Long short-term memory (LSTM) neural networks for predicting dynamic responses and application in piezoelectric energy harvesting

Long Short-Time Memory (LSTM) deep neural networks are capable of learning order dependence in sequence problems and capturing long-term, non-linear temporal dependencies between the input and out of a system. With the long-term vision to model dynamical systems to which analytical or numerical methods are impossible or difficult to apply, this paper presents a study of modeling system dynamics and predicting responses using the LSTM networks, which have demonstrated excellent capability in predicting single-mode responses in a prior study. However, the LSTM network exhibits difficulties in modeling and predicting multi-mode responses accurately. To resolve the multi-mode issue, this paper presents an approach that obtains an equivalent network consisting of a set of sub-networks learned on isolated modes, and demonstrates its effectiveness on a simulated 2-degree-of-freedom mass-spring-damper system of nonlinear Duffing springs. The second part of the paper is focused on the application of the proposed approach in piezoelectric energy harvesting. Experiments are conducted on a harvester subjected to random base-motion excitation and exhibiting nonlinearity in its multi-mode response. Both the direct and mode-separation LSTM modeling approaches are applied to predict the output voltage given a random base-motion excitation. The mode-separation approach outperforms the direct approach significantly, and yields an excellent match between the actual and predicted responses. Specifically, for a test electrical voltage response of RMS value 0.2241 V, the difference between the actual test and predicted responses by using the mode-separation approach has an RMS value of 0.0504 V, compared to 0.1645 V obtained by using the direct LSTM approach. It is also much lower than the RMS value of 0.1835 V obtained by using the attention-based LSTM network, another comparison method. Leveraging a deep learning strategy, the validated approach opens up opportunities for accurately modeling energy harvesting systems of high complexities and/or strong nonlinearities.