An interpretable Dahl-LRN neural-network for accurately modelling the systems with rate-dependent asymmetric hysteresis

The motion accuracy and stability of piezoelectric positioning systems are significantly compromised by inherent hysteresis and other nonlinearities. This paper presents an innovative method integrating the Dahl model with Layer Recurrent Neural Networks (LRN) to model piezoelectric actuators accurately. Initially, the Dahl model is reformulated into a neural network structure, resulting in the Dahl Neural Network (DahlNN), which strictly adheres to the underlying mathematical equations. The weights of this network directly correspond to the parameters of the Dahl equations, thereby creating a transparent neural network architecture with clear physical significance and interpretability. Subsequently, the DahlNN is enhanced by incorporating feedback mechanisms and recurrent effects from LRN, improving its ability to describe asymmetric and rate-dependent hysteresis characteristics. Extensive experiments demonstrate that, compared to LRN models without physical knowledge guidance, the proposed Dahl-LRN model reduces peak-to-valley fluctuations by 70 % and decreases the average error by approximately 97.3 %, with only a 5 % increase in computational time while maintaining interpretability and achieving superior modelling performance. Through this approach, this paper aims to provide a novel perspective on leveraging physical information to advance the application of deep learning in modelling complex physical phenomena.