlinear

Key Equations / Quotes

From the introduction (§1), read off the corpus text extraction:

“…the precise formulation of questions of robustness with respect to disturbances, formulated in the paradigm of input to state stability.”

The desired-property table (§2.1) motivates ISS as the mapping $u(\cdot )\mapsto y(\cdot )$ that carries bounded / (eventually) small / (integrally) small / convergent inputs to outputs with the corresponding property.

Open Questions

• The corpus PDF is a chapter offprint: markitdown loses inter-word spaces in the justified body (searchable but cosmetically degraded), and the host-volume editors + LNM volume number are not printed on the offprint (omitted from the .bib). Fill in from the resolved DOI 10.1007/978-3-540-77653-6_3 if a complete record is wanted.