Vibration Control of Piezoelectric Cantilever Beam with Physics-Informed Neural Networks

Intelligent actuators, particularly piezoelectric actuators, are widely used for vibration control of engineering structures like beams, plates, and shells due to their advantages of good linearity, high precision, and simple configuration. However, traditional control methods often suffer from limited adaptability to complex dynamic environments. This paper proposes a Physics-Informed Neural Networks (PINNs) enhanced Deep Reinforcement Learning (DRL) framework for high-precision vibration control of piezoelectric cantilever beams. Unlike conventional model-based methods, our approach integrates the Euler-Bernoulli beam dynamics directly into the DRL training process, generating voltage control strategies under physical constraints through joint optimization of data-driven loss and partial differential equation (PDE) residuals. A Double Deep Q-Network (DDQN) agent observes real-time tip displacement and velocity, then outputs voltage actions. In the paper, the fundamental electromechanical coupling mechanism is established based on the cantilever beam's governing equations and sensor equations of general shell structures. Employing modal expansion methods, we derive both the modal voltage expression and the modal force formulation for the piezoelectric actuator. The architecture of the Deep Reinforcement Learning (DRL) controller—specifically a Physics-Informed Double Deep Q-Network and its underlying neural network structure are subsequently detailed. Evaluations across the first three vibration modes under free decay, sinusoidal excitation and white noise loads demonstrate that the PINNs-DRL controller significantly outperforms conventional negative velocity feedback (NVF) in suppressing transient oscillations and residual vibrations.