A Real-Time Monitoring System for Cable Tension with Vibration Signals Based on an Automated Algorithm to Sieve Out Reliable Modal Frequencies

Cables or suspenders are the critical force-transmitting components of cable-supported bridges, and their timely tension monitoring is consequently the most important issue in the corresponding structural health monitoring. However, very few works regarding the full automation of vibration-based tension estimation have been reported in the literature. To develop a monitoring system of cable tension based on real-time vibration signals, this research frst employs an efcient stochastic subspace iden-tifcation (SSI) method with tailored parameter selection to continuously identify the three frequencies of adjacent modes for the cables of Mao-Luo-Hsi Bridge. More importantly, an automated sieving algorithm is delicately established to obtain the stable modal frequencies by making the best of the specifc modal frequency distribution for cables. Te ratios between the frequency values identifed from SSI analysis are exhaustively checked to systematically extract the qualifed cable frequencies and decide their corresponding mode orders. Te tension is fnally computed with one available cable frequency according to the priority order predetermined by the statistics of identifcation rate. Demonstrated by analyzing the vibration signals measured from the stay cable of Mao-Luo-Hsi Bridge in real time for two full years, the efectiveness and robustness of this real-time monitoring system have been extensively testifed. Te long-term success rates for the immediate determination of dependable tension are found to be perfect for 15 of the 18 investigated cables. As for the other three cables, their corresponding success rates are still higher than 99.99% with very few cases of absent or false tension values.


Introduction
Cable-supported bridges generally include suspension bridges, cable-stayed bridges, and through-type arch bridges. Due to the demand for longer spans and aesthetic consideration, these types of bridges have become increasingly popular worldwide to rise as the mainstream in modern bridge engineering. Cables or suspenders are the critical force-transmitting components of those cablesupported bridges, and their timely tension monitoring is consequently the most important issue in the corresponding structural health monitoring (SHM). Te sudden collapse of Nan-Fang-Ao Bridge occurred in Taiwan in 2019 can be taken as a typical example. Such a tragic accident caused the casualties of six deaths and twelve injured people together with heavy damage of three fshing vessels and one tank truck. Te suspenders of this arch bridge successively broke in a very short time to result in the loss of suspension for the main bridge girder and its subsequent fall into the water. After the cruel lessons learned from the incident of Nan-Fang-Ao Bridge, much more resources from the central and local governments of Taiwan have since been invested to conduct extensive inspection and SHM of cable-supported bridges.
In practical tension estimation and monitoring, the ambient vibration method [1][2][3] is most commonly employed because of its expedient operation and the unsophisticated analysis benefting from the one-dimensional geometry of cable. Application of the vibration-based tension estimation method typically starts with identifying the modal frequencies of a cable from its ambient vibration measurements and then calculates the cable tension based on an analytical formula requiring at least a cable frequency, the given vibration length, and its mass per unit length. More involved efects such as fexural rigidity [4,5] and gravity sag [6,7] have also been well studied in the literature to improve the accuracy of this method. Nevertheless, the rubber constraints and special anchorage systems are normally designed near the boundaries of modern cables. Tese devices can unfriendly obstruct the accurate determination of their vibration lengths and lead to non-negligible errors in tension estimation. To tackle this difculty and avoid the troubles of complex modelling in the neighborhood of cable boundaries, a new methodology has been frst propelled and systematically developed by the authors [8][9][10][11][12]. An original idea was adopted in those studies to directly obtain the efective vibration lengths for desired modes by further combining the information of mode shapes. Collection of multiple synchronized measurements on a cable is required in this method for the identifcation of mode shape values at diferent sensor locations. Te efective vibration length corresponding to each interested mode can then be independently attained by optimally ftting the sinusoidal component of the mode shape. With the identifed modal frequencies and the determined efective vibration lengths for a few selected modes, the cable tension and fexural rigidity are eventually solved by simple linear regression techniques. Tis innovative method has so far been applied to conduct the cable tension estimation for more than 30 cable-supported bridges in Taiwan. A number of other researchers have also joined to explore the same concept of efective vibration length [13][14][15].
Following the galloping progress of sensor technology in recent years, much more cable-supported bridges have been equipped with permanent vibration sensors on their cables to carry out real-time and long-term tension monitoring. For establishing an accurate cable tension monitoring system, it needs to conduct the abovementioned method based on multiple vibration measurements only once in the initial stage. With the corresponding efective vibration lengths determined, one sensor installed on each cable would be adequate to supply the vibration measurement, from which the cable frequencies can be immediately identifed for continuous tension monitoring. Under normal operating conditions of bridges, it is usually sufcient to accurately identify the modal frequencies of cables by applying the conventional discrete Fourier transform (DFT) technique. To achieve real-time monitoring in practical applications, however, the continuously stable identifcation of cable frequencies becomes a great challenge. Te ambient vibration signals can be very weak and highly polluted by noise during the hours without trafc. Accordingly, a more resilient modal identifcation method with its guaranteed performance is necessary in such long-term operations.
Operational modal analysis (OMA) has been well advanced in the past few decades to yield several robust algorithms for obtaining reliable modal parameters simply from the collected output signals. Among all these methods, stochastic subspace identifcation (SSI) may attract the most popular applications on civil engineering structures [16][17][18][19]. Tis technique has been successfully practiced for the longterm SHM of historical architectures [20,21], arch dams [22], skyscrapers [23,24], and long-span bridges [25,26]. In these cases, a regularly encountered problem is that diferent values of the time lag parameter and the system order parameter prescribed in SSI analysis would produce varied modal parameters. Te conventional stabilization diagram and proper discrimination criteria according to theories like clustering analysis [27][28][29] have been established to deal with such a difculty.
During the past decade, the development is further focused on automated SSI approaches that are capable of processing large datasets from long-term monitoring systems without the need of human interventions [30][31][32]. Particularly in the last few years, many eforts have been made to accomplish automatic SSI-based monitoring. Zonno et al. [33] proposed a multistage methodology combining the data-driven SSI together with the techniques of hierarchical clustering and adaptive modal tracking for the automatic modal parameter identifcation of historical buildings in Portugal and Peru. Li et al. [34] developed a three-stage automated algorithm by incorporating the second-order blind identifcation with the covariance-driven SSI to conduct continuous monitoring for a concrete arch gravity dam in China. Sadeqi et al. [35] established an automated procedure integrating a simplifed SSI algorithm in conjunction with the statistical feature extraction of identifed modal parameters. In this research, three years of continuous data from a modern tower in Iran were processed to efectively perform its SHM. Sun et al. [36] used a multistage clustering approach by defning the hierarchical clustering threshold to handle the results from the covariance-driven SSI and automatically identify the modal parameters of a cable-stayed bridge in Australia. Zhang et al. [37] lately amalgamated a fast density peak clustering algorithm with the covariance-driven SSI to automatically select the appropriate structural modes from the stabilization diagram and applied it to a cable-stayed bridge in China.
After the abovementioned accident of Nan-Fang-Ao Bridge, the authors earned an opportunity to help develop a real-time tension monitoring system based on the vibration signals of Mao-Luo-Hsi Bridge that will be introduced in the next section. While a few new techniques [38][39][40] were recently suggested to efectively identify the time-varying cable tension, very few works regarding the full automation of vibration-based tension estimation have been reported in the literature. Jin et al. [41] frst applied a modifed automatic multiscale-based peak detection algorithm on the power spectral density function. Te median absolute deviation following baseline corrections is then determined to extract the modal frequencies of cables for autonomous tension monitoring. Jeong et al. [42] constructed an automated cable tension monitoring system by proposing a fully automated peakpicking algorithm using a region-based convolution neural network. Even so, only short durations of vibration data from real stay cables were tested in these two recent studies to demonstrate the efectiveness of their proposed methods. No long-term results were actually presented to unambiguously verify the feasibility of a totally automatic cable tension monitoring system in practice.
Enhanced with an alternative stabilization diagram and a hierarchical sifting process, a novel covariance-driven SSI methodology was also proposed in a recent work by the authors [43]. In addition to its original application in stay cables [43], this improved SSI algorithm has been further demonstrated to sturdily identify accurate modal parameters of cable-stayed bridges [44] and ofce buildings [45,46] from long-term vibration measurements. On the basis of employing such an efective SSI algorithm to continuously identify the modal frequencies, a real-time tension monitoring system is developed and applied to the cables of Mao-Luo-Hsi Bridge. In long-term SHM, the occurrence of abnormal signals due to various possible sources is virtually inevitable and may signifcantly obstruct the subsequent analyses. Te frst phase of this tension monitoring system aimed to automatically detect and process the signal anomaly of cable vibration measurements has been reported in a recent article [47]. Te current paper will focus on elaborating the most critical ingredient of the tension monitoring system, which is an automated algorithm innovatively created to robustly sieve out the stable modal frequencies of the cable based on the characteristic of their steady ratios. Furthermore, the real-time cable tension monitoring results of Mao-Luo-Hsi Bridge for a long duration of two years will be also presented to solidly attest the robustness and performance of the developed tension monitoring system in real applications.

Mao-Luo-Hsi Bridge and Its Cable Vibration Measurements
With its photo displayed in Figure 1(a), Mao-Luo-Hsi Bridge transverses Mao-Luo River to connect National Highway 3 and the east end of Bagua Mountain Tunnel. Open to trafc since 2002, this bridge has been serving as a main artery between the two towns of Yuanlin of Changhua County and Caotun of Nantou County in Central Taiwan. It is an unsymmetrical single-pylon cable-stayed bridge containing a main span with a length of 119 m and a side span with a length of 51 m, both built with steel box girders. A steel arch crossing over the bridge deck and with a height of 60 m is adopted as its pylon. As illustrated in Figure 1( Table 1 for reference. It should be noticed that all the steel strands of each stay cable are encased in a high-density polyethylene (HDPE) tube, but no grouting material is flled between them. A typical cable cross section is depicted in Figure 1(b), and the mass per unit length for each cable is also listed in Table 1 by considering both the steel strands  and the HDPE tube. To ensure the structural safety of Mao-Luo-Hsi Bridge, an SHM system was installed by the Directorate General of Highways and started its operation in November of 2020. Te permanent sensors deployed on the bridge include one anemometer, 11 temperature sensors, one seismograph, one biaxial inclinometer, two displacement transducers, seven settlement gauges, 16 strain gauges, and 36 accelerometers. Te detailed description given here will be limited to the accelerometers which are directly relevant to the investigation of this study. On each stay cable, one force balance accelerometer SA-10 produced by Sara Electronic Instruments in Italy was installed at the position around 5 m high above the deck to continuously conduct its in-plane acceleration measurements. For the convenience of timely cable tension monitoring, the monitoring system is designed to output the measured acceleration signals for all cables every 15 minutes and then store them into a separate data fle for subsequent analyses. Terefore, it would generate 96 measurement data fles every day for each stay cable to attain real-time tension monitoring.

Identification of Cable Frequencies with Stochastic Subspace Methodology
Multiple vibration measurements on each cable of Mao-Luo-Hsi Bridge were conducted in the initial stage to decide its fexural rigidity and efective vibration lengths for the dominant modes [8][9][10][11][12]. It was found that the identifed fexural rigidity of each cable is consistently larger than the lower limit estimated by considering a circular cross section with an area equal to the sum of all its strands. Besides, the corresponding values for diferent cables follow the same order as that sorted by their numbers of steel strands. Both tendencies certify that the cable's fexural rigidity such determined is reasonable and can be confdently used as a known parameter in long-term monitoring. Merely one modal frequency is required to estimate the real-time tension value for each cable with the given fexural rigidity and efective vibration length. Te SSI methodology recently developed by the authors [43][44][45][46] is employed in the current work aimed to continuously identify three cable frequencies of adjacent modes. Such a strategy is adopted to guarantee the identifcation of at least one cable frequency at any time considering the wide variability possibly occurring in longterm monitoring. Tis SSI algorithm is succinctly reviewed in the frst subsection. Based on the preliminary analysis with one week of cable vibration measurements for Mao-Luo-Hsi Bridge before the ofcial operation of its SHM system, discussions on the determination of frequency range and the selection of SSI parameters are also presented in the second and third subsections.

Alternative Stabilization Diagram and Hierarchical
Shifting Process. Te improved SSI algorithm follows the covariance-driven approach, which generally starts from the Structural Control and Health Monitoring state space description for a linear dynamic system with n degrees of freedom (DOF). It assumed that the output measurements for such a system are conducted using the sampling time increment ∆t to have the l × 1 output vector y (j) at the time instant j∆t. Te output vectors measured at M consecutive time instants (j � 0, 1, 2, · · · · · · , M − 1) Multiplication of the two equally divided submatrices Y p and Y f of the Hankel matrix would result in the approximation of the so-called Toeplitz matrix T. Singular value decomposition (SVD) of T can next be carried out to obtain where n 1 � il − 2n, U and V are orthogonal matrices, and S is a quasi-diagonal matrix with positive diagonal elements sorted in a decreasing order. Further taking then the discretized system matrix A in the state space can be solved by where ⊕ symbolizes the pseudo-inverse operation.
According to the theory of linear systems, the modal frequencies ω j 's, the damping ratios ξ j 's, and the mode shape vectors at the output measurement locations φ j 's can be ultimately determined from the eigenvalues and eigenvectors of A. From the above review, it is evident that the time lag parameter i in equation (1) and the system order parameter n in equation (2) need to be specifed in conducting the SSI analysis. A value no less than the number of physical modes within the interested frequency range is normally required for the system order parameter to ensure the incorporation of unambiguously contributing modes. As for the time lag parameter, the identifed modal parameters should be insensitive to it if the excitation is close to the white noise and stationary assumptions on which the derivation of SSI is based. Consequently, the determination of modal parameters by the SSI technique is usually accompanied with the stabilization diagram to check the stability of the identifed frequency values with the increasing value of n under a designated value of i. In applications of large-scale civil structures typically subjected to narrowly banded or non-stationary excitations, however, the performance of conventional stabilization diagrams may be signifcantly altered by diferent selections of the time lag parameter [16]. Te authors [43] recently tackled such a difculty by frst establishing a threshold i c of the time lag parameter to ensure stable identifcation results: where s � 1/∆t denotes the sampling rate of measurement and F 1 signifes the fundamental frequency of structure.
With the criterion of equation (5), an alternative stabilization diagram was then proposed by displaying the identifed frequency values with the increasing value of i under a fxed value of n [43]. Te lower limit i min of the time lag parameter is suggested to be set as the critical threshold i c presented in equation (5), and its upper limit i max is decided by the need in the subsequent processes to extract reliable modal parameters.
Even with the construction of an alternative stabilization diagram, further eforts are needed to complete the task of automatically and consistently extracting clustered modal parameters from this diagram. A three-stage hierarchical sifting process in the order of modal frequency, damping ratio, and mode shape vector was additionally advanced by Structural Control and Health Monitoring the authors [43,48]. Te following review will simply focus on the frst sifting stage because merely the modal frequencies are necessary for the estimation of the cable tension monitored in the current study. With the alternative stabilization diagram where the time lag parameter is increased from i min to i max � i min + ∆i, the conduction of SSI analysis for all these ∆i + 1 cases would produce multiple sets of modal parameters: In equation (6), the modal frequency f j � ω j /2π is in the unit of Hz and N represents the total number of frequency values falling in the inspected frequency range. Te frst stage of the hierarchical sifting process begins with sorting all frequency values appearing in the stabilization diagram in ascending order: It is designed to conveniently extract the clustered frequency values for each mode using a simple clustering index: to indicate the span of any adjacent K frequency values. Te extraction of all locally clustered frequency values can then be easily accomplished by just looking for all local minima can be further checked to confrm that each extracted group of frequency values would satisfy a prescribed threshold of concentration.

Determination of Target Modes and Frequency Range.
Te improved SSI methodology reviewed in the previous subsection is applied in this research to conduct tension monitoring for stay cables. Teoretically, any modal frequency F j can be used to estimate the cable tension by the following formula: with predetermined efective vibration length L j and fexural rigidity EI (E signifes Young's modulus and I stands for the cross-sectional area moment of inertia). But for long-term monitoring, the selection of a particular or a few stably identifable modes needs to be made such that systematic operations can be implemented. Since the steady identifcation of three consecutive modal frequencies of a cable is aimed in the current work, the examined frequency range for each cable is determined by confdently covering its three chosen modal frequencies. In other words, a corresponding band-pass flter is exerted on the acceleration signal of each cable before conducting the SSI analysis.
With 7 days of cable measurements from 2020/10/21 to 2020/10/27, the characteristics of mode contribution for each cable are frst investigated in this study to decide its three consecutive modes as the identifcation target and the corresponding frequency range. Taking Cables 107  . It is apparently demonstrated in these two fgures that the modal frequencies of a stay cable regularly hold values approximately in integer multiples of its fundamental frequency and are accordingly easy to be distinguished. More importantly, the second, third, and fourth modes are the three most contributing modes with three highest peaks for both cables. Te trend shown in Figures 2 and 3, however, is valid only for one data fle including the selected 15 minutes of cable measurements. A complete picture can be more distinctly observed if the results for all 672 measurement data fles from 2020/10/21 to 2020/10/27 are scrutinized. According to the peak values exhibited in the FAS of each 15-minute time interval, statistics of the three most dominantly contributing modes in one whole week are illustrated in Figure 4. For Cable 107, the percentage of its second mode to rank the frst three is 22% + 25% + 19% � 66%, that of its third mode is 51% + 29% + 9% � 89%, and that of its fourth mode is 12% + 22% + 29% � 63%. Regarding Cable 201b, the percentage of its second mode to rank the frst three is 65% + 22% + 7% � 94%, that of its third mode is 22% + 29% + 24% � 75%, and that of its fourth mode is 7% + 25% + 49% � 81%. All other modes of either Cable 107 or Cable 201b are with less percentage to rank the frst three than the three modes listed above. Terefore, the best choice of three consecutive modes would be from the second to the fourth mode for both cables.
Based on the aforementioned statistics, the frequency range of the band-pass flter for each cable can be subsequently determined. As indicated in Figure 2(b) by the interval between two vertical red lines, the considered frequency range for Cable 107 is taken from f min � 1.8 Hz (roughly the middle value for the frequencies of the frst and second modes) to f max � 5.4 Hz (approximately the middle value for the frequencies of the fourth and ffth modes). Likewise, the investigated frequency range for Cable 201b is also shown in Figure 3(b) between the two vertical red lines at 2.2 Hz and 6.5 Hz. Te three consecutive modes and the frequency range selected for each cable in the frst two cable planes of Mao-Luo-Hsi Bridge are listed in Table 2. It can be found from this table that the second to fourth modes are identically chosen for the longer cables labelled with 01a, 01b, 02, 06, 07, and 08 in each cable plane. On the other hand, the frst to third modes are consistently picked for the shorter cables labelled with 03, 04, and 05 in each cable plane. Such a trend is the same for the cables in the other two cable planes, whose results are not comprehensively included in the current paper due to the limitation of length. 6 Structural Control and Health Monitoring

Parameter Selection for Real-Time SSI Analysis.
Following the guidelines described in Section 3.1, the system order parameter n, the range of time lag parameter i, and the parameters for the frst sifting stage are selected for the realtime SSI analysis of each cable. Inspecting the FAS for the measurement on Cable 107 exhibited in Figure 2(b), it is obvious that three prominent peaks corresponding to three consecutive cable modes appear in the examined frequency range between 1.8 Hz and 5.4 Hz. As a result, it is proper to fx the system order parameter at n � 3 in this case for establishing the alternative stabilization diagram. Further checking the FAS in Figure 3(b) obtained from the measurement on Cable 201b, there exist four noticeable peaks including one clearly not belonging to a cable mode. In this case, n � 4 would no doubt be an appropriate choice in constructing the alternative stabilization diagram for more  steady identifcation of three cable frequencies. It is noteworthy that higher values of n can actually be employed in these two cases to obtain stable identifcation as well. Nevertheless, more clustered frequency values associated with higher values of n in the alternative stabilization diagram would induce severer challenges to sieve out reliable cable frequencies as will be dealt with in the next section.
Since the frequencies of three consecutive modes are targeted for each cable, the assigned value of n � 3 is the frst priority taken in this study as long as three signifcant peaks are typically observed in its examined frequency range like the case of Cable 107. But for a few cases such as Cable 201b where additional peaks may also occur, the selection of n � 4 is found to be sufcient as will be seen in the next section.
With the fundamental frequency F 1 ≈ 1.2Hz of Cable 107 and the sampling rate of measurement s = 200 Hz, the corresponding critical threshold can be determined from equation (5)   As for the assigned values of n, i min , and i max for all cables in the frst two cable planes, they are also summarized in Table 2.

Automated Sieving Algorithm for Reliable Cable Frequencies
After going through the frst stage of sifting, K � 51 survivors in each clustered group of Figure 5 Figure 7. As for the case of Cable 201b where n � 4 is adopted in the SSI analysis, Figure 8 Figures 7 and 8) need to be efectively sieved out. In addition, the correct determination has to be made for the mode orders corresponding to the remaining cable frequencies. Taking advantage of the specifc modal frequency distribution of cables, an automated sieving algorithm will be developed in this section to satisfy the above two essential features such that the accuracy and robustness of the subsequent cable tension estimation can be secured in long-term monitoring.

Difculties and General
Ideas. It may not seem complicated for human beings to pick out the anomalous red dots in Figures 7 and 8 and decide the mode orders corresponding to the three or fewer cable frequencies in each interval. To attain fully automatic cable tension monitoring, however, a comprehensive discrimination and classifcation algorithm has to be rigorously established. It should be especially pointed out that this task cannot be conveniently handled by simply setting a range of variation for each cable frequency. A wide range would not suit the purpose to efectively sieve out the frequency values not corresponding to cable modes such as the red dot in Figure 7. On the other hand, a narrow range may not reasonably cover the frequency variation induced by environmental factors such as temperature and live load. More importantly, the occurrence of bridge damage can lead to signifcant variations of cable tension and consequently substantial changes in cable frequency. Under such circumstances, it is nothing but fshing in the air to conduct the discrimination of cable frequencies based on a fxed range of variation. Te special characteristic for the modal frequencies of a stay cable to hold values close to an arithmetic sequence (particularly for the frequencies of lower modes where the Structural Control and Health Monitoring efect of the fexural rigidity is relatively trivial) is exploited in this study to tackle the above difculty. In other words, the ratio between any two obtained frequency values after the SSI analysis can be checked to systematically determine whether a frequency value is associated with a cable mode or not. Taking again Cables 107 and 201b as examples, the targets for both cases are similarly the three consecutive frequencies from the second to the fourth mode (F 2 , F 3 , and F 4 ) as mentioned in Section 3.2. Tus, it should be feasible to efciently sieve out the red dots in Figures 7 and 8 with a numerical algorithm based on the three conditions F 4 /F 2 ≈ 2.0, F 4 /F 3 ≈ 1.333, and F 3 /F 2 ≈ 1.5 from reasonable frequency ratios. Even if the situation for Cable 107 in the interval from 00:15 to 00:30 does occur, it is still not difcult to judge that the two remaining frequency values after the sieving algorithm belong to the second and third modes. Similarly, the other situation that occurred in the interval from 01:15 to 01:30 can also be well handled to confrm that the two sieved frequency values correspond to the third and fourth modes.   Overall, the SSI analysis reviewed in the previous section would be frst performed on each interval of 15minute cable acceleration measurement to obtain the identifed frequency values at each interval. Te sieving algorithm elaborated in this section is then applied to exclude the possibly occurring red dots as shown in Figures 7 and 8 and specify the mode orders corresponding to the cable frequencies remaining. Finally, the cable tension at each interval is calculated by equation (10) according to the prescribed mode priority order that will be discussed in the next section. For establishing an automatic algorithm, criteria on the ratios between frequency values are imposed with a threshold of tolerance δ determined from the statistical tests described in the next subsection. Furthermore, the extreme situations where only one frequency value is identifed after the SSI analysis with no frequency ratio available are also considered in the current work. Te solely identifed frequency value in such cases will be compared with the average of each targeted cable frequency in a certain period (say, a week or a month). Another threshold of tolerance c also decided by the statistical tests is additionally adopted to search for the corresponding mode under such circumstances. All details of the sieving algorithm will be given in the next subsection.

Detailed Descriptions of Algorithm.
For the convenience of explanation, the whole automated sieving algorithm to extract reliable cable frequencies and determine their mode orders can be frst illustrated with the fowchart for the mainstream process shown in Figure 9. Tis main fowchart also includes three branch processes whose fowcharts are displayed in Figures 10-12. If m 1 < m 2 < m 3 denote the three consecutive mode orders selected for each cable as listed in Table 2, the three corresponding theoretical frequency ratios can then be directly computed by α 31 � m 3 /m 1 , α 32 � m 3 /m 2 , and α 21 � m 2 /m 1 . As discussed in Section 3.3 and listed in Table 2, the assigned value of the system    Structural Control and Health Monitoring order parameter in conducting the SSI analysis for each cable can be either n = 3 or n = 4. Tat is to say, there are at most three identifed frequency values after the SSI analysis when n = 3 or four when n = 4. Accordingly, two entrances corresponding to these two cases are designed for the main fowchart in Figure 9. In each time interval, assume that four frequency values f � f 1 f 2 f 3 f 4 are identifed when n = 4 and three frequency values f � f 1 f 2 f 3 are obtained when n = 3, both arranged in an ascending order. If the number of identifed frequency values is less than three for the case of n = 3 or less than four for the case of n = 4 in any interval, the last one or more elements in the vector f would be flled with the value zero to serve as an indicator for the subsequent automatic checking. No matter starting from the entrance for n = 4 or n = 3, the frst step is always to input the identifed frequency vector f.
For the case of n = 4, the condition of f(4) ≠ 0 is frst inspected to confrm the identifcation of four frequency values after conducting the SSI analysis for the considered time interval. If the answer is positive, the matrix P 4 �

12
Structural Control and Health Monitoring associated with the arrangement of the six frequency ratios in equation (11) can also be defned and will be used later. P 4 and Q 4 can be subsequently input into the branch process (a) as illustrated in Figure 10 to deal with the cases possessing four identifed frequency values or three. When the four identifed frequency values are confrmed, this branch process will frst carry out the following operation on P 4 : Tis operation takes the absolute value for each element in P 4 and then stores the minimum value of each column at X = [X 1 X 2 X 3 ] together with recording the row positions of these minimum values at w = [w 1 w 2 w 3 ]. It should be particularly noted that X 1 , X 2 , and X 3 actually represent the smallest relative errors for the frequency ratios compared to α 31 , α 32 , and α 21 , respectively. Hence, sequential inspection on whether X 1 , X 2 , or X 3 exceeds the threshold of tolerance δ as schemed in Figure 10 would explicitly discriminate which three or which two identifed values are cable frequencies.
With the help of the allocation matrix Q 4 , the corresponding mode orders for these cable frequencies can also be handily matched to output the determined cable frequencies F m 1 , F m 2 , and F m 3 . A zero value for any of these output cable frequencies would imply that no identifed value can correspond to that mode.
But if X 1 , X 2 , and X 3 all go beyond δ, then there is at most one among the four identifed values to be possibly taken as the modal frequency of the cable and this part of sieving is handled by the branch process (c) demonstrated in Figure 12. Under such circumstances, the check of frequency ratios would not provide any useful information and the only left approach may be to compare with the reference values of cable frequencies. Tis research adopts the average frequency values F m 1 , F m 2 , and F m 3 for the three selected modes in the past one week to serve as the reference. By computing the matrix

f(1) f(1) f(1) f(2) f(2) f(2) f(3) f(3) f(3) f(4) f(4) f(4)
the four elements in each column of R 4 would indicate the relative errors for the four identifed frequency values compared with the average frequency value of each selected mode. R 4 is then input into the branch process (c) to perform the following operation: Similar to equation (13), equation (15) takes the absolute value for each element in R 4 , collects the minimum value of each column at Y = [Y 1 Y 2 Y 3 ], and also saves the row positions of these minimum values at u = [u 1 u 2 u 3 ]. Again, Y 1 , Y 2 , and Y 3 obtained from equation (15) represent the smallest relative errors for the four identifed frequency values compared to F m 1 , F m 2 , and F m 3 , respectively. Sequential examination on whether Y 1 , Y 2 , or Y 3 is no larger than the threshold of tolerance c as arranged in Figure 12 f, R would fnd the only identifed value to be qualifed as a cable frequency and its corresponding mode. Such a value can then be output to the only appropriate choice among F m 1 , F m 2 , and F m 3 . Te zero value is fnally assigned to the other two left candidates. In addition to the situation with four identifed frequency values considered above, the circumstance with three identifed frequency values after conducting the SSI analysis for the examined time interval will be discussed next. Te occurrence of such a situation can be verifed by satisfying f(4) � 0 and f(3) ≠ 0 in the case of n = 4 or checking f(3) ≠ 0 in the case of n = 3. Te branch process (a) in Figure 10 is also applicable if simply three frequency values are identifed, but the inputs need to be replaced with

f(3)/f(1) f(3)/f(1) f(3)/f(1) f(3)/f(2) f(3)/f(2) f(3)/f(2) f(2)/f(1) f(2)/f(1) f(2)/f(1)
and a new allocation matrix: Te three elements in each column of P 3 represent the relative errors for all possible three ratios produced by any two of the three identifed frequency values compared with the theoretical frequency ratios α 31 , α 32 , and α 21 . Furthermore, R 4 defned in equation (14) has to be also substituted by when it is necessary to enter the branch process (c) in Figure 12 for dealing with the sieving of only one possible cable frequency. As for the situation with only two identifed frequency values after the SSI analysis, its appearance can be detected if f(4) � f(3) � 0 and f(2) ≠ 0 in the case of n = 4 or if f(3) � 0 and f(2) ≠ 0 in the case of n = 3. Te branch process (b) in Figure 11 is designed for such a circumstance and requires an even simpler input matrix:  Te single element in each column of P 2 indicates the relative error for the ratio of two identifed frequency values compared with the theoretical frequency ratios α 31 , α 32 , and α 21 . Tis branch process will frst execute the following operation on P 2 : i.e., taking the absolute value for each element in P 2 and then storing the minimum value of all three elements at Z together with recording its original column position at v. Te obtained value of Z represents the smallest relative error between the single frequency ratio and its closest partner among α 31 , α 32 , and α 21 . If the condition check of Z ≤ δ is positive, it means that the two identifed values would correspond to two cable frequencies. Further with the value of v, these two cable frequencies can fnd their corresponding mode orders and are output to the appropriate two of F m 1 , F m 2 , and F m 3 . A zero value will also be assigned to the remaining one without any correspondence. If the value of Z exceeds the tolerance δ, on the other hand, then at most one of the two identifed values is possibly regarded as a cable frequency and this part of sieving can again be conducted by the branch process (c) in Figure 12. But in this case, the input should be changed as which is similar to R 4 defned in equation (14) and R 3 defned in equation (18) Figure 12 is also applicable in this case but requires a new input matrix: In the automatic sieving algorithm consisting of the mainstream process of Figure 9 and the three branch processes of Figures 10-12, the selection of the tolerances δ and c also plays a pivoting role for its success in long-term SHM. It is always a great challenge to oscillate between a conservative choice that may mistakenly sieve out the modal frequencies of the cable and a loose criterion that could fail to efectively exclude a few unqualifed frequency values. For a more reasonable determination of the thresholds of tolerance δ and c, a trial value of 5% is frst assigned for both to conduct the proposed sieving algorithm after carrying out the SSI analysis on the cable measurements from 2020/10/21 to 2020/10/27. Te maximum and minimum values for the relative errors of the three sieved frequency ratios F m 3 /F m 1 , F m 3 /F m 2 , and F m 2 /F m 1 with respect to their corresponding mode order ratios α 31 , α 32 , and α 21 in one week are arranged in Table 3 for the cables in the frst two cable planes. Besides, similar statistics for the relative errors of the three sieved cable frequencies F m 1 , F m 2 , and F m 3 with respect to their corresponding weekly average values F m 1 , F m 2 , and F m 3 are also listed in Table 4. More specifcally, Tables 3 and 4 are prepared to testify the legitimacy of the choices of δ and c, respectively. It is evident from both tables that there are rather few cases to hold the relative errors slightly larger than 3% and these values are highlighted with Structural Control and Health Monitoring 15 slanted bold numbers. Tis means that very limited frequency values will be further sieved out if both tolerances are lowered to 3%. In other words, δ � c � 3% should be an excellent choice that can keep a good balance between identifcation rate and frequency variation. Consequently, such thresholds are taken in the subsequent long-term monitoring of this study and the results reported in the next section would demonstrate its efectiveness.
With the automatic sieving algorithm developed in the current section, the complete analysis for one week of acceleration measurements from 2020/10/21 to 2020/10/27 is frst conducted to investigate its applicability in all 36 stay cables of Mao-Luo-Hsi Bridge. Even though various types of sieving results can be observed, they all lucidly refect the efcacy of the proposed algorithm. In the case of Cable 107 (n � 3) as illustrated in Figure 13, only four frequency values indicated by red dots are sieved out in one week and three cable frequencies can be typically obtained in most time intervals. Regarding the case of Cable 201b (n � 4) as plotted in Figure 14, a number of inappropriate frequency values denoted by red dots cannot pass the sieving algorithm and three cable frequencies are eventually yielded in the majority of time intervals. Another sieving type can be found in Figure 15 for Cable 104 (n � 3), where a few unqualifed frequency values are excluded and two cable frequencies are fnally attained in a greater part of instances. It is especially noteworthy that several values very close to the frequency of the frst cable mode are removed Table 3: Relative errors of sieved frequency ratios for cables in the frst two cable planes.   by the sieving algorithm in this case. Closer examination reveals that there are actually two identifed frequency values from the SSI analysis for these time intervals to locate in the neighborhood of the fundamental frequency and one of them defeated by a narrow margin is appropriately sieved out by the algorithm. Tis can be directly recognized from the amplifed illustration for the part near 00:00 of 10/23 in Figure 15(a).

Long-Term Monitoring Results of Cable Tension
Other than a reliable cable frequency identifcation methodology and an automated sieving algorithm as described in the previous two sections, the robustness of long-term cable tension monitoring may also rely on a proper choice of cable frequency to compute the tension value from equation (10). In this section, the priority order of cable frequencies for tension determination will be discussed in the frst subsection, followed by presenting the cable tension monitoring results for a total of two years together with several observations in the second subsection.

Priority Order of Cable Frequencies for Tension
Determination. In the current work, the decision of priority order in three target modes for each cable to compute the tension value is also based on one week of analyzed results from 2020/10/21 to 2020/10/27. For example, the identifcation rates for the sieved cable frequencies shown in Figures 13(b)-15(b) are reckoned and then ranked as graphed in Figure 16.  choice of Mode 2 (frst), Mode 4 (second), and Mode 3 (third). Te priority order of mode selection for all cables in the frst two cable planes is also listed in Table 2.
According to the determined priority order, the tension value at each time interval can then be estimated with equation (10) using the mode with the highest priority. Te tension histories of Cables 107, 201b, and 107 in one week are plotted in Figure 17 following from the sieved cable frequencies demonstrated in Figures 13(b)-15(b). From Figure 17, it is apparent that the tension values at all 672 time intervals can be consistently acquired and they all fall in a reasonable range of variation for each cable. It can be further observed that there exist diferent tendencies of variation for these three cables. For Cable 201b, its tension variation in one week can reach 30 tons (about 10%) and the daily maximum normally occurs during the daytime. Tis suggests that live load may be a dominant factor to induce the tension variation of a main cable like Cable 201b in this bridge. Regarding Cable 104, its weekly tension variation is only restricted to 7 tons (around 5%) and the daily minimum clearly happens at the daytime. In this case, temperature may be a crucial environmental factor to cause the change of cable tension, which is also supported by the smoother trend of variation. As for Cable 107, its tension variation in one week is also limited and shows no strong clues for the main infuencing factor.

Long-Term Results of Cable Tension and Observations.
Combining the previously reported anomaly processing algorithm [47] and the robust tension determination algorithm developed in this study, a real-time monitoring system for cable tension has ofcially started its operation on Mao-Luo-Hsi Bridge since 2020/11/1. It has lasted for more than two years so far. To convincingly verify the efectiveness of this monitoring system, the monitored tension histories of Cables 107, 201b, and 104 from 2020/11/01 to 2022/10/31 are comprehensively displayed in Figure 18. It can be evidently seen that reliable daily and long-term variations in cable tension are steadily obtained in real time for 24 months except for a duration of around 7 days in August of 2022 due to a problem of frmware. Similar to the trend observed in Figure 17 for one week, the long-term tension variation of Cable 201b is also noticeably greater than those of Cables 107 and 104. Moreover, the long-term tension variation of Cable 104 undoubtedly exhibits its yearly minimum in summer and yearly maximum in winter. Such a phenomenon further endorses that temperature is the major source to induce the tension variation of Cable 104.
To more extensively examine the practical performance of the real-time monitoring system for cable tension presented in the current work, the detailed statistics of tension monitoring for cables in the frst two cable planes of Mao-Luo-Hsi Bridge in two full years are     Table 5. Tere are totally 730 days from 2020/ 11/01 to 2022/10/31 such that 70080 15-minute measurement data fles are potentially generated. Te number of 15-minute time intervals with abnormal or no collected signals for each cable is frst listed in Table 5. Te reason for Cable 201a to have clearly more time intervals in this category is that the accelerometer installed on it together with the transmission wire was broken by strong winds in August of 2022 and had not been entirely repaired until the end of October. As for Cable 205, its extra data missing occurs in one period of around 5 days from 2021/9/4 to 2021/9/9 and another period of approximately 26 days from 2022/2/6 to 2022/3/4, both due to technical problems.
For each valid 15-minute measurement, the real-time monitoring system is applied to conduct tension estimation for the corresponding cable. Similar to the results plotted in Figure 18 for Cables 107, 201b, and 104, those of the other 15 cables also are found to be as successful. More specifcally, the success rates for the determination of reliable tension in two years as long as the vibration signal is available are 100% for 15     Structural Control and Health Monitoring an expected ratio with the merely available cable frequency. In fact, the only false tension observed in the case of Cable 208 is caused by the identical scenario. Even so, such a problem can be considered as a trivial defect of the proposed sieving algorithm based on fxed ratios of cable frequencies because it occurs in very rare and isolated occasions. Regarding the case of Cable 201a, there are eight time intervals in which no cable frequency can be identifed by the sieving algorithm. Further inspection discloses that these intervals all fall between midnight and early morning. Particular weak vibration signals in situations with almost no trafc bring about this type of difculty in the identifcation of cable frequencies.
From the statistics of mode priorities for each cable also listed in Table 5, it is clear that the tension values are overwhelmingly determined by the mode with the frst priority in most cases for all the cables. Particularly for Cables 102, 105, and 202, all of their monitored tension values in two full years are decided by the most dominant mode. Te mode with the second priority plays the most important role in the case of Cable 205 where the tension values of 1538 15-minute time intervals (2.4%) are calculated from this mode. As for the mode with the third priority, it appears on the stage only in the cases of 5 stay cables and is most active for Cable 108 where the tension values of 12 15minute time intervals (0.02%) are obtained from this mode. Even though the marginal utility of including the third cable mode seems to be limited from these statistics, it should be especially noted that merely one frequency ratio would be available for the subsequent sieving algorithm if simply two cable frequencies are targeted in the identifcation process. Under such circumstances, the potential for the occurrence of false tension values as discussed in the previous paragraph may be substantially raised. On the other hand, the choice of more cable frequencies to further complicate the automatic sieving algorithm is also meaningless considering the excellent success rates of tension determination in all the examined cables. Accordingly, the selection of three cable frequencies as the identifcation target can be persuasively justifed in the application of Mao-Luo-Hsi Bridge.

Conclusions
To develop a monitoring system of cable tension based on real-time vibration signals, this research frst employs an efcient SSI method with tailored parameter selection to continuously identify the three frequencies of adjacent modes for the cables of Mao-Luo-Hsi Bridge. More importantly, an automated algorithm is delicately established to sieve out the stable modal frequencies by making the best of the specifc modal frequency distribution for cables. Te ratios between the frequency values identifed from SSI analysis are exhaustively checked to systematically extract the qualifed cable frequencies and decide their corresponding mode orders. Te tension is fnally computed with one available cable frequency according to the priority order predetermined by the statistics of identifcation rate. Overall, the current study achieves a real-time cable tension monitoring system by completing several imperative works in the analysis of longterm cable vibration signals, the selection of suitable parameters, the choice of balanced criteria, and the verifcation of proposed numerical procedures.
Demonstrated by analyzing the vibration signals measured from the stay cable of Mao-Luo-Hsi Bridge in real time for two full years, the efectiveness and robustness of this real-time monitoring system have been extensively testifed. Te success rates for the immediate determination of dependable tension are found to be perfect in two years for 15 of the 18 investigated cables. As for the other three cables, their corresponding success rates are still higher than 99.99% with eight cases of absent tension values due to very weak Table 5: Statistics of tension monitoring for cables in the frst two cable planes in two years. signals and six cases of false tension values owing to the coincident sifting of a proportioned non-cable frequency value. Contrasting to the practical validation with relatively short durations of vibration measurements in other similar studies [38,39], much more persuasive testimony has been provided for the real-time cable tension monitoring system developed in this work with such long-term results. Another interesting observation from these long-term cable tension histories of Mao-Luo-Hsi Bridge is that the tension variations of the main cables are usually more signifcant and possibly created by live load. On the other hand, the tension variations for other cables can be apparently more restricted and may be either infuenced by temperature or without dominant environmental factors.
It is expected that the cable tension monitoring system currently applied in Mao-Luo-Hsi Bridge can be easily generalized to conduct real-time monitoring for the other cable-supported bridges with similar selections of parameters in the SSI analysis and the same automated sieving algorithm. Eforts are presently made to utilize the proposed methodology on another cable-stayed bridge and the other four arch bridges in Taiwan. Te success in the application of Mao-Luo-Hsi Bridge, however, may be at least partly attributed to the high-quality measurements on its stay cables. Tis advantage keeps the cases with absent or false tension values to a minimum amount. If extended to the tension monitoring for the suspenders of arch bridges, the quality of the corresponding ambient vibration signals may be highly fuctuating and a few issues need to be more carefully explored. For example, the more frequent occurrence of coincidently proportioned frequency values as a result of inferior measurements in this case would deteriorate the faw existing in the sieving algorithm to generate more false tension values. In fact, research work has been under way to further set up impeccable safety nets for the sieving algorithm under the interference of proportioned frequency values. Another efort attempting to utilize deep learning techniques such as the convolution neural network for replacing the sieving algorithm and increasing the application fexibility is also ongoing. It is hoped that these advanced works can be completed and reported in the near future.

Data Availability
Te data used to support the fndings of this study are available from the corresponding author upon request.

Conflicts of Interest
Te authors declare that they have no conficts of interest.