The Duffing-Holmes Oscillator Under Hybrid Position Feedback Controller: Response Type and Basin of Attraction Analyses

Abstract

This paper presents a detailed response type and basin of attraction (BOA) analyses of a linear mass-spring-damper oscillator and a Duffing-Holmes (D-H) oscillator controlled by a class of position feedback controllers. First, the response-type comparison of both linear and D-H systems subjected to a, Negative Position Feedback (NPF), and Positive Position Feedback (PPF) controllers, and Hybrid Position Feedback (HPF) controller (which combines the previous two) is analyzed individually. Initially, the bistable system is expressed as two linear models around the stable equilibriums, and shows similar dynamic characteristics near the vicinity of stable equilibria. Three relevant response types are identified for the controlled D-H oscillator. These are the intra-well, single cross-well, and multiple cross-well response types describing all possible responses. With the BOA analyses, three response convergence types are defined. These are the convergence to state-1, convergence to state-2, and no convergence. The overall behavior of the bistable system under the hybrid controller is examined and described using these response- and convergence-type analyses. In this paper, it is shown that the HPF control concept provides desirable response for a wider range of systems and initial conditions when compared to the other simpler control schemes. The range of desirable controller parameters is identified.