Identification of abrupt stiffness changes of structures with tuned mass dampers under sudden events

This paper presents a recursive system identification method for MDoF structures with tuned mass dampers (TMDs) considering abrupt stiffness changes in case of sudden events, such as earthquakes. Due to supplementary nonclassical damping of the TMDs, the system identification of MDoF + TMD systems disposes a challenge, in particular, in case of sudden events. These identification methods may be helpful for structural health monitoring of MDoF structures controlled by TMDs. A new adaptation formulation of the unscented Kalman filter allows the identification method to track abrupt stiffness changes. The paper, firstly, describes the theoretical background of the proposed system identification method and afterwards presents three parametric studies regarding the performance of the method. The first study shows the augmented state identification by the presented system identification method applied on a MDoF + TMD system. In this study, the abrupt stiffness changes of the system are successfully detected and localized under earthquake, impulse, and white noise excitations. The second study investigates the effects of the state covariance and its relevance for the system identification of MDoF + TMD systems. The results of this study show the necessity of an adaptive definition of the state covariance as applied in the proposed method. The third study investigates the effects of modeling on the performance of the identification method. Mathematical models with discretization of different orders of convergence and system noise levels are studied. The results show that, in particular, MDoF + TMD systems require higher order mathematical models for an accurate identification of abrupt changes.


INTRODUCTION
In the past decades, the importance of system identification in civil engineering has grown continously. Response measurements using accelerations, velocities, etc., are common and can be applied for the identification of important parameters, such as the natural frequencies and damping ratios of linear structures. However, the identification of nonlinear structures, including system damages, got more attention recently. As Devin and Fanning 1 mentioned, even partition algorithms for the UKF either require a completed system response time window in a nonrecursive manner or include highly sensitive nonrobust calculation procedures. Since for real-time measurement scenarios, the response data are available only stepwise, and signals are biased by noise, a robust recursive adaptation criterion is required.
This paper presents an UKF-based system identification method for MDoF + TMD systems. In Section 2, an adaptive approach with robust, recursive algorithms is proposed, which enables the detection of abrupt stiffness changes of MDoF + TMD systems during sudden events with a high-level accuracy. In particular, the proposed approach needs no special knowledge of a posteriori system responses. Thus, a constant adaptation criterion based on known sensor properties is driven by statistical signal properties. In Section 3, the presented method is investigated on a MDoF + TMD system by three parametric studies. In the first study, using the proposed system identification method a stiffness and state estimation considering abrupt stiffness changes is performed for several load scenarios. The remaining two studies focus on the filter setup of the system identification method and its influence regarding the identification performance. Therefore, the state covariance influence regarding the identification speed, especially in terms of TMD equipped structures, is investigated in the second study. Finally, the accuracy of four Taylor expansion-based mathematical models of different orders of convergences is analyzed in the third study. In particular, the relationship between the system noise level and mathematical model is explored. A conclusion of the work is presented in Section 4.

System identification method
For the system identification of MDoF + TMD systems, a UKF 12 -based system identification method is proposed. Similar to the linear KF, the UKF consists of a prediction as well as a correction step. To cover the nonlinearities, the UT is applied. In the UT, sampling pointsX i k of size n, where· denotes corrected values, are created on the basis of the known meañ x k and the current state covarianceP k , which is assumed to be Gaussian distributed. For the calculation of the sampling points, weighting factors W i m and W i c are introduced for mean and respectively covariance values. The scaling parameters , , , and are standard values and are mostly chosen based on a Gaussian distribution 14 : Using a nonlinear time variant state equation (·), all sampling points are transformed to the estimatesX i k+1 at the next time step k + 1, where· denotes estimation values. Under assumption of a Gaussian distribution for both time steps k and k + 1, the state estimatex k+1 , as well as the state covariance estimateP k+1 at time step k + 1, can be predicted by summing up all weighted sampling points. Applying the observation equation h(·) at first, the sampling points of the output vector Y i k+1 at time k + 1 can be found and finally weighted to the estimated output vectorŷ k+1 as well: On basis of the innovation error e k+1 = y k+1 −ŷ k+1 , the estimated statex k+1 is corrected using the Kalman gain K k+1 , which is calculated by the covariancesP ,k+1 andP x ,k+1 . Finally, it yields the predicted and corrected statex k+1 and covarianceP k+1 , respectively:P For the parameter estimation, the previous state vector x k , including displacement, velocity, or acceleration information, has to be augmented by a parameter vector k , containing all to be identified parameters. It yields the augmented state vector x a k . Afterwards, the state equation has to be changed, and the UKF can be applied using x a k .

Adaptation scheme for the system identification method
For systems with both initial nonlinearities and abrupt changes, an adaptation procedure is presented as follows. As the corrected state covarianceP k describes the confidence of the estimated and corrected statex k , it can be used to influence the upcoming parameter estimation step, that is, a high state covariance yield more sensitive system identification and nonlinearities can be identified better. For this purpose, firstly, similar to Bisht and Singh, 22 the trigger parameter based on the innovation error e k+1 is introduced. In contrast to Bisht and Singh, the innovation error is normalized by the measurement noise covariance R instead of the measurement covariance P , allowing the trigger parameter to be independent from the measurement noise level: For m sensors with the identical constant measurement noise covariance R i = R, reads To detect system changes, a threshold 0 is defined as an adaptation criterion, which has to be exceeded by in adaptation cases: Bisht and Singh 22 proposed to choose a constant threshold based on the a posteriori known covariance of the measurement signal y k , whereas Rahimi et al. 23 compensated the unknown a posteriori information by introducing variable thresholds over time. However, different than the previous approaches, in this paper, the proposed threshold avoids both the necessity of a posteriori knowledge as well as the high sensitivity resulting from variable thresholds. On basis of known sensor numbers m and measurement noise covariance R, the threshold 0 is calculated. Figure 1 shows the individual steps of the derivation of the threshold, including the approximation of the innovation error with noise terms, the corresponding statistics, the realization of the trigger parameter , and the threshold 0 .
The innovation error e k+1 contains both the system and measurement errors. System changes correspond to system errors. For the identification of system changes from the innovation error, the threshold 0 must cover with a certain probability the measurement error portion of the innovation error. If the trigger parameter exceeds this threshold 0 , the existence of a system error, that is, system change, is ensured. For the definition of such a threshold, we consider firstly a constant system behavior without any changes, that is, no system errors. In this case, the innovation error can be solely approximated by the measurement error, Figure 1 (Step 1: Approximation of innovation error). For the sake of simplicity, the time steps are not explicitly given for each parameter, since each parameter corresponds to the same time step. The measurement error is computed from the difference between the true measurement signal i and the predicted measurement signal̂i. The predicted measurement signal is calculated by the observation equation by the output matrix  with the predicted statex and the transition matrix  with the input u (Section 2.4). The predicted state is independent from the measurement error. Accordingly, output n and the input n u govern the measurement error. If the system motion is observed by displacement and velocity sensors, the transition matrix  becomes zero, so that the innovation error e i solely depends on the output measurement noise n . If acceleration sensors are used,  is an identity matrix, and, consequently, e i is approximated by the difference of both the output n and the input n u measurement noises.
Both measurement noises are assumed to be Gaussian, and each has a covariance of R. Accordingly, their superposition can be treated as Gaussian as well, 24 Figure 1 (Step 2: Statistical properties of innovation error). Consequently, e i has a mean of zero, and its variance R e can be written as the sum of both variances as 2R. The innovation error e i can now be expressed for each sensor by a standard normally distributed variable z e instead of e i and R.
As shown in Figure 1 (Step 3: Trigger parameter and threshold), substituting z e instead of e i and R i in Equation (19) yields for the case of accelerometers ≈ 2mz 2 e , which solely depends on the number of sensors m, the variable z e , and the Scalar 2, which results from the choice of accelerometers. The scalar changes to ≈ mz 2 e for displacement and velocity sensors. Consequently, a parameter is introduced in the calculation of the trigger parameter as with = 1 for displacement and velocity sensors and = 2 for accelerometers, respectively. Accordingly, the corresponding threshold is given by Now, since and m are system dependent preset parameters, z 0 governs the threshold based on the exceeding probability of the Gaussian distribution. For the variable z 0 = 3 √ 2, which corresponds to an exceeding probability of 99.998%, 25 the threshold yields 0 = 72 for two accelerometers. This threshold value will be used in the performance studies in Section 3.1. The presented threshold, Equation (22), is valid for monitoring systems consisting of either only displacement and velocity sensors or only accelerometers. Considering mixed sensor types in the monitoring system, instead, the threshold has to be derived individually as shown above. After detecting abrupt system changes, a localization algorithm has to follow. For this purpose, the localization scheme of Bisht and Singh 22 is extended as shown in Figure 2. The flowchart of the proposed adaptive UKF presents besides the detection of system changes the adaptation step, in particular, consisting of localization and covariance adaptation. In the following paragraph, the subscript denotes covariances P, which are dependent on the system parameters only, and subscript x analogously denotes the state dependent covariances only. To localize system changes, an additional UKF estimation step is shown in Figure 2 for the next time step k + 1. The state covariance componentP ,k+1 [i, i] is set to P adapt for each i = 1, … , n individually, where P adapt is a high constant covariance value, which is introduced to increase the sensitivity of the parameter identification. Since only stiffness degradations are expected, each parameter i with corresponding index i is additionally decreased by 5 %, different than previous studies, in order to facilitate the localization. Accordingly, for each index i, now a different set ofP k+1 and̃k +1 exists. For each of these sets and otherwise unchanged conditions a single calculation step of the UKF is executed and finally the trigger parameter i of Equation (18) is recalculated. Now assuming that the lowest value of i describes the lowest system error and, thus, yields the best estimate for the system properties, the related index i belongs to the degrading parameter i . For the next simulation step k + 1, solely the state covariance componentP ,k+1 [i, i] of the localized index i is substituted by the new state covariance value P adapt , which has to be chosen in advance and is highly dependent on the chosen system noise covariance Q and the present measurement noise covariance R. The parameter has to be chosen as high as possible to enable a system identification of abrupt changes. Section 3.3 will give a detailed simulation example of how to choose P adapt .

Application of the system identification method on MDoF + TMD systems
The theory is introduced using the example case, at which a TMD is attached at the top DoF of a MDoF frame structure, Figure 3. The structure is instrumented with a monitoring system and the proposed system identification method will be implemented on this system to obtain its abrupt stiffness changes. Thus, the equation of motion 26 with stiffness, damping, and mass matrices (K, C, M) can be set up for a seismic ground excitation distributed equally over the height of the system as The stiffness matrix K is assumed to be time variant with abrupt changes assembling a nonlinear structural behavior. Whereas the to be identified stiffness parameters k 1 (t), … , k n (t) are time variant, the damping constants c 1 , … , c n and masses m 1 , … , m n remain constant during the simulation as well as the initially adjusted TMD parameters k d , c d , m d .
A monitoring system is set up to observe the actual system responses. As it is shown in Figure 3, sensors are assumed to be placed on each DoF i = 1, … , n. However, for other systems, e.g. high-rise structures with a large number of DoFs, a different sensor layout with a reduced amount of sensors is possible with a reduced accuracy. For the monitoring system all motion sensors (e.g., displacement, velocity, acceleration, forces) are possible. In the scope of this paper, accelerometers are used only, since they are the most commonly used sensor types for vibration measurements. Sensor properties, such as offset and RMS value of the measurement noise, are required for later system identification steps and the adaptation step, in particular. Using the response y and input signals u, the adaptive system identification method, introduced in Sections 2.1 and 2.2, is applied for the joint state and parameter estimation computing the corrected and estimated state vectors, consisting of displacements, velocities, and system stiffnesses.
The challenge for identification of highly damped systems (e.g., TMD) is to deal with rapidly decreasing vibration amplitudes compared to lightly damped systems. For such a system, a system identification is, therefore, only possible during a significantly smaller time period. In particular, for strongly (nonclassically) damped systems, special attention has to be FIGURE 3 System identification scheme for MDoF + TMD systems. TMD, tuned mass damper paid on the sensitivity or filter settings, respectively, of the system identification as well as the used mathematical models. This aspect will be elaborated in Section 3 by three parametric studies on a MDoF structure with and without TMD.

Modeling of the MDoF + TMD systems
As described in Section 2.1, the UKF requires a state equation (·) and an observation equation h(·), which are herein assumed as stepwise linear state-space representations. Starting with the state equation, the equation of motion of the previously in Section 2.3 described system can be rewritten to a differential equation of 1st order as follows: The system matrix (t) and input matrix  contain the nonlinear system properties and information of input signals, respectively. An additive noise w(t) is added to the state equation describing the system noise, including errors of the mathematical model. The input signal u(t) is given as ground acceleration with the incidence vector from Equation (23).
The monitoring system is transferred to the observation equation, where the output matrix  and transition matrix  describe the sensor layout of number, type, and position. The result is finally enhanced by the noise component v(t), representing measurement noise: Both state and measurement equations are given in continuous time so far. However, the system identification method requires a discrete time formulation, since the measurement data has a discrete form. A discretization can be realized by many methods, e.g. Euler or 4th-order Runge Kutta method. Although every method has different characteristics and calculation rules, all of them can be compared by the order of convergence p, defined by discretization errors. Higher orders of convergences generally yield more accurate results but also have higher computational costs. In case of the explicit Euler method, the order of convergence is p = 1 and for 4th order Runge Kutta p = 4, respectively. In this paper, however, the discretization is done by a Taylor expansion developed from the analytical solution with orders of convergence p = 1 − 4. This approach is preferred here, since all p = 1 − 4 easily can be implemented based on one model only allowing a parametric study of the influence of model accuracy, Section 3.4. The discretization yields the matrices  d and  d : Using the above discretization, the state and observation equation can be easily transformed into the discrete domain assuming real sampling, with the sampling time T s :

PERFORMANCE STUDIES
In this section, investigations on a 2-DoF structure with and without TMD will be presented under seismic, white noise and impulse excitations. Detailed parameter studies are done regarding the system accuracy and convergence behavior of the system identification method considering TMD influence and abrupt stiffness changes of the structure.
Recommendations to the filter and model setup are given for the investigated systems.

Description of the investigated MDoF + TMD systems
Two different systems are investigated: solely a 2-DoF structure as well as the same 2-DoF structure with a TMD attached at the top DoF, Figure 4. Stiffness, damping, and mass matrices (K, C, M) can be set up corresponding to the system parameters, 26 listed in Table 1, as The masses (m 1 , m 2 ) and the damping constants (c 1 , c 2 ) are time invariant values. The stiffness values (k 1 , k 2 ) consider abrupt system changes of 10% at each DoF corresponding to a 5% decrease of the 1st natural frequency, which are realistic values to observe during load cases, such as earthquakes. The initial stiffness values are chosen as k 1 = 12 kN∕m and k 2 = 10kN∕m. For each load case, abrupt stiffness changes appear at a defined time step t according to high interstory drifts between the individual DoFs.
The attached TMD is defined by the time invariant parameters k d , c d , m d , which are chosen in the initial time step. The undamaged 2-DoF structure has a natural frequency of 1 = 0.33 Hz and a damping ratio of D 1 = 0.92% for the 1st eigenmode and analogously for the 2nd eigenmode 2 = 0.84 Hz and D 2 = 2.49%. Accordingly, the structure (2-DoF) is assumed to be classically damped and with supplementary damping of the TMD the damping matrix of the system (2-DoF + TMD) becomes nonclassically damped. Table 1 provides the remaining natural frequencies of the 2-DoF + TMD system. To adjust the damper parameters, several possible approaches are proposed in the literature. In this paper, we System parameter (a) 2-DoF    27 Assuming the structure to be lightly damped (D 1 = 0.92%), an application of Warburton is reasonable. The TMD is tuned to the 1st natural frequency of the 2-DoF structure. Therefore, the mass ratio , describing the relation of damper mass m d and generalized mass of the 1st modem 1 ; the optimal damper frequency opt , dependent on 1 and ; and finally the optimal damping ratio D opt , dependent on , are calculated: Subsequently, all damper parameters can be calculated using fundamental SDoF relations, Table 1. The detuning effect of the TMD due to abrupt stiffness changes can be observed in Figure 5, where for each presented damage pattern, the corresponding deformation response factor is shown. Here, the structure is excited with the harmonic force F = F 0 sin(Ωt) applied on the top floor. Accordingly, the TMD is tuned to Den Hartog's criteria. 28 In a final step, K, C, and M are transformed into the state-space representation. The time variant system matrix (t) and the input matrix  can be set up according to the nonlinear system properties. We formulate the representation for a ground accelerationẍ g as input. Furthermore, the output matrix  and transition matrix  can be calculated as follows, describing a monitoring system of two accelerometers on both DoFs x 1 and x 2 and one accelerometer for the ground motionẍ g : with the input vector u and the output vector y: (34)  Both state and observation equations are calculated for the joint state and parameter estimation, that is, the state vector x extends to an augmented state vector x a , including displacements, velocities, and stiffnesses of the system: For the investigations, one far-field and one near-field earthquake acceleration history is considered according to those input signals in Ohtori et al. 29 :

Study 1: Identification of the augmented state including abrupt stiffness changes
The chosen filter setup of the identification algorithm applied herein is presented in Table 2. The state covariance P is chosen corresponding to the results of the Parametric Study 2, which will be presented in Section 3.3. According to the results of the parametric study, the new state covariance value after adaptation is chosen as high as possible as P adapt = 10 0 . The initial covariance value, however, is chosen as low as possible P 0 = 10 −6 I 8×8 , so that the identification algorithm does not show oversensitive reactions regarding the stiffness estimation before the abrupt stiffness change. The system noise covariance Q is chosen corresponding to the results of the Parametric Study 3, which will be presented in Section 3.4. Both 2-DoF and 2-DoF + TMD systems are modeled based on 3rd-order Taylor expansion. Effects of the used order for the Taylor expansion, in particular, will be also shown in Section 3.4. The measurement noise covariance R is calculated from the square of the RMS value for the present noise, according to 2% RMS noise of El Centro earthquake. The initial stiffness estimationsk 1 andk 2 correspond to the real stiffness values k 1 and k 2 , as shown in Table 1.
In Figure 7 (left), the time history of the trigger parameter is shown. The curve is calculated by the previously, in Section 2.2, introduced Equation (18). In addition, the right diagram shows the time history for a time window around the abrupt stiffness change. The trigger parameter shows a peak value corresponding to the time step of the abrupt stiffness change t = 9 s. After comparing the threshold 0 , which is calculated from Equation (22), the state covariance is adapted. In Figure 7, three different thresholds 0 = 10.8, 0 = 26.5, and 0 = 72 are shown according to the exceeding probability of 90%, 99% and 99.998% respectively. For both probabilities 90% and 99% the threshold is exceeded several times with significantly high values (e.g., = 53 at t = 52 s). Best result is achieved with the threshold value of 0 = 72, which is exceeded only during the abrupt stiffness change.
In the first part of the study, we consider the El Centro earthquake excitation, including a stiffness degradation of 10% at the 1st DoF after t = 9 s due to large story drift. Figure 8 compares the true values of the motion (displacement, Abrupt stiffness change at 1st DoF velocity, and acceleration) of both DoF of the structure and the stiffness time histories with those time histories, which are estimated by the proposed system identification method. Both cases with and without TMD are presented in the graphics. The abrupt stiffness change can be directly seen from the time histories of k 1 andk 1 . Both estimated and true values match with each other. The abrupt stiffness change is identified for both systems.
In the second part of the study, we enhance our investigation by considering besides the both El Centro and Northridge earthquakes also impulse and white noise excitations. Furthermore, we allow an abrupt stiffness change on the 2nd DoF as well. The occurrence times also in this second part of the study correspond to the interstory drift between the 1st and 2nd DoF. In Figures 9 and 10, the corresponding time histories of the estimated and true values of the displacements and stiffness values are shown. A high accuracy of the estimated results is also observed here.

Study 2: Effects of the state covariance
The supplementary damping introduced by the TMD as well as abrupt stiffness changes of the structure shorten the time window, in which the proposed system identification method must complete its estimation. In this regard, the most powerful parameter is the state covariance P. By increasing the state covariance, the reaction time of the identification method can be reduced. On the other hand, too high P values can decrease the estimation accuracy. To clarify this effect, this study performs calculations with different constant P values between 10 −8 and 10 0 . Further filter parameters are shown in Table 3. Calculations are performed using 3rd-order Taylor expansion-based models of 2-DoF and 2-DoF + TMD systems under the Northridge earthquake. The initial stiffness estimates of the structure are assigned ask 1 = 14.4 kN∕m and k 2 = 12 kN∕m, which are 20% higher than the true stiffness values of k 1 = 12 kN∕m and k 1 = 10 kN∕m. Figure 11 (left) shows the true and estimated values of the 1st DoF stiffness k 1 andk 1 . On the right side, in Figure 11, we see the true and estimated values of the displacement of the 1st DoF x 1 andx 1 . The displacement time histories are shown for the selected state covariance values of 10 −8 and 10 0 . From the comparison of the displacement time histories, the effect of the TMD can be clearly observed from the short vibration duration. Already after 35 s, the vibration of the 2-DoF + TMD In the time histories of the stiffness, we observe, in particular for lower P values, that as soon as the vibrations vanish the estimated stiffness of the 2-DoF + TMD system converges to a constant value, which is far away from the real stiffness value. For instance, the estimated stiffness value of 2-DoF + TMD system is for P = 10 −8 approximately 14 kN∕m, which does not match the true stiffness value of 12 kN∕m. For the same P value of 10 −8 , the estimated stiffness of the 2-DoF structure without TMD converges slowly to the true stiffness value as the structure is still continuing to oscillate.
By increasing the P value, we observe from the results that both systems can be identified with high accuracy. For the 2-DoF + TMD system, the correct stiffness value is estimated with P = 10 −4 . On the other hand, as stated before, the 2-DoF structure is estimated already with P = 10 −8 . A further increase of the P value causes the system identification method to behave oversensitive and the estimated stiffness course begins for both systems to fluctuate. With high P values, we observe at the beginning of the both time histories initially underestimated stiffness values.
Accordingly, the P value must be chosen depending on the expected abrupt changes and the type of the system, which is a challenge for all UKF-based system identification methods. To overcome this effect, as introduced in Section 2.2, the proposed parameter identification algorithm tunes the state covariance in an adaptive manner.

Matrix
Taylor expansions

Study 3: Modeling effects
The accuracy of recursive system identification methods is directly related with the accuracy of the chosen mathematical model describing the system properties. The error inherent in the chosen mathematical model is considered in the proposed UKF-based identification method by the system noise covariance Q. However, due to additional damping of TMDs, the accuracy sensitivity of the identification process increases. Therefore, Q struggles to realize the desired identification efficiency. Accordingly, the necessity of an accurate mathematical model increases for MDoF + TMD systems.
In this section, to show the modeling effect, four mathematical models are investigated using  d and  d discretization, introduced in Section 2.4, by Taylor expansions of 1st to 4th order of convergence, Table 4. During the study, different Q matrices, which are constant over simulation time, are introduced varying from 10 −8 I 8×8 to 10 −15 I 8×8 . Two load scenarios are investigated: the El Centro and the Northridge earthquakes. To determine the accuracy of the final stiffness estimation, the deviation parameter Δk i is introduced, which defines the percentage deviation of the final estimated stiffnessk i to the true value k i : In this study, the initial stiffness estimates are chosen to be k 1 = 14.4 kN∕m and k 2 = 12 kN∕m, which are 20% higher than the true stiffness values of k 1 = 12 kN∕m and k 1 = 10 kN∕m. Accordingly, a nonlinear parameter identification is required. Besides this fact, in this study, the structure is assumed to behave linearly during the earthquake excitation without any abrupt stiffness changes. All remaining filter setup parameters are shown in Table 5. These results show the increased sensitivity of the system identification due to supplementary TMD. By increasing the covariance level to Q [i, i] = 10 −9 (right) the deviation reduces. In Figure 12, the other investigated higher order models do not show any dependency with the covariance level.  The study is expended in Figure 13 for further Q [i, i] values. Here, we observe that the Taylor 1st-order expansion-based model of the 2-DoF structure allows for system noise covariance level values higher than Q [i, i] = 10 −9 a high accuracy system identification with Δk 1 < 0.001%. With the same order of the model, the system identification accuracy of the 2-DoF + TMD system also increases by increasing the system noise covariance. However, after reaching its minimum deviation at Q [i, i] = 10 −9 with increasing system noise covariance, the deviation of the stiffness estimation increases again. This effect exists invisible small also for the 2-DoF structure without TMD.
Corresponding to the results of Figure 12, also in Figure 13, we see again for higher order models that the accuracy is independent from the system noise covariance level. Accordingly, as introduced before, we emphasize also with these results the necessity of higher order mathematical models for the identification MDoF + TMD systems.
In Figure 14, the study is repeated for the near-field Northridge earthquake. The performance results of the investigated models conform with the conclusions of the in Figures 12 and 13 shown El Centro results. Also, here, the course of the deviation parameter Δk 1 shows for the 1st-order Taylor expansion model of the 2-DoF structure a stable accuracy after a certain system noise level. On the other hand, for the same order 2-DoF + TMD model the deviation Δk 1 fluctuates depending on the system noise level. For the estimated 2nd DoF stiffnessk 2 , we get similar results, which we do not include here for the sake of brevity.

CONCLUSIONS
In this paper, for MDoF structures with TMDs a recursive system identification method is presented, which is able to detect and localize abrupt stiffness changes during sudden events, such as earthquakes. The method enhances the UKF by a new adaptation formulation, which is modifying the state covariance initiated by a trigger parameter. The proposed adaptation algorithm operates in a recursive manner and calculates the trigger parameter depending on the innovation error, which is normalized by the measurement noise covariance. A constant threshold is formulated based on the sensors. Three parametric studies are conducted on a 2-DoF + TMD system to investigate the performance of the system identification method. In the first study, earthquake, impulse, and white noise excitations are applied. Single and combined abrupt stiffness changes of the DoFs of the structure are simulated. Time histories of estimated and true values of structural motion and stiffness changes are compared. Results show that the proposed identification method is able to detect and localize the abrupt stiffness changes. The estimated state conforms with the true values. The second study investigates the effects of the state covariance. On the 2-DoF + TMD system, an earthquake excitation is applied. An increase of the state covariance improves the parameter estimation performance. However, after a certain value, a further increase causes the identification method to behave oversensitive and loose its accuracy. The results conclude the necessity of an adaptive formulation of the state covariance as applied in the proposed approach. In the third study, the effects of the modeling accuracy are investigated on the 2-DoF + TMD system under earthquake excitation. Besides the effects of the system noise covariance, the study considers also the effects of convergence orders for discretization using Taylor expansion. The results confirm that the identification of abrupt stiffness changes requires a high-level accuracy of the method, in particular, for the identification of MDoF structures with supplementary TMDs.