[Paper Review] Integral-Input-Output to State Stability

Brian Ingalls|ArXiv.org|Jun 15, 2001

Fault Detection and Control Systems25 references3 citations

TL;DR

This paper introduces and characterizes integral-input-output-to-state stability (iIOSS), a detectability notion for nonlinear systems using integral (L1-type) norms instead of supremum (L∞) norms. It proves that iIOSS is equivalent to the existence of a continuous Lyapunov function satisfying a specific dissipation inequality, with an extension to smooth Lyapunov functions when the input set is compact. The key contribution is a Lyapunov characterization of iIOSS, enabling observer design and robustness analysis for systems with energy-like signal measurements.

ABSTRACT

A notion of detectability for nonlinear systems is discussed. Within the framework of ``input to state stability'' (ISS), a dual notion of ``output to state stability'' (OSS), and a more complete detectability notion, ``input-output to state stability'' (IOSS) have appeared in the literature. This note addresses a variant of the IOSS property, using an integral norm to measure signals, as opposed to the standard supremum norm that appears in ISS theory.

Motivation & Objective

• To formalize a detectability concept for nonlinear systems based on integral (L1-type) signal norms, extending classical IOSS theory.
• To establish a Lyapunov function characterization for the iIOSS property, providing a constructive framework for stability analysis.
• To demonstrate that iIOSS systems admit norm-observers, enabling state bound estimation from output measurements.
• To extend the Lyapunov construction to smooth functions under compact input constraints, enhancing applicability in control design.

Proposed method

• Proposes iIOSS as a variant of IOSS using integral norms to measure input and output signals, modeling total signal energy.
• Defines iIOSS via existence of functions α ∈ K∞, β ∈ KL, and γ1, γ2 ∈ K such that state norm is bounded by initial state and integrals of output and input signals.
• Establishes equivalence between iIOSS and existence of a continuous Lyapunov function V satisfying a dissipation inequality involving time-varying gains.
• Uses viscosity subgradients to handle nonsmooth Lyapunov functions and derive the necessary decrease condition in the generalized sense.
• Extends the result to smooth Lyapunov functions by constructing a smooth function eV on Rn\{0}, then composing with a smooth K∞ function ρ to ensure global smoothness.
• Applies results from [11] and [15] to ensure existence of a smooth Lyapunov function under compact input set assumptions.

Experimental results

Research questions

• RQ1 Can a detectability notion based on integral norms be rigorously defined for nonlinear systems, and how does it differ from standard IOSS?
• RQ2 Is there a Lyapunov function characterization for the iIOSS property, and if so, what are the regularity requirements on the function?
• RQ3 How does iIOSS relate to existing concepts like iISS and passive systems, and what are the implications for observer design?
• RQ4 Can the iIOSS Lyapunov function be constructed to be smooth under mild assumptions, such as compact input sets?
• RQ5 What is the role of the viscosity subgradient in formulating the Lyapunov condition for nonsmooth functions?

Key findings

• The iIOSS property is equivalent to the existence of a continuous Lyapunov function V satisfying a dissipation inequality involving the integral of output and input signals.
• A viscosity subgradient formulation allows the derivation of the necessary Lyapunov decrease condition even when the Lyapunov function is not differentiable.
• For systems with compact input sets, the iIOSS Lyapunov function can be constructed to be smooth, enabling stronger analytical and design tools.
• The Lyapunov function satisfies a bound of the form V(x(τ,ξ,v)) − V(ξ) ≤ ∫₀^τ [−c₁λ(Tξ + r) + c₂(γ₁(|y|) + 2eγ₂(|v|))] dr for small τ, ensuring state decay.
• The existence of such a Lyapunov function implies that the system admits a norm-observer, allowing estimation of the state's distance from the origin.
• The construction yields a smooth iIOSS Lyapunov function V = ρ∘eV, where ρ is a smooth K∞ function, ensuring global smoothness and positive definiteness.