Some Remarks on Positive/Negative Feedback

TL;DR

This paper challenges the traditional view that positive and negative feedback have opposite effects on stability in linear control systems, showing that in nonlinear systems, this dichotomy can fail.

Contribution

It demonstrates through examples that the intuitive stability effects of feedback types in linear systems do not necessarily extend to nonlinear systems.

Findings

Positive feedback can sometimes enhance stability in nonlinear systems.

Negative feedback does not always guarantee stability in nonlinear contexts.

The classical dichotomy between positive and negative feedback stability effects is not universal.

Abstract

In the context of linear control systems, a commonly-held intuition is that negative and positive feedback cannot both be stability enhancing. The canonical linear prototype is the scalar system $\dot x=u$ which, under negative linear feedback $u=-kx$ ($k >0$) is exponentially stable for all $k >0 $, whereas the lack of exponential instability of the (marginally stable) uncontrolled system is amplified by positive feedback $u=kx$ ($k >0)$. By contrast, for nonlinear systems it is shown, by example, that this intuitive dichotomy may fail to hold.