Effects of Delays, Poles, and Zeros on Time Domain Waterbed Tradeoffs and Oscillations

Abstract

his letter aims for a simple and accessible explanation as to why oscillations naturally arise due to tradeoffs in feedback systems, and how these can be aggravated by delays and unstable poles and zeros. Such results have been standard for decades using frequency domain methods, which yield a rich variety of familiar “waterbed” tradeoffs. While almost trivial for control experts, frequency domain methods are less familiar to many scientists and engineers who could benefit from the insights such tradeoffs can provide. So here we present an entirely time domain model using discrete time dynamics and l1 norm performance. A simple waterbed effect is that imposing zero steady state response to a step naturally create oscillations that double the response to periodic disturbances. We also show how this tradeoff is further aggravated not only by unstable poles and zeros, but also delays, in a way clearer than in the frequency domain versions.