Isola in a linear one-degree-of-freedom feedback system with actuator rate saturation

Abstract

This short communication uses numerical continuation to highlight the existence of an isola in a simple one-degree-of-freedom harmonically forced feedback system with actuator rate limiting as its only nonlinear element. It was found that the isola (1) contains only rate-limited responses, (2) merges with the main branch when the forcing amplitude is sufficiently large, and (3) includes stable solutions that create a second attractor in regions where rate limiting is not expected. Furthermore, the isola is composed of two solutions for a given forcing frequency. These solutions have the same amplitudes in the state (pitch rate) projection; however, they have distinct phases, and their amplitudes are also distinct when projected onto the integrator state in the controller. The rich dynamics observed in such a simple example underlines the impact of rate limiting on feedback systems. Specifically, the combination of feedback and rate limiting can create detrimental dynamics that is hard to predict and requires careful analysis.