Controlling Coexisting Attractors of a Simplified Model for the Micro-Vibration Molding Machine via Linear Feedback Based on BP Neural Network

In order to reveal the coexistence mechanism of the attractors in the micro-vibration molding machine and to implement a rational switching strategy for generating these coexisting attractors, we have developed a spring–mass–damping equivalent model that includes clearance and elastic constraints. The Jacobi matrix of the system Poincaré mapping is obtained by constructing a local mapping. The stability of periodic motions and the type of bifurcation are analyzed based on Floquet theory and Lyapunov exponent. The transition laws of the basic periodic motions of the system are analyzed by using the shooting method, the cell mapping method, and the parameter continuation algorithm, and the coexistence of attractors caused by different types of bifurcations and boundary crises is analyzed. The appearance of a saddle-node bifurcation changes the structure of the system’s basin of attraction, which is the main reason for the appearance or disappearance of coexisting attractors. The boundary crisis is the main reason for the disappearance of chaotic attractors. Using the continuous linear feedback control method, a linear feedback controller based on a BP neural network (BP-LFC) is designed to control the coexistence attractors of the system. The linear feedback control method can simultaneously apply continuous disturbances to the displacement and velocity of the controlled periodic motion, and the BP neural network enables the gains of the Linear Feedback Controller (LFC) to be dynamically adjusted according to the distance between the desired trajectory and the controlled trajectory, which effectively reduces energy consumption while improving control accuracy. The performance index functions are constructed to analyze the control performance of BP-LFC. Compared with the fixed-gain LFC, BP-LFC has a smaller control error while consuming less energy, offering new insights for the stable control of the motion state and parameter optimization design in micro-vibration molding machines.