The earthquake simulation shaking table is a key laboratory tool for studying and evaluating the seismic resistance of structures. It generates horizontal, vertical, and multidimensional accelerations through a driven platform, simulating the impact of seismic waves on buildings and other structures [1]. The earthquake simulation shaking table array comprises multiple independent earthquake simulation shaking tables that work together to simulate more realistic and complex ground motions. Each table can also be controlled independently to achieve more accurate earthquake wave simulations [2]. Due to factors such as high investment, expensive maintenance and experimental costs, and long construction periods, it is clearly unreasonable to infinitely increase the size and scale of shaking table. Additionally, due to similarity ratios, simply enlarging the shaking table cannot fully meet the requirements. For large-span structures such as bridges, pipelines, aqueducts, and transmission lines, combining multiple small shaking table arrays can be used for testing. The construction and research of shaking table array systems are becoming a trend in both domestic and international research. Gao Chunhua [3] conducted a survey and comparative analysis of various algorithms for domestic shaking tables, summarized the construction forms and loading methods of shaking table substructure tests, as well as the construction and control technical difficulties of shaking table array systems. Ji Jinbao [4, 5] et al. summarized and introduced the functions and characteristics of the shaking table array control system using the nine sub array of Beijing University of Technology as an example, and conducted large-span spatial structure model tests using the equipment. They pointed out the issues and areas that need improvement in using the shaking table array system and explored the relevant research and development of control technology for multi-shaking table array systems. Tao Dehuai [6] conducted a dynamic analysis of the foundation of a dual-array earthquake simulation shaking table and found that the impact of shaking on the foundation remained essentially unchanged under different load conditions. Guan Guangfeng [7] et al. conducted a detailed analysis of different control strategies for a dual array shaking table system and verified the effectiveness of the array system controller through experiments.

The nonlinear influence of shaking table system has always existed in earthquake simulation shaking table test, and has seriously affected its reproduction accuracy and waveform reproduction ability [8, 9]. For more complex shaking table array systems, due to the simultaneous operation of a large number of actuators, the existing control methods cannot meet the requirements for system stability and synchronization [10]. This means that more advanced intelligent control algorithms are needed. As a type of intelligent algorithm, neural network algorithms have good adaptive and generalization abilities, can model and handle nonlinear problems [11–13], and perform well in the control of seismic simulation shaking tables. Gao Chunhua [14, 15] et al. carried out parameter optimization and parameter identification for seismic simulation shaker by intelligent control algorithm, and simulation results showed that the intelligent control algorithm could optimize control parameters, identify multiple parameters, and improve the control effect of the shaker. Yu Shipin [16] et al. used a BP neural network to optimize the control instructions so that the control peak and valley values reached the expected values. Byung Kwan Oh [17] et al. proposed a new model for earthquake response prediction of buildings based on the correlation between ground motion and structure using neural networks. They verified the effectiveness of the proposed neural network model by studying its response prediction performance. A. Zeroual [18] et al. proposed an artificial neural network model, applied it to the prediction of the safety factor for a new earth dam dataset, and compared the predicted results with the stability calculation results of different limit equilibrium slopes. The comparison proved that the prediction ability of the artificial neural network model for the safety factor is satisfactory. Long Short-Term Memory network (LSTM) is a special type of recurrent neural network (RNN) that efficiently processes and predicts sequence data by introducing gating mechanisms. LSTM is designed to solve the problem of gradient vanishing or gradient explosion encountered by traditional RNNs when processing long sequences of data [19]. LSTM can learn long-term dependencies, which is difficult for traditional RNNs to achieve. It is easy to integrate into network structures and is suitable for various time series prediction tasks. Compared with other methods for solving long series problems (such as bidirectional RNNs), LSTM requires fewer parameters. Zhang Wenpeng [20] et al. proposed a three-parameter control parameter tuning algorithm for shakers based on LSTM and adopted the gradient descent method for offline tuning of control parameters. This was combined with the original parameters of the control system for real machine verification. The results showed that the proposed tuning method can achieve better results than manual tuning, and the tuning process is completed offline by the system model without real machine operation, offering advantages of high efficiency and good effect. Ruiyang Zhang [21] et al. proposed two long short-term memory (LSTM) network schemes aimed at data-driven structural seismic response modeling. The verification results show that the proposed LSTM network is a promising, reliable, and computationally efficient method for nonlinear structural response prediction. It has great potential in the reliability assessment of seismic vulnerability analysis of buildings.

System identification is the process of analyzing the input and output data of a system to obtain the mathematical model or dynamic characteristics of the system. This process includes determining the transfer function, state-space model, or other mathematical descriptions of the system. Effective system identification is the key to realize high performance control of shaking table [22, 23]. Zhan Pengyun [24] et al. used the least squares method to identify the model parameters of the shaking table model for seismic simulation. The research shows that the identified model can well reproduce the characteristics of the shaking table system itself, and the least squares identification method can be used to identify the hydraulic and control systems of the shaking table for seismic simulation. Ji Jinbao [25] et al. trained and tested a constructed LSTM network model based on the shaking table system model. The test results show that the LSTM network can reproduce the characteristics of a single-axis open-loop system and can be used as the control object for system simulation and algorithm testing. Febina Christudas [26] et al. utilized input-output data of long short-term memory recurrent neural networks (LSTM-RNN) to model real-time CTS. Compared with empirical models, the LSTM-RNN model achieved better modeling results. Wei Guo [27] et al. developed a physics-guided long short-term memory (PhyLSTM) network for system identification of shaking tables. After detailed hyperparameter testing, the performance of the PhyLSTM model significantly outperformed that of traditional transfer function models.

Based on the above analysis, this paper designs a system identification scheme for a dual-array seismic simulation shaking table based on LSTM neural networks. The LSTM is used to identify the multi-parameter control closed-loop system of the dual array. By comparing the predicted results after neural network training with theoretical results, the feasibility and high research value of this identification scheme are verified.

The LSTM network structure used in this paper specifically includes: an input layer, a hidden layer with 15 neurons, and an output layer. The momentum factor is set to 0.0005, the initial weights of the network are randomly chosen within the range [−1,1], and the learning algorithm employed is the “gradient descent algorithm.” The maximum number of training iterations is set to 20,000, and the loss function is calculated using the MSE formula. The training process for system identification is conducted offline. In the results presented below, the curve labeled “Actual Result” represents the output of the shaking table, which serves as the network’s label. The curve labeled “Forecast Result” represents the output of the neural network. The smaller the deviation between the forecast results and the actual results, the better the training performance of the neural network and the higher the accuracy of the system identification. We will first present intuitive result display graphs, and finally evaluate the performance of this method for system identification using the similarity between the actual and forecast waveforms. Assuming the two waveforms are denoted as x and y, their correlation coefficient can be expressed as Equation 18: c o r r x , y = ∑ n = 1 N x n − x ¯ y n − y ¯ ∑ n = 1 N x n − x ¯ 2 ∑ n = 1 N y n − y ¯ 2 (18)

We used different seismic waves as input waveforms, with a sampling time of 0.02 s, corresponding to a sampling frequency of 50 Hz. The simulation time is set to 40 s, resulting in two input datasets each containing 4,000 data points. The first 2000 data points are used as the training set, and the remaining 2000 data points are used as the test set. For the dual-array system, the output data is modeled as a two-dimensional matrix, corresponding to the response waveforms of Shaking Table 1 and Shaking Table 2, respectively. We obtained the output of the dual-array closed-loop system through simulation, which serves as the labels for neural network training. Using the EL Centro wave to analyze the training set, the training results of the dual-array closed-loop system are shown in Figure 7.

Figures 7A, B are comparison charts of the output waveforms and network training results for Shaking Table 1 and Table 2, respectively. After calculation, the root mean square error (RMSE) between the network output waveform and the actual waveform is −52.9 dB for Table 1 and -52.8 dB for Table 2. The correlation coefficients are 0.9998 for Table 1 and 0.9998 for Table 2. The network training results match the actual results very well.

For the test set, we used three different seismic waves and one artificial wave for analysis, performing both time domain and frequency domain analyses. First, we analyzed the EL Centro wave, with the time domain chart shown in Figure 8.

The calculated root mean square error (RMSE) between the network output waveform and the actual waveform is −60.4 dB for Table 1 and -60.9 dB for Table 2. The correlation coefficients are 0.9998 for Table 1 and 0.9998 for Table 2. After performing a Fourier transform on the waveform data, the frequency spectrum characteristics of the seismic wave can be obtained, as shown in Figure 9. It can be seen that the frequency domain performance of the network output waveform also almost perfectly replicates the actual frequency domain waveform.

Next, we performed time and frequency domain analyses of the Wenchuan surface wave, as shown in Figures 10, 11. The calculated results indicate that the root mean square error (RMSE) between the network output waveform and the actual waveform is −42.3 dB for Table 1 and -56.2 dB for Table 2. The correlation coefficients are 0.9996 for Table 1 and 0.9997 for Table 2.

Next, we performed time and frequency domain analyses using the Traf wave, as shown in Figures 12, 13. The calculated results indicate that the root mean square error (RMSE) between the network output waveform and the actual waveform is −51.74 dB for Table 1 and -51.20 dB for Table 2. The correlation coefficients are 0.9988 for Table 1 and 0.9986 for Table 2.

Finally, we performed time and frequency domain analyses using the artificial wave, as shown in Figures 14, 15. The calculated results indicate that the root mean square error (RMSE) between the network output waveform and the actual waveform is −49.62 dB for Table 1 and -49.43 dB for Table 2. The correlation coefficients are 0.9996 for Table 1 and 0.9996 for Table 2.

Based on the analysis of these four waveforms, the root mean square error (RMSE) and correlation coefficients between the network output waveforms and the actual time domain waveforms are summarized in Supplementary Table 2. Both the time domain and frequency domain waveform correlations exceed 0.99, indicating that the output closely matches the actual waveforms. This demonstrates that the identification scheme has excellent performance, and the network trained with LSTM can achieve the dual-array system identification task with high accuracy.

This paper primarily designs an identification scheme for a dual-array closed-loop system based on an LSTM network. First, the dual-array system of a seismic simulation shaking table is modeled. Then, a dual-array system identification method based on LSTM is proposed. Using the dataset from the constructed Simulink simulation model, the neural network is trained and tested. The LSTM network is modified to adapt to the system output, considering the characteristics of the dual-array system. This identification scheme is highly applicable, efficient in terms of parameters, and reflects the true characteristics of the system. The main conclusions drawn are as follows:

(1) In training the neural network model, the gradient descent algorithm is highly efficient and adaptable. It effectively handles the nonlinear problems in seismic simulation shaking tables and has broad application prospects in optimizing control systems.

The research work in this paper is based on closed-loop system and single-degree-of-freedom structure. The subsequent research may consider verifying the applicability of the proposed LSTM neural network identification to open-loop system and multi-degree-of-freedom structure, and further improve the efficiency and accuracy of system identification by combining with other intelligent control algorithms.