Modelling Hysteresis in Shape Memory Alloys Using LSTM Recurrent Neural Network - Mohammad Reza Zakerzadeh; Seyedkeivan Naseri; Payam Naseri (Journal of Applied Mathematics)
The complex behavior of shape memory alloys (SMAs), characterized by hysteresis and nonlinear dynamics, results in complex constitutive equations. To circumvent the complexity of solving these equations, a black box neural network (NN) has been employed in this research to model a rotary actuator actuated by an SMA wire. Considering the historical dependence of the pulley’s rotational angle on the applied voltage, a recurrent neural network (RNN) is suitable for capturing past information. Specifically, a long short-term memory (LSTM) neural network is selected due to its ability to address issues encountered in standard recurrent networks. There are major drawbacks with modelling hysteresis with NNs that do not account for historical behavior. Traditional NNs, characterized by a one-to-one mapping, struggle to capture hysteresis loops wherein system behavior varies during loading and unloading cycles. Therefore, a single-tag data is used to determine the loading or unloading state, but tag signal causes discontinuity in network and omits various aspects of hysteresis in SMA, particularly within minor loops. In contrast, NNs incorporating past data to predict hysteresis behavior alleviate the need for tag data. However, such networks tend to have complex structures with a substantial number of neurons to effectively capture the inherent nonlinearity in SMAs. The long short-term memory (LSTM) neural network employed in this research, characterized by a simpler structure, achieves high accuracy in predicting hysteresis in SMAs without the need for tag data. In the proposed LSTM model, data related to the pulley’s rotational angle and the wire’s applied voltage from the current moment and the two previous moments serve as input. The data passes through a layer comprising three LSTM cells, and the output from the last LSTM cell is fed into a fully connected layer to predict the pulley’s rotational angle for the next moment. Training data are obtained by applying voltage at various frequencies and formats to the SMA wire while simultaneously recording the pulley’s angle with an encoder. Evaluation of the LSTM model is conducted in two configurations: online prediction (one-step ahead) and offline prediction (multistep ahead). In the online configuration where the model uses encoder data as angular inputs, the root mean square error (RMSE) of predictions for various input voltages is significantly low at about 0.1 degrees where the maximum rotational angle of pulley is 8 degrees. In the offline configuration when using the model’s predictions as angular inputs instead of encoder data, the RMSE rises to 0.3 degrees. To provide a clear demonstration of the LSTM model’s ability in this particular configuration, a comparison has been conducted between LSTM model and a rate-dependent Prandtl-Ishlinskii (RDPI) hysteresis model for predicting the pulley’s angle. The LSTM model outperforms the RDPI model by 70% in terms of accuracy. Overall, the LSTM model demonstrates capability in effectively modeling SMA hysteresis in both online and offline configurations.