Machine learning evaluation of PI control effects on neutral equilibrium in bridge virtual pier systems

Abstract

This study investigates the application of Neutral Equilibrium Mechanism (NEM) in active control systems for bridge structures, with a focus on analyzing the effects of proportional gain (GP) and integral gain (GI) parameters on vertical displacement stability. A scaled bridge model equipped with dual NEMs, displacement sensors, and servo motors was used to simulate dynamic loading responses in a closed-loop control system. Machine learning techniques, including Random Forest Regression and Neural Networks, were employed to develop nonlinear predictive models. These were supplemented by K-means clustering and feature sensitivity analysis to evaluate control strategies and identify optimal parameter settings. The experiment collected over 21.3 million high-resolution time-series data points across four PI control parameter combinations. Results demonstrated that the optimal parameter configuration (GP = 1.0, GI = 0.010) significantly reduced maximum vertical displacement from 5.02 mm and 5.23 mm (at points A and B) to 0.39 mm and 0.38 mm, respectively, while cutting stabilization time to 9.8 s. The Neural Network model achieved excellent predictive performance with an R² of 0.934 and RMSE of 0.038. Clustering and sensitivity analyses revealed that medium-gain settings (GP = 1.0, GI = 0.010) optimally balanced system stability and structural symmetry. This research confirms the feasibility of machine learning-based analytical models for bridge displacement control and provides data-driven guidance for parameter optimization, offering valuable insights for future intelligent bridge control system design.

Introduction

Bridges represent one of the most critical infrastructures in modern transportation systems, with their structural safety and seismic performance directly impacting public safety and regional economic stability. As traffic volumes continue to rise and extreme weather events become increasingly frequent, the dynamic loads acting on bridges—such as vehicular impacts, earthquakes, and strong winds—are growing in complexity and variability. Traditional passive structural designs are becoming insufficient to cope with rapidly changing external disturbances. In response, active control technologies have emerged as a prominent research direction in civil engineering, aiming to enhance the adaptability and stability of bridges under extreme environmental conditions.

Among various control strategies, the Neutral Equilibrium Mechanism (NEM), proposed by Shih and Sung in 2019^1,2, has recently garnered attention due to its low energy consumption, high output force, and geometric stability. NEM functions as a simulated virtual pier to improve bridge dynamic stability, utilizing controllable rotating arms to generate directional reaction forces that emulate structural support. This mechanism effectively regulates bridge deformations and suppresses oscillations^3,4,5,6,7. However, the effectiveness of NEM heavily depends on the proper configuration of its controller parameters, particularly the proportional gain (GP) and integral gain (GI). While Sung and Shih⁸ previously explored the influence of different gain combinations on displacement control from a statistical perspective, there remains a lack of systematic analysis regarding the dynamic behavior of the displacement system, the synchronization of multiple NEMs, and their interactive effects.

To address this research gap, the present study introduces a novel concept and framework leveraging machine learning techniques to explore the optimal control parameter configurations when multiple NEMs are deployed to simulate multiple virtual piers on a bridge.

While previous studies have primarily focused on traditional active control methods for civil structures—such as hydraulic actuators, dampers, and magnetorheological systems—these approaches have largely relied on classical control algorithms like PID^9,10,11 or Linear Quadratic Regulators (LQR)^12,13,14,15. Although such strategies are well established in structural engineering, the integration of machine learning for parameter optimization remains relatively underexplored. Building upon the foundational work of Shih and Sung^1,2 on NEMs, this study advances the field by incorporating neural networks and clustering techniques to enable data-driven optimization of control parameters. In terms of applying the concepts of virtual support and geometric control forces to bridges, recent efforts have investigated the use of steel tendons and rotational mechanisms to create adjustable reaction forces^16,17. However, these works mainly emphasize mechanical simulation and geometric analysis, with experimental validation of physical devices and evaluation of control parameter sensitivity still in early exploratory stages.

On the other hand, machine learning techniques are increasingly being applied in structural engineering, particularly in areas such as Structural Health Monitoring (SHM), damage identification, and optimization of control strategies^18,19,20. Notably, in modeling high-dimensional, nonlinear, and time-series data, approaches such as Artificial Neural Networks (ANN), Random Forests (RF), and Reinforcement Learning (RL) have demonstrated strong performance^21,22,23,24. Nevertheless, the application of machine learning to PI parameter tuning and performance prediction in bridge active control systems remains largely unexplored, underscoring the urgent need for data-driven parameter optimization and decision-making models.

In light of the above, this study proposes an integrated framework that combines NEM-based bridge control systems with machine learning methodologies. Through physical experimentation and data-driven model analysis, the study aims to achieve the following objectives:

• 1. To analyze the influence of varying proportional (GP) and integral (GI) control gains on the vertical displacement stability of bridges;

• 2. To construct predictive models of bridge dynamic displacement using neural networks and random forest algorithms;

• 3. To implement K-means clustering and feature sensitivity analysis for classification of control strategies and identification of optimal parameters;

• 4. To develop a parameter recommendation and control performance evaluation system with potential for real-time applications, providing guidance for future smart bridge control system design.

This research is expected to offer an evidence-based decision support framework by integrating structural mechanics, control theory, and artificial intelligence. Furthermore, it seeks to enhance the practical implementation of NEM-based control systems in the field of bridge engineering.

Theoretical analysis and experimental method of dual virtual piers

This study investigates the performance of the NEMs under various control parameters using a scaled bridge model in dynamic experiments. Machine learning techniques are employed to create an analytical framework for evaluating control effectiveness. By analyzing the vertical displacement responses of the bridge with different PI controller settings, the study aims to validate the interactive effects of parameter combinations and the use of dual NEM units on dynamic control.

Operating principle of the neutralequilibrium control mechanism

The NEM consists of high-tensile steel tendons anchored at both ends of a simply supported bridge, connected by a rotatable arm mechanism at specific locations. This forms a spatial three-force equilibrium structure, shown in Fig. 1. The arm’s rotation axis is parallel to the bridge’s longitudinal axis and aligned with the tendon anchor points for stable force transmission. Adjusting the rotating arm changes the vertical force exerted by the tendons, providing actively controllable vertical force. According to static equilibrium theory, the pre-tensioned tendons generate a resultant force (R) at the rotating arm’s location. When the arm is perpendicular to the bridge deck, R becomes vertically upward, simulating a virtual support (virtual pier). This support resists structural deformation and counteracts displacements caused by external loads, enhancing the overall stability and disturbance resistance of the bridge system.

Fig. 1

Experimental setup of dual NEMs formed as virtual piers of scaled bridge.

Mathematical analysis model of the control force in the dual neutral equilibrium mechanisms (NEMs)

This section establishes a mathematical framework for analyzing the control force generated by the Dual NEM system installed symmetrically on a bridge (Fig. 1). Focusing on one NEM unit, the model considers geometric and mechanical relationships as follows: Let T be the pre-tension in each steel tendon, and θ₁ and θ₂ the angles between the left and right tendons and the horizontal plane. The control mechanism is installed at a horizontal distance of 1/3 Lc from both anchorage points, where Lc is the bridge span. The rotating arm length is l. Assuming symmetry and that the rotating arm is vertically aligned at the center of the installation point, the angles θ₁ and θ₂ can be analytically derived based on the system geometry. These angles are critical in determining the vertical component of the control force generated by the pre-tensioned tendons, can be expressed as follows:

$$\theta_{1} = \tan^{ - 1} \left( {\frac{l}{{\frac{1}{3}L_{c} }}} \right)$$

(1)

$$\theta_{2} = \tan^{ - 1} \left( {\frac{l}{{1/3L_{c} }}} \right)$$

(2)

where l is the length of the rotating arm (i.e., the vertical distance between the tendon anchoring point on the arm and the bridge deck); 1/3Lc is the horizontal distance from the rotating arm to either anchorage point (assuming a symmetric setup and total span Lc).

Given the angles θ₁ and θ₂, the resultant force R_i acting on the rotating arm of the first NEM unit can be determined by resolving the vertical components of the pre-tensioned tendons on both sides. Assuming the tendons have identical pre-tension force T, the vertical components of the forces exerted by the left and right tendons are Tsin θ₁ and Tsin θ₂, respectively. Since the system is symmetric and the rotating arm is centrally positioned, the total vertical resultant force R_i acting upward on the arm can be expressed as:

$$R_{i} = T \cdot \sin \theta_{1} + T \cdot sin\theta_{2} = {2}T \, sin \, \theta$$

(3)

Under a symmetric configuration where θ₁ = θ₂, the vertical control force simplifies to R_i = 2Tsinθ. This upward force acts as a virtual support at the control point, effectively serving as a virtual pier that resists external loads and suppresses vertical displacements, thereby improving dynamic stability and control performance. The rotating arm of each NEM unit is driven by a servo motor, allowing precise rotation within a plane parallel to the bridge’s cross-section (Fig. 2). This setup enables real-time adjustment of the arm’s orientation, thereby dynamically modulating the direction and magnitude of R_i. The result is enhanced adaptability to varying load conditions and improved precision in structural response control.

Fig. 2

Front view of the neutral equilibrium mechanism (NEM).

The rotating arm of each NEM unit is servo-driven, enabling rotation within a plane parallel to the bridge’s cross-section (Fig. 2). The second NEM unit is modeled using the same analytical approach, allowing consistent derivation of control forces.

By adjusting the angle $\emptyset$ between the arm and the bridge deck, the direction of the resultant force changes, altering both vertical and horizontal components. However, due to the symmetric configuration, the horizontal forces cancel out, leaving only the vertical component as the effective control force. This upward force opposes deflections from external loads, and its magnitude is directly governed by the arm’s angle ∅, providing a mechanism for fine-tuned vertical force control.

Assuming the arm of the first NEM unit forms an angle $\emptyset$ with respect to the bridge deck, the net vertical control force u_i generated by this NEM unit can be expressed as:

$$u_{i} = (T \cdot \sin \theta_{1} + T \cdot sin\theta_{2} ) \times sin\emptyset \ldots$$

(4)

In this study, the NEM control strategy adopts a linear control law, wherein the rotational angle of the swing arm is calculated based on the displacement response at specific control points of the bridge structure. This rotational angle is defined as a linear combination of the displacement and the integral of the displacement at the NEM device location, expressed as follows:

$$\emptyset \left( t \right) = GP \cdot x\left( t \right) + GI\mathop \smallint \limits_{0}^{t} x\left( t \right)dt$$

(5)

where GP represents the displacement gain coefficient; GI denotes the integral gain coefficient, also referred to as the adaptive coefficient; x(t) is the displacement function at the structural control point; t denotes time.

This control law enables the swing arm angle to automatically adjust in response to the dynamic behavior of the structure. Consequently, it generates a control force that effectively counterbalances external excitations, thereby enhancing the stability and seismic resilience of the bridge.

Experimental setup and parameters

To evaluate the feasibility and control effectiveness of the Neutral Equilibrium Mechanism (NEM) in dynamic bridge systems, a series of scaled physical experiments were conducted, incorporating machine learning for data analysis and performance evaluation. Due to the complexity introduced by dual NEM systems—including control logic, signal acquisition, and real-time feedback—a scaled bridge model was adopted to enable practical testing and comprehensive data collection.

The test model is a simply supported bridge with a 1000 mm span, fabricated using a polypropylene (PP) deck (density: 0.895–0.92 g/cm³; Young’s modulus: 1.3–1.8 GPa) with dimensions of 1100 × 120 × 15 mm. Two NEM systems were installed at the 1/3 and 2/3 span positions. Each NEM consists of two rotary cantilever arms mounted on aluminum bases, actuated by digital RC servo motors (maximum torque: 250 N·cm). Pre-stressing forces were applied through polyethylene #30 tendons (tensile strength: 250 N), anchored with 120 × 30 × 20 mm aluminum blocks.

Key material and equipment specifications are listed in Table 1, with the overall experimental configuration illustrated in Fig. 3.

Table 1 Specifications of materials and equipment for the scaled bridge model experiment.

Fig. 3

Locations of NEM devices at sensor points A and B for dynamic displacement data acquisition.

Experimental method

The bridge span used in testing measured 1020 mm, with four carbon fiber tendons (tensile strength: 500 N each) simulating prestressing. A 196.25 N steel block was applied to reproduce equivalent static loading. Each NEM system employed two 70 mm rotary arms, driven by servo motors (maximum torque: 200 N·cm; control accuracy: ± 1.2°). Sensing Devices as follows:

• 1. Displacement measurement Resistive-type displacement sensors (potentiometers) were installed at the cantilever connection points to capture bridge deck displacements under dynamic loads.

• 2. Angle measurement Two MPU-9250 accelerometers monitored the rotation of the cantilever arms.

• 3. Data recording All signals were sampled by an Arduino DUE microcontroller (MCU) and logged via an SD module. Calibration involved zeroing displacement and angle readings, with repeated baseline checks to minimize drift and noise.

Control architecture

The control system was structured around the Arduino DUE MCU, which integrated signal acquisition, control computation, and servo actuation:

• 1. RC Servo Motors were controlled via Pulse Width Modulation (PWM) signals, providing precise angle adjustments (1.0 ms pulse for 0°, 2.0 ms for 90°).

• 2. The MCU logic processed real-time input from displacement and angle sensors, adjusted the rotation of cantilever arms, and output corresponding PWM commands.

• 3. The system integration ensured synchronized control of the dual NEM systems while simultaneously recording displacement and rotational responses.

Experimental reliability

To ensure repeatability and accuracy of the experimental results, each test was conducted multiple times under identical loading conditions. Signal noise was carefully minimized through a combination of hardware and software filtering techniques. Prior to each run, the servo system was calibrated to verify its proper functioning and maintain measurement precision. Additionally, environmental conditions were strictly controlled throughout the experiments to avoid any external variability that could influence the results. These measures collectively ensured the reliability and consistency of the collected data.

Machine learning methodology

This study employs machine learning techniques to predict and analyze the dynamic displacement behavior of bridge structures equipped with dual NEMs under varying proportional (GP) and integral (GI) gain parameters. The primary objective is to develop an optimized controller that enhances overall system stability and dynamic response. A nonlinear predictive model was developed using the Neural Designer AI platform²⁵ to conduct parameter sensitivity analysis and control strategy optimization. The methodology includes: (1) simulating the bridge’s dynamic response under different GP and GI combinations; (2) constructing a neural network model to predict displacement at key sensor locations; and (3) evaluating control performance and identifying optimal parameter sets via model-based analysis.

Experimental data collection

Experimental data were obtained from dynamic control tests conducted under constant loading of 24.51 N and a pulley movement speed of 5.963 cm/s. Data acquisition spanned 266.338 s with a high-resolution sampling interval of 0.001 s. The bridge model, shown in Fig. 3, was equipped with two displacement sensors. By applying four pairs of control gain coefficients (A0–B0, A1–B1, A2–B2, A3–B3) to the two sensors, a total of eight dynamic displacement response datasets were obtained. Displacement values (in mm) served as output features, while the corresponding GP and GI values were used as input features. Four distinct PI parameter sets were tested, and for each set, displacements were recorded at two sensor points over time. Each experiment yielded 266,338 time points, resulting in a comprehensive dataset: 4 experiments × two sets of displacement data × 266,338 samples = 21,330,704 entries. This extensive and high-precision dataset provided a robust foundation for training and validating the neural network model, enabling accurate predictions of bridge dynamics and facilitating the identification of optimal PI control parameters for enhanced precision and performance.

Machine learning model development process

The predictive model was developed using an Artificial Neural Network (ANN) architecture, which is well-suited for modeling nonlinear dynamic systems^21,22,23,24. In this study, Neural Designer was employed to construct the ANN-based predictive model. Particularly effective for time-series analysis and nonlinear system modeling, Neural Designer allowed for accurate representation of the complex dynamic interactions between the NEMs and bridge displacement responses. The model development process was systematically organized into four key phases, as outlined follows.

Data preprocessing

The preprocessing of experimental data in this study consisted of two key components:

• 1. Time-series data structuring Time-displacement pairings were established by synchronizing each time point (t) with the corresponding sensor-measured displacement values. This organization ensured the integrity of the temporal sequence and enabled accurate tracking of dynamic system responses.

• 2. Normalization To mitigate the influence of differing feature magnitudes and enhance model learning efficiency, all input and output variables were standardized using Z-score normalization. The transformation was applied using the formula:

  $$x^{*} = \frac{x - \mu }{\sigma }$$

  (6)

  where x represents the original data, μ is the mean, σ is the standard deviation, and x^∗ denotes the normalized value.

Feature engineering

The model’s input and output features, expressed in Eq. (7), were defined as follows:

• (1) Input features Proportional Gain (GP), Integral Gain (GI) and Time variable (t)

• (2) Output features Displacement values at each sensor location on the bridge structure

This feature set enables the ANN to learn the relationship between control parameters and the resulting dynamic displacements, facilitating accurate prediction and control optimization within the NEM-integrated bridge system.

$$D_{out} = \left\{ {D_{A0} ,D_{B0} , \, D_{A1} ,D_{B1} , \, D_{A2} ,D_{B2} , \, D_{A3} ,D_{B3} } \right\}$$

(7)

where each D_Ai and D_Bi denotes the displacement time series at points A and B, respectively, under the i-th control parameter set.

Data splitting

In accordance with standard machine learning practices, the complete dataset was partitioned into three subsets to facilitate model development and evaluation: (1) Training Set: 60% of the data, used to train the model by adjusting internal parameters. (2) Validation Set: 20% of the data, used to tune hyperparameters and monitor overfitting during training. (3) Testing Set: 20% of the data, reserved for evaluating the model’s generalization performance on unseen data. This data splitting strategy ensures that the model achieves effective learning while maintaining robust generalization capability, thereby reducing the risk of overfitting.

Model configuration and training

The architecture adopted in this study is a Feedforward Neural Network (FFNN), which incorporates one or more hidden layers to capture complex nonlinear relationships inherent in the system dynamics. The Rectified Linear Unit (ReLU) function was selected as the activation function, defined as:

$$f\left( x \right) = {\text{max}}\left( {0,x} \right)$$

(8)

To guide the model’s learning process, the Mean Squared Error (MSE) was used as the loss function:

$$MSE = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left( {y_{i} - \hat{y}_{i} } \right)^{2}$$

(9)

where y_i denotes the actual observed value, $\hat{y}_{i}$ represents the model-predicted value, and n is the number of samples.

Model training was carried out using the Adam (Adaptive Moment Estimation) optimizer, which integrates both momentum and adaptive learning rate mechanisms. Adam is particularly advantageous for high-dimensional and nonlinear optimization problems due to its fast convergence and stable performance. To further mitigate the risk of overfitting, an Early Stopping Criterion was implemented during the training process. Specifically, if the validation loss fails to exhibit significant improvement over a predefined number of consecutive iterations, the training is halted automatically. The model checkpoint corresponding to the lowest validation loss is then selected as the final model. This model selection strategy enhances the network’s generalization ability and ensures that the resulting model maintains high predictive stability and accuracy when applied to previously unseen data.

Model evaluation

To verify the accuracy and robustness of the neural network model developed in this study for predicting dynamic bridge displacements, a systematic model evaluation was conducted after training completion. The evaluation employed three key performance metrics: Mean Squared Error (MSE), Coefficient of Determination (R²), and Residual Analysis, described as follows:

• (1) Mean squared error (MSE) previously defined in Eq. (9), quantifies the average squared difference between predicted and actual values. It serves as a fundamental indicator of the model’s prediction accuracy.

• (2) Coefficient of determination (R²) The Coefficient of Determination, denoted as R², measures the proportion of the variance in the target variable that is explained by the model. It evaluates the goodness-of-fit between the predicted and actual outputs. The formula is given as:

  $$R^{2} = 1 - \frac{{\mathop \sum \nolimits_{i = 1}^{n} (y_{i} - \widehat{{y_{i} }})^{2} }}{{\mathop \sum \nolimits_{i = 1}^{n} (y_{i} - \overline{y})^{2} }}$$

  (10)

  where $\overline{y}$ is the mean of the observed output values. An R² value approaching 1 indicates strong explanatory power of the model with respect to the data variance, whereas a value closer to 0 suggests limited predictive effectiveness.

• (3) Residual analysis: Residuals (ε_i) are defined as the difference between the observed and predicted values:

  $$\varepsilon_{i} = y_{i} - \hat{y}_{i}$$

  (11)

Residual analysis is performed to assess the distributional characteristics of these residuals, such as normality and time-based patterns. This helps identify potential model deficiencies, including bias or heteroscedasticity (non-constant variance). A random and homoscedastic residual distribution indicates that the model is well-calibrated and free from systematic errors.

Model application and performance evaluation

Once the neural network model has been successfully trained, it can be employed to predict the dynamic displacement response $\hat{D}_{i} \left( t \right)$ of the bridge system at various sensor locations, given any combination of PI control parameters (GP, GI) and time point t. This predictive capability enables both the evaluation of control strategies and the optimization of parameter settings. The model facilitates an in-depth analysis of the impact of different control parameters on system stability, thereby supporting inverse identification of optimal control configurations to minimize displacement deviations, oscillations, and ultimately enhance control performance.

Inverse parameter identification strategy

To identify the optimal PI control parameter combination, this study utilizes the Optimization module embedded in Neural Designer. The objective is to minimize the steady-state displacement errors across all sensor points. The objective function is formulated as:

$${\text{Objective}} = \mathop \sum \limits_{i = 1}^{8} \left| {\hat{D}_{i} \left( {t = 1.2s} \right)} \right|$$

(12)

where $\hat{D}_{i}$ denotes the predicted displacement at sensor point i at the stabilization time t = 1.2 s.

The time t = 1.2 s was chosen as the stabilization time based on empirical data, which indicated that the system reached steady-state behavior within this period under optimal control conditions. The goal is to minimize the absolute values of these displacements to approach zero. By leveraging built-in optimization algorithms—such as evolutionary search or gradient-based methods—the model can systematically explore the parameter space and identify the optimal GP and GI values for precise and efficient control.

Control performance metrics

To quantify the effectiveness of various control parameter combinations, the following four performance metrics are defined:

• (1) Steady-state error, $E_{ss}^{i}$: Measures the absolute displacement at the stabilization point t = 1.2 s:

  $$E_{ss}^{i} = \left| {D_{i} \left( {t = 1.2s} \right)} \right|, i = 1, \ldots ,8$$

  (13)

• (2) Maximum overshoot, $M_{p}^{i}$: Represents the peak displacement magnitude over time:

  $$M_{p}^{i} = max_{t} \left| {D_{i} \left( t \right)} \right|$$

  (14)

• (3) Settling time, $t_{settle}^{i}$ Defined as the shortest time at which the system output enters and remains within a ± 0.5 mm tolerance band:

  $$t_{settle}^{i} = min\left\{ {\left. t \right|\forall_{\tau } > t, \left| {D_{i} \left( \tau \right)} \right| < 0.5} \right\}$$

  (15)

• (4) Integral of absolute error, IAEⁱ Quantifies the cumulative absolute error over time:

  $$IAE^{i} = \mathop \sum \limits_{j} \left| {D_{i} \left( {t_{i} } \right)} \right| \times \Delta t$$

  (16)

  where t_j denotes the j-th time point and Δt is the sampling interval (0.001 s in this study).

Model evaluation metrics

To rigorously assess the predictive performance of the developed neural network model, two widely accepted statistical metrics were employed:

• (1) Coefficient of determination (R²) The coefficient of determination quantifies the proportion of variance in the observed data that is explained by the model. It serves as an indicator of the correlation between the predicted and actual values and is defined as:

  $$R^{2} = 1 - \frac{{\mathop \sum \nolimits_{i = 1}^{n} \left( {y_{i} - \hat{y}_{i} } \right)^{2} }}{{\mathop \sum \nolimits_{i = 1}^{n} (y_{i} - \overline{y}_{i} )^{2} }}$$

  (17)

  where y_i denotes the actual value, $\hat{y}_{i}$ the predicted value, and $\overline{y}_{i}$ the mean of the observed values. An R² value approaching 1 indicates strong predictive capability, whereas a value near 0 implies poor model fit.

• (2) Root mean squared error (RMSE) The RMSE measures the average magnitude of the prediction error and reflects the overall deviation between the predicted and actual values. It is calculated as:

  $$RMSE = \sqrt {\frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left( {y_{i} - \hat{y}_{i} } \right)^{2} }$$

  (18)

This metric is sensitive to large errors and thus provides insight into the model’s robustness.

In conjunction with residual analysis, these two indicators offer a comprehensive evaluation of the model’s prediction accuracy and stability across different sensor locations and time domains. This facilitates a holistic understanding of the model’s reliability in practical applications.

Machine learning-based analysis and discussion

This study investigates the effectiveness of the Neutral Equilibrium Mechanism (NEM) in controlling vertical displacement at two critical points (A and B) on a model bridge under dynamic loading. Various combinations of proportional (GP) and integral (GI) gains were tested to assess system performance. Utilizing time-series data and machine learning techniques, the analysis focuses on structural stability, dual-point coordination, and prediction accuracy. The evaluation consists of two key phases:

• 1. Model training on raw data—Neural networks were trained using unprocessed displacement time-series data to examine the direct impact of control parameters on dynamic behavior.

• 2. Normalization and feature analysis—Data normalization was applied to improve model generalizability, followed by statistical and sensitivity analyses to compare control group performance.

This dual-phase approach enables a robust interpretation of control dynamics and supports the development of data-driven optimization strategies for NEM-based bridge control systems.

Experimental design and data analysis

Experimental setup and data collection

A dynamic loading experiment was conducted on a simply supported bridge model using a pulley mechanism moving at 5.963 cm/s and applying a constant load of 24.51 N to induce vertical displacement. Four control conditions were tested by varying proportional (GP) and integral (GI) gains: (1) No control: GP = 0.0, GI = 0.000; (2) Low gain: GP = 0.5, GI = 0.005; (3) Moderate gain: GP = 1.0, GI = 0.010; (4) High gain: GP = 2.0, GI = 0.020. Each trial ran for approximately 30 s, with continuous monitoring of vertical displacements at Points A and B. The comprehensive experimental results at measurement points A and B are illustrated in Fig. 4, while the corresponding time-history responses are displayed in Fig. 5a,b. Complete time-series data were collected under each control condition, yielding eight principal variable sets across all experiments for subsequent analysis.

Fig. 4

Experimental results of the bridge system with and without NEM control under different gain combinations.

Fig. 5

Time-history plots of vertical displacement at monitoring Points A and B on the bridge during dynamic testing.

Feature extraction and machine learning models

Four key structural response metrics were extracted from the experimental data to characterize system performance: (1) Peak Displacement, (2) Settling Time, (3) Root Mean Square Error (RMSE), and (4) Steady-State Offset. These features provide a comprehensive assessment of dynamic suppression and stability. Random Forest Regression was selected for modeling due to its robustness and ability to capture nonlinear relationships between control parameters and structural responses. K-means clustering was additionally employed to classify control scenarios and identify performance patterns under varying gain settings. To enhance predictive capability for time-series data, a multilayer perceptron neural network was developed using Neural Designer. This enabled accurate forecasting, sensitivity analysis, and parameter optimization, supporting data-driven improvements in control design.

Model training and prediction results using raw time-history displacement data

Control effectiveness and displacement trend analysis

This experiment analyzed the dynamic displacement responses of the bridge structure under different proportional gain (GP) and integral gain (GI) combinations, evaluating the controller’s impact on system stability and oscillation suppression. The maximum displacement and settling time at Points A and B under each control condition are summarized in Table 2:

Table 2 Summary of maximum displacement and settling time at points A and B under different control conditions.

Evaluation of prediction model accuracy

This study compares the performance of Random Forest Regression and Linear Regression models in the task of predicting bridge displacement control. The evaluation metrics include the Coefficient of Determination (R² Score) and Mean Squared Error (MSE). The analysis revealed that the Random Forest Regression model achieved a higher R² Score of 0.934 and a lower MSE of 2.31, indicating better predictive accuracy. In contrast, the Linear Regression model had a lower R² Score of 0.792 and a higher MSE of 5.44, reflecting comparatively reduced performance. These results demonstrate that the Random Forest Regression model is more effective for this specific prediction task.

K-means clustering analysis results

To further investigate the impact of different proportional gain (GP) and integral gain (GI) combinations on the control performance of bridge vertical displacement, this study employed the K-means clustering algorithm for automatic classification analysis based on system response characteristics, including maximum displacement, settling time, and Root Mean Square Error (RMSE). Using unsupervised learning, the model categorized experimental data under different control conditions into three main clusters. The clustering results are presented in Table 3.

Table 3 Clustering results under different control conditions.

Discussion of model training and prediction analysis results using raw temporal displacement response data

Control effectiveness and displacement trend analysis

Experimental results (Table 2) reveal that varying proportional gain (GP) and integral gain (GI) significantly impact bridge system stability and control effectiveness. Under the uncontrolled condition (GP = 0, GI = 0), the system showed high sensitivity to disturbances, with maximum displacements exceeding 5 mm and no natural decay, indicating poor stability. Group 3 (GP = 1.0, GI = 0.010) demonstrated optimal control, significantly reducing displacement and maintaining settling time under 10 s. This balance of stability and efficiency makes it the most promising configuration. In contrast, Group 4 (GP = 2.0, GI = 0.020) achieved the shortest settling time (7.1 s) and smallest maximum displacement but exhibited overshoot and spikes, suggesting an overly rapid response that could lead to stress and fatigue, making it unsuitable for long-term use. Group 2 (GP = 0.5, GI = 0.005) offered partial stabilization but failed to achieve optimal control, indicating its use for system calibration or low-demand scenarios. While GP = 1.0 and GI = 0.010 yielded the best performance among the tested values, further studies using finer parameter intervals (e.g., GP = 1.2, 1.5, 1.8) are recommended to more accurately determine the global optimum.

Prediction model accuracy evaluation—performance comparison and analysis of model predictions

This study compares the performance of Random Forest Regression and Linear Regression in predicting the bridge control system’s dynamic displacement response. The evaluation results uses the R² Score and Mean Squared Error (MSE), which reveal that the Random Forest model outperforms the Linear Regression model. With an R² score of 0.934, the Random Forest model explains 93.4% of data variance, indicating excellent fitting and stability. In contrast, the Linear Regression model has an R² score of 0.792, demonstrating limited explanatory power and difficulty in capturing complex control behaviors. Additionally, the Random Forest model’s MSE is 2.31, significantly lower than the Linear Regression model’s MSE of 5.44, further confirming its ability to model nonlinear relationships effectively. Overall, Random Forest excels in handling nonlinear dynamics and noise, making it ideal for complex structural control problems. Future bridge control and prediction applications should prioritize nonlinear machine learning models like Random Forest for improved accuracy and stability.

K-means clustering analysis results

Based on the results in Table 3, the characteristics of the clusters are as follows:

• 1. Cluster A (stable control) Corresponding to the control parameter combination GP = 1.0, GI = 0.010, this group exhibits the most balanced performance. Not only is the maximum displacement significantly reduced, but the settling time is also maintained within a reasonable range, with no overshoot observed. This cluster represents the optimal control effectiveness in this study.

• 2. Cluster B (insufficient control) Including the combinations GP = 0.0, GI = 0.000 and GP = 0.5, GI = 0.005, the system under these control conditions still demonstrates significant oscillations. Both maximum displacement and settling time are elevated, indicating insufficient control power and an inability to effectively suppress dynamic disturbances.

• 3. Cluster C (Overcorrection) Corresponding to the control parameter combination GP = 2.0, GI = 0.020, this group shows outstanding performance in terms of settling time and displacement control. However, the time history response curve reveals significant overshoot and rapid correction phenomena, which could lead to localized stress concentrations in the structure and unnecessary energy consumption.

Significance of clustering analysis and control strategy recommendations

K-means clustering was employed to assess bridge control performance under various GP and GI combinations, integrating multiple structural response features such as RMSE, settling time, and peak displacement. This approach provides a more holistic and objective evaluation of control strategies compared with traditional methods based on individual metrics. The clustering results identified GP = 1.0 and GI = 0.010 (Cluster A) as the optimal control combination, achieving a balance between system stability and performance by effectively reducing oscillations while avoiding overshoot—making it ideal for critical balance control strategies.

In contrast, sensitivity analysis was applied to determine which input features most strongly influenced model predictions, highlighting displacements under GP = 1.0 and 2.0 as particularly predictive. Both methods consistently pointed to GP = 1.0 as a robust setting, demonstrating the complementary value of clustering for strategy categorization and sensitivity analysis for feature importance assessment.

The analysis also revealed two extreme control conditions:

• 1. Cluster B (insufficient control) GP = 0.0, GI = 0.000 and GP = 0.5, GI = 0.005, which result in prolonged oscillations and inadequate stability, potentially increasing structural fatigue.

• 2. Cluster C (overcorrection) GP = 2.0, GI = 0.020, which stabilizes quickly but induces overshoot and excessive stresses, making it unsuitable for long-term operation.

These findings provide valuable insights for optimizing control strategies, guiding sensor placement, and avoiding settings that are either ineffective or potentially damaging, highlighting the practical relevance of combining K-means clustering with sensitivity analysis in engineering applications.

Application value and engineering implications of clustering analysis

• 1. Optimizing control parameter selection K-means clustering aids engineers by identifying optimal control parameter combinations efficiently. Cluster A, the “Stable Control Zone,” serves as a baseline for parameter optimization and initial setup in automatic control systems.

• 2. Risk identification and early warning Clustering analysis highlights “insufficient control” and “overcontrol” scenarios, providing a mechanism to avoid system instability and stress concentrations, ensuring safe and efficient system operation.

• 3. Smart control system enhancement The study’s findings can inform the development of a “Control Performance Identification Module” for smart systems. By real-time clustering of system responses, this module can automatically adjust parameters to optimize control dynamically.

In summary, K-means clustering offers a comprehensive approach for stability design, risk management, and the development of intelligent control strategies, ensuring efficient and safe operation of bridge control systems.

Normalized statistics and synchronization analysis

To improve the objectivity and comparability of control performance evaluation, this study applied Min–Max Normalization to the displacement response data at points A and B. This method removes scale discrepancies, allowing for a clearer assessment of how control strategies influence stability and coordination between the points. After normalization, standard deviations (σ) were calculated for the displacements at points A and B to evaluate response stability. Additionally, the Pearson correlation coefficient was used to assess synchronization between the displacements at both points. The results of this analysis are shown in Table 4.

Table 4 Statistical and synchronization analysis of normalized displacement data.

Discussion of normalized statistical and synchronization analysis

As seen in Table 4, increasing the controller gain parameters (GP, GI) significantly reduces the standard deviations of displacement at points A and B, indicating improved system stability. The lowest displacement fluctuation and highest stability were observed at GP = 2.0, corresponding to the lowest standard deviation. However, while local stability improves, the correlation between displacements at points A and B decreased with higher control intensity—from 0.798 in the uncontrolled state to 0.415 at GP = 2.0. This suggests that stronger control may lead to inconsistent responses between control points, compromising structural symmetry and coordination.

Engineering implications and design recommendations

This analysis highlights a critical trade-off: stronger control enhances node stability but may reduce coordination and synchronization. If minimizing local displacement is the goal, a high-gain setting (e.g., GP = 2.0) is suitable. However, for maintaining global structural symmetry, the parameter set GP = 1.0, GI = 0.010 offers a more balanced solution, ensuring stability and synchronization between control points. In conclusion, control parameter design should focus on both displacement minimization and synchronization for more reliable dynamic control strategies for bridges.

Neural network prediction performance and sensitivity analysis

To improve prediction accuracy for displacement variations in the bridge’s dual-point critical equilibrium control system, a Multilayer Perceptron (MLP) neural network was utilized. This model is ideal for capturing nonlinear behavior and performing regression analysis, providing a data-driven approach to understanding the complex dynamics of active structural control. Additionally, a sensitivity analysis was performed to assess how different input variables—across varying control parameters—impact the MLP model’s output predictions. This analysis offers key insights into the influence of specific control parameters on displacement responses, aiding in the development of more effective control strategies. Figure 6 illustrates the architecture of the MLP model used in this study.

Fig. 6

Architecture of the multilayer perceptron (MLP) neural network model used in this study.

Model configuration and performance evaluation

The predictive model employed a four-layer Multilayer Perceptron (MLP) architecture with Rectified Linear Unit (ReLU) activation functions in the hidden layers and a linear activation function in the output layer, as shown in Fig. 6. The Adam optimizer was used during training, with a Mean Squared Error (MSE) loss function and an L2 regularization term to reduce overfitting. The model utilized nine input features, including time sequences and displacement responses under various PI control settings (D_A0 through D_B3). The output was the predicted displacement response under these conditions.

The model exhibited consistently low error values across all datasets, with an MSE of 0.001 and a Root Mean Squared Error (RMSE) of 0.029 on the training set. Both the validation set and the test set also achieved MSE values of 0.001, with RMSEs of 0.038. These results indicate high predictive accuracy and strong generalization ability. The consistently low error across different data partitions confirms the MLP model’s effectiveness in capturing the nonlinear and temporal dynamics of the bridge control system, demonstrating its potential for real-world engineering deployment.

Feature sensitivity analysis

A feature sensitivity analysis was performed to examine how individual input features influence the model’s output, specifically the final displacement at control point B (D_B3). The analysis aimed to identify key control variables and time-dependent features that contribute to control effectiveness, providing a foundation for parameter optimization. The setup for the analysis was as follows: (1)Target Output Variable: D_B3 (final normalized displacement at point B under GP = 2.0 control setting); (2) Input Variables: Nine inputs, including displacement features at points A and B under various control conditions (8 variables) and one time variable; (3) Model Architecture: Four-layer MLP with ReLU and Linear activation functions, trained using Adam optimization with L2 regularization; (4) Sensitivity Metric: Feature contributions were assessed using the Neural Designer sensitivity analysis module, which computes the relative impact of each input variable on the output. Table 5 summarizes the sensitivity scores and their importance rankings.

Table 5 Feature sensitivity results.

Discussion

Key findings from the model configuration, performance evaluation, and feature sensitivity analysis (Fig. 6, Tables 4 and 5) are summarized as follows:

• 1. Influence of late-stage control groups (GP = 1.0 and GP = 2.0) (1) D_A3 is the most influential input feature, suggesting a strong relationship between the dynamic response at Point A and stability at Point B when the control gain is set to GP = 2.0. (2) D_B2 and D_A2 also show significant sensitivity scores, indicating the importance of GP = 1.0 in predicting final displacement outcomes.

• 2. Minimal contribution from low-gain control groups (GP = 0.0 and 0.5) Sensitivity scores for D_A0 and D_B0 are much lower, confirming that data from uncontrolled or weakly controlled conditions provide limited value for predicting system behavior.

• 3. Limited influence of time variable Despite its temporal nature, the time variable’s low sensitivity score suggests that the other features sufficiently capture the system’s dynamics, reducing the need for time as a standalone predictor.

• 4. Practical recommendations (1) Data Collection: Prioritize gathering data under GP = 1.0 and GP = 2.0 for improved model accuracy. (2) Simplification for Real-time Applications: Exclude non-critical features like D_A0, D_B0, and Time to streamline the model and enhance computational efficiency. (3) Focus on D_A3: Given its dominant influence, D_A3 should be incorporated as a core feedback variable in future active control strategies.

Integrated comparison and optimal parameter recommendation

To comprehensively assess the influence of control parameters on system behavior, a systematic comparative analysis was conducted across different control parameter combinations. The summary is presented in Table 6:

Table 6 Comparative performance analysis of control parameter combinations.

Based on the results presented in Table 6, the following key findings are summarized:

• 1. Uncontrolled group (condition Code 0) The system exhibited significant oscillations with maximum displacement standard deviations (σ) of 0.282 (Point A) and 0.287 (Point B). Despite high synchronization (correlation coefficient = 0.798), the overall system stability was poor and classified as “Outlying,” indicating the need for active control to reduce oscillations.

• 2. Low-gain control group (condition code 1) With GP = 0.5 and GI = 0.005, displacement σ decreased to 0.231 (Point A) and 0.226 (Point B), but synchronization dropped to 0.546, and the system was classified as "Warning." While some stability improvement was observed, control effectiveness remained limited.

• 3. Medium-gain control group (condition code 2) With GP = 1.0 and GI = 0.010, maximum displacement σ was reduced to 0.183 (Point A) and 0.215 (Point B), while synchronization decreased to 0.424. The system was classified as "Stable," providing a favorable balance between stability and predictive accuracy, suitable for practical applications.

• 4. High-gain control group (condition code 3) With GP = 2.0 and GI = 0.020, maximum displacement σ was reduced to 0.124 (Point A) and 0.141 (Point B), the lowest across all groups. However, synchronization decreased slightly to 0.415. This configuration demonstrated the best stability but may require further optimization due to reduced synchronization.

Conclusions and recommendations

This study demonstrates the successful integration of the Neutral Equilibrium Mechanism (NEM) with machine learning to develop a data-driven analytical framework for real-time suppression and evaluation of bridge vertical displacements. Through dynamic experiments on a scaled bridge model and the analysis of over 20 million time-series data points, the effectiveness of NEM in structural control is validated. Neural networks and statistical learning techniques are employed to construct accurate predictive models, while parameter clustering and sensitivity analysis are used to identify optimal PI parameters. The findings not only highlight the practicality of the proposed approach for intelligent bridge control systems but also provide evidence-based guidelines for control parameter optimization. Future research will focus on full-scale experimental validation, extension to multi-point control strategies, and the incorporation of adaptive and reinforcement learning methods to further enhance system robustness and applicability.

The main conclusions of this study and recommendations for future research are summarized as follows:

Conclusions

• 1. Feasibility of NEM in engineering applications The Neutral Equilibrium Mechanism (NEM) was validated through dynamic experiments on a scaled bridge model, demonstrating its ability to provide stable, low-energy vertical control forces. This improvement enhances the structural stability and seismic performance of the bridge, demonstrating strong potential for practical engineering applications.

• 2. Strong Predictive Performance of Machine Learning Models Nonlinear machine learning models, including feedforward neural networks and random forest regression, achieved high accuracy (R² = 0.934, RMSE = 0.038), confirming the reliability of data-driven models in capturing complex dynamics of bridge control systems for intelligent control platforms.

• 3. Optimal control with medium-gain parameters The control setting GP = 1.0 and GI = 0.010 achieved the best balance, reducing displacement to 0.39 mm at point A and 0.38 mm at point B, with faster stabilization and preserved symmetry. This combination is recommended as the standard for future applications.

• 4. Clustering and sensitivity analysis for control optimization K-means clustering and sensitivity analysis identified key parameters for control optimization. High-gain settings (GP = 2.0) provided optimal stability but affected synchronization. Key input features like D_A3 and D_B2 are critical for model simplification and real-time optimization.

Finally, while this study validates the methodology at laboratory scale, the principles are designed for scalability. To explicitly address practical relevance, a full-scale bridge model has been constructed and experimental validation is underway. This critical next step will demonstrate the transition of the NEM-based machine learning control system from a conceptual framework to a field-ready technology.

Recommendations

• 1. Integrate machine learning in bridge control systems and validate with full-scale testing Machine learning should be incorporated into the selection of control parameters for smart bridge systems, optimizing the trade-off between stability and synchronization. As a direct continuation of this work, we are currently conducting experimental validation on a full-scale bridge model. This step is crucial for transitioning the proposed methodology from laboratory proof-of-concept to field-ready technology. The results from these large-scale tests will provide definitive evidence of the system’s practicality and serve as a benchmark for enhancing both new and existing bridges.

• 2. Develop synchronization compensation and adaptive control To mitigate synchronization issues in high-gain control, adaptive feedback and collaborative adjustment mechanisms should be developed to improve overall system coordination and stability.

• 3. Broaden application and explore reinforcement learning Future research should expand to multi-point control networks and various bridge types, incorporating techniques like Deep Reinforcement Learning (DRL) to enhance the adaptability and robustness of dynamic stability strategies for bridges.