Some Remarks on Positive/Negative Feedback

TL;DR

The paper challenges the linear intuition that negative feedback always enhances stability while positive feedback can destabilize, by analyzing a nonlinear scalar system and introducing a nonlinear notion of control direction via a $\chi$-functional. It presents a funnel-control based static feedback framework with $h(t,\xi)$ and a bijective $\varphi$ that yields global stabilization ($x(t)\to 0$) and bounded input for systems possessing a control direction. Crucially, it constructs an explicit nonlinear example with non-unique control direction, $G(\rho,\xi,v)= f(\rho,\xi) + b g(v)$ with $g(v)= v\sin(\ln(1+|v|))$, showing that both polarity feedbacks stabilize by making $\sup_{s\ge 0}\chi(\eta s)=\infty$ for $\eta=\pm 1$. This existential result situates the work relative to universal adaptive control (Morse/Nussbaum) and highlights that, in nonlinear settings, the stabilizing effect of feedback polarity can be more nuanced than in linear theory.

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.

Figures (3)

• Figure 1: Domain $\mathcal{F}$.
• Figure 2: Schematic construction of the sequence $(\kappa_n)$.
• Figure 3: Schematic construction of the sequence $(\sigma_n)$.

Theorems & Definitions (8)

• Definition 1.1
• Proposition 1.2
• proof
• Proposition 2.1
• proof
• Corollary 2.2
• Proposition 3.1
• proof