Modeling, response and BPNN-PID control of symmetric multistable galloping energy harvesters

In this paper, the modeling and response characteristics of a symmetric multistable galloping energy harvester (GEH) are studied, meanwhile, the controlled performance with the BPNN-PID control method is explored. The improved Lindstedt-Poincaré method is used to derive the relationship between amplitude and frequency, which is then verified through multi-scale analysis. The results indicate that under different excitation amplitudes, the GEH presents nine different solutions, demonstrating four unstable and five stable solutions. The stability analysis reveals the variation of response region and its complexity under different excitation levels. Wind speed and excitation amplitude are optimized to achieve higher output over a wider frequency range. Then, to realize the high-energy output, the PID controller and BPNN-PID controller are designed to guide the trajectories of the GEH from low-energy orbit to high-energy orbit accurately. Compared with the traditional PID control, the BPNN-PID controller demonstrates superior adaptability and robustness by adjusting PID parameters through the online learning ability of the neural network. Although the BPNN-PID controller has certain fluctuations during the initial learning phase, this fluctuation is due to the transition phenomenon in the learning process of the neural network. After the controller is shut down, the GEH can successfully stabilize on the target orbit, which indicates that the network can effectively learn the dynamic characteristics of the GEH, and shows the excellent performance of the controller in long-term stability. The BPNN-PID control not only achieves precise control but also adapts to the nonlinear dynamics of the GEH, making it more suitable for complex and variable operating conditions compared to the traditional PID control.