Stability, Contraction, and Controllers for Affine Systems

TL;DR

This paper develops representation-independent conditions for stability and control of affine systems, extending data-driven behavioral methods beyond linear models to include nonlinear effects.

Contribution

It introduces necessary and sufficient conditions for stability and control in affine behaviors, including Lyapunov theorems and controller implementability criteria.

Findings

Converse Lyapunov theorems for contraction of input-output affine systems.

Conditions for implementing prescribed contractive references.

Linear controllers suffice for contractive closed-loop, affine controllers for equilibrium placement.

Abstract

Recent developments in data-driven control have revived interest in the behavioral approach to systems theory, where systems are defined as sets of trajectories rather than being described by a specific model or representation. However, most available results remain confined to linear systems, limiting the applicability of recent methods to complex behaviors. Affine systems form a natural intermediate class: they arise from linearization, capture essential nonlinear effects, and retain sufficient structure for analysis and design. This paper derives necessary and sufficient conditions independent of any particular representation for three fundamental stability problems for affine behaviors: (i) converse Lyapunov theorems for contraction of input-output systems; (ii) implementability and existence of prescribed contractive references; and (iii) whether these references can be implemented with linear or affine feedback control. For the latter, we show that linear controllers suffice for implementing contractive closed-loop, and and affine controllers are needed for equilibrium placement.