Research on intelligent semi-active control algorithms and seismic reliability based on machine learning

Abstract

Aiming to address the shortcomings of existing semi-active control algorithms with poor robustness and the limited generalization ability of current evaluation methods based on deterministic analysis, a novel approach based on probability density evolution is proposed. This method is designed to assess the seismic reliability, enabling a more comprehensive evaluation of the control effectiveness of aqueduct structures. Building upon this, an intelligent semi-active control algorithm leveraging machine learning is introduced. The algorithm is further validated through engineering case studies to investigate semi-active control strategies in response to random seismic events. The results show that the seismic reliability of the machine learning-based semi-active control algorithm is significantly higher than that of the uncontrolled state for the same failure threshold under random seismic effects.

Introduction

The aqueduct, a vital component of water conservancy projects, plays a pivotal role in ensuring the ongoing stability of water transfer operations. The expansion of water diversion and transfer projects into China’s central and western regions has heightened concerns over the seismic safety of aqueducts, drawing the interest of the research community. Take, for instance, the recently constructed Dianzhong water diversion project, which features an aqueduct traversing a highly seismic area. Earthquake-induced damage to the aqueduct poses a risk of disrupting the entire water supply system. Furthermore, the consequences of such damage at the project site can be particularly grave.

The premise and foundation of structural vibration control is the control algorithm. Therefore, an essential aspect of structural vibration control is to apply the actual control force based on the characteristics of external disturbances through predefined control algorithms to ensure that the structural response is maintained within a safe range. Optimal control algorithms have been widely used in civil structure control. However, the optimal control problem of forced vibration of the aqueduct structure caused by seismic dynamic loading has not yet established effective optimal control measures and methods to meet the extreme value condition. Some scholars have conducted research on this issue using classical linear optimal control (COC), instantaneous optimal control (IOC), and linear quadratic Gaussian control (LQG), among others. However, the aforementioned algorithms still suffer from significant system parameter perturbations and are overly sensitive to external disturbances. Developing and applying strong and robust structural vibration control algorithms is of great importance. In recent years, the rapid development of machine learning has greatly expanded its application possibilities in structural engineering. These applications include design optimization, structural system identification, structural assessment and monitoring, structural control, and finite element mesh generation. Therefore, it is particularly important to develop a semi-active control algorithm based on machine learning and explore the effect of its application to vibration control of aqueduct structures.

Zhou H et al.¹ a control algorithm, comprising a small number of parameters, is proposed using a machine learning framework to identify the most crucial feedback measurements for semi-active control and to derive the optimal policy from the data. Qing W et al.² propose that in order to mitigate the detrimental effects of surface motion on building structures, a semi-active non-smooth control algorithm based on deep learning can be applied to the vibration control of building structures. The results show that the designed control algorithm has strong robustness and anti-jamming properties.

Shao-Yong Xu et al.³ proposed a machine learning method based on fuzzy optimal control and the actual model of the vehicle to improve the smoothness and control performance of the vehicle’s semi-active air suspension on different road surfaces. The control objectives were defined as the weighted root-mean-square (RMS) values of acceleration for the driver’s seat and the pitch angle of the vehicle.

In recent years, the theoretical framework for the evolution of probability density has emerged as a powerful method for addressing reliability challenges in engineering structures. The method of probability density evolution offers a cohesive approach to tackling the initial conditions within stochastic systems, the variability in structural parameters, and the inherent randomness of external forces. It provides a scientific and comprehensive description of the stochastic system’s internal physical mechanisms, particularly those pertaining to randomness. Conducting a seismic reliability study on the aqueduct structure is of significant engineering importance for an objective assessment of its seismic performance and for minimizing potential earthquake damage.

Chen Z et al.⁴ first conducted shaker tests to verify the numerical model. Subsequently, they performed stochastic seismic analysis of the model using the Probability Density Evolution Method (PDEM). It was found that the stochastic analysis method can further obtain the probability of each value of the seismic performance evaluation index, thus making the evaluation results more comprehensive.

Zhang C et al.⁵ established a random parameter structural model, a random ground shaking model, and a parameter-excitation composite stochastic model based on the damage mechanics of the concrete continuum and the probability density evolution theory. They conducted seismic response analysis and seismic reliability analysis of random parameter structures under random ground shaking. Huiquan M et al.⁶ proposed a new strategy for assessing the seismic suitability of water supply networks (WSNs) by considering their operational physical mechanisms. Li H et al.⁷ explored and demonstrated the favorable and unfavorable effects of seismic isolation on high-speed railroad bridges from a stochastic perspective. They compared the stochastic seismic response of non-isolated bridges with that of isolated bridges with varying parameters under the influence of random ground shaking. Jie Li⁸ utilized a generalized density evolution equation to solve the probability distribution of the annual maximum typhoon surface wind speed by introducing the probability density evolution theory. Miao Huiquan⁹ developed a stochastic seismic response analysis of buried pipeline networks with nonlinear characteristics based on the physical model of ground vibration random field at the engineering site. This was achieved by introducing a seismic response analysis model for buried pipeline networks and integrating it with the probability density evolution method. This work laid the groundwork for studying the reliability of the seismic function of water supply pipeline networks. Zhang et al.¹⁰ improved the traditional damping control scheme and proposed a semi-active control scheme considering fault tolerance. Yan Long et al.¹¹ conducted a comparative analysis of the seismic response and damping rate of bridges under various control schemes. They also analyzed the seismic reliability of the structure under random seismic excitation with different control schemes and threshold conditions. Zhang Wei et al.¹² developed a framework for random dynamic response analysis and reliability assessment of large-scale aqueduct structures.

From the current state of research summarized above, it is evident that there is a significant lack of studies on intelligent semi-active control algorithm analysis for aqueduct structures and the verification of their seismic reliability, Considering the strong randomness of ground motion, the method of using a specific deterministic natural or artificial seismic wave for deterministic analysis of aqueduct structures has its limitations and flaws. This paper introduces an intelligent semi-active control algorithm based on machine learning, integrating the evolution of probability density under random earthquakes to study the seismic reliability of semi-active control for aqueduct structures, and provides a generalized assessment, with the aim of offering a novel perspective for the research on seismic vibration control and seismic reliability of aqueduct structures.

Vibration control algorithms

Establishment and validation of the dynamic analysis model of the aqueduct channel

Modeling of the dynamic analysis of the aqueduct

The aqueduct’s main body is a thin-walled beam structure partitioned longitudinally¹³. The cross-section of the aqueduct and its finite element model are shown in Figs. 1 and 2, respectively. The element nodes are located at the transverse bracing, vertical stiffeners, and supports on the aqueduct body to measure vibration displacements.

Fig. 1

Cross-sectional view of the aqueduct.

Fig. 2

Finite element model of the aqueduct.

The large aqueduct structure is discretized along its span using dynamic thin-walled beam segment elements (BSE), with each BSE node located at the aqueduct’s transverse bracing, supports, and vertical stiffeners. Thin-walled cross-sectional properties: Torsion center at ‘s’; Centroid at ‘c’; The coordinate origin ‘c’ is defined using the right-hand coordinate system, with the longitudinal origin of the element at the left end of the aqueduct body. The aqueduct’s supports are modeled using spatial beam elements to accurately represent their structural behavior in three-dimensional space. The designed compressive strength of the concrete is C40. The base of the bridge pier is rigidly connected to the ground, while the upper part of the pier is simply supported on the main beam.

Vibration table model validation

In order to verify the feasibility of the aqueduct model proposed in this paper, shaking table experiments are carried out, and the aqueduct model is taken as 1/16 of the model in “Engineering examples” section. The shaker experimental aqueduct model is shown in Fig. 3.

Fig. 3

Vibration table model.

To validate the reliability of the finite element model, the aqueduct scaled test model was constructed using the aforementioned method, simulating the lateral displacement response of the aqueduct under El-Centro waves at 0.2 g and 0.4 g, Fig. 4 displays the mid-span displacement of the aqueduct as obtained from both finite element analysis and testing. The test results closely match the numerical results, with consistent overall trends and peak values within the allowable error range. This indicates that the numerical model used in this study has sufficient reliability.

Fig. 4

The mid-span displacement of the aqueduct.

LQR optimal control algorithm

The sample data in this paper is selected from the LQR optimal control algorithm, and “r” controllers are added to the structure with “n” degrees of freedom. The equations of motion governing the semi-active control system of the structure under the consistent input of ground vibration are as follows:

$$\begin{gathered} M\ddot {X}(t)+C\dot {X}(t)+KX(t)={B_s}U(t)+M{H_0}{{\ddot {X}}_g}(t) \hfill \\ X({t_0})={X_0} \hfill \\ \dot {X}({t_0})={{\dot {X}}_0} \hfill \\ \end{gathered}$$

(1)

In formula, M, C and K are the mass, damping, and stiffness matrices of the aqueduct structure in n × n dimensions, respectively. X(t), $\dot {X}(t)$, $\ddot {X}(t)$ are the n-dimensional displacement, velocity, and acceleration vectors of the aqueduct structure, respectively.${\ddot {X}_g}(t)$ is the time scale of ground shaking acceleration. U(t) is the r × 1 dimensional control force vector. H₀ is the n × 1 dimensional ground shaking acceleration position matrix. B_s is the n × r dimensional semiactive control damper position matrix.

Introducing state vectors $Z=\left[ {\begin{array}{*{20}{c}} X \\ {\dot {X}} \end{array}} \right]$, The equation of motion of the structure can be expressed as:

$$\begin{gathered} \dot {Z}(t)=AZ(t)+BU(t)+H{{\dot {X}}_g}(t) \hfill \\ Z({t_0})={Z_0} \hfill \\ \end{gathered}$$

(2)

Where:

$$A=\left[ {\begin{array}{*{20}{c}} 0&I \\ { - {M^{ - 1}}K}&{ - {M^{ - 1}}C} \end{array}} \right]$$

(3)

$$B=\left[ {\begin{array}{*{20}{c}} 0 \\ {{M^{ - 1}}{B_s}} \end{array}} \right]$$

(4)

$$H=\left[ {\begin{array}{*{20}{c}} 0 \\ {{M^{ - 1}}{H_0}} \end{array}} \right]$$

(5)

The Linear Quadratic Regulator (LQR) classical optimal control algorithm with full state feedback^14,15 is utilized for active control calculations. The performance objective function of the system is defined as:

$$J=\int\limits_{0}^{\infty } {[{Z^T}QZ+{U^T}RU]dt}$$

(6)

Where, Q and R are the weight matrices, and the performance objective function represents the concept of energy in physical terms. By minimizing the performance objective function, the optimal control force vector can be found as:

$$U(t)= - GZ(t)$$

(7)

In formula:

$${\text{G=}}{{\text{R}}^{{\text{-1}}}}{{\text{B}}^{\text{T}}}{\text{P}}$$

(8)

R is the r × 2n dimensional state feedback gain matrix.

P is the 2n × 2n dimensional matrix, Solved by the Riccati^15,16 number Eq. (7).

$$- PA - {A^T}P+PB{R^{ - 1}}{B^T}P - Q=0$$

(9)

The state equation of the control system is as follows:

$$\begin{gathered} \dot {Z}(t)=(A - BG)Z(t)+H{{\ddot {X}}_g}(t) \hfill \\ Z({t_0})={Z_0} \hfill \\ \end{gathered}$$

(10)

The selection of the weight matrices Q and R significantly impacts the response of the control structure and the magnitude of the control force. Determining the optimal forms and sizes of Q and R to achieve the globally optimal control force remains a challenging task. Therefore, trial calculations are typically employed during the design process of the optimal control force. The forms and sizes of Q and R are continuously adjusted to attain the optimal active control force that combines the control effect and control force effectively. This process further establishes a foundation for the design of semi-active control systems. This provides a basis for the design of semi-active control.

BP neural network

The Back Propagation (BP) neural network is a widely used model of artificial neural networks, particularly effective for solving complex nonlinear mapping and pattern recognition problems.Known for its excellent nonlinear modeling capabilities and strong learning ability, the BP neural network is extensively applied in various machine learning tasks such as classification, regression, and clustering, especially for problems where traditional methods fail to directly decipher the mathematical relationship between inputs and outputs, the BP neural network provides an effective solution.

The learning process of a Back Propagation (BP) neural network can be summarized in the following steps:

• 1. Initialization of Weights and Thresholds: Prior to training, all connection weights (W¹ and W²) and thresholds (θ¹ and θ²) must be randomly initialized.

• 2. Forward Propagation: Input data is fed into the network through the input layer, where each input element is multiplied by its corresponding weight, summed, and then added to the respective threshold. After processing through the activation function (Sigmoid), the output s¹ of the hidden layer is obtained. The output s¹ of the hidden layer then undergoes similar operations of weighting, summing, and activation to produce the output s² of the output layer, which represents the network’s preliminary prediction for the input data.

• 3. Error Calculation: The discrepancy between the actual output s² of the output layer and the desired target output is calculated, typically using Mean Squared Error (MSE) or another suitable loss function.

• 4. Backward Propagation: Starting from the output layer, the error signal is calculated based on the error and propagated backward through the weight matrix W² to the hidden layer. In the hidden layer, the error signals propagated backward from the output layer, along with the weights W¹ of the hidden layer itself, are used to continue calculating the error signals for each neuron in the hidden layer. Through the backward propagation of error, the gradient contribution of each weight and threshold to the total error is computed.

• 5. Weight Update: Based on the gradient descent algorithm (or other optimization algorithms), the gradients obtained from backward propagation are used to update all weight matrices (W¹ and W² and threshold matrices (θ¹ and θ²). The update rule typically follows the formula weight = weight - learning_rate * gradient, where learning_rate represents the learning rate.

• 6. Iterative Repetition: The aforementioned four steps constitute a complete training iteration, and the network will continuously perform such iterations on the training dataset until the output error of the network is reduced to an acceptable level or the preset maximum number of iterations is reached.

Through this iterative process of forward and backward propagation, the BP neural network can automatically adjust the weights and thresholds, improving its fit to the training data and acquiring a certain degree of generalization ability, enabling it to handle complex nonlinear mapping problems. The network structure is shown in Fig. 5.

Fig. 5

BP neural network architecture.

Sand cat swarm optimization algorithm

Sand Cat Swarm Optimization Algorithm(SCSO)¹⁷ is an intelligent optimization algorithm that mimics the survival behavior of sand cats in nature. The algorithm offers advantages such as easy implementation, global search capabilities, and fast convergence speed. The BP neural network randomly initializes the connection weights of each layer to a value between 0 and 1 before starting the training. This unoptimized random initialization tends to slow down the convergence speed of the BP neural network and may lead to a non-optimal final result. To solve this problem, the BP neural network is initially trained to determine the optimal number of hidden layers. Subsequently, SCSO is employed to optimize the initial weights and thresholds of the network, enhancing the network’s accuracy and overcoming the limitations of using the BP network in isolation.

Initialization

The SCSO algorithm is initiated by creating a problem candidate matrix¹⁸ ($\:{N}_{pop}$×$\:{N}_{d}$),(pop = 1,…,n) based on the problem’s size. Subsequently, the fitness of each dune cat is evaluated with respect to the objective function. The optimal individual is selected, and all other individuals move toward that individual.

$$Sand{\text{ Ca}}{{\text{t}}_i}={\left[ {\begin{array}{*{20}{c}} {{X_1}} \\ \vdots \\ {{X_i}} \\ \vdots \\ {{X_n}} \end{array}} \right]_{n \times d}}={\left[ {\begin{array}{*{20}{c}} {{x_{11}}}& \cdots &{{x_{1j}}}& \cdots &{{x_{1d}}} \\ \vdots & \ddots & \vdots &{}& \vdots \\ {{x_{i1}}}& \cdots &{{x_{ij}}}& \cdots &{{x_{id}}} \\ \vdots &{}& \vdots & \ddots & \vdots \\ {{x_{n1}}}& \cdots &{{x_{nj}}}& \ldots &{{x_{nd}}} \end{array}} \right]_{n \times d}}$$

(11)

$$Fitness=f\left( {S{\text{and Cat}}} \right)=f\left( {{x_1},{x_2}, \cdots {x_d}} \right);\forall {x_i}\left( {is{\text{ calculated for n time}}} \right)$$

(12)

$$Fitness={\left[ {\begin{array}{*{20}{c}} {{F_1}} \\ \vdots \\ {{F_i}} \\ \vdots \\ {{F_n}} \end{array}} \right]_{n \times 1}}={\left[ {\begin{array}{*{20}{c}} {f\left( {{x_{11}},{x_{12}}, \cdots {x_{1d}}} \right)} \\ \vdots \\ {f({x_{i1}},{x_{i2}}, \cdots {x_{id}})} \\ \vdots \\ {f({x_{n1}},{x_{n2}}, \cdots {x_{nd}})} \end{array}} \right]_{n \times 1}}$$

(13)

In formula:$\:Sand\:\:{\text{C}\text{a}\text{t}}_{\text{i}}$ represents the population matrix of the sand cat population;$\:{X}_{i}\:$represents the ith cat colony; $\:{X}_{i,j}$ represents the dimension of the ith population; Fitness represents the fitness function of the sand cat population.

Search for prey (exploration phase)

Sand cats search for prey by emitting low-frequency sounds. In the mathematical model, the variable linearly $\:\overrightarrow{{\:r}_{G}}$decreases from 2 to 0 as the iteration progresses, allowing for a gradual approach to the prey without the risk of losing or overlooking it. The definition of the aforementioned mathematical model is as follows:

$$\overrightarrow {{r_G}} ={s_M} - \left( {\frac{{2 \times {S_M} \times ite{r_c}}}{{ite{r_{Max}}}}} \right)$$

(14)

$$\overrightarrow R =2 \times \overrightarrow {{r_G}} \times rand\left( {0,1} \right) - \overrightarrow {{r_G}}$$

(15)

$$\overrightarrow r =\overrightarrow {{r_G}} \times rand\left( {0,1} \right)$$

(16)

In formula:$\:{S}_{M}$ simulated the auditory properties of a dune cat;$\:\overrightarrow{R}\:$denotes the main parameter controlling the transition between exploration and utilization; ite$\:{r}_{c}$ is the current iteration count; ite$\:{r}_{Max}$ is the maximum number of iterations.

$$\overrightarrow {Po{s^ \circ }} \left( {t+1} \right)=\overrightarrow r \cdot \left( {\overrightarrow {Po{s_{bc}}} \left( t \right) - rand\left( {0,1} \right) \cdot \overrightarrow {Po{s_c}} \left( t \right)} \right)$$

(17)

In formula: $\:\overrightarrow{{Pos}_{bc}}$(t) is the optimal position at iteration;$\:\overrightarrow{{Pos}_{c}}$(t) is the current position at iteration t; $\:\overrightarrow{r}\:$is the sensitivity range.

Attacking prey (development phase)

The attack phase of the Sand Cat Swarm Optimization (SCSO) algorithm is mathematically modeled to represent the distance between the optimal position and the current position, as follows:

$$\overrightarrow {Po{s_{md}}} =\left| {rand\left( {0,1} \right) \cdot \overrightarrow {Po{s_b}} \left( t \right) - \overrightarrow {Po{s_c}} \left( t \right)} \right|,$$

(18)

$$\overrightarrow {Pos} \left( {t+1} \right)=\overrightarrow {Po{s_b}} \left( t \right) - \overrightarrow r \cdot \overrightarrow {Po{s_{md}}} \cdot \cos \left( \theta \right).$$

(19)

In the formula:$\:\overrightarrow{{Pos}_{b}}$ represents the best position (optimal solution); $\:\overrightarrow{{Pos}_{c}}$ represents the current position;$\:\overrightarrow{{Pos}_{rnd}}$ represents a random position. The SCSO algorithm uses a roulette wheel selection method to choose a random angle for each sand cat, allowing the sand cats to approach the hunting position.

The Sand Cat Swarm Optimization (SCSO) algorithm employs a roulette wheel selection mechanism to randomly determine the angle for each sand cat, thereby enabling them to approach the hunting position. Figure 6 illustrates the iterative positions of the sand cat population.

Fig. 6

Sand cat colony location.

Explore and utilize

Exploration and exploitation are ensured by the adaptive $\:\overrightarrow{{r}_{G}}$ and R. The SCSO algorithm enforces search agents to exploit when R is less than or equal to 1, otherwise, search agents are compelled to explore and seek out prey. The aforementioned process is mathematically modeled, as shown in Eq. (20):

$$\overrightarrow X \left( {t+1} \right)=\left\{ {\begin{array}{*{20}{c}} {{\text{ }}\overrightarrow {Po{s_b}\left( t \right){\text{ }}} - \overrightarrow {Po{s_{rnd}}} \cdot \cos \left( \theta \right) \cdot \overrightarrow r {\text{ }}} \\ {\overrightarrow r \cdot \left( {\overrightarrow {Po{s_{bc}}} \left( t \right) - rand\left( {0,1} \right) \cdot \overrightarrow {Po{s_c}} \left( t \right)} \right)} \end{array}} \right.{\text{ }}\begin{array}{*{20}{c}} {\left| R \right| \leqslant 1;\exp loitation} \\ {\left| R \right|>1;\exp loration} \end{array}$$

(20)

In the initial phase of searching for solutions, the adaptable search area of each sand cat helps to avoid becoming trapped in less optimal solutions. Figure 7 shows how the position of each sand cat changes as it moves through the exploration and then the exploitation phase. When R ≤ 1, the sand cats concentrate on their current targets; beyond this point, they search for new potential solutions across the broader search area. The algorithm’s balanced approach, along with its tendency to explore other possibly rewarding areas within the wider search space, results in a quick and accurate identification of the best solution.

Fig. 7

Attacking prey (exploitation) vs. finding prey.

Modified Sand Cat Swarm Smart optimization algorithm (ISCSO)

The conflict between exploration and exploitation also exists in the computational process of the Sand Cat Swarm Optimization Algorithm. The distribution of the population during initialization determines the convergence accuracy and speed of the algorithm. In order to achieve a uniform distribution of the sand cat population in the search space, this study incorporates chaotic mapping to enhance the diversity of population initialization. Additionally, it introduces the mutualistic symbiosis strategy and the Lévy flight strategy to enhance information exchange among individuals and optimize individuals, thereby broadening the scope of exploitation and enhancing the algorithm’s accuracy and speed in optimization search.

Chaotic mapping initialization

Chaotic mapping¹⁹ exhibits good randomness, regularity, and ergodicity. This not only enhances the diversity of the population but also improves the algorithm’s global search ability, convergence speed, and convergence accuracy. The improved formula is as follows

$$\:{C}_{k+1}=\left\{\begin{array}{c}\frac{{C}_{k}}{a},\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:0\le\:{C}_{k}\le\:a\\\:{(1-C}_{k})/(1-a),\:\:\:\:a\le\:{C}_{k}\le\:1\end{array}\right.$$

(21)

$$\:{\overrightarrow{Pos}}_{new}=lb+\left(ub-lb\right)\times\:{C}_{id}$$

(22)

Introducing mutually beneficial symbiotic strategies

Sand Cat Swarm Optimization Algorithm involves randomly selecting the angle to approach and attack prey using a roulette mechanism. However, such an attack is more random and may lead to local optimization. By enhancing the information exchange between individual sand cats and optimizing their strategies, the mutual benefit symbiosis strategy can mitigate the negative impact of sand cats attacking prey. This approach also enhances the accuracy of sand cats’ optimization search and speeds up convergence. The improved formula is as follows.

$$\:{\overrightarrow{Pos}}_{new}={\overrightarrow{Pos}}_{bc}+\left({\overrightarrow{Pos}}_{bc}-{bf\times\:R}_{MV}\right)$$

(23)

$$\:{R}_{MV}=\frac{{\overrightarrow{Pos}}_{bc}+{\overrightarrow{Pos}}_{rnd}}{2}$$

(24)

In formula, $\:{\overrightarrow{Pos}}_{new}\:$is the position after update;$\:{\overrightarrow{Pos}}_{bc}$ is the location of the optimal individual:$\:{\overrightarrow{Pos}}_{rnd\:}$is the position of the random individual; $\:bf$ indicates a random selection of 1 or 2 for the benefit factor, indicating the possibility of partial or full benefit; $\:{R}_{MV}$denotes the information exchange between the optimal and random individuals.

Introducing the Levi’s Flight Strategy

The Sand Cat Swarm Optimization (SCSO) algorithm, in the initial process of searching and locating optimal solutions, may rely on movement strategies with a high degree of randomness, which may lead to an insufficient comprehensive exploration of the optimal solution space, making it easy to miss some better solutions, especially for highly complex, multimodal optimization problems, it is more likely to fall into local optimal solutions. Introducing the Lévy flight strategy²⁰ can increase the algorithm’s global coverage of the solution space, and improve the thoroughness of the search process, which is conducive to finding the global optimal solution or higher-quality solutions, thereby significantly enhancing the algorithm’s convergence accuracy and efficiency. To enhance the thoroughness of the search and eliminate the negative impact of local optimal solutions, the improved formula is as follows:

$$\:{\overrightarrow{Pos}}_{new}={\overrightarrow{Pos}}_{bc}+\left|{\overrightarrow{Pos}}_{bc}-{\overrightarrow{Pos}}_{Levy}\right|lcos\left(2\pi\:l\right)$$

(25)

$$\:{\overrightarrow{Pos}}_{Levy}=\frac{\mu\:}{{\left|\alpha\:\right|}^{\frac{1}{\beta\:}}}\left({\overrightarrow{Pos}}_{rnd}-{\overrightarrow{Pos}}_{bc}\right)$$

(26)

$$\:{\sigma\:}_{\mu\:}=\left\{\frac{\varGamma\:\left(1+\beta\:\right)\text{s}\text{i}\text{n}\left(\frac{\pi\:\beta\:}{2}\right)}{\varGamma\:\left[\frac{\left(1+{\upbeta\:}\right)}{2}\right]{\upbeta\:}\bullet\:{2}^{\frac{{\upbeta\:}-1}{2}}}\right\}$$

(27)

In the equation, $\:{\overrightarrow{Pos}}_{new}$ represents the updated position of an individual;$\:{\overrightarrow{Pos}}_{bc}$indicates the position of the best individual; $\:{\overrightarrow{Pos}}_{Levy}$ signifies the Levy flight position;$\:l\:$is a random number between (0,1);$\:{\overrightarrow{Pos}}_{rnd}$ denotes the position of a random individual; µ is a random number following a normal distribution µ~N(0,$\:{\sigma\:}_{\mu\:}^{2}$), where $\:\varGamma\:\:$is the gamma function, $\:\beta\:\:$and is a random number between (0,2). Figure 8 shows the step-by-step process of our enhanced optimization method, inspired by the behavior of sand cats.

Fig. 8

Flowchart of improved sand cat swarm intelligent.

Model training and validation

This paper constructs a three-layer Backpropagation (BP) neural network model and applies it to the control of vibrations in aqueduct structures under seismic action. The BP neural network strives to fully reflect the characteristics of the full-state feedback LQR optimal control algorithm during the simulation process, ensuring that the optimal control force it maps out highly matches the results directly calculated using the LQR algorithm. The designed neural network has excellent robustness, maintaining good control performance even in the face of external environmental changes or internal parameter perturbations, ensuring effective control of vibrations in the aqueduct structure under seismic action. The network is specifically optimized for the vibration control problem of aqueduct structures under seismic action, making it suitable for the control requirements of such complex nonlinear systems. The network is trained with data from a semi-active control algorithm based on energy balance, with input data including 49,700 sets of continuous structural status parameters: displacement, velocity, acceleration, and corresponding optimal control force values. At the same time, the corresponding 49,700 magnetorheological damper control voltages are extracted as the expected neural network outputs. During the training process, the sample data is divided with 80% for the training set and 20% for the validation set, the maximum number of training iterations for the neural network is set to 1000, and the error training target is set to 0.001 to ensure the training accuracy and generalization capability of the model. The training results show that after only 20 training iterations, the neural network’s error target function has reached the pre-set training target, demonstrating that the optimized BP neural network model can quickly converge in a short time and efficiently simulate the optimal control strategy required for vibration control of aqueduct structures under seismic action.

To evaluate the performance of the model before and after optimization, the comparison between predicted and expected values is depicted as shown in Fig. 9 and the schematic diagram of errors is shown in Fig. 10.

Fig. 9

Comparison of predicted and expected values before and after model optimization.

Fig. 10

Schematic representation of the error before and after model optimization.

.

The error of the BP neural network model and the error of the model optimized by the ISCSO algorithm are compared using the mean absolute error and the root mean square error as evaluation metrics, and the results are listed in Table 1.

Table 1 Model prediction error.

As can be seen from Figs. 9 and 10; Table 1,the values obtained by the ISCSO-BP model are essentially in line with the expected values. The two types of errors are significantly reduced compared to the BP model, indicating that the ISCSO algorithm has a significant optimization effect on the BP neural network model. After the training is completed, a Simulink control module is generated, which responds extremely quickly during program operation, meeting the requirements of practical engineering applications.

Probability density evolution theory

Physical modeling of stochastic ground shaking

Jie Li and colleagues^21,22,23,24 I proposed the idea of modeling random ground shaking based on physical properties. I deeply considered the influence of the physical mechanism of the earthquake source and the seismic propagation pathway on the random ground shaking model. I established a physical stochastic function model of earthquakes and proposed a method for generating the narrow-band wave swarm superposition of the seismic sample time course from the model. Ding Y²⁵ has devoted himself to identifying stochastic parameters of the model and has proposed a statistical method for classifying earthquakes. Thus, a comprehensive engineering stochastic ground vibration physical model is developed, capable of considering various magnitudes, site types, and propagation distances. Considering the factors mentioned above, the seismic acceleration time course can be expressed as:

$${\tilde {a}_R}(t)= - \frac{1}{{2\pi }}\int_{{ - \infty }}^{{+\infty }} {{A_R}(\xi ,\omega ) \cdot \cos } [\omega t+{\Phi _R}(\xi ,\omega )]d\omega$$

(28)

Where: ${A_R}(\xi ,\omega )$ represents the Fourier amplitude spectrum, ${\Phi _R}(\xi ,\omega )$ represents the Fourier phase spectrum and ${\Phi _R}(\xi ,\omega )$ represents the Fourier amplitude spectrum. The narrowband wave group superposition approach is used to correct the equation above since the earthquake has non-stationary characteristics:

$${a_R}(t)= - \sum\limits_{{j=1}}^{N} {{A_j}{F_j}(t) \cdot \cos ({\omega _j}t+{\Phi _j}) \cdot \Delta \omega }$$

(29)

In Eq. (29),${\omega _j}$ is the j th frequency component, and ${A_j}$ is the j th wave group’s amplitude. The expression for the modified Fourier amplitude spectrum model is:

$${A_j}=\frac{2}{\pi }\frac{{{A_0}{\omega _j} \cdot {e^{ - KR{\omega _j}}}}}{{\sqrt {{\omega _j}^{2}+{{\left( {\frac{1}{T}} \right)}^2}} }} \times \sqrt {\frac{{1+4{\xi _g}^{2}{{\left( {{{{\omega _j}} \mathord{\left/ {\vphantom {{{\omega _j}} {{\omega _g}}}} \right. \kern-0pt} {{\omega _g}}}} \right)}^2}}}{{{{\left[ {1 - {{\left( {{{{\omega _j}} \mathord{\left/ {\vphantom {{{\omega _j}} {{\omega _g}}}} \right. \kern-0pt} {{\omega _g}}}} \right)}^2}} \right]}^2}+4{\xi _g}^{2}{{\left( {{{{\omega _j}} \mathord{\left/ {\vphantom {{{\omega _j}} {{\omega _g}}}} \right. \kern-0pt} {{\omega _g}}}} \right)}^2}}}}$$

(30)

Where ${\xi _g}$ is the equivalent damping ratio of the site, which depicts the dampening influence of the site soil layer on the seismic propagation process, ${A_0}$ is the amplitude coefficient, T is the Brune source coefficient, which is used to depict the process of dislocation formation.and ${\omega _g}$ is the comparable excellence circle frequency of the site. The four factors are all fundamental physical random factors. Li J et al. demonstrated that the statistical analysis of the aforementioned four physical random variables may be used to quantify the stochastic character of the time course of the ground shaking. R is the distances between the epicentres, and K is the attenuation coefficient of the seismic waves as they travel. Depending on the engineering circumstance, K and R are general deterministic variables that can be chosen. Consider K to be 10^− 5s and R to be 20 km in this example.

${F_j}$ is the jth wave group’s time-energy envelope function, with the formula:

$${F_j}(t)=\frac{{\sin \left[ {\left( {t - \frac{R}{{{c_j}}}} \right)\Delta {\omega _j}} \right]}}{{\left( {t - \frac{R}{{{c_j}}}} \right)}}$$

(31)

${c_j}$ represents the jth wave group’s equivalent group velocity.

$${c_j}=\frac{{\left( {a+0.5} \right){\omega _j}+b+\frac{1}{{4c}}\sin \left( {2c{\omega _j}} \right)}}{{d \cdot \left[ {a+{{\cos }^2}\left( {c{\omega _j}} \right)} \right]}}$$

(32)

${\Phi _j}$ represents the phase of the jth wave group. The modified Fourier phase spectrum model is expressed as:

$${\Phi _j}=\arctan \left( {\frac{1}{{T{\omega _j}}}} \right) - R \cdot d\ln \left[ {\left( {a+0.5} \right){\omega _j}+b+\frac{1}{{4c}}\sin \left( {2c{\omega _j}} \right)} \right]$$

(33)

Where, during the propagation of seismic waves, a, b, c, and d are empirical coefficients describing the connection between wave speed and frequency. Their values have a significant impact on the phase spectrum modelling. Relevant studies have shown that the value of the phase spectrum has the most critical influence on the spectral characteristics of the ground-shaking process²⁶. The aforementioned four coefficients may be calculated as a =  1.02, b = 403${\text{rad/s}}$, c = 1.89${\text{s/rad}}$, d = 130${\text{rad/km}}$ based on engineering experience.

Therefore, a physical model of stochastic ground shaking that considers the source, propagation path, and site conditions can be modelled based on four physical stochastic parameters. Scholars identify the stochastic parameters in the model and give suggested values for the corresponding engineering parameters. Table 2 contains a list of the pertinent parameters. The parameters $\sigma$and $\mu$in the table are the log standard deviation parameter and the log mean parameter, respectively, with the distribution type of lognormal; $1/\theta$ is the size parameter, k is the shape parameter, and the distribution types of $1/\theta$and k are gamma distributions. $\alpha$ is the probability of error of the hypothesis distribution given using the Kolmogorov-Smirnov test, which characterises the possibility of the null hypothesis being incorrectly rejected.

. The recommended values in Table 2 are used for the ground shaking model site parameters for the reliability analysis in this work because there aren’t any field-measured experimental data. The model is helpful for the reliability analysis of aqueduct structures and random seismic response analysis.

Table 2 Parameter values of random parameter probability distribution functions for different site types.

To evaluate the stability and examine the stochastic dynamic response properties of the aqueduct structure under seismic loading. The GF deviation technique is used to choose a discrete representative point set of various random variables and give values to the model’s physical random parameters in order to create the required number of seismic acceleration time-range samples with specified probabilities. A hundred seismic acceleration time-range samples are generated based on the model, and 0.4 g is added to the peak acceleration time range of the random seismic wave. The time-range curves for the 100 seismic waves produced at random are displayed in Fig. 11 below.

Fig. 11

Ground shaking acceleration time course and acceleration response spectrum.

Probability density evolution equations

The dynamical system’s equation of motion when stochasticity is considered is denoted by the following.

$$M\left( \Theta \right)\ddot {U}\left( t \right)+C\left( \Theta \right)\dot {U}\left( t \right)+f\left( {\Theta ,U\left( t \right)} \right)=F\left( {\Theta ,U\left( t \right)} \right)$$

(34)

Where M and C are the mass and damping matrices, $f(U,t)$, $f(t)$are the restoring forces and input seismic vectors, respectively; $\Theta =({\Theta _1},{\Theta _2}, \ldots, {\Theta _n})$characterises n random parameters in the system; $\ddot {U}\left( t \right)$,$\dot {U}\left( t \right)$ and $U\left( t \right)$ represent acceleration, velocity and displacement, respectively.

Let $Z={({Z_1},{Z_2}, \ldots, {Z_m})^T}$,characterise the m essential physical response quantities in the system. $\left( {Z,\Theta } \right)$ Being the augmented system taking into account the stochastic factor. The following equation represents the generalized probability density evolution equation based on stochastic occurrences and stated by the principle of conservation of probability:

$$\frac{{\partial {p_{z\Theta }}\left( {z,\theta ,t} \right)}}{{\partial t}}+\sum\limits_{{i=1}}^{m} {{{\dot {Z}}_i}\left( {\theta ,t} \right)} \frac{{\partial {p_{z\Theta }}\left( {z,\theta ,t} \right)}}{{\partial {z_i}}}=0$$

(35)

The ${p_{z\Theta }}\left( {z,\theta ,t} \right)$ in Eq. (45) represents the joint probability density function of $\left( {Z,\Theta } \right)$. In the initial case, ${p_{z\Theta }}\left( {z,\theta ,t} \right)$ can be expressed as a Dirac-δ function:

$${p_{z\Theta }}\left( {z,\theta ,t} \right){|_{t={t_0}}}=\delta \left( {z - {z_0}} \right){p_\Theta }\left( \theta \right)$$

(36)

${z_0}$ is the deterministic initial value. The finite difference method is used to solve Eq. (32) to (34)

$${p_z}\left( {z,t} \right)=\int_{{{\Omega _\Theta }}} {{p_{z\Theta }}} \left( {z,\theta ,t} \right)d\theta$$

(37)

${\Omega _\Theta }$ is the probability space. If only one critical physical response quantity is explored, Eq. (35) can be simplified to:

$$\frac{{\partial {p_{z\Theta }}\left( {z,\theta ,t} \right)}}{{\partial t}}+\dot {Z}\left( {\theta ,t} \right)\frac{{\partial {p_{z\Theta }}\left( {z,\theta ,t} \right)}}{{\partial z}}=0$$

(38)

The probability density evolution equation creates connects stochastic and deterministic studies and illustrates how probability density evolution works, i.e., how it is dependent on changes in the physical state.

Point evolution method solution

In this section, we solve the probability density evolution equation using the point evolution method. The point evolution method addresses the problem by conducting multiple deterministic analyses and merging these with the probability density evolution equation, accurately reflecting the theoretical process of probability density evolution. By integrating the probability density evolution approach with the system’s physical equations via numerical simulation, we can determine the probability density function of the key physical response variables. The solution method encompasses four fundamental steps:

• 1. Since probabilities are additive, choose r representative points ${\theta _q}=({\theta _{1,q}},{\theta _{2,q}},…,{\theta _{s,q}})$, $q=1,2,…,r$. Divide ${\Omega _\Theta }$ into a series of subspaces. Calculate the assigned probability of each point with the following formula:

  $${P_q}=\int\limits_{{{V_q}}} {{P_\Theta }\left( \theta \right)d\theta }$$

  (39)

  ${V_q}$ is the representative volume. The GF bias method²⁷ was used to select the points to ensure that the expected points were evenly distributed in ${\Omega _\Theta }$.

• 2. A deterministic dynamic analysis is carried out to solve the physical equations from the representative point $\Theta ={\theta _q}$. The time derivatives of the critical physical response quantities, ${\dot Z_i}\left( {{\theta _q},t} \right)$, are solved for i=1,2,…m.

• 3. After selecting the representative points and assigning assigned probabilities, the probability density evolution equation of Eq. (39) is updated as:

  $$\frac{{\partial {p_{z\Theta }}\left( {z,{\theta _q},t} \right)}}{{\partial t}}+\sum\limits_{{i=1}}^{m} {{{\dot {Z}}_i}} \left( {\theta ,t} \right)\frac{{\partial {p_{z\Theta }}\left( {z,{\theta _q},t} \right)}}{{\partial {z_i}}}=0,{\text{ }}q=1,2,.,r$$

  (40)

  Equation (37) is updated to:

  $${p_{z\Theta }}\left( {z,{\theta _q},t} \right)\left| {_{{t={t_0}}}} \right.=\delta \left( {z - {z_0}} \right){p_q}$$

  (41)

  Substituting the solved ${\dot Z_i}\left( {{\theta _q},t} \right)$ into ${p_{z\Theta }}\left( {z,{\theta _q},t} \right)$Eqs. (40) and (41) to calculate the joint probability density function $\left( {Z,\Theta } \right)$ of corresponding to each representative point.

• 4. Accumulating all the ${p_{z\Theta }}\left( {z,{\theta _q},t} \right)$ obtained in the previous step, the joint probability density function Z(t) of P_z(z,t) can be found as follows:

  $${P_z}\left( {z,t} \right)=\sum\nolimits_{{q=1}}^{n} {{P_{z\Theta }}} \left( {z,{\theta _q},t} \right)$$

  (42)

Results and discussion

Engineering examples

Selection of aqueduct structure

Taking the Guandimiao Flood Discharge Aqueduct of the Central Route Project of the South-to-North Water Diversion as the engineering background, the aqueduct has a total length of 84 m, divided into 3 spans, each 28 m long; the body of the aqueduct is composed of a base slab and side plates, water stop belts are connected between each span, the body of the aqueduct is connected to the support with a basin-type rubber bearing, the support adopts an H-shaped frame, with a circular cross-section of 0.8 m in diameter; the cross-sectional area of the aqueduct is 3.32 m², the eccentricity distance is 0.863 m, and the torsion center is situated 1.3175 m from the centerline of the aqueduct’s base plate, the moment of inertia of the cross-section about the X-axis is 4.133 m⁴, about the Y-axis is 9.550 m⁴, the torsional moment of inertia of the cross-section is 0.146 m⁴, the sectorial moment of inertia of the cross-section is 12.702 m⁴. The concrete has a mass density of 2500 kg/m³, the elastic modulus of the material is 2.55 × 10¹⁰ Pa, and the Poisson’s ratio is 0.3. The aqueduct is constrained with simply supported ends and the support bottom is fixed. The structural schematic is shown in Fig. 12:

Fig. 12

Sketch of the elevation of the aqueduct structure (unit: m).

Arrangement of controls

As shown in Fig. 13, for the aqueduct structure, taking control of the down-channel item (i.e. longitudinal direction) as an example, the control device can be installed primarily in the expansion joints between the two ends of the channel body and between the channel body and the top of the pier.

Fig. 13

Diagrammatic representation of the control unit’s installation position.

Seismic wave input

This study analyzes and compares the seismic response of the structure under various operating situations utilizing two different types of seismic waves that are input along the trench’s longitudinal bridge direction. The results are then used to weigh the benefits and drawbacks of the suggested semi-active control scheme. Elcen-NS and Taft-NS seismic waves, two representative vital seismic records, are input, and the seismic waves’ peak acceleration is increased to 400 gal.

Analyzing the efficiency of semi-active control techniques

This research develops a machine learning-based semi-active control algorithm. The algorithm above is simulated in the MATLAB application to determine the aqueduct structure’s ground vibration response. The voltage variation of the magnetorheological damper is illustrated in Fig. 14. The damper at location No. 1 is labeled as 1, and the damper at location No. 4 is labeled as 2. The displacement time curve is presented in Fig. 15 and obtained using the displacement time histories from the aqueduct’s spans 2 and 3.

Fig. 14

Intelligent semi-active control programme to control signal changes.

Fig. 15

Displacement time course curves for three different working conditions.

The control method is effective at dampening the earthquake, as seen in Fig. 15, where the seismic response of the proposed semi-active control is much lower than that of the uncontrolled condition.

The peak displacement values of various key parts of the aqueduct are selected and listed in Tables 3 and 4.

Table 3 Peak displacement of various parts under Elcen’s longitudinal effect.

Table 4 Peak displacement of various parts under Taft’s longitudinal effect.

$$J=\frac{{Y_{{\hbox{max} }}^{{uncontrol}} - Y_{{\hbox{max} }}^{{control}}}}{{Y_{{\hbox{max} }}^{{uncontrol}}}} \times 100\%$$

(43)

Where $Y_{{\hbox{max} }}^{{uncontrol}}$ and $Y_{{\hbox{max} }}^{{control}}$ represent the maximum values of the structure’s dynamic reaction in the applied control state and the uncontrolled state, respectively (displacement, velocity, acceleration, etc.). The continuous girder bridge’s seismic reaction in each of the two states determines the damping rate value, and the bigger the value of J, the better the damping effect is shown.

The centre of the crossing’s span is where the largest displacement under seismic action can be located in relation to the direction of the seismic wave action. Therefore, data are collected at the mid-span site for analysis and comparison of the seismic response of aqueduct. The Table 5 lists the average damping rates for the suggested semi-active control condition under the influence of two natural seismic waves.

Table 5 Comparison of damping rates by part of semi-active control.

The results in Table 5 demonstrate that the proposed semi-active control algorithm has strong adaptability and robustness, and exhibits significant control effects on the seismic resistance of aqueduct structures.

Analysis of the stochastic seismic response

Utilizing the stochastic ground shaking model described in the previous section, we initially generate random seismic waves. We generate ground-shaking acceleration time-history samples utilizing the Voronoi diagram, select discrete representative point sets via the GF deviation method, and calculate the probabilities assigned to each seismic wave. Introducing 100 seismic waves into the aqueduct’s finite element model, we perform a series of seismic response analyses. It is crucial to note that the values of random variable parameters may not share the same order of magnitude, potentially leading to the “large number overshadows small number” issue when identifying the representative volume. To avert these inaccuracies, the probability space should first be normalized. The procedure is as follows normalized: let $\Theta$ be the random variable under investigation, and the mean and standard deviation of the random variable can be expressed as ${\mu _\Theta }$ and ${\sigma _\Theta }$. Let:

$$\bar {\Theta }=\frac{{\Theta - {\mu _\Theta }}}{{{\sigma _\Theta }}}$$

(44)

$\bar {\Theta }$ is the normalised random variable, which is randomly sampled and then reduced according to the above equation can again yield $\Theta$.

The required response quantities’ probability density evolution information may be obtained by combining the probability density evolution approach with the response data obtained from the solution of the ferries under random seismic actions. Seismic reliability assessments for the aqueduct can be conducted by incorporating absorbing boundary conditions, and the probability density evolution equations are resolved through the finite difference method, ensuring control over total variance. Figure 16 illustrates the solution flowchart for the aqueduct’s reliability study and the analysis of its stochastic seismic response.

Fig. 16

Process for analyzing the dependability of aqueduct structures under stochastic seismic reaction.

The study’s primary physical response quantity is the aqueduct’s pier top displacement, along with its probability density evolution analysis and corresponding reliability assessment. To investigate the stochastic dynamic properties of the aqueduct structure and to evaluate its seismic performance under more intense environmental shaking than the design basis, the peak ground acceleration for the random ground motions is specified at 0.4 g. The complete probability distribution information of the system, including the reaction’s probability distribution function, has garnered significant interest in practical engineering applications. In the subsequent section, we conduct a probability density evolution analysis for the aqueduct structure and develop its probability density function. The second pier of the aqueduct is scrutinized. Figure 17 illustrates the progression of displacement at the pier’s top under 100 instances of randomly generated ground motions with a peak acceleration of 0.4 g.

Fig. 17

100 piers’ tops saw a 0.4 g time course displacement during seismic activity.

Figures 18 and 19 show the statistical mean and statistical standard deviation of the displacements at the top of the aqueduct piers during the 0.4 g earthquake.

Fig. 18

Statistical mean values of the displacement response of the top of the aqueduct pier.

Fig. 19

Statistical mean values of the displacement response of the top of the aqueduct pier.

For comparative analysis, the statistical mean and standard deviation values are presented together in a single figure, The results of this comparison are displayed in Figs. 20 and 21, illustrating significant variability in the aqueduct’s seismic response. The variability in ground shaking excitation can result in structural response variability, potentially up to sixfold.

Fig. 20

Comparison of the statistical standard deviation of the top displacement caused by a 0.4 g earthquake at the aqueduct pier.

Fig. 21

Probability density surfaces for typical periods.

Figures 21 and 22 for the uncontrolled and controlled states show surface plots of the probability density of the pier top displacements of pier 2 of the aqueduct over time for a typical period.

Fig. 22

Probability density contours for typical periods.

The two charts, characterized by their distinct shapes and fluctuating trends reminiscent of mountains and rivers, are termed “peaks” and “rivers,” respectively. The maps reveal that the contours exhibit significant irregularity, akin to the many eddies in a river, especially near the “boundary layer.” If the contours are aligned like parallel lines or exhibit laminar flow, the probability density remains relatively stable over time. Conversely, the observed inconsistencies indicate that the probability density is non-stationary, fluctuating significantly over time. The probability density evolution approach, when applied to examine the stochastic response of structures like ferries, can provide a wealth of probabilistic information regarding the structural response. In the study and assessment of aqueduct dependability, utilizing probabilistic information to represent the temporal variations in overall performance can be both more straightforward and innovative. Figure 23 displays the probability density curves for select representative moments, plotted beneath their respective instances.

Fig. 23

Probability density function for typical moments.

The probability density function distribution in this scenario is time-variant and lacks a consistent shape and range. The seismic response of the structure exhibits random fluctuations, reflecting the unpredictability of seismic accelerations. Unlike traditional functions, the one utilized here illustrates that the seismic response of the aqueduct structure is a complex and evolving process. To assess the system’s seismic performance and the effectiveness of mitigation strategies, the design of the aqueduct channel’s seismic mitigation plan must include analysis of the response to random ground vibrations.

Seismic reliability analysis

The preceding assessments were used to gather data on the probability density of the aqueduct’s seismic reaction. Based on this, the defined failure levels’ matching absorbing border conditions are imposed. The calculation is then used to determine the aqueduct s’ dynamic reliability. In this chapter, the study uses the “Seismic Design Code for Highway Aqueduct s JTG/T 2231-01-2020”²⁸as a specification for reference. The aqueduct in Guandimiao village of Xingyang is 84 m, divided into three spans, each span 28 m, and the seismic defence category is B¹⁸. Under seismic action, the maximum removal allowed for the dock ${\Delta _u}$ cannot be exceeded by the displacement of the aqueduct pier top ${\Delta _d}$. The following formula can be used to determine the port’s permissible displacement:

$${\Delta _u}=\frac{1}{3}{H^2} \times {\phi _y}+\left( {H - \frac{{{L_p}}}{2}} \right) \times {\theta _u}$$

(45)

$${\theta _u}={L_p}\left( {{\phi _u} - {\phi _y}} \right)/{K_{ds}}$$

(46)

In the formula, ${\theta _u}$ is the maximum permissible turning angle, ${\phi _y}$ is the equivalent yield curvature of the cross-Sect. (1/cm), ${\phi _u}$ is the curvature capacity of the ultimate damage state (1/cm), for the pier cross-section of the rectangular channel, and with the cross-section size, the yield strain of the reinforcement constraints on the ultimate compressive stress of the concrete. The equivalent plastic hinge length (cm), ${K_{ds}}$ is the ductile safety coefficient, which can be 2.0.

Combined with the cross-section and material data of the aqueduct channel in Guandimiao village of Xingyang, after calculation according to the specification, the thresholds for the permissible displacement ${\Delta _u}$ of the pier top under the excitation of ground vibration acceleration of 0.4 g are set to be 0.15 m, 0.12 m, 0.08 m, 0.06 m, and 0.04 m. The corresponding dynamic reliabilities are calculated for the different thresholds set.

Table 6 lists the aqueduct structure’s dependability for each failure threshold index for various control strategies. Figure 24 displays the aqueduct structure’s seismic reliability curves for each control strategy.

Table 6 Aqueduct structure seismic resistance under various control strategies and failure levels.

Fig. 24

Aqueduct structures’ seismic reliability curves at different failure threshold indicators.

When Table 6; Fig. 24 are taken into account, it is evident that the seismic reliability of the aqueduct structure varies significantly and declines stepwise with a decrease in the failure threshold index. Comparing the seismic reliability statistics of the aqueduct structure under two operating situations with distinct failure thresholds. As observed, the semi-control technique based on machine learning is more reliable than the uncontrolled state for any given failure threshold. For different failure thresholds, the difference between the two reliabilities rises from 5.48 to 23.37%. It demonstrates that the semi-active control strategy can continue to keep the aqueduct operating safely even when the seismic effect is more than the intensity of the aqueduct structure itself and that the improvement effect is more pronounced the lower the failure threshold is. Figure 25 depicts the trend of the aqueduct structure’s dynamic reliability with a lowering of the chosen failure threshold under two operational scenarios.

Fig. 25

Reliability trends with failure thresholds.

Figure 25 illustrates that in both scenarios, as the failure threshold decreases, the dynamic dependability of the aqueduct exhibits a trend of accelerated deterioration. This occurs because, during seismic events, instances of significant displacement or exceedance of the maximum allowable displacement at the pier top are still a minority of the overall seismic process. Furthermore, as the failure threshold decreases, the probability of the pier top’s displacement exceeding this threshold during seismic events rises, leading to a swift decline in dynamic dependability. The semi-active control scheme exhibits the most gradual reduction in dynamic dependability, in contrast to the uncontrolled condition, which shows the steepest decline. A comparison of the seismic reliability between the two scenarios demonstrates the effectiveness of the semi-active control system in managing the aqueduct.

Conclusion

This paper initially proposes an intelligent semi-active control algorithm based on machine learning and conducts simulation tests with examples. Subsequently, to verify its generalization and reliability, random dynamic response analysis and seismic reliability assessment of the aqueduct structure are carried out based on the physical model of random ground motion and the theory of probability density evolution, applying reliability analysis methods to the seismic research and vibration reduction optimization design of the aqueduct structure. The following conclusions were drawn:

• 1. Leveraging the high nonlinear mapping and approximation capabilities of neural networks, a BP neural network was designed and trained that inherits the characteristics of the LQR optimal control algorithm, optimized with an improved sand cat swarm algorithm, specifically for the response characteristics of aqueduct structures under seismic action. Achieved an effective mapping from the structural dynamic response of the aqueduct to the optimal control force, thereby implementing semi-active seismic control of the aqueduct structure, overcoming the shortcomings of traditional control algorithms that are inapplicable to complex nonlinear structural vibration control.

• 2. A numerical simulation of the vibration control effect of this method was conducted using a typical aqueduct structure as an example. The simulation results indicate that this control method can be applied to the vibration control of large aqueduct structures, effectively reducing displacement and velocity responses, with a vibration reduction of over 43%.

• 3. A comparative analysis of the seismic performance under random earthquake action was conducted for aqueduct structures under intelligent semi-active control and uncontrolled conditions. The results show that in uncontrolled aqueduct structures, the evolution characteristics of probability density exhibit stronger randomness, and the structural response also demonstrates a greater degree of dispersion. Under different failure criteria, the seismic reliability of the aqueduct structure with intelligent semi-active control can be increased by up to 20.89% compared to the uncontrolled state. The semi-active control strategy based on the BP neural network can effectively enhance the ability of the aqueduct structure to withstand intense random earthquakes to a certain extent.