LQR controller is one of the most widely used classical control algorithms for active control (Moghaddasie and Jalaeefar, 2019). Since its inception, LQR has evolved to address complex challenges in structural dynamics, enhancing the ability to mitigate vibrations and improve the stability of structures under various dynamic loads. The LQR controller aims to minimize the quadratic performance index (J) (Equation 14) defined as: J = 1 2 ∫ 0 t f z T t Q z t + f u T t R f u t d t (14) where Q and R ( Q ≥ 0), ( R > 0), are two weighting matrices that weight the system state z and the active control force vector f u . The effectiveness of LQR depends on the proper selection of the weighting matrices Q and R , which determine the trade-off between system performance and control effort. The matrix Q imposes a penalty on the system response, where larger values lead to greater response reduction but require higher control forces. Conversely, the matrix R penalizes the control effort, where larger values result in lower control forces, potentially limiting the reduction in structural response. The dimensions and shape of the R depend essentially on the number of actuators and their location on the structure (Moghaddasie and Jalaeefar, 2019). Matrix Q (Equation 15) is detailed below: Q = q 1 , 1 q 2 , 2 ⋱ q 2 n + 2 , 2 n + 2 (15)

The matrix R in this study is a scalar due to the use of a single actuator acting only on the movable mass of the ATMD while being grounded at its second end.

The resulting actuator optimal control force (Equation 16) is evaluated by: f u t = G z t (16) where G (Equation 18) is the gain matrix determined by solving the Riccati equation (Equation 17) as: P A + A T P + Q − P B R − 1 B T P = 0 (17) G = R − 1 B T P (18)

The block diagram of the LQR controller is illustrated in Figure 2.

ANN controllers play a pivotal role in structural control systems, offering a robust approach to optimize control strategies using active and semi-active dampers (Kim, 2020). Their application in managing seismic responses has led to significant advancements, particularly in reducing structural oscillations and enhancingare two weighting matricesthat weight the system state overall system performance. A key strength of ANN controllers is their ability to accurately approximate nonlinear functions, making them ideally suited for complex systems like multistory buildings. ANNs enable the training of controllers to adapt to varying conditions, thereby minimizing dynamic responses during seismic events.

In this study, the developed ANN model is designed to predict the active control force f u t replicating the behavior of an ideal LQR controller, where the main goal is to drive the ATMD to reduce the structural response at the base level of the isolated MDOF structure. The ANN achieves this prediction by learning the relationship between selected input variables and the corresponding control force through a training process. This involves adjusting internal weights, biases, and activation functions to minimize the prediction error.

The process of training and deploying the ANN model for driving the ATMD can be summarized as follows:

Architecture Definition: select the optimal ANN architecture by determining the number of hidden layers and neurons per layer.

A flowchart detailing the major steps of the procedure can be seen in Figure 3.

The current study employed a multilayer ANN architecture, visually represented in Figure 4, structured with three layers: an input layer, a hidden layer, and an output layer.

The input data utilized in the input layer include the base displacement and its velocity, respectively x b t and  x ˙ b t , and the ground motion acceleration x g ¨ t taken from an earthquake record characterized by a wide frequency range. Optimized over iterations, the hidden layer has six nodes, and the output layer has one node Y ¯ representing the LQR control force f u t trained with target data of an optimal LQR control force. Therefore, the model is represented as Fisco and Adeli (2011a), Fisco and Adeli (2011b), Djerouni et al. (2020) and Elia et al. (2017). The layers in the neural network are interconnected by weights, which are adjusted during the training process to minimize the error between the ANN model’s output and the desired target LQR control force. The connections between the input and hidden layers are represented by weight terms w i j 1 , where i corresponds to the index of the hidden neuron and j to the input neuron. Similarly, the connections between the hidden and output layers are defined by the weight terms w 1 i 2 , where i denotes the hidden neuron index. The hidden layer consists of activation neurons a i , which processes the weighted sum of the inputs through a nonlinear activation function f g ¯ i , enhancing the network’s ability to capture complex relationships in the data. This architecture allows the ANN to learn and replicate the control strategy of the LQR controller effectively.

For training and optimization, the network configuration implemented the backpropagation algorithm, a widely used technique for training neural networks, in conjunction with the TANSIG (hyperbolic tangent sigmoid) activation function (Sibi et al., 2013). These choices were pivotal in the model’s ability to effectively learn and adapt to the complexities of the dataset under investigation. The equation of the activation (Equation 19) function is written as follows: f g ¯ i = 2 1 + e − 2 g ¯ i − 1 (19)

The output of the first layer is obtained by applying the activation function (Equation 20): a i = f g ¯ i = f ∑ j = 1 3 w i j 1 x ¯ j + b i 1 (20)

Finally, the final output is determined as (Equation 21): Y ¯ = ∑ i = 1 6 w 1 i 2 a i + b 2 (21)

In this scenario, x j ¯ represents the input features, while w 1 and w 2 correspond to the weights associated with the first and hidden layers, respectively. Similarly, b 1 and b 2 denote the biases for the first and second layers.

To guarantee the robustness of the model during the training phase, the dataset was divided into training and validation subsets, with 80% dedicated to training the neural network. This allowed for effective pattern recognition. The remaining 20% was used for validation, providing an evaluation of the model’s accuracy on new data. The Levenberg-Marquardt backpropagation technique (trainlm) (Reynaldi et al., 2012) from MATLAB’s Neural Network Toolbox was selected for its proficiency in optimizing parameters and reducing training errors.

According to the model training process, the mean squared error (MSE) (Equation 22) and the correlation coefficient R (Equation 23) are utilized as performance metrics and evaluated as: M S E = 1 N ∑ i = 1 N Y t a r , i − Y o u t , i 2   0 < M S E < ∞ (22) R = ∑ i = 1 N Y t a r , i − Y t a r ¯ Y o u t , i − Y o u t ¯ ∑ i = 1 N Y t a r , i − Y t a r ¯ 2 Y o u t , i − Y o u t ¯ 2   − 1 < R < 1 (23) where Y t a r , i , Y o u t , i , Y t a r ¯ , Y o u t ¯ correspond to the target, output, mean target, and mean output, respectively, whereas N represents the dataset size to measure the performance of the model and assess the error for each split. According to Smith (1986), the evaluation criteria for R include the following limit:

R < 0.2 Weak correlation.

Following the selection of the optimal model with the optimal node’s number according to statistical performance, the K-fold cross-validation approach (Kohavi, 1995) was employed to assess the model’s capability prediction. K-fold cross-validation is a method used to evaluate a machine learning model’s performance by dividing the dataset into k equally sized subsets. The model is trained k times, each time using k − 1 folds for training and the remaining fold for validation. This ensures that each fold serves as the validation set exactly once.

In this study, 5-fold cross-validation was adopted, as it offers a good compromise between computational efficiency, bias, and variance, making it a standard choice for model evaluation.

Figure 5 illustrates the active control feedback, in which the ANN controller actuates the ATMD to mitigate structural vibrations. The system comprises three primary sensors: one measuring the base layer displacement, another measuring the base layer velocity, and a third measuring the ground acceleration. The sensor data are fed into the ANN controller, which processes the inputs and determines the appropriate control force. This control force is applied to the structure to counteract the vibrations. The structure’s response is continuously monitored and fed back into the neuro controller, creating a closed-loop system. The ANN controller dynamically adjusts the damping force based on real-time feedback, ensuring optimal vibration suppression and enhancing structural stability and performance under various dynamic conditions. The ANN controller dynamically adjusts the damping force in real-time based on feedback, ensuring effective vibration suppression and enhancing structural stability and performance under various dynamic loading conditions.

An 8-story benchmark isolated building, proposed by Yang et al. (1994), is utilized to implement the hybrid control system. The structure is equipped with a lead core rubber-bearing isolation system and an ATMD located on the base floor. The structural properties of each floor are summarized in Table 1.

An ATMD was coupled to the BIS to enhance its performance during seismic events and reduce structural vibrations, specifically the base isolation layer displacement. The tuning parameters for the passive part of the ATMD are determined using equations provided by Sadek et al. (1997) in terms of the non-dimensional parameters (Equation 24): µ = m d M T , γ = 1 1 + μ 1 − ξ s μ 1 + μ , ξ d = ξ s 1 1 + μ + μ 1 + μ (24) m d = μ M T (25) c d = 2 ξ d m d ω d (26) k d = ω d 2 m d (27) where µ is the mass ratio between the ATMD mass, m s , and the structure’s total mass, M T , γ = ω d ω s is the frequency ratio between the TMD and the structure’s first frequency and ξ d is the TMD damping ratio calculated with respect to the structure damping ratio ξ s = 5 % . Being the total mass of the adopted base-isolated structure equal to M T = 3214.8  tons , while its first natural frequency is ω s = 2.2  rad / s , having assumed a mass ratio μ = 0.05 , the TMD proprieties (Equations 25–27) are: m d = 160.74  tons , k d = 694.59  kN / m , and c d = 177.62  kNs / m . It is worth noticing that the assumed mass ratio is in accordance with typical values adopted in literature for conventional TMD systems (Djedoui et al., 2016).

The properties of the BIS are taken from Yang et al. (1994) and are detailed in Table 2.

The inputs to the Artificial Neural Network (ANN) are derived from the structural response generated by a Linear Quadratic Regulator (LQR) controller, which requires the definition of the weighting matrices Q and R. These matrices play a crucial role in shaping the control performance by penalizing deviations in system states and excessive control effort, respectively. n this study, the system includes a single actuator associated with the ATMD. Consequently, the R matrix is reduced to a scalar. A value of R = 10⁻³ was selected to ensure a suitable trade-off between minimizing the control effort and achieving effective vibration mitigation.

The weighting matrix Q is used to penalize deviations in selected state variables. Since the ATMD is directly connected to the BIS, the base displacement x b t and base velocity x ˙ b t are key states to be regulated. To emphasize their importance in the control design, the diagonal elements corresponding to these variables, specifically q 1 , 1 and q 11 , 11 , were set to 10³. This choice reflects the objective of reducing base motion without completely constraining it, consistent with the dynamic behavior expected in seismically isolated systems.

The values of Q and R were fine-tuned through an iterative trial-and-error process. Multiple numerical simulations were performed to assess the effectiveness of different combinations of weights. The configuration yielding the best compromise between vibration reduction and control effort was selected. The optimal control forces computed through the tuned LQR controller were then used as reference outputs for training the ANN model, enabling it to emulate the LQR behavior in real-time without relying on an explicit dynamic model during operation.

The ANN-based controller is designed to regulate the ATMD system in real-time, with the objective of improving the seismic performance of an eight-story nonlinear base-isolated structure. The network is trained using data derived exclusively from a single natural earthquake record. Specifically, the ground motion from the 1940 El Centro earthquake was recorded at the Imperial Valley station, with a peak ground acceleration (PGA) of 0.32 g measured on the North/South component. The El Centro record was selected due to its broadband frequency content, which makes it a representative and widely adopted benchmark for training purposes. Once the optimal ANN model was identified through training, its generalization capability and robustness were evaluated by subjecting the structure to six additional earthquake records. These included both far-field and near-field events, with and without velocity pulses, to comprehensively assess the model’s performance under a diverse range of seismic scenarios.

The process is illustrated in Figure 6, which outlines the step-by-step development and validation of the ANN-based controller. The procedure begins with programming a shear frame structure in MATLAB and applying an LQR controller, where the weighting matrices ( Q and R ) are optimized for improved performance. The ANN is then trained using the collected data, and the best-performing model is selected based on accuracy and error minimization. Finally, the trained ANN model is tested with six different earthquake records to evaluate its generalization and effectiveness. Training with historical earthquake data allows the ANN to learn, adjust, and reduce the differences between the LQR control force and the desired one, thereby enabling the ANN to approximate the optimal control strategy for which the LQR is known.

To assess the effectiveness of the ANN and ensure its emulation to the LQR, its structural response will be compared with that of the LQR controller itself, as well as with scenarios where no active control is applied.

The effectiveness of the ANN model within the hybrid system is evaluated by assessing it with six different real seismic recordings, each featuring diverse frequency components and peak accelerations. The seismic data for these six earthquakes, as sourced from the Federal Emergency Management Agency (FEMA) P659 (Council, 2009), are presented in Table 3. To keep the structural behavior within a reasonable range, all the used records were scaled to a PGA of 0.2 g, as seen in Figure 7.

The utilized ground acceleration records are depicted in Figure 7.

In this section, the evaluation of the ANN model’s performance is made, employing the two key statistical metrics defined in Equations 22, 23 the MSE and the correlation coefficient R, are evaluated. Furthermore, the model’s predictive capability is assessed through the implementation of K-fold cross-validation.

Figure 8 shows the MSE over 31 epochs, which highlights the model’s accuracy and error dynamics. Initially, the MSE decreases sharply across training and validation datasets, indicating a reduction in prediction error as the model learns from the data. The close alignment of MSE values across datasets underscores the model’s robustness. The best validation performance is observed at epoch 25, where the MSE reaches its minimum value of 3.3 · 10 − 4 . This epoch represents the point at which the model generalizes best to unseen data, striking an optimal balance between underfitting and overfitting, ensuring maximum predictive performance and minimal error between the output and the target. Beyond this point, MSE slightly increases, which may lead to overfitting.

Figure 9 shows the error histogram from the simulation of the optimal ANN model. The green bars indicate validation data, and the blue bars represent training data. Errors between the output and target values can be observed to lie within the range of −0.04 to 0.04.

As defined in Equation 23, the correlation coefficient R measures the strength and direction of the linear relationship between predicted and actual values, with higher values closer to 1 indicating stronger correlation and better predictive performance. Figure 10 displays scatter plots comparing the output and the target values of the control force in the optimal ANN model in the training phase, validation phase, and both phases. The results show strong correlation coefficients, with R values of 0.99115 for training and 0.99118 for validation, which demonstrates the effective learning capability of the model.

The performance measures of the best ANN model are shown in Figure 11, revealing an R range of 0.98–0.99, where each column in the figure corresponds to a different fold in the K-fold cross-validation process, representing the model’s performance across multiple training-validation splits. This underscores the success of K-fold cross-validation in showcasing the model’s robustness. The model consistently performs well across all five folds, effectively mitigating overfitting and underfitting risks, showcasing its remarkable ability to learn from training data and generate new data.

After identifying the most effective neural network training model, it is utilized to control the ATMD through a feedback loop. This will enable computing the required control force to mitigate the effect of seismic activity on several ground motions to evaluate and validate the performance of the ANN model in adapting to new seismic conditions. For this purpose, the six ground motion records from the FEMA database not utilized during the training phase are selected (Table 3; Figure 7).

Various control strategies are considered to assess the performance of the ANN-driven ATMD. The control strategies adopted are:

The base-isolated structure is equipped with a classical TMD, where the actuator force is set to null; this strategy is denoted BIS + TMD.

The first one has been compared with a hybrid control system employing only passive strategies. The performances of each control strategy are compared with those of BIS considered alone, and the results are reported in terms of reduction rate.

The dynamical parameters of interest investigated are the peak base floor displacement, the root mean square (RMS) of the base displacement, the peak control device stroke, the base floor acceleration, velocity and displacement time histories, the maximum drift and inter-story drift of all the floors and the hysteresis loops of the nonlinear isolator.

Table 4 presents the peak base displacement and the base displacement RMS for each control strategy under each earthquake, along with their respective reduction rate (RR) percentages evaluated across the six different earthquake scenarios. The passive TMD system provides moderate displacement reductions, ranging from 17% to 36%, showing its ability to mitigate vibrations to a certain extent. The active control strategy using LQR shows a more substantial improvement, reducing displacements by 63%–81%. Notably, the ANN-based ATMD demonstrates the highest effectiveness, consistently achieving displacement reductions between 81% and 87%, indicating its effectiveness and successful learning from the LQR controller. Further, the passive TMD system achieves moderate reductions in RMS values, ranging from 27% to 52%, indicating its limited yet consistent damping capacity. The ANN demonstrates reduction rates ranging from 91% to 94%, surpassing LQR’s rates of 86%–92%. Similar to peak base displacement results, the ANN highlights its ability to learn from and surpass LQR’s performance.

In addition to displacement and RMS evaluations, it is crucial to assess the stroke response of the damper systems to ensure their practical applicability. The stroke, defined as the peak displacement of the damper mass, determines the device’s required motion range. Excessive stroke demands may render a control system impractical despite its effectiveness in reducing vibrations. Therefore, Table 5 presents a comparative analysis of the peak stroke displacements for the TMD, the LQR-controlled ATMD, and the ANN-controlled ATMD under the various earthquake records, offering insights into their mechanical feasibility alongside their control performance.

As shown in Table 5, the TMD system exhibits the highest stroke demands across all earthquake records, with peak values reaching up to 0.711 m. In contrast, the LQR-controlled ATMD significantly reduces the required stroke, and the ANN-based ATMD achieves even lower stroke values in most cases. This demonstrates not only the superior control capability of the ATMD systems but also their mechanical efficiency. Notably, the ANN consistently maintains a lower stroke than LQR, confirming its intelligent control ability and practical feasibility.

Figure 12 illustrates for the six ground motions the base acceleration response of the nonlinear base-isolated structure comparing four scenarios: BIS, BIS and TMD, BIS and ATMD driven by the LQR controller, BIS and ATMD trained with the ANN controller. Applying TMD to the BIS is able to reduce the response better for far fault earthquakes than near fault ones. The graph reveals that the response can be further decreased by adopting the ATMD: a close performance between the ANN system and the LQR system in mitigating structural vibration for all the records is observed.

Figures 13, 14 illustrate the time history of base velocity and base displacement, respectively; it is evident that the ANN control system consistently achieves lower peak responses than the LQR control system; the adjustments of the control action in real time make this strategy more effective than applying only a TMD to limit base displacement and velocity. The response reductions occur across a wide range of earthquakes with different intensities and frequencies, which indicates that the ANN effectively adapts to the nonlinear behavior of the structure, providing better damping and stability.

The base shear force is shown in Figure 15 for the cases examined. The reduction observed with the hybrid control with ATMD helps to decrease seismic forces, thereby minimizing stress on structural components and reducing the risk of failure. Figure 15 illustrates how well active control methods are capable of minimizing the base shear across all seismic events and how this performance increases, especially for near fault earthquakes, compared to classical passive TMD. Notably, the ANN control exhibited nearly identical behavior to that of the LQR control, with only a slight increase observed in the maximum peak levels.

Figure 16 presents the maximum drift at all floors. The isolated structure exhibits significant drift for some earthquakes (Chi Chi, and Kocaeli earthquakes), indicating high susceptibility to seismic forces and potential structural damage. Applying a TMD to the base isolation layer can reduce the response to some extent; however, the LQR system shows markedly improved performance. ANN system performs even better, highlighting its superior efficiency in managing seismic responses. This reduced drift implies less structural damage and better protection.

The inter-story drift, depicted in Figure 17, shows that, again, both control methods based on ATMD effectively reduce the responses, especially at the lower floors, where the inter-story drift values are significantly minimized. While the LQR system shows superior performance, the ANN system also achieves notable reductions, closely miming the behavior of the LQR. This underscores the potential of ANN as a viable control strategy also compared to passive control with TMD.

Figure 18 illustrates the normalized restoring force of the BIS versus the normalized base displacement for the control strategies under six different testing earthquakes. Normalization is performed by dividing all the values by the maximum absolute value of the reference BIS responses, namely the restoring force and the base displacement. The addition of an ATMD to the BIS significantly reduces the energy dissipated by the isolator compared to the case where a classical TMD is added. This reduction is evident from the smaller peaks in the restoring force for both LQR and ANN controllers compared to the isolated structure. The decrease in the restoring force indicates lower hysteretic energy dissipation during seismic events, highlighting the ATMD’s effectiveness in minimizing the relative motion between the building and its foundation.

Moreover, the ANN controller exhibits relatively smaller restoring force values, suggesting that it provides a more adaptive and efficient control strategy compared to the LQR controller. The ANN’s ability to learn from data and predict optimal control actions allows it to handle the nonlinear behavior of the base isolation system more effectively.

In order to emulate real-world conditions more accurately in the model, time delays were incorporated into the control loop of the model. Specifically, the structural responses sent to both LQR and ANN-based controllers were delayed by a one-time step. Additionally, the control forces generated by each controller were delayed by a one-time step before being applied via the actuator. This configuration resulted in an overall delay of two time steps within the control loop, effectively emulating the inherent delays present in practical applications.

The introduction of these delays revealed marked differences between the LQR and ANN approaches. When operating with a delayed full-state vector, the LQR encountered significant challenges in producing an effective control force. This difficulty highlighted concerns about the LQR’s robustness and reliability in environments characterized by time delays, as the delayed response compromised its ability to maintain optimal control performance. The inherent limitations of the LQR in handling time delays emphasize the need for alternative control strategies that can better cope with such conditions.

Conversely, the ANN demonstrated an excellent adaptability to time delays, consistently generating effective control forces and swiftly responding to mitigate structural responses. The ANN’s performance remained robust despite the delays, showcasing its ability to adapt and learn from the delayed input data. This adaptability is attributed to the ANN’s inherent capability to model complex, nonlinear relationships and its flexibility in handling dynamic changes within the system. The ANN’s superior performance in the presence of time delays underscores its potential as a more reliable and effective control strategy in scenarios where delays are inevitable. These findings are confirmed in Table 6, which presents the root mean square (RMS) displacement results.

These findings underscore the potential advantages of employing ANN control strategies in scenarios where time delays are a critical factor. The adaptability and resilience of the ANN in the face of delays suggest that it can maintain high levels of control performance due to its natural ability to detect and react to nonlinearities, offering a significant improvement over traditional LQR methods. Therefore, incorporating ANN-based controllers in systems where time delays are prevalent could lead to more robust and efficient control solutions, ultimately enhancing the overall stability and performance of nonlinear systems.

Accurate measurement of dynamic response parameters such as base peak displacement, acceleration, velocity, inter-story drift, maximum drift, and base shear is critical for effective structural control. However, in real-world applications, sensor data is often contaminated with noise, which can compromise the accuracy and reliability of the measurements.

To address this challenge, this study investigates the impact of noise contamination on the performance of both the ANN controller and the LQR controller. A white noise disturbance (Chen and Xu, 2008; Abdeddaim et al., 2017) is randomly generated and then integrated to obtain velocity and displacement signals. The obtained signals are summed with the displacement and velocity data during both the LQR simulation and the testing phase of the ANN to simulate noise contamination in sensors. The white noise peak acceleration is set to be 30 times less than the PGA of the earthquake.

The noise-affected responses were fed into both the LQR and ANN control systems within the feedback loop. Table 7 presents the base floor root mean square (RMS) under both LQR and ANN controllers with and without noise contamination for all the used records.

After introducing noise, a slight increase in the RMS displacement response was observed for both the LQR and ANN control methods. However, it was found that an effective solution to mitigate this increase is to adjust the control force. The robustness of the ANN controller was evaluated based on its ability to maintain control performance despite the presence of noise. Finally, this consistent effectiveness across different responses underscores ANN’s potential to enhance structural resilience against seismic impacts.