The Silver Ratio and its Relation to Controllability

TL;DR

This paper explores how the controllability of certain unstable coupled systems, like inverted pendulums, is maximized when their time constants relate to the silver ratio, linking system dynamics to a mathematical constant.

Contribution

It demonstrates that the controllability of coupled second-order systems is optimized at the silver ratio of their time constants, providing a novel insight into system design.

Findings

Controllability is maximized at the silver ratio of time constants.

Controllability is measured by the volume of reachable states with unit energy.

The study applies to various inverted-pendulum systems.

Abstract

This note investigates the controllability of two unstable second-order systems that are coupled through a common input. These dynamics occur for different types of inverted-pendulum systems. Controllability is quantified by the volume of the state-space that can be reached with unit energy, provided that the system starts and ends at the origin. It is shown that controllability is maximized when the ratio between the time constants amounts to the silver ratio.