The railway catenary model discussed in this paper is shown in Fig. 1. Where La is the distance between the weights and the catenary, and Lb is the distance between the bottom of the weights and the ground. The changes in La and Lb can both serve as displacement changes in the railway catenary. However, considering that in practice, the weights will cause slight vibration of the arc with the pulley as the origin, which has a significant impact on the Lb . Therefore, the displacement value of the catenary is defined as the La .

Metal cables have a specific coefficient of thermal expansion, which means that changes in ambient temperature can lead to variations in the catenary length [22]. The coefficient of thermal expansion of metals [23], La can be calculated as follows:(1) La=Lmin+γLcφ(Tc−Tmin) where Lmin is the minimum length, γ represents the transmission coefficient of the compensating pulley, and Lc represents the distance from the center anchor to the compensator. φ indicates the expansion coefficient of the contact network, Tc represents the current temperature, Tmin represents the lowest temperature.

When no train passes by, the trend of ambient temperature and catenary displacement over time plotted based on data collected by sensors is shown in Fig. 2. The temperature curve is the temperature data within 24 h, the theory curve is the displacement change trend calculated by Eq. (1), and the actual curve is the data collected by the displacement sensor. The data sampling period is 10 min. From the statistical results, it can be seen that the theory value of the catenary displacement is correlated with temperature. Due to environmental interference, there is a certain dissimilarity between the displacement collected by the sensor and the theory value, but the correlation is still obvious.

After being affected by sudden interference like pantograph passing, the displacement of the catenary will experience significant fluctuations. As shown in Fig. 3, the pantograph passing effect occurred between 1000 and 2000 s. The catenary experiences significant interference resulting in a substantial deviation from the theoretical expectations. However, once the interference diminishes, they gradually realign with the theoretical predictions.

As the catenary ages, the physical properties are altered due to cable fractures. In Fig. 4, a comparison between the displacement of the aged catenary and that of a normal one is made, with the differences illustrated in Fig. 5. The results indicate a substantial disparity in the displacement distribution between the aged and normal catenaries, and this difference does not spontaneously revert to the normal state for a certain period. When the detected difference surpasses a predefined threshold and persists for a specific duration, it serves as an indicator to determine whether the catenary is indeed aged [24].

The values obtained from displacement sensors are susceptible to numerous influencing factors, including rain, wind, physical disturbances, etc. Without proper filtering of these observations, it can result in erroneous assessments. The aging prediction method employed in this paper relies on catenary displacement, making it crucial to account for significant interfering factors. Other factors like temperature and corrosion can manifest as changes in displacement. However, factors that do not have an abrupt impact on displacement data do not necessitate technical consideration.

The traditional KF architecture can be described as Fig. 6. The item R represents the measurement error covariance. The prior estimate error covariance is(2) Pt−=FPt−1FT+Q where F indicates the state transition matrix, Pt−1 represents the posterior estimate error covariance, and Q is the process noise. Kt represents the KF gain. z indicates the actual measurement. The prior state estimate is

where xt−1 is the last posterior state estimate, B indicates the control matrix and ut−1 represents the input state.

As seen in Eqs. (2) and (3), the optimal state of the KF is based on the output from the previous time, which exhibits hysteresis and is unable to effectively mitigate nonlinear interference. To enhance the anti-interference performance, this paper proposes replacing the prior state estimate equation of the KF with a T-S FNN model. The suggested structure is depicted in Fig. 7. Furthermore, the corresponding prior estimate error covariance has also been substituted. Additionally, the measurement error R has been replaced by Rv , which will be discussed in Section 3.5. The design of a T-S FNN to learn how to compute xt− and Pt− as integral components of an overall KF framework involves addressing three pivotal questions:

1) What are the inputs and outputs of the network?

2) How to design the structure of the network?

3) What methods are needed for training the network?

Compared with the traditional Mamdani fuzzy model, the T-S FNN adopts multiple linear functions for defuzzification. This enables the T-S fuzzy model combined with a back propagation neural network to reduce fuzzy rules and achieve better training results [25]. Based on the on-site collected data characteristics, this paper proposes to design the T-S fuzzy model as a three-input single-output model. As shown in Fig. 8, the model consists of the Premise Network and the Latter Network.

The lower part shown in Fig. 8 is the Premise Network. The first layer is the input, represented by(4) x=[x1x2x3]

where x1 , x2 and x3 indicate time, temperature, and the history of filtered displacement value, respectively. The input membership function is designed as a bell curve

(5) μij=e(xi−cij)σij2, i=1, …, nj=1, …, m

where cij is the center of the bell curve, σij represents the width of a bell curve, n indicate the number of input, m is the number of membership function. In the discussion of this paper, n=m=3 .

The second layer of the Premise Network is the fuzzy language variable layer. This layer plays a primary role in computing the membership function for each component input from the first layer, which corresponds to its respective fuzzy set. This computation is carried out as follows:

(6) μij=μAij(xi) where Aij represents the j-th language variable of the x-th input.

The third layer represents the matching fuzzy rule layer, where each node corresponds to a specific fuzzy rule. The primary function of this layer is to match the fuzzy rules and compute the fitness of each rule. The fitness value is determined through the following calculation:

(7) αj=min{μ1i1, μ2i2, μ3i3}

The third layer of the Premise Network also needs normalization. The specific calculation method is as follows:

(8) α¯j=αj∑i=1nαi

After taking the minimum value and normalization, the coefficient vector ν can be obtained

(9) ν=[f1f2f3]

(10) fj=[ajbj] where a and b are the coefficients. Convert the coefficients into a linear function(11) yfj=ajα¯j+bj

Finally, take the minimum value to obtain the output

(12) D=min{yf1yf2yf3}

The Later Network is a fusion method based on the Premise Network and fuzzy system. It receives input signals from the Premise Network and outputs the results using fuzzy inference processing based on these signals. The network structure proposed in this paper is shown in the upper half of Fig. 8. The Later Network can be trained based on real-time data to improve the performance of fuzzy control. It consists of n subnetworks with the same structure, each of which generates an output.

The first layer of the subnetwork is the input layer, the same as the Premise Network. The second layer consists of m nodes, each of which can represent a fuzzy rule in a fuzzy model, and its main function is to calculate the consequences of each rule. The calculation method for the output of this layer is as follows:(13) Wij=∑i=1n∑j=1mpijxi where p indicate the usage weights of various fuzzy rules in the second layer.

The third layer is the output of the Later Network, which is the weighted sum of the Later Network of each rule. The weighted coefficients are the weights of each fuzzy rule, which means that the output of the Premise Network is used for the connection weights of the third layer of the Later Network:(14) Wi=∑j=1mα¯jWij

In summary, FNN combines the advantages of fuzzy systems and neural networks, enabling the model to not only have self-learning and parallel processing capabilities but also handle fuzzy knowledge and utilize expert experience. In the Premise Network, the use of bell curves as input membership functions reduces training parameters while improving accuracy. Use Eqs. (7) and (8) to calculate the output of the membership function and normalize it. Output the result through Eq. (11), which has higher accuracy compared to constants. Using Eqs. (13) and (14) in the Later Network, parallel computation of the weights corresponding to each fuzzy rule is achieved.

After all, the output of the proposed T-S FNN model is Dt . The process noise of the proposed KF is as follows:(15) Q=∑n=1k(Dn−dn)2n×Dt where k indicates training data volume, d represents the historical displacement of filtered. Eq. (15) elucidates that the process noise of the KF is transformed through the prediction process error of the T-S FNN model. The prior estimate error covariance and priori state estimate are defined as follows:(16) Pt−=Pt−1+Q

(17) xt−=Dt

Considering that the model of each catenary on-site is not the same, an adaptive algorithm can certainly improve the practicality of the algorithm. There are many learning algorithms for neural networks, such as Unsupervised learning algorithms based on iterative learning [26], Supervised learning algorithms [27], deep learning [28], etc. Supervised learning is suitable for big data-intensive and purposeful training. Deep learning requires a larger number of data for specific functional training. Unsupervised learning can use a small amount of data to quickly train the model. Considering real-time requirements, using an iterative Unsupervised learning algorithm to train the model can have a better effect.

The input membership function of the proposed fuzzy system is set to a bell curve. So the parameters that need to be adaptively updated in the Premise Network can be defined as the center value cij and the width value σij corresponding to each node in the second layer. The Later Network needs to adaptively update the usage weights of various fuzzy rules in the second layer. Therefore, define the error cost function as follows:(18) E=12∑i=1n(Wi~−Wi)2 where Wi~ represents the expected output of the neural network. First, fix the usage weights of various fuzzy rules pij , and discuss the Premise Network. Then the output of each fuzzy rule is equal to the weight of the third layer of the Later Network. Let Wij=pij , the activation function of the neural network can be defined as follows:(19) δi(5)=Wi~−Wi

(20) δj(4)=∑i=1nδi(5)Wi

(21) δj(3)=δj(4)∑i=1,i≠jnαi(∑i=1nαi)2

(22) δij(2)=∑k=1mδk(3)sije(xi−cij)2σij2 where sij represents adaptive parameters, which is defined as follows:(23) sji={1, μij is the minimum value for the k‐th rule0, otherwise

Above all, the partial derivative of the error concerning the central and width of the membership function is as follows:(24) ∂E∂cij=−δij(2)2(xi−cij)σij

(25) ∂E∂σij=−δij(2)2(xi−cij)2σij3

The learning algorithm for adjusting the parameters of the bell curve is as follows:(26) cij(k+1)=cij(k)−β∂E∂cij

(27) σij(k+1)=σij(k)−β∂E∂σij

where β is the learning rate. Then the usage weights pij can be calculated as follows:(28) ∂E∂pij=−(Wi~−Wi)α¯jxi

(29) pij(k+1)=pij(k)+β(Wi~−Wi)α¯jxi

In summary, when designing adaptive algorithms for the FNN to select different fuzzy rules based on varying states, the inclusion of parameter sij becomes crucial. This parameter plays a pivotal role in the selection of specific fuzzy rules by matching them with the training data.

When the accuracy of the sensor is high, its interior error range is too small compared to the error range of the system measurement error, which cannot truly reflect the system error information. Therefore, the vibration data of the catenary is proposed as the KF measurement error. Then the measurement error covariance is calculated as follows:

(30) Rv=∑n=1kSn2n where S is the vibration data. Then the KF gain is as follows:

(31) Kt=1Pt−HT(HPt−HT+Rv) where H is the observation matrix. Finally, the output state estimate of the KF is

(32) xt=xt−+Kt(z−Hxt−)

In the fields of neural networks and artificial intelligence, to validate algorithms and collect data, it is essential to establish a reasonable experimental platform [29–32]. To obtain sufficient and accurate data, as well as the authenticity of the experiment, the equipment structure used in this experiment is shown in Fig. 10. The control unit adopts a combination of MCU and FPGA, SHT30 is a high-precision temperature and humidity sensor, and 4GCAT1 is an IoT module using the CAT1 frequency band. The product is shown in Fig. 11. MCU is mainly responsible for network communication and peripheral control, and FPGA is mainly responsible for the implementation and acceleration of T-S FNN algorithms.

To show the effectiveness of the proposed method, under the coordination of the local railway department, experimental equipment was installed in a local operating railway section for experiments. The experimental setting is shown in Fig. 12. The device within the red outline is the catenary weights, and the distance between the weights and the top pulley is the displacement of the catenary. Firstly, collect catenary data in a static state to study the vibration characteristics of the catenary. For this purpose, the equipment is installed on the weights at the end of the catenary and uses high-precision thermal resistance as a sensor to obtain temperature data. Then, a rope sensor modified by an encoder is used to measure the displacement of the catenary.

Due to provisions, the complete dataset used in this experiment cannot be fully disclosed. However, for reference, some samples have been provided in the supplementary materials. The uploaded dataset sample can be found in Table S1.

To illustrate the effectiveness of the proposed training algorithm, we compare the performance of the proposed adaptive algorithm with the traditional backpropagation (BP) algorithm. The experimental dataset consists of 24-h data collected at a railway site. The experimental results are presented in Table 1, and the comparison of iteration times is depicted in Fig. 13.

The results indicate that the suggested algorithm exhibits rapid convergence, with an increase in error of only 0.15 mm. Despite the slight growth in error, it has minimal effect on the prediction of aging. The purpose of this effect is to eliminate vague rules that bear minimal influence on the ultimate output result, thus enhancing computational efficiency.

To validate the effectiveness of replacing the KF measurement error, we conduct a comparison between using sensor internal noise and vibration information within the same dataset. The parameter of comparison is the disparity between the filtered and theoretical values. As illustrated in Fig. 14, the comparison showcases the filtering performance on three days of random data. Further details regarding the experimental data comparison can be found in Table 2, and visual comparisons of the filtered data are available in Figs. S1–S3.

It can be seen from the results that using vibration information as the measurement error covariance of the KF can significantly reduce the amplitude and fluctuation of the displacement difference. The comparison of the standard deviation shows that using vibration data can significantly reduce the fluctuation of the catenary displacement. The reason for this result is that using catenary vibration information as observation noise can more accurately reflect the interference carried by sensors when detecting catenary displacement.

To assess the progressive nature of the proposed algorithm, we compare it with the unscented Kalman filter (UKF) in [10] and the KalmanNet in [13]. The test set comprises three days of data collected from sensors, with the comparison parameter remaining as the difference between the filtered and theoretical values. The results of this comparison are depicted in Fig. 15, and detailed experimental data can be found in Table 3. For a visual representation of the original filtered data comparison, refer to Figs. S4–S6.

From the experimental results, compared to the traditional KF, the UKF, and the KalmanNet have a certain filtering effect. However, the filtering effect of the proposed algorithm is the most visible because it has made proprietary improvements to the problem discussed in this paper. From Fig. 15c, it can be seen that even for relatively less obvious interference, the proposed algorithm can still achieve better filtering performance.

To further verify the actual aging prediction ability of the proposed algorithm, a prediction accuracy comparison is also conducted. In this experiment, the fault diagnosis threshold is set to the displacement difference of the catenary greater than 20 mm and the duration exceeding 10 min. The comparative experimental results are shown in Table 4. The results show that the proposed algorithm greatly reduces the false alarm rate of faults caused by environmental factors or pantograph influence in the railway catenary system.