Constrained Control of Coexisting Attractors in Impact Oscillator with Delay

Abstract

The coexistence of attractors is a characteristic of a range of nonlinear systems that results in systems exhibiting different dynamics for varying initial conditions with invariant system parameters. This characteristic often provides an effective means to jump between different behaviours by employing control methods. However, control signals are influenced by delay and constraint, which affect controllers' performance. In this paper, we investigate the effects of actuator and memory delay, start time and actuator constraints on the performance of the timedelayed feedback (TDF) control scheme enabling the switching between coexisting attractors of the impact oscillator. Looking at two potential applications, we focused on two case studies with two desired stable orbits: Case 1: Non-impacting attractor and Case 2: High amplitude attractor. In both case studies, the effects of the actuator and memory delay in the control signal are investigated. Then, within a meaningful range of delay, their effects on the controller performance are predicted and compared. One notable observation is that by increasing the delay, settling time follows similar patterns in two case studies, experiencing a decrease followed by an increase and then failure of the controller to jump between the targeted coexisting attractors. Furthermore, the effects of the start time and the actuator constraints (force limit) on the controllers' performances in these case studies are also investigated.