[Paper Review] A Small-Gain Theorem with Applications to Input/Output Systems, Incremental Stability, Detectability, and Interconnections

Brian Ingalls, Eduardo D. Sontag|ArXiv.org|Oct 30, 2001

Control and Stability of Dynamical Systems25 references4 citations

TL;DR

This paper presents a general small-gain theorem for input/output systems that unifies and extends existing results on input-to-state stability (ISS), input-to-output stability (IOS), incremental stability, detectability, and interconnections of stable systems. The theorem uses abstract gain functions and provides conditions under which interconnected systems remain stable, with explicit bounds on transient behavior and steady-state responses via KL and K functions, recovering classical and recent small-gain results as special cases.

ABSTRACT

A general ISS-type small-gain result is presented. It specializes to a small-gain theorem for ISS operators, and it also recovers the classical statement for ISS systems in state-space form. In addition, we highlight applications to incrementally stable systems, detectable systems, and to interconnections of stable systems.

Motivation & Objective

• To develop a unified, abstract small-gain theorem applicable to diverse stability notions in nonlinear control systems.
• To extend existing ISS and IOS small-gain results to include transient behavior and incremental stability.
• To provide a framework that recovers classical small-gain theorems and applies to detectability and interconnection problems.
• To establish conditions under which interconnected systems remain input-to-output stable using nonlinear gain functions.

Proposed method

• Formalizing trajectories as quadruples (τ, u(·), x(·), y(·)) over a time set T, allowing abstract treatment of inputs, states, and outputs.
• Defining the (µ, C)-KL-practical-IOS property using KL and K functions to capture asymptotic decay and bounded overshoot.
• Introducing a generalized small-gain condition based on composition of gain functions: γ₁(γ₂(s)) < s or γ₂(γ₁(s)) < s for all s > 0.
• Applying a technical lemma involving KL functions and gain bounds to derive stability estimates under the small-gain condition.
• Using abstract gain functions and functional inequalities to generalize stability analysis beyond state-space models.
• Proving stability via comparison with KL functions that bound output trajectories in terms of initial state and input norms.

Experimental results

Research questions

• RQ1 Under what conditions does a feedback interconnection of two systems remain input-to-output stable?
• RQ2 How can small-gain theorems be generalized to systems without state-space representations, such as pure input/output systems?
• RQ3 Can the small-gain condition be expressed in terms of nonlinear gain functions that capture transient behavior?
• RQ4 How does the proposed theorem unify ISS, IOS, incremental stability, and detectability in a single framework?
• RQ5 What abstract conditions ensure practical stability and attractivity in interconnected systems with bounded gains?

Key findings

• The small-gain theorem guarantees input-to-output stability (IOS) for interconnected systems when the composition of gain functions satisfies γ₁(γ₂(s)) < s or γ₂(γ₁(s)) < s for all s > 0.
• The theorem recovers classical small-gain results for ISS systems in state-space form and extends them to systems with abstract inputs and outputs.
• For incrementally stable systems, the theorem ensures that trajectories converge to each other, with bounds depending on initial differences and input norms.
• The result applies to detectable systems via the IOSS and IMES properties, providing conditions under which estimation errors decay asymptotically.
• The stability bound is expressed via a KL function β and a K function γ, ensuring that output trajectories are bounded by initial state size, input energy, and a transient decay term.
• The proof relies on a technical lemma involving KL functions and gain functions, establishing that β(r,t) can be constructed to satisfy min{ε−2C, δ−1(r)} ≤ β(r,t) under specific conditions on T(ε,r).