A Novel Adaptive Square Root UKF with Forgetting Factor for the Time-Variant Parameter Identification

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Introduction
During the service life of civil engineering structures, when subjected to extreme loads, the structural parameters may experience sudden or gradual changes, including stifness degradation and an increase in damping. Te challenge of accurately tracking parameter changes has been a topic of ongoing interest in the feld of civil engineering [1,2]. Timely and accurate understanding of the changing characteristics of structural parameters holds great signifcance for tasks such as structural optimization design, maintenance, and reinforcement, as well as the selection of postdisaster rescue routes. When civil engineering structures are subjected to extremely large external excitations, such as earthquakes on buildings or overweight vehicles on bridges, they often exhibit nonlinear behavior. Furthermore, the identifcation of time-variant parameters in structures remains challenging due to factors such as incomplete test information, model errors, measurement noise, and other interferences.
Te LSE method optimizes parameters by minimizing the errors between measured and simulated responses [22]. However, the LSE requires measurements of all displacement and velocity values. Te identifcation process of the MCF method often necessitates a signifcant number of sampling points, leading to high computational demands [2]. Compared to the LSE and MCF methods, the EKF and UKF ofer a signifcant advantage as fast online identifcation methods with recursive characteristics. When new data are measured, the EKF and UKF can efciently update the parameters through Bayesian data fusion technology [23,24], which is known for its space-saving and highefciency capabilities. However, the EKF is typically employed for handling weakly nonlinear problems and requires the computation of complex Jacobian matrices when linearizing nonlinear functions using the frst-order Taylor series expansion technique. In addition, the EKF algorithm can diverge if the sampling interval is not small enough. To address the limitations of the EKF, the UKF was introduced by Julier and Uhlmann in 1995, specifcally designed to handle strongly nonlinear problems [25,26]. In the UKF, it is assumed that the system state vector and noise follow Gaussian probability distributions. Te systematic mean and covariance are deterministically sampled using the unscented transformation (UT) to generate sigma points. By approximating the probability density distribution of the nonlinear function, the UKF avoids errors caused by directly linearizing the nonlinear function and achieves accuracy beyond second-order. Currently, the UKF has gained widespread adoption in the feld of constant parameter identifcation and load identifcation [27,28]. However, in practical applications, the UT process of the UKF requires the covariance matrix to be positive defnite. Due to the adverse infuence of rounding error and noise, the covariance matrix can become nonpositive defnite during the actual recursive process, leading to the divergence of the recursive algorithm. To address this issue, Van der Merwe et al. [29] proposed the SRUKF method, building upon the UKF framework. Te SRUKF directly recurses the square root of the covariance matrix, eliminating the need for square root operations and enhancing the numerical stability of the UKF algorithm to a certain extent.
Some research studies further show that both the UKF and SRUKF methods are inefective in accurately identifying time-variant parameters [1,2,30]. Te values of the state covariance, modeling error covariance (Q in Section 2.1), and measurement error covariance (R in Section 2.1) matrices signifcantly impact the stability and accuracy of the identifcation process as they represent model uncertainty and measurement noise levels. It is crucial to determine appropriate and reasonable values for these matrices. As a result, most existing adaptive methods related to the UKF are modifcations that focus on adjusting one or more of these three variables. Representative research in this area includes the state covariance adaptive flter [1,12,[30][31][32][33], the Sage-Husa adaptive flter [2,30,[34][35][36][37], the dual adaptive flter [38], the forgetting factor flter [39,40], and the moving window adaptive flter [40,41]. In the state covariance adaptive flter, as described in [30], the prior state covariance in the time update step was uniformly enlarged using an adaptive weighting coefcient λ k . Tis coefcient was calculated based on a fading factor formula, where the choice of the forgetful factor was determined through empirical experience. However, in [1,12,31], only specifc major diagonal elements corresponding to time-variant parameters of the state covariance matrix were expanded, while the other elements remained unchanged. Further analysis reveals that in [1,12] and [31], the decision regarding which major diagonal element needs to be expanded is made by comparing the sensitive parameter with a threshold. Tis threshold is calculated based on transcendental probability. In the context of the strong tracking flter, the authors in [32,33] introduced a correction to the prior state covariance using a fading factor that can be approximated through a suboptimal algorithm. Te derivation process of this fading factor is known to be highly complex. After verifcation, the abovementioned state covariance correction technologies are all applicable to timevariant parameter identifcation. In the case of the Sage--Husa adaptive flter, the authors in [2] focused on modifying the measurement error covariance, while the authors in [30,[34][35][36][37] simultaneously addressed the revision of both the modeling error covariance and the measurement error covariance. Furthermore, the authors in [34,37] introduced further simplifcations to the correction formula. Regrettably, among the mentioned Sage-Husa adaptive flter studies, only the authors in [30] took into account the timevariant characteristics of structural parameters. Furthermore, according to the research conducted by Yang and Gao [42], when the statistical characteristics of modeling error 2 Structural Control and Health Monitoring and measurement error are unknown, simultaneous estimation of them based on the Sage-Husa noise estimator is susceptible to algorithm divergence. Regarding the other adaptive flters, the studies [38,40] primarily focused on addressing the impact of model uncertainty. Tey tackled the issue of updating the nonlinear fnite element model when uncertainties existed in parameters such as geometry, node mass, dead load, damping coefcient, and the number of integration points. However, it should be noted that the studies [38,40] did not specifcally address the time-variant problem of parameters. Similarly, the methods proposed in [39,41] are primarily utilized for fault diagnosis purposes. Although these methods are efective in handling time-varying noise systems, they do not specifcally address the time-variant parameter identifcation problem. By the way, the measurement error covariance was modifed in [38][39][40], while the modeling error covariance and measurement error covariance were improved simultaneously in [41]. Based on the analysis provided, it can be concluded that methods based on state covariance correction technology, as well as a limited number of Sage-Husa adaptive flters, have shown efectiveness in identifying time-variant parameters. Nevertheless, it should be noted that the proposed Sage--Husa adaptive methods may encounter numerical stability issues in certain scenarios. Furthermore, it is worth mentioning that the adaptive flters proposed in [30,32,33] are based on the SRUKF method [29], which still relies on the rank 1 update to Cholesky factorization process. Tis requirement for positive-defniteness in the matrix indicates that the SRUKF does not fundamentally resolve the issue of numerical instability in calculations. Indeed, the authors in [34] proposed a modifed SRUKF method; however, it is acknowledged that the derivation of the adaptive algorithm in this approach can be relatively complex. Terefore, taking inspiration from the concept presented in literature [34], this paper modifes the standard SRUKF by incorporating QR decomposition. Tis modifcation aims to ensure the unconditional stability of the algorithm. Subsequently, the state covariance correction technology is applied to the MSRUKF algorithm, leading to the proposal of the ASRUKF-FF method. In contrast to the aforementioned methods, where the covariance correction coefcient is typically an empirical constant or determined through experience, this paper introduces a novel approach. Te correction coefcient of the state covariance can be adaptively adjusted based on the forgetting factor, which follows a clear mathematical derivation process. Furthermore, the proposed adaptive algorithm is characterized by its simplicity in implementation, ease of programming, and robustness. Te specifc arrangement of this paper is as follows: the algorithms used are described in Section 2. Te identifcation of time-variant stifness and damping of a three-degree-of-freedom frame structure is presented in Section 3. Te identifcation of time-variant stifness of a simply supported bridge in the vehicle bridge system is considered in Section 4. Finally, the conclusion is obtained in Section 5.

Te Standard Square Root UKF.
Te SRUKF method is a model-driven algorithm that enables the direct utilization of optimization algorithms for the identifcation of structural parameters. Tis can be accomplished by leveraging the equations in state space. Te state and measurement equations of the nonlinear discrete-time system can be expressed in the following equations, respectively: where k is the discrete time, X is the system state vector, u is the system input matrix, Y is the system measurement vector, w and v are the modeling error and measurement error, respectively, and f(•) and h(•) represent the nonlinear function; both w and v are assumed to follow a Gaussian distribution and satisfy the following relationship: w ∼ N(0, Q) and v ∼ N(0, R). Te algorithm procedure of the standard SRUKF method can be described as follows: (1) Initialization of the state vector and square root of the state covariance matrix: where X + 0 is the initial state vector and X 0 denotes the initial values determined by experience, fnite element analysis, or design drawings. P + 0 is the initial state covariance matrix that is composed of uncorrelated diagonal elements [43]. P + 0 represents the confdence in the initial state estimates and must be specifed a prior. In the absence of any prior knowledge of X 0 , it is common to assume high values for P + 0 [44,45]. chol{•} denotes Cholesky factorization.
(2) Time update: (9) where n is the dimension of the state vector, qr{•} denotes the QR decomposition that returns the upper triangular part of the matrix, cholupdate{•} denotes the rank 1 update to Cholesky factorization which also returns the upper triangular Cholesky factor, and sgn(•) is the sign function. As an illustration, chol-update{A, B, ±c} gives the Cholesky factor of (D ± c BB T ), where A � chol{D}. For more operation details of (8) and (9), please refer to [29,46]. (3) Measurement prediction:

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(4) Measurement update: where y k is the measurement at step k.
Weights can be expressed as where λ � α 2 (n + κ) − n is a compound scaling parameter. 10 −4 ≤ α ≤ 1 is the primary scaling factor determining the extent of the spread of the sigma points around the prior mean. b is a secondary scaling factor used to emphasize the weighting on the zeroth sigma point for the posterior covariance calculation. For Gaussian priors, b � 2 is optimal. κ is a tertiary scaling factor and is usually set equal to 3 − n.

Te Modifed Square Root UKF.
Te standard SRUKF method updates the square root of the covariance matrix by utilizing the rank 1 update to Cholesky factorization. Taking (9) as an illustration, the equivalent form of (9) can be written as where P − k is the prior state covariance calculated by the time update step of the conventional UKF method, and the zeroorder weight W (0) c contains positive or negative signs. To calculate S k using equation (9), the matrix on the right-hand side of equation (22) must be positive defnite, indicating that the standard SRUKF method does not fundamentally eliminate the requirement for a positivedefnite covariance matrix. Te presence of a negative zero-order weight can indeed contribute to the matrix on the right-hand side of equation (22) not being positive defnite. From equation (20), when κ � 3−n and β � 2, W (0) c � 4 −n/3α 2 − α 2 . Because the value of α is usually small, when the dimension of the state vector is large, the zero-order weight may have a negative value, which will cause the matrix on the right-hand side equation (22) to be nonpositive, afecting the stability of the recursive algorithm.
According to Cholesky decomposition, the prior state covariance of the time update step of UKF is P − k � S T k S k . Let S k � qr, where q is an orthogonal square matrix and r is an upper triangular matrix, then P − k � S T k S k � (qr) T (qr) � r T r. Terefore, the following relationship can be obtained: 4 Structural Control and Health Monitoring Te UT process typically generates (2n + 1) sigma points using symmetrical sampling. Te sigma points are symmetrically distributed around the origin (zero point) to maintain the consistency of the sample's mean and covariance with the original state's distribution. Because removing the sigma point at the origin position does not impact the mean and covariance of the remaining sample points, based on this idea and according to equation (23), equations (8) and (9) can be replaced as follows: Similarly, equations (13) and (14) can be replaced as follows: To update equations (18) and (19), it is necessary to derive them using the conventional UKF algorithm and the defnition of covariance. First, the equivalent form of equations (18) and (19) is expressed as equation (28). Te matrix on the right-hand side of equation (28) still needs to be positive defnite, which is not conducive to the stability of the recursive algorithm.
where P + k is the posterior state covariance of the measurement update step of the conventional UKF method.
Second, the following relationship is obtained based on the posterior state covariance defnition and equation (17): By considering the measurement equation of (1), we can establish the following relationship: Substituting equations (30) and (31) into equation (29), we obtain the following equation: where H k is the Jacobian matrix of the nonlinear function at the kth recursive step, and it is assumed that the measurement noise v k is unrelated to other quantities. Tird, utilizing the defnition of cross-covariance and referring to equations (30) and (31), we deduce the following relationship: Te calculation method for obtaining H k from equation (33) is as follows: where equation (34) uses the property that the transposition of a symmetric matrix is equal to itself. Finally, equations (18) and (19) can be replaced by the following expressions: Te calculation procedure of the MSRUKF algorithm is as follows: (1) Initialization with equations (3) and (4) (2) Time update with equations (5)-(7) and (24) and (25) (3) Measurement prediction with equations (10)- (12), (26), (27), and (15) (4) Measurement update with equations (16), (17), (35), and Structural Control and Health Monitoring

Te Adaptive Square Root UKF with Forgetting Factor.
Although the MSRUKF algorithm alleviates the condition of requiring a positive-defnite covariance matrix, ensuring numerical stability throughout the calculation process, its recursive nature still involves performing the UT to approximate the probability density distribution of the nonlinear function. In each recursive step, the weight values of UT remain consistent, resulting in the data at each step exerting an identical impact on fltering. As the fltering progresses, the amount of acquired data steadily increases. However, the accumulation of older data diminishes the updating efect of new data on the estimates. Consequently, the state covariance matrix loses its corrective infuence on the state vector, leading the fltering process to approach stability. Once the fltering reaches a state of stability, the state vector cannot be updated by new measurements. As a result, the MSRUKF becomes incapable of tracking changes in the time-variant parameters.
To enhance the capability of the MSRUKF algorithm in identifying time-variant parameters, this article proposes the following three steps for algorithm modifcation. First, drawing inspiration from the concept of utilizing a scalar in literature [1] to detect the occurrence of parameter changes, this paper also incorporates a scalar value, referred to as η (a sensitivity parameter), to pinpoint the moment of parameter change. Te discrete form of η is depicted in the following: where ε k � y k − y k , in which y k is the measurement at step k.
Second, a threshold value η 0 is introduced to intelligently trigger the adaptive algorithm when the sensitive parameter value η t exceeds η 0 , where η t denotes the value of the sensitive parameter at time instant t. Te authors in the literature [1,31,47] suggest that η 0 follows a chi-square distribution with degrees of freedom equal to the number of measurements for zero-mean Gaussian innovation. However, in practice, η 0 remains interconnected with various factors, including measurement noise, modeling error covariance, measurement error covariance, initial state covariance, initial state vector, and sampling frequency. At times, relying solely on the number of measurements makes it challenging to calculate a reasonable η 0 . Considering the complexity and the involvement of multiple factors, it becomes difcult to calculate η 0 using a precise mathematical formula. Terefore, this paper proposes a more practical and reasonable approach to determine η 0 . It primarily involves the following three steps: (1) calculate the time history curve of η using MSRUKF; (2) identify the maximum value η t before the initial curve pulse emerges, for example, η t = 6.966 at 10 s, as shown in Figure 1(b); and (3) set the threshold slightly higher than η t , such as η 0 = 7 in Figure 1(b), and further refne if necessary. Attention. If the time history curve of η appears stable and it becomes challenging to determine the exact moment of parameter change, it indicates two possible situations: (1) the parameters may remain unchanged throughout the identifcation process and (2) the measurement noise is sufciently high, making it difcult to discern the changes accurately.   Structural Control and Health Monitoring Tird, a forgetting factor α is defned, as shown in (38), where α belongs to the interval (0, 1) and "tr" represents the trace of the matrix. In contrast to the constant coefcient correction method suggested in [1,12,31,47], the determination of the forgetting factor in this approach relies on the residual information at each recursive step. Tis adaptive approach enables the correction factor to be dynamically determined based on the uncertainty associated with each step.
Finally, leveraging the forgetting factor α, the update of the square root of the posterior state covariance matrix, as depicted in (36), is performed according to the following equation: Te specifc algorithm fow of the ASRUKF-FF method is illustrated in Figure 2.

Te Mathematical Explanation of the Forgetting Factor.
During the recursive process, the state covariance plays a crucial role in quantifying the level of uncertainty associated with the estimated state. A well-selected state covariance is a prerequisite for accurately identifying the time-variant parameters of the system. As discussed in Section 2.1, the initial state covariance can be set to a small value if the initial state values are highly reliable. However, if there is uncertainty in the initial state values, it should be chosen to be sufciently large to encompass the actual state uncertainty. Furthermore, in the intermediate steps of the recursive process, a more precise estimation of the state covariance is required. Adjusting the state covariance appropriately can expedite the convergence to the correct value, as discussed in [1]. Terefore, the core principle of adaptive identifcation is to automatically adjust the state covariance by taking into account the forgetting factor and parameter sensitivity throughout the identifcation process.
To elucidate the origin of the forgetting factor, a mathematical explanation is provided based on the SRUKF method. When the structural parameters undergo changes during the identifcation process, it becomes necessary to expand the state covariance to encompass the uncertainty of the state [1]. Building upon this concept, a coefcient α is introduced, where 0 < α < 1. According to (18) and (19), the a posteriori estimate of the state covariance can be defned as follows: where P + k is the posterior state covariance corrected by the forgetting factor, and the wave sign above the character represents the quantity corrected by the forgetting factor.
Substituting (16) into (40), we get the following equation: Based on the identical equation principle, equation (41) can be modifed as  Structural Control and Health Monitoring Since the measurement noise is infuenced by sensor accuracy and the testing environment, excessive noise can cause the identifcation process to diverge. Terefore, the measurement noise is not expanded. By utilizing equations (13) and (41)∼(15), the following equation can be formed: where P yy,k is the innovation covariance matrix in the measurement update step of the conventional UKF method. If the forgetting factor correction is not considered, equation (43) is mathematically equivalent to equations (13) and (14). Te form of equation (43) is expressed to facilitate subsequent derivations.
Substituting equations (43) and (44) into equation (41), we get the following equation: where K k � P xy,k (P yy,k ) − 1 . Te theoretical presentation above demonstrates that the ASRUKF-FF algorithm can be derived by combining the standard SRUKF with Equations (43)- (45). However, equation (45) poses challenges when applying it to the MSRUKF due to the altered calculation method for the posterior state covariance. Te application of the forgetting factor to equation (35) requires additional mathematical derivation and verifcation, which will not be discussed in detail here. To elaborate on the origin of the forgetting factor, only equation (43) will be further discussed.
For further analysis, modify equation (43) as Add R k to both sides of equation (46) and form as Take the trace of equation (48) as Note that equation (49) is an approximate forgetting factor due to the omission of R k . Tis operation makes sense as both the numerator and denominator contain the same covariance R k . Moreover, as the denominator of equation (49) emphasizes the impact of the new measurements, the corrected innovation covariance in the denominator is computed using the predicted measurement mean (y k ) and the actual measurements (y k ). Terefore, equation (49) can be written as Te mathematical derivation provided above allows us to obtain the rationale behind the forgetting factor and its application in the standard SRUKF form. While the MSRUKF has introduced changes in the calculation method for obtaining the square root of the posterior state covariance, as shown in equations (35) and (36), making it difcult to determine the exact location of the forgetting factor, we can still draw inspiration from equation (40). By correcting the square root of the posterior state covariance matrix, we can achieve the adaptive adjustment of the state covariance. In addition, due to the recursive calculation of the covariance in the MSRUKF method, the correction coefcient also necessitates the square root form of the forgetting factor, as demonstrated in equation (39).

Te Tree-Degree-of-Freedom Frame Model.
Te frame structure is a widely utilized construction system in civil engineering. In this section, a three-story frame structure is chosen as the subject of investigation to assess the performance and efectiveness of the proposed method. To facilitate the study, the actual frame structure is simplifed into a shear model, as shown in Figure 3(a), where m i , c i , and k i represent the mass, interlayer damping, and interlayer stifness, respectively, for i � 1, 2, and 3, and € x g represents the seismic excitation, and its time history curve is shown in Figure 3(b).
Te equations of motion for this system can be derived from the interlayer responses, as described in reference [33]. Furthermore, to simulate real-world conditions, random noise is introduced into the measurements. Te noise type is given as follows: where E P is the percentage of the RMS noise; N noise is a stochastic process following a standard normal distribution with zero mean and unit standard deviation; and σ is the standard deviation of the response without noise.

Te Efect Comparison of Diferent Algorithms.
In this case study, the structural parameter values used are m 1 � m 2 � m 3 � 1000 kg, k 1 � k 2 � 120 kN/m, k 3 � 60 kN/m, and c 1 � c 2 � c 3 � 0.6 kNs/m. To depict the variations in parameters and evaluate the efcacy of identifying time-variant parameters, it is assumed that the stifness parameters k 1 ∼k 3 undergo a sudden reduction to 80 kN/m, 80 kN/m, and 40 kN/m, respectively, at 10 s. Meanwhile, at 10 s, the damping parameters c 1 , c 2 , and c 3 are assumed to undergo a sudden change to 0.7 kNs/m, 0.65 kNs/m, and 0.65 kNs/m, respectively. Te percentage of the RMS noise is 5%. Te initial values of the state vector and state covariance are provided in Table 1. In addition, the covariance matrix Q, representing the modeling error, is set to 1 × 10 −8 I, and the measurement error covariance matrix R is set to 5 × 10 −2 I. Furthermore, in this section, the acceleration responses of the frst, second, and third stories are considered as measurements. Te sampling frequency is 100Hz.
Before applying the ASRUKF-FF approach, it is essential to establish a threshold for the sensitive parameter. Te time history curve of this parameter, computed using both the MSRUKF and ASRUKF-FF algorithms, is presented in Figure 1. As depicted in Figure 1(a), when structural parameters change, the time history curve of the sensitive parameter would exhibit a pulse response at the moment of change occurrence. Based on the calculation method proposed in Section 2.3 and the results shown in Figure 1(b), the threshold for the sensitive parameter can be determined as η 0 � 7. Furthermore, the ASRUKF-FF algorithm demonstrates its efectiveness in diminishing the impulse response of the sensitive parameter after 10 seconds. A lower value of the sensitive parameter indicates decreased uncertainty in the identifcation process, thus highlighting the enhanced accuracy of the ASRUKF-FF method.
To comprehensively demonstrate the efectiveness of the method proposed in this paper, a comparison is conducted between the ASRUKF-FF method and other existing methods, namely, the MSRUKF, adaptive UKF (AUKF) as described in literature [1], and modifed strong tracking SRUKF (MSTSRUKF) as described in literature [33]. It is worth noting that the MSTSRUKF method requires the determination of a gradual forgetting factor ρ, typically satisfying 0 < ρ ≤ 1. Te identifcation results of various algorithms are illustrated in Figure 4. Within the fgure, the threshold β 0 for AUKF is determined as 16.3 using transcendental probability [1] and a gradual forgetting factor of 0.95 is applied to MSTSRUKF. For a detailed analysis of the errors in the fnal identifcation results, refer to Table 2.
Based on the analysis of Figure 4 and Table 2, the following conclusions can be drawn: (1) Te MSRUKF method exhibits limited efectiveness in identifying time-variant parameters of Note. for convenience, dimensional units are omitted in Table 1 and x i is the ith relative displacement, i � 1, 2, and 3. (2) Te ASRUKF-FF method demonstrates a signifcant improvement in identifcation accuracy when the sensitive parameter threshold is set at η 0 � 7. It achieves a satisfactory identifcation efect with a maximum identifcation error of less than 3%. Furthermore, based on the damping identifcation results (Figures 4(d)-4(f)), the ASRUKF-FF method exhibits a tendency to generate pulse fuctuations precisely at the moment of structural parameter mutation. Tis characteristic can be utilized to estimate the exact timing of parameter changes. In comparison to other methods, the ASRUKF-FF algorithm exhibits the fastest convergence speed prior to parameter changes. (3) In the scenario, where the number of measurement values is 3 and the transcendental probability is set at 0.001, the threshold value of AUKF is calculated as β 0 � 16.3 using the chi-square inverse cumulative distribution function. However, it is observed that when β 0 is set to 16.3, the identifcation performance of AUKF is poor, resulting in a maximum error of 16.59% in the damping identifcation. By utilizing the threshold value β 0 � 7 calculated using the proposed method in Section 2.3, the identifcation efect of AUKF has signifcantly improved. Nevertheless, there is still a maximum error of 9.10%, and the convergence to the true values prior to parameter changes has not been achieved. In addition, when compared with the ASRUKF-FF and MSTSRUKF methods, the AUKF method exhibits slower convergence speed. (4) Te MSTSRUKF method achieves a maximum identifcation error of 0.17% for stifness and 1.94% for damping, aligning with the conclusion drawn in literature [33] that the identifcation error for stifness and damping should not exceed 0.2% and 4%, respectively. Furthermore, as the literature [33] does not discuss the identifcation efect of diferent gradual forgetting factors, a brief discussion on the corresponding parameter ρ is presented here. Based on the fndings in Table 2, it can be observed that the MSTSRUKF method is not signifcantly afected by variations in the gradual forgetting factor. Te identifcation results remain relatively stable across diferent values of the parameter. (5) In terms of identifcation accuracy, both the ASRUKF and MSTSRUKF methods have demonstrated excellent performance in this particular case.

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To ofer a more comprehensive explanation for the ASRUKF-FF algorithm's ability to identify time-variant parameters, an investigation was conducted on the time history curves of stifness and damping parameters in the square root of the state covariance matrix (SRSCM). Te comparative results of this study are presented in Figure 5, where the solid line denotes the outcomes computed by the MSRUKF algorithm and the dashed line represents the outcomes computed by the ASRUKF-FF algorithm. Figure 5 clearly indicates that the MSRUKF algorithm, lacking an adaptive function, mistakenly assumed that the identifcation result had converged to the true value at the time of parameter mutation (10 s). However, due to the failure to properly adjust the algorithm, the identifcation result was ultimately unsuccessful, as depicted in Figure 4. In contrast, the ASRUKF-FF algorithm efectively expands the SRSCM using the adaptive forgetting factor at the time of parameter change. Tis expansion increases the search threshold for the parameters to be identifed, enabling the successful identifcation of time-variant parameters. Please note that due to the dynamic adjustment of the adaptive forgetting factor based on the residual information at each recursion step, the time history curves of stifness or damping parameters in the SRSCM calculated using the ASRUKF-FF algorithm exhibit distinct peak pulses after the occurrence of parameter mutations. Tese peak pulses represent the results of diferent forgetting factor corrections.

Discussion on the Identifcation Efect of Diferent
Sensitive Parameter Tresholds. Tis section discusses the identifcation efect of diferent sensitive parameter thresholds to demonstrate the stability and reliability of the proposed adaptive method. Te comparative analysis adopts the control variable method, wherein all parameter settings remain the same as those in Section 3.2.1, except for the variation in sensitive parameter thresholds. Tis ensures a fair and controlled comparison of the results. Te specifc identifcation results are depicted in Figure 6, while the error analysis of the fnal identifcation values is presented in Table 2.

Algorithms
Error     As described in Section 2.3, the calculation method for the sensitivity parameter threshold involves selecting a value slightly larger than the maximum value before the initial pulse in the curve. Considering Figure 1, it can be observed that the threshold value of 20 is signifcantly higher than the maximum value of 6.966 (Figure 1(b)). Terefore, the range of sensitivity parameter thresholds discussed in this context varies from 7 to 20, with the maximum threshold being nearly three times higher than the minimum threshold. Analyzing Figure 6 and Table 2, it can be observed that the stifness identifcation results remain consistent across diferent sensitive parameter thresholds. However, there are slight variations in the damping identifcation results. For instance, when the threshold value is set to 15, the maximum identifcation error for damping is 6.56%, whereas the identifcation errors for other thresholds do not exceed 4%. Moreover, it can be observed that when the threshold is set to 7, the proposed method exhibits a relatively high identifcation accuracy. Tis confrms the efectiveness of the proposed method as outlined in Section 2.3. Terefore, it is possible to choose a smaller threshold while ensuring the convergence of the algorithm. Tis can be done to improve the identifcation accuracy of the method. Although there may be slight variations in the identifcation results for diferent sensitive parameter thresholds, the overall accuracy of the estimation results remains high. Tis indicates the stability of the proposed algorithm in consistently producing reliable identifcation outcomes.

Discussion on the Identifcation Efect of the
Values of X 0 , P 0 , Q, and R. Te identifcation results are notably afected by the initial values of the state vector (X 0 ), state covariance (P 0 ), modeling error covariance (Q), and measurement error covariance (R). Tus, in this section, we explore the infuence of diferent X 0 , P 0 , Q, and R values on the identifcation performance of time-variant structural parameters through the utilization of the ASRUKF-FF method. Te comparative analysis employs the control variable method, where parameters other than the ones under study are set according to Section 3.2.1. Furthermore, the initial state variables, X 0 and P 0 , primarily consist of displacement, velocity, stifness, and damping parameters. Typically, the initial displacement and velocity can be assumed to be zero, indicating relatively low uncertainty in these values. Conversely, the initial uncertainty associated with the stifness and damping parameters is relatively high. Terefore, this section primarily focuses on the stifness and damping parameters within X 0 and P 0 . Te specifc identifcation results are illustrated in Figures 7-10, while the fnal identifcation error is presented in Table 3.
Based on the analysis of the results in Figures 7-10 and Table 3, for the shear model, in comparison to the original intact state values of [k 1 , k 2 , k 3 , c 1 , c 2 , c 3 ] � [120, 120, 60, 0.6, 0.6, 0.6], as mentioned in the frst paragraph of Section 3.2, when the stifness and damping values in the state vector X 0 are smaller [50, 50, 10, 0.01, 0.015, 0.02] or larger [220, 220, 220, 1.5, 1.5, 1.5], they exert a more signifcant infuence on the accuracy of damping parameter identifcation. Te corresponding maximum identifcation errors for the damping parameters are 5.9% and 3.7%, respectively. Tere are two possible reasons for this observation. First, stifness plays a more signifcant role in the response of the frame structure, and acceleration measurements are more sensitive to the identifcation of stifness. Second, there is a disparity in numerical magnitudes between stifness and damping. Hence, it is advisable to select more appropriate values for the stifness and damping parameters in X 0 , in order to mitigate the negative impact caused by excessively large or small initial values on the identifcation results. Te values of the stifness and damping parameters in the initial state covariance P 0 have a signifcant infuence on the accuracy of damping identifcation. For example, in the " [20000, 20000, 20000, 1, 1, 1]" operating condition, the maximum identifcation error for damping reaches 8.37%. Covariance represents the level of uncertainty in the data, and overestimating the degree of uncertainty can potentially impact the accuracy of identifcation. Furthermore, according to Figure 8, it can be inferred that appropriately increasing the values of the parameters to be identifed in P 0 helps speed up convergence and reduce the time required for the parameters to converge to their true values. Te value of the modeling error covariance Q has a signifcant impact on the identifcation of damping. While smaller Q values may lead to higher accuracy in the fnal identifcation results, they might also result in slower convergence rates. In certain cases, the algorithm might not converge to the true parameter values before parameter mutations occur, as indicated by the red line in Figure 9. Furthermore, it is worth noting that the maximum identifcation error for damping is higher at Q � 1 × 10 −10 I compared to Q � 1 × 10 −9 I. On the other side, as the value of Q increases (1 × 10 −9 I⟶1 × 10 −8 I⟶1×10 −7 I⟶1 × 10 −6 I), the identifcation error for damping also progressively increases. Te main reason behind this observation is that Q serves as a compensation term for modeling errors. Overestimating the value of Q can introduce larger model errors, which in turn can result in inaccurate parameter identifcation. Indeed, the measurement error covariance R also plays a signifcant role in the identifcation of damping. R value plays a crucial role in the measurement equation, and any underestimation or overestimation of its value can result in inaccurate estimation outcomes. Based on Table 3, it is evident that underestimating R (2 × 10 −2 I) has a more signifcant impact. Furthermore, despite the minimal impact of a lower R value (2 × 10 −2 I) on the fnal identifcation errors of structural stifness parameters, it leads to a highly oscillatory identifcation process, as depicted by the red line in Figure 10. To summarize, variant X 0 , P 0 , Q, and R values have a negligible efect on the identifcation of stifness parameters for frame structures. However, they exert a notable infuence on the identifcation of damping, with R having the most substantial impact. Furthermore, in terms of value rationality, smaller or larger values of X 0 were not selected as the focus of the research.

Discussion on the Identifcation Efect of Diferent
Measurements. Tis section primarily examines the impact of various measurement combinations on the identifcation outcomes. Te control variable method is employed for conducting comparative analysis. Te specifc settings for algorithm parameters X 0 , P 0 , Q, and R can be found in Section 3.2.1, and the noise level is 5%. Te sensitivity parameter thresholds may vary due to diferent measurement values.

Structural Control and Health Monitoring
Terefore, the calculation process for each working condition is as follows: frst, calculate the sensitivity parameter threshold using MSRUKF; then, utilize the sensitive parameter threshold calculated in the previous step along with the ASRUKF-FF algorithm to identify the structural parameters and record the identifcation results; fnally, perform data processing, analysis, and summary. Te specifc identifcation results are depicted in Figure 11, while the comparison of identifcation errors in the fnal outcomes is presented in Table 4. Te meanings of the labels in Figure 11 and Table 4 are explained as follows: (1) "ACC-1-2-3" indicates the utilization of all acceleration measurements, with a total of 3 measurements representing the accelerations of the frst, second, and third stories, respectively.
(2) "ACC-2 and DIS-1" signifes the simultaneous adoption of acceleration and displacement as measurements, with the second-story acceleration and frst-story displacement being specifcally selected.
Based on the information presented in Figure 11 and Table 4, it can be concluded that when using a single measurement value (ACC-1, ACC-2, or ACC-3), the ASRUKF-FF algorithm exhibits a noticeable identifcation error and fails to accurately identify all stifness and damping parameters of the structure simultaneously. Nevertheless, it is observed that the identifcation efect for stifness is superior to that of damping. In addition, the identifcation efect of "ACC-1" and "ACC-3" outperforms that of "ACC-2," suggesting that using the acceleration measurements from the frst or third story alone yields better identifcation 14 Structural Control and Health Monitoring results compared to using the acceleration measurement from the second story alone. When two measurements are utilized, the identifcation performance of the ASRUKF-FF algorithm demonstrates a notable improvement compared to the case of using a single measurement value. Specifcally, the "ACC-1-2" combination achieves better identifcation performance than the other two measurement combinations. Furthermore, upon comparing "ACC-2" and "ACC-2 and DIS-1," it is observed that incorporating the frst-story displacement as an additional measurement can enhance the identifcation accuracy of the algorithm. In practical applications, direct measurement of displacement is feasible, such as utilizing ground-based radar with an accuracy of 0.1 mm [48]. Terefore, it is reasonable to include displacement as a measurement value. Te details regarding displacement measurements will be thoroughly discussed in Section 4, while no in-depth research will be conducted in this section. In addition, considering the maximum         identifcation error, the ranking of identifcation efects for the measurement combinations in Table 4 is as follows: ACC-1-2-3 > ACC-1-2 > ACC-2-3 > ACC-1-3 > ACC-2 and DIS-1 > ACC-1 > ACC-3 > ACC-2. Tis implies that utilizing three accelerations as measurement values yields the most favorable identifcation efect.

Discussion on the Identifcation Efect of Diferent
Modeling Errors. Tis section examines the infuence of parameter modeling errors on the performance of identifcation. Based on the model analysis presented in Section 3.1, the focus of this study is on the mass parameter, while the parameter settings for other variables are referenced from Section 3.2.1. Furthermore, diferent modeling errors can also lead to variations in sensitivity parameter thresholds. Terefore, the specifc implementation process is referenced from Section 3.2.4. It is worth noting that the measurement noise is set at 5%. Te specifc identifcation results are depicted in Figure 12, while the fnal identifcation error is presented in Table 5.
Based on the simulation analysis, it is found that the location of the maximum error for the stifness parameter corresponds directly to the location of the mass change. Tis correlation can be visually observed from the light green or light blue cells in Table 5. However, there is no such relationship between the location of the maximum error for the damping parameter and the location of the mass change. Trough the analysis of the simulation results for working 10     conditions 1 to 3, it is observed that the modeling error of the third-story mass parameter has the most signifcant impact on the identifcation outcomes. It is followed by the secondstory mass parameter, while the modeling error of the frststory mass parameter has the least impact on the identifcation results. Furthermore, in working conditions 4 to 6, it is observed that the simultaneous modeling error caused by the mass parameters of the second and third stories leads to the poorest identifcation efect. Trough the analysis of the simulation results for working conditions 7 to 9, it is observed that the higher the uncertainty of the mass parameter, the larger the error in parameter identifcation. Tis  Table 4: Estimation errors of the fnal identifcation results with diferent measurements based on the three-story frame structure.

Working condition
Type of measurements

Vehicle Bridge System.
To the best of the author's knowledge, there are few cases of directly identifying the time-variant stifness of bridges using methods related to the UKF. Hence, to thoroughly demonstrate the efectiveness of the proposed method, this section selects a simply supported bridge as the object of verifcation. To simplify the calculation process and improve the computation efciency, the bridge structure is simulated using a beam model. Te purpose is to determine the stifness parameters of the bridge by the vehicle bridge interaction force when the vehicle crosses the bridge. Te bridge section is assumed to be constant, as illustrated in Figure 13. Te specifc parameters are bridge span L � 21 m, cross-sectional area A � 1.2 m 2 , section moment of inertia I � 0.12 m 4 , elastic modulus E � 2.4 × 10 4 MPa, and density ρ � 2000 kg/m 3 . Furthermore, pavement roughness is taken into consideration and is characterized by random numbers following a normal distribution. Te fnite element model of the bridge is constructed using Euler-Bernoulli beam elements, which are divided into six elements, represented as beams ① to ⑥ in Figure 13. Te stifness and mass matrices of the beam element are given in the following equations, respectively: where M e and K e are the element mass and stifness matrices, respectively; x i and x j represent the starting and ending coordinate of one element; N is the shape function of element i; and B is the strain matrix of element i. Te specifc expression of N and B is given in the following equations: where ξ is a scalar and ξ � x/l i , x is the distance the vehicle moves on a beam element, and l i is the length of element i.

Structural Control and Health Monitoring
Te quarter-car model is generally used to demonstrate the theoretical basis of the vehicle bridge interaction model [49,50] and has been successfully applied to numerous applications [51][52][53]. Terefore, a moving quarter-car model is chosen as the external excitation for the analysis. Te vehicle system has two degrees of freedom (Figure 13), including the body mass m 1 � 3.6 × 10 4 kg, bogie mass m 2 � 2.5 × 10 2 kg, secondary suspension stifness k 1 � 6.0 × 10 5 N/m, secondary suspension damping c � 1.0 × 10 3 Ns/m, and primary suspension stifness k 2 � 8.5 × 10 5 N/m. Te moving speed is v � 30.24 km/h. Based on the coupling relationship between contact force and displacement, the equations of motion for the VBS are formulated as (57) M b , K b , and C b are the mass, stifness, and damping matrices of the bridge, respectively; u b is the displacement vector of the bridge; L is the mapping matrix for the input force, such as L � [0, 0, . . ., N i (t),. . ., 0, 0] T ; y 1 is the vertical displacement of the vehicle body; y 2 is the vertical displacement of the vehicle bogie; and r(x(t)) is the pavement roughness at position x(t) which is the vehicle position at time t.
In this case study, the Rayleigh damping is selected as where a 1 and a 2 are the Rayleigh damping coefcients of the structure, respectively; ω m and ω n represent the mth and nth modal circular frequencies of the structure, respectively; and τ m and τ n are the mth and nth modal damping ratios, respectively. In this simulation, take τ m � τ n � 0.015, ω m � ω 1 , and ω n � ω 2 .

Te Efect Comparison of Diferent Algorithms.
In this case study, it is assumed that the stifness of the ② to ⑤ beam elements of the bridge is unknown. It is further assumed that the ② and ③ beam elements exhibit gradual and abrupt changes, respectively. In addition, the beam stifness is assumed to be solely dependent on the elastic modulus. Te specifc parameter changes are illustrated in Table 6. Because each element of the Euler-Bernoulli beam has 2 nodes and each node has 2 degrees of freedom, the total number of degrees of freedom of the bridge is n � 14. As there are four elastic modulus parameters to be identifed, the state vector is chosen to be of order (2n + m). Te state vector can be written as where subscript "p × q" represents a vector or matrix with p row(s) and q column(s) and the quantities appearing in the equation are defned as u b � u b1 u b2 u b3 u b4 u b5 u b6 u b7 u b8 u b9 u b10 u b11 u b12 u b13 u b14 T .
(61) In this case study, it is observed that using displacements as measurements yields better results. Terefore, the vertical displacements of nodes 2, 3, 4, 5, and 6, as shown in Figure 13, are selected as the measurements. Te measurement equation can be expressed as follows:

Structural Control and Health Monitoring
Te modeling error covariance matrix Q is set as 1 × 10 −8 I, and the measurement error covariance matrix R is set as 1 × 10 −8 I. Furthermore, the initial state vector and covariance are given by where the initial values of E 2 to E 5 are set to 0.264, respectively, to reduce the diference in the order of magnitude between state variables; however, the corresponding missing order of magnitude should be considered when solving the actual equation; 1 × 10 −8 diag (14) represents a diagonal matrix with the order of 14 and the diagonal elements are all 1 × 10 −8 . In this section, the RMS noise percentage is set to 2%, and the sampling frequency is 100Hz. Te threshold β 0 of the AUKF is calculated as 21 using transcendental probability [1]. In addition, the gradual forgetting factor of the MSTSRUKF is set to 0.95. Te identifcation results of diferent algorithms are shown in Figure 14, and the error analysis of the fnal identifcation results is described in Table 7.
Based on Figure 14 and Table 7, it can be observed that the MSRUKF method is unable to accurately identify the time-variant parameters of the beam structure, resulting in a maximum identifcation error of 77.63%. Te MSTSRUKF method demonstrates a better identifcation efect on timevariant parameters compared to constant parameters. Te maximum error in time-variant parameter identifcation is 6.2%, while the maximum error in constant parameter identifcation is 16.67%. Furthermore, the identifcation process of MSTSRUKF exhibits nonsmoothness and signifcant fuctuations. Te AUKF and ASRUKF-FF methods exhibit satisfactory performance in identifying time-variant parameters of beam structures, with a maximum identifcation error of no more than 3%. However, the ASRUKF-FF method achieves higher accuracy in identifying time-variant parameters.

Algorithm Robustness Analysis.
To evaluate the robustness of the algorithm and its adaptability to larger uncertainties, the noise level is increased to 5% in this section, while keeping the other parameters the same as in Section 4.2.1. Due to the difculty of achieving convergence with the MSTSRUKF method in the presence of high levels of noise in the vehicle bridge system, the remaining three algorithms are compared in this scenario. In addition, it is important to note that when the noise level is set to 5%, the AUKF algorithm Structural Control and Health Monitoring

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Structural Control and Health Monitoring cannot use a threshold β 0 � 21 to complete the identifcation task, and a low threshold value would cause identifcation divergence. Hence, in this analysis section, the AUKF algorithm utilizes the threshold calculated through the method proposed in this article to accomplish the identifcation of stifness parameters. Te specifc identifcation results are presented in Figure 15, and a detailed error analysis of the fnal identifcation results is provided in Table 8. From Figure 15 and Table 8, it is evident that as the system uncertainty increases, the maximum error of the MSRUKF method in identifying time-variant parameters of the beam structure also increases, with the maximum identifcation error reaching 95.21%. Furthermore, as the noise level increases, both the AUKF and ASRUKF-FF methods exhibit a reduced capability to identify time-variant parameters of the beam structures. However, it is worth noting that the ASRUKF-FF method outperforms the AUKF method in terms of identifcation efectiveness. As indicated in Table 8, the AUKF method exhibits a maximum identifcation error of 7.61%, whereas the ASRUKF-FF method achieves a maximum error of 4.14%. Tis demonstrates that the proposed ASRUKF-FF method exhibits greater robustness in the identifcation of time-variant parameters.

Discussion on the Identifcation Efect of the Values of
X 0 , P 0 , Q, and R. Similar to Section 3.2.3, this section focuses on examining the identifcation performance of time-variant structural parameters using the ASRUKF-FF method under diferent values of X 0 , P 0 , Q, and R. Te aim is to assess the impact of these algorithm parameters on the accuracy and efectiveness of parameter identifcation. Te comparative analysis adopts the control variable method, and except for the parameters to be studied, all other parameter settings are the same as those in Section 4.2.1. Furthermore, this section primarily considers the parameters to be identifed in X 0 and P 0 . Te specifc identifcation results are depicted in Figures 16-19, while the fnal identifcation error is presented in Table 9.
Based on simulation analysis research conducted on the beam model, it has been observed that when solely considering variations in the initial state vector, there exists a specifc range requirement for the optimal elastic modulus value within X 0 . Values outside this range can result in the matrix becoming illconditioned, which in turn leads to divergent identifcation. Te appropriate range for this case is about X 0 ∈ [0.09 × 10 11 , 1 × 10 11 ], and within this range, the specifc value of X 0 has minimal impact on the identifcation results. Likewise, there is an upper limit requirement for the initial values to be identifed in P 0 . Values outside this range can also result in the matrix becoming ill-conditioned, which in turn leads to computational divergence. Furthermore, an excessively small value for P 0 , such as P 0 � 1 × 10 −8 I, can decrease the convergence rate, as illustrated by the red line in Figure 17. Tis observation aligns with the conclusion stated in Section 3.2.3. Simultaneously, excessive values for P 0 can result in signifcant jitter during the initial stages of identifcation (as depicted by the yellow line in Figure 17), leading to algorithmic instability. However, within a certain range, the value of P 0 has minimal infuence on the accuracy of stifness parameter identifcation. When solely considering variations in the modeling error covariance Q, both excessively large and excessively small Q values (such as 1 × 10 −7 I or 1 × 10 −10 I) would result in an increase in identifcation error. Tis observation aligns with the conclusion stated in Section 3.2.3. Te reasoning behind this observation can also be found in Section 3.2.3. When solely considering variations in the measurement error covariance R, an increase in R would correspondingly lead to a further increase in the identifcation error of stifness parameters. Tis observation aligns with the conclusion stated in Section 3.2.3. However, in this case, when the R value is excessively small, it can cause the matrix to become ill-conditioned, leading to divergent identifcation. Based on the above analysis, it can be concluded that within a certain range, the values of X 0 and P 0 have minimal impact on the identifcation performance of the ASRUKF-FF algorithm. Te initial values of Q and R have a substantial impact on the identifcation results, with particular sensitivity observed in the R value, which signifcantly afects the accuracy of stifness parameter identifcation for the beam structure.

Discussion on the Identifcation Efect of Diferent
Measurements. Analogous to Section 3.2.4, this section primarily focuses on the examination of the infuence of diferent combinations of measurements on the identifcation outcomes. Te comparative analysis employs the control variable method, where all parameter settings remain the same as those in Section 4.2.1, except for the variations in measurements. Furthermore, the calculation process for each working condition can be referenced from Section 3.2.4. Te specifc identifcation results are illustrated in Figures 20-22, while the comparison of fnal identifcation errors is provided in Table 10. Te meanings of the characters in   Table 10 are explained as follows: taking "DIS-3-5-7-9-11" as an example, "DIS-3-5-7-9-11" indicates that all displacement measurements are utilized, with a total of 5 measurement values included. Te measurement values used are the vertical displacements of the simply supported bridge, as depicted in Figure 13, with degrees of freedom of 3, 5, 7, 9, and 11. Furthermore, for the purpose of facilitating comparative observation, a structural diagram of the simply supported bridge is depicted in Figure 13 is provided in Figure 23. It is worth noting that during the simulation analysis process, it has been observed that the time history curve of the sensitive parameter based on the "DIS-7-9" working condition does not exhibit a distinct pulse response characteristic. In addition, the working condition based on "DIS-2-4-8" fails to converge. Terefore, the subsequent discussion and analysis will exclude these two working conditions. Based on the fndings from Figures 20-22 and Table 10, it can be observed that, for the simply supported bridge structure depicted in Figure 13, a minimum of 3 measurements is sufcient to accurately identify the middle four stifness parameters simultaneously. However, it should be noted that not every combination of three measurements can successfully accomplish the identifcation task in this particular case. Trough data analysis, no evident pattern of measurement combinations has been identifed. However, among the combinations of working conditions with three measurements, those that include degrees of freedom 7 and 9, and where the third degree of freedom is located in the left half of the bridge (such as DIS-3-7-9 and DIS-5-7-9), higher identifcation accuracy is exhibited. In addition, it is worth noting that in this case study, there is no positive correlation between the number of measurements and identifcation  accuracy. Having a large number of measurement values does not necessarily guarantee high identifcation accuracy. For instance, in this case, the identifcation performance of the "DIS-3-5-9-11" and "DIS-3-7-9-11" working conditions is superior to that of the original fve measured values. Te combination of measurements is an optimization problem, which is related to the type of structure, environmental noise, response sensitivity, and modal shape. In this case, due to the relatively small number of measurement value combinations, a comprehensive study is conducted to examine the identifcation efect of each combination. In the future, further research is needed to explore the determination method of the optimal measurement value combination scheme, incorporating optimization algorithms for enhanced accuracy.  measurement noise is set at 2%. In addition, diferent modeling errors can result in variations in sensitivity parameter thresholds. Terefore, the specifc implementation process is referenced from Section 3.2.4. Te specifc identifcation results can be observed in Figures 24-27, while the fnal identifcation error is presented in Table 11.
Based on the simulation analysis, it has been determined that the section moment of inertia (I) is the modeling parameter with the most signifcant impact on the identifcation results for the beam model. Furthermore, it has been observed that the maximum identifcation error of the stifness parameters can reach 9.05% when a negative 5% modeling error is present. Meanwhile, the modal damping ratio τ is identifed as the modeling parameter that has the least impact on the identifcation efect.
Moreover, the maximum identifcation errors of stifness parameters are 2.16% and 2.50% under a ± 30% modeling error. When considering only the modeling error of the elastic modulus, it has been observed that the maximum identifcation error of the stifness parameters gradually increases with higher modeling errors. Notably, negative modeling errors have a greater impact on the identifcation efect. Furthermore, it is important to note that the modeling errors of the mass and modal damping ratio parameters do not exhibit a positive correlation with the identifcation errors of the stifness parameters. Interestingly, there is a peculiar phenomenon, where a large modeling error is present, yet the identifcation accuracy remains high. It is speculated that the modeling parameters may compensate for the uncertainties introduced by

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Structural Control and Health Monitoring  Structural Control and Health Monitoring certain random noise components. In summary, even with 2% measurement noise, the proposed adaptive algorithm exhibits a certain ability to mitigate modeling errors for beam structures. However, it is crucial to emphasize the strict control of the parameter value for the section moment of inertia.
(1) Cell data with background color represents the maximum error in each row. (2) A positive modeling error represents an increase in the parameter, while a negative modeling error represents a decrease in the parameter. If the true parameter value is 10, considering a modeling error of ±5%, the parameter value becomes 10 × (1 ± 5%).

Conclusion
In this paper, the standard SRUKF method is modifed by incorporating QR decomposition technology. Tis modifcation aims to eliminate the requirements for a positivedefnite covariance matrix in the unscented transform process and ensure the numerical stability of the algorithm unconditionally. Building upon this modifcation, the ASRUKF-FF method is proposed in this study by incorporating an adaptive forgetting factor. Tis enhancement aims to improve the algorithm's capability to identify timevariant parameters. Te main conclusions are summarized as follows: (1) Te proposed ASRUKF-FF method utilizes a sensitive parameter threshold to intelligently determine whether to invoke the adaptive algorithm. Numerical simulation verifcation, as depicted in Figure 4 and Table 2, demonstrates that the method of determining the threshold based on the time history curve of sensitive parameters calculated by MSRUKF is reliable and accurate. Further simulation results indicate that the proposed ASRUKF-FF method exhibits insensitivity to the threshold within a certain range. Tis characteristic ensures the stability of the algorithm.
(2) In contrast to adaptive methods, where the covariance correction coefcient is typically an empirical constant or determined through experience, the forgetting factor correction coefcient used in this paper is calculated based on the residual information obtained at each recursive step. Tis approach allows for automatic adjustment of the coefcient based on the system's uncertainty. In addition, the forgetting factor can be mathematically derived, providing high reliability.
(3) Te proposed ASRUKF-FF method ofers a broader range of applications and can be efectively utilized in research on inverse problems in the feld of building and bridge structures. Moreover, the method is characterized by its simplicity in implementation, ease of programming, and robustness.
(4) For shear models, variations in X 0 , P 0 , Q, and R values have a minimal impact on the identifcation of stifness parameters. However, they do have a signifcant efect on the identifcation of damping parameters, with R having the most pronounced impact. In the case of stifness parameter identifcation for beam models, the values of X 0 and P 0 have a minor infuence on the identifcation performance. However, the initial values of Q and R play a crucial role in determining the identifcation results, particularly with regard to the sensitivity of the R value in achieving accurate identifcation of the stifness parameters of the beam structure. In addition, for the beam model, it is important to consider the parameter range requirements for X 0 , P 0 , and R. Exceeding the specifed value range can lead to ill-conditioned matrices and ultimately result in identifcation failure. (5) Te proposed algorithm demonstrates high identifcation accuracy even when dealing with incomplete measurement values. In the case of the shear model examined in this paper, as the number of measured values increases, the identifcation accuracy is further improved. In the case of the more complex beam model, the number of measurements is not positively correlated with identifcation accuracy. Te determination method of the optimal measurement value combination scheme still requires further research, particularly in combination with optimization algorithms. (6) In the case of the shear model, when there is a 10% modeling error in the mass parameter, the maximum identifcation errors of the structural stifness and damping parameters are 10.07% and 11.71%, respectively. In the case of the beam model, the section moment of inertia is identifed as the most sensitive modeling parameter, while the modal damping ratio is identifed as the least sensitive modeling parameter. In summary, the proposed ASRUKF-FF algorithm exhibits a certain level of resilience against the infuence of modeling errors.

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

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