Generalized Necessary and Sufficient Robust Boundedness Results for Feedback Systems

TL;DR

This paper generalizes classical robust stability conditions for feedback systems to arbitrary semi-inner product spaces, providing algebraic results that apply beyond traditional dynamical systems and clarifying the conditions for necessity.

Contribution

It introduces a purely algebraic, generalized framework for robust stability conditions applicable to nonlinear relations in semi-inner product spaces, extending beyond classical dynamical systems.

Findings

Provides algebraic conditions for robust stability in generalized spaces

Clarifies when sufficient conditions are also necessary

Explains the need for linearity and time-invariance for necessity

Abstract

Classical sufficient conditions for ensuring the robust stability of a dynamical system in feedback with a nonlinearity include passivity, small gain, circle, and conicity theorems. We present a generalized version of these results for arbitrary semi-inner product spaces. Our result is purely algebraic, and holds even when the conventional discrete or continuous-time causal dynamical systems are replaced by general nonlinear relations, where there need not exist a notion of time. Our result clarifies when the sufficient conditions for robust stability are also necessary, and explains why stronger assumptions such as linearity and time-invariance are typically needed to prove necessity in the conventional dynamical systems setting.