Constrained control of coexisting attractors in impact oscillator with delay

Purpose
In this paper, we investigate the effects of actuator and memory delay, start time and actuator constraints on the performance of the time-delayed 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.

Methods
In this study, the dynamics of the impact oscillator are investigated through direct simulations in MATLAB. We illustrate the coexistence of period-one and period-two and the amplitude contrast of these responses. We introduce a closed-loop system by forming the weighted difference between the current and delayed states of the system, enabling the exchange between these attractors. We predict and compare the effects of different types of delay, the existence of constraints, and the start time of the controller on its performance within a meaningful range.

Results
One notable observation is that increasing the delay leads to similar patterns in settling time for both case studies. There is initially a decrease, followed by an increase, and ultimately the failure of the controller to jump between the targeted coexisting attractors.

Conclusions
The results indicate that the system responds to the delay with a decrease in settling time, followed by an increase and eventual failure of the controller. Therefore, it would be interesting to investigate the current delayed scenarios using another stability criterion.