# Simultaneous Identification of Time-Varying Parameters and External Loads Based on Extended Kalman Filter: Approach and Validation

Authors: Xiaoxiong Zhang; Jia He; Xugang Hua; Zhengqing Chen; Zhouquan Feng
Journal: Structural Control and Health Monitoring
 (2023)

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## Abstract

Online identification of time-variant parameters without knowledge of external loads is an important but challenging task for structural health monitoring and vibration control. In this study, a two-stage approach, named extended Kalman filter with forgetting factor matrix under unknown inputs (EKF-FFM-UI), is proposed for simultaneously identifying the time-variant parameters and external loads. In stage 1, an extended Kalman filter under unknown inputs (EKF-UI) approach previously proposed by the authors is employed for estimating the structural states and unknown loads. This EKF-UI approach is solely suitable for time-invariant system identification. Therefore, the aim of stage 2 is to improve this approach for the purpose of possessing tracking capability. In this stage, the acceleration responses are first reconstructed by using the differential equation of motion and employed for improving the accuracy of estimated structural states. A forgetting factor matrix is introduced into the priori estimation error covariance matrix to track time-varying parameters. The square errors between the measurements and the corresponding estimates are defined as an index and used for detecting the damage time instant. Then, a covariance resetting technique is employed to assure that such changes in structural parameters can be efficiently captured. A shear-type building structure without/with magneto-rheological (MR) dampers and a fixed beam structure are used as numerical examples for validating the effectiveness of the proposed approach. Experimental tests on a six-story building model are also conducted. Results show the time-varying parameters and unknown inputs can be simultaneously identified with acceptable accuracy.


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## Body

## 1. Introduction

The occurrence and growth of structural damages (e.g., stiffness degradation) caused by severe loads or environmental effects are inevitable during the service life of infrastructure. There is no doubt that such phenomena are detrimental to structural safety and durability. Identification of these time-varying parameters by using real-time data is an important but still challenging task for structural health monitoring and vibration control. Much effort has been made in this field, and many literature reviews on parameter identification with different emphases can be found [1–3].In general, real-time identification methods are performed in a recursive manner and enable stepwise parameter identification to detect system changes online. These methods include but do not limit to wavelet transform [4–6], recursive least-squares [7, 8], Hilbert transform [9, 10], deep learning [11, 12], and Bayesian filter [13, 14]. This paper is not intended to provide a literature review on real-time parameter identification methodologies. Instead, the following introduction will focus on the online estimation of structural systems based on the Kalman filter (KF) technique.As known, the KF is a powerful tool for recursively estimating structural states, such as displacement and velocity. However, the classic KF is only suitable for linear systems whose parameters are assumed to be known. On this account, with the aid of a linearization procedure and Jacobian matrix, the extended Kalman filter (EKF) is developed for parameter identification by adding the unknown parameters into structural states. Considering the linearized approximation may not be accurate enough for highly nonlinear systems. The unscented Kalman filter (UKF) is further developed with the aid of unscented transformation. Both EKF and UKF can be used to identify time-invariant parameters of linear and nonlinear systems.As mentioned before, due to various man-made or natural factors, the physical parameters of structures often display time-dependent characteristics. To cover the case of time-varying parameter identification, the EKF or UKF-based methods should be improved to possess adaptive tracking capabilities. In this regard, some attempts have been made in the past several years. For example, the tracking ability of adaptive fading KF was investigated by Loh et al. [15, 16]. By using an adaptive factor matrix, Yang et al. [17] proposed an EKF-based time-varying parameter identification approach and compared it with other tracking techniques using experimental data [18]. The derivation of this approach was complicated, and then Lei et al. [19] proposed a three-stage EKF approach to improve the computational efficiency. Based on the appropriate adjustment of the state covariance matrix, Bisht and Singh [20] proposed an adaptive UKF for tracking sudden stiffness changes. By using an innovation error and the corresponding trigger parameter, Schleiter and Altay [21] proposed a UKF-based approach for the identification of abrupt stiffness changes in controlled structures. With the combination of KF and wavelet decomposition, Chen et al. [22] proposed an approach for identifying time-variant parameters. Based on the concept of pre-estimation of system parameters, Ciolek et al. [23] proposed a decoupled KF approach for the estimation of time-varying systems. To enhance the convergence, Yu et al. [24] proposed a modified EKF for identifying time-variant Hammerstein–Wiener nonlinear systems. More recently, the KF-based online identification of cable force [25], precast segmental columns [26], or sensor fault [27] was also conducted.Although the methods mentioned above can deal with time-varying systems, they are performed on the premise that the external loads are known for the identification. However, in many practical situations, it may be difficult or impossible to obtain the measured information of external excitation. Thus, based on the framework of KF, some effort has also been made for developing the so-called joint input-state estimation techniques. For example, Eftekhar Azam et al. [28] proposed a dual KF approach for estimating the unknown forces and structural states. This method was then validated and compared with other filters via a four-story frame structure [29]. Liu et al. [30] proposed an improved KF approach under unknown inputs by fusing acceleration and displacement together. The corresponding modal KF approach was then presented by Lei et al. [31]. Based on a modified observation equation, He et al. [32] proposed a KF-based approach for estimating nonlinear systems with unknown inputs. This approach was then extended for multiscale response reconstruction with the aid of data fusion techniques [33]. In general, the structural states to be estimated in the KF algorithm are composed of displacement and velocity. By adding the unknown inputs into the state vector, Lourens et al. [34] developed an augmented Kalman filter (AKF) for joint input-state estimation. This idea is attractive because the unknown inputs can be recursively estimated in conjunction with the states. To date, AKF is still widely investigated by many researchers. For example, under the framework of the AKF, Cumbo et al. [35] proposed an approach for optimal sensor placement. Using modal expansion, Eftekhar Azam et al. [36] proposed an AKF-based approach in the modal domain for input-state estimation of steel railway bridges. Based on sparse constraint theory, Wei et al. [37] proposed an AKF approach for improved force identification.The KF-based force identification methods introduced before are developed based on prior knowledge of structural parameters. The simultaneous identification of structural parameters and external forces will be more complicated, especially in the case of parameters being time-variant. To the authors’ knowledge, the idea of EKF under unknown inputs (EKF-UI) for adaptively capturing time-variant parameters was first presented by Yang et al. [38]. Since then, some efforts have been made. For example, with the combination of wavelet multiresolution analysis and nonlinear least-squares optimization, Lei and Yang [39] proposed a KF-based approach for identifying time-variant parameters under unknown inputs. By employing wavelet multiresolution analysis, Yang et al. [40] proposed a fading-factor UKF method for the identification of time-variant nonlinear systems under unknown inputs. Lei et al. [41] proposed a generalized EKF approach for identifying and controlling time-variant systems under unknown seismic inputs. Based on the idea of substructure and fading factor, Yang et al. [42] proposed an EKF-UI approach for time-varying parameter identification in large-scale structures. With the usage of an online tracking matrix, Zhang et al. [43] proposed an EKF-UI approach for real-time structural identification. By modelling the unknown inputs as modulated colored noise and using Bayesian methodology to recursively update the noise covariance matrices, Huang et al. [44] proposed an EKF-based real-time input-state-parameter estimation approach.Online identification without prior knowledge of inputs is quite a challenging task. Under this circumstance, the KF-based approaches mentioned above are basically conducted with the combination of other techniques, such as wavelet transformation, least-squares estimation, or nonlinear optimization. However, these procedures may be time-consuming, resulting in the difficulty of achieving real-time identification. Therefore, by solely using the EKF principle, a two-stage approach, named extended Kalman filter with forgetting factor matrix under unknown inputs (EKF-FFM-UI), is proposed in this study for simultaneously identifying time-variant parameters and external loads. In stage 1, an EKF-UI approach previously proposed by the authors [45] is employed for estimating the structural states and unknown loads. This EKF-UI approach is solely suitable for time-invariant system identification. Therefore, the aim of stage 2 is to improve this approach for the purpose of possessing tracking capability. In this stage, the acceleration responses are first reconstructed based on the differential equation of motion and used for updating the estimated structural states. A forgetting factor matrix is introduced into the priori estimation error covariance matrix to track time-varying parameters. The square errors between the measured acceleration responses and those estimated ones are defined as an index and used for detecting the damage time instant. The abrupt increase in square errors can be viewed as an indication that variations in structural parameters are occurring. Then, a covariance resetting technique is employed at this time to assure that such changes in structural parameters can be efficiently captured.This paper is organized as follows: First, the formulas of the proposed two-stage EKF-FFM-UI approach are introduced in Section2. Then, a shear-type building structure without/with MR damper and a fixed beam structure are employed as numerical examples in Section 3 for validating the effectiveness of the proposed approach. In Section 4, dynamic tests on a six-story shear-type building model are conducted and used for demonstrating the effectiveness of the proposed approach for a real structure. In Section 5, some conclusions are presented.


## 2. The Formulas of the Proposed Two-Stage EKF-FFM-UI Approach

### 2.1. Stage 1: The Previously Proposed EKF-UI Approach

In this stage, the EKF-UI approach previously developed by the authors [45] is employed for estimating the structural parameters and unknown excitation. This approach is briefly introduced herein.The equation of motion of ann degrees-of-freedom (DOFs) system can be expressed as(1)Mx¨t+Fx˙t,xt,θt=φufut,where x¨t,x˙t, and x(t) denote structural acceleration, velocity, and displacement, respectively, M is the mass matrix and assumed to be known, θ(t) is the m-dimensional parameter vector to be identified, Fx˙t,xt,θt represents the linear/nonlinear restoring force, and fut and φu are the unknown input and the corresponding influence matrix, respectively.An extended state vector is defined as(2)Zt=xtT,x˙tT,θtTT.Then, one can obtain(3)dZtdt=x˙tM−1−Fx˙t,xt,θ+φufutθ˙t=uZt,fut,t+wt,where wt is the process noise vector with zero mean and a covariance matrix Q(t). If the time-independent structural parameters are considered, θ˙t in equation (3) will be zero.Here, limited acceleration and displacement responses are measured and fused into the observation equation. The observation equation at thek-th time instant can be discretized as(4)y¯k=ykayks=φax¨k+vkaφsxk+vks=−φaM−1Fx˙k,xk,θk−φfufku+vkaφsxk+vks,where yka and yks are acceleration and displacement measurements, respectively, φa and φs are associated with the placement of accelerometers and displacement transducers, respectively, and vka and vks are the corresponding measurement noise vectors with zero mean and covariance matrices Rka and Rks, respectively.A modified version of the observation equation can be derived with the aid of projection matrix [45],(5)Φ¯y¯k=Φ¯h¯Zk+Φ¯v¯k,where Φ¯=Φ00I;Φ=In−DDTD−1DT;D=−φaM−1φu;v¯k=vkavks;(6)h¯Zk=hZkφsxk;hZk=−φaM−1Fx˙k,xk,θk.Based on the aforementioned state equation and modified observation equation, the EKF-UI approach can be performed as follows, including priori estimation, posteriori estimation, and load identification. More details can be found in Ref. [45].Priori estimation:(7)Z^k+1k=Z^kk+∫k∆tk+1∆tuZ^kk,f^ku,k∆tdt,(8)Pk+1k=A1PkkA1T+A2RkaA2T+∆t2Qk,where ∆t is the time interval; Z^kk and f^ku are the estimates of Zk and fku, respectively, and Z^k+1k and Pk+1k are the priori state estimate and the corresponding error covariance matrix, respectively;(9)A1=I+∆tUkk−∆t0I−ΦHkk0;A2=0−∆tI−Φ0;Hk+1k=∂hZk+1∂Zk+1Zk+1=Z^k+1k;Ukk=∂uZk,fku,tk∂ZkZk=Z^kk,fku=f^ku.Posteriori estimation:(10)Z^k+1k+1=Z^k+1k+Gk+1Φ¯y¯k+1−Φ¯h¯Z^k+1k,(11)Pk+1k+1=I−Gk+1Φ¯H¯k+1kPk+1kI−Gk+1Φ¯H¯k+1kT+Gk+1Φ¯R¯k+1Φ¯TGk+1T,(12)Gk+1=Pk+1kH¯k+1kTΦ¯TΦ¯H¯k+1kPk+1kH¯k+1kT+R¯k+1Φ¯T−1,where Z^k+1k+1 and Pk+1k+1 are the posteriori state estimate and the corresponding error covariance matrix, respectively, Gk+1 is the gain matrix, R¯k+1 is the covariance matrix of measurements noise vector v¯k+1, and H¯k+1k=∂h¯Zk+1/∂Zk+1Zk+1=Z^k+1k.Load identification:(13)f^k+1u=DTD−1DThZ^k+1k+1−yk+1a.


### 2.2. Stage 2: Tracking Capability Improvement

As mentioned before, the EKF-UI approach introduced in the previous subsection is only suitable for time-invariant system and unable to identify time-variant parameters. In order to capture time-varying parameters, the following procedures are developed for the improvement of tracking capability.Based on the identified results shown in equations (10) and (13), the structural acceleration can be calculated according to equation (1).(14)x¨^k+1=−M−1Fx˙^k+1,x^k+1,θ^k+1−φuf^k+1u,where x˙^k+1 and x^k+1 are the estimates of velocity and displacement obtained from equation (10), respectively. This calculated acceleration shown in equation (14) can be divided into two parts. The first part named as x¨^k+1a is associated with the DOFs where the accelerometers are installed, whereas the other part named as x¨^k+1u corresponds to the unmeasured DOFs.By assumingx¨^k+1u as “fictitious measurements,” the observation equation shown in equation (4) can be updated as(15)y˜k+1=y˜k+1ayk+1s+v˜k+1avk+1s,where y˜k+1a=x¨^k+1uyk+1a and v˜k+1a is the noise vector of y˜k+1a with zero mean and covariance matrices R˜k+1a.Similar to equation (5), a modified observation equation can be found as(16)Φ˜y˜k+1=Φ˜h˜Zk+1+Φ˜v˜k+1,where Φ˜=Φ′00I;Φ′=In−D˜D˜TD˜−1D˜T;D˜=−M−1φu;h˜Zk+1=h′Zk+1φsxk+1;(17)h′Zk+1=−M−1Fx˙k+1,xk+1,θk+1v˜k+1=v˜k+1avk+1s.As can be seen from equations (15) and (16), all the acceleration responses including the measured and reconstructed ones are considered in the observation equation. The usage of these reconstructed responses is helpful to assure convergence speed and stability.For the purpose of tracking time-variant parameters, a forgetting factor matrixΛk+1 is introduced into the priori estimation error covariance matrix. The aim of using the forgetting factor is to place more importance on recent errors than older ones. As stated by Yang et al. [17], if the forgetting factor is set to be large, it has less tracking capability but is robust to noise. On the contrary, if a small value is adopted, it has better tracking capability, but is vulnerable to noise pollution. Λk+1 is a (2n + m)×(2n + m) diagonal matrix. The first 2n diagonal elements associated with displacement and velocity are assumed to be one. The last m diagonal elements corresponding to the unknown parameters are set to be close to one.Then, equation (8) can be updated as(18)P˜k+1k=Λk+1A˜1P˜kkA˜1TΛk+1T+A˜2R˜kaA˜2T+∆t2Qk,where P˜k+1k is the updated priori estimation error covariance matrix.(19)A˜1=I+∆tUkk−∆t0I−Φ′Hkk′0;A˜2=0−∆tI−Φ′0;Hk+1k′=∂h′Zk+1∂Zk+1Zk+1=Z˜k+1k.According to equation (8), the updated prior state estimate Z˜k+1k can be obtained as(20)Z˜k+1k=Z˜kk+∫k∆tk+1∆tuZ˜kk,f˜ku,k∆tdt,where f˜ku means the updated value of the identified load at the k-th time step.Based on equation (16), the updated posteriori state estimate Z˜k+1k+1 can be found as(21)Z˜k+1k+1=Z˜k+1k+G˜k+1Φ˜y˜k+1−Φ˜h˜Z˜k+1k,where G˜k+1 is the updated gain matrix.Similarly, the updated posteriori estimation error covarianceP˜k+1k+1 can be obtained basing on equation (11),(22)P˜k+1k+1=I−G˜k+1Φ˜H˜k+1kΛk+1P˜k+1kΛk+1TI−G˜k+1Φ˜H˜k+1kT+G˜k+1Φ˜R˜k+1Φ˜TG˜k+1T,where R˜k+1 is the covariance matrix of noise vector v˜k+1; H˜k+1k=∂h˜Zk+1/∂Zk+1Zk+1=Z˜k+1k.The updated gain matrix can be found as(23)G˜k+1=Λk+1P˜k+1kΛk+1TH˜k+1kTΦ˜TΦ˜H˜k+1kΛk+1P˜k+1kΛk+1TH˜k+1kT+R˜k+1Φ˜T−1.Finally, the identified force can be updated as(24)f˜k+1u=D˜TD˜−1D˜Th′Z˜k+1k+1−y˜k+1a.More details on the derivation of equations (16)–(24) can be found in the Appendix. Although the procedures introduced above make the EKF-UI approach possess tracking capability, the limitation of using a constant forgetting factor matrix may be especially evident for identifying abrupt changes in structural parameters. Therefore, in order to ensure the variations of structural parameters that can be efficiently captured, a covariance resetting technique is further employed.As stated by other researchers [19, 46, 47], abrupt changes in parameters introduce immediate discontinuities in structural acceleration responses. To clearly display these discontinuities in acceleration signals, the square errors between the measured accelerations and those reconstructed ones are calculated and considered an index.(25)∆i,k+1=yi,k+1a−x¨^i,k+1a2,where ∆i,k+1 is the square error of the acceleration at the i-th DOF; yi,k+1a is the measured acceleration at the i-th DOF; x¨^i,k+1a is the reconstructed acceleration as mentioned in equation (14).A jump increase of∆i,k+1 indicates the onset of parameter variations such as stiffness degradation. As this damage time instant is detected, a covariance resetting technique is used immediately. The posteriori estimate error covariance matrix is reset as follows:(26)P¯=diag01×2n,P112n+1,2n+1,…,P112n+m,2n+m.where P¯ is called as reset error covariance matrix; and P1|1 is the initial value of error covariance matrix which is given at the beginning. It can be seen that P¯ is a (2n + m)×(2n + m) diagonal matrix. The first 2n diagonal elements are set to be zero to reduce the oscillation of the estimated responses, and the last m diagonal elements are equal to the corresponding initial value of P1|1. As the usage of P¯ at the damage time instant, the error covariance of structural parameters is reset to a larger value which means more powerful estimation is conducted at this time. This makes the time-variant parameters can be captured more efficiently.The flowchart of the proposed approach is plotted in Figure1 for ease of understanding.Figure 1
Flowchart of the proposed two-stage EKF-FFM-UI approach.


## 2.1. Stage 1: The Previously Proposed EKF-UI Approach

In this stage, the EKF-UI approach previously developed by the authors [45] is employed for estimating the structural parameters and unknown excitation. This approach is briefly introduced herein.The equation of motion of ann degrees-of-freedom (DOFs) system can be expressed as(1)Mx¨t+Fx˙t,xt,θt=φufut,where x¨t,x˙t, and x(t) denote structural acceleration, velocity, and displacement, respectively, M is the mass matrix and assumed to be known, θ(t) is the m-dimensional parameter vector to be identified, Fx˙t,xt,θt represents the linear/nonlinear restoring force, and fut and φu are the unknown input and the corresponding influence matrix, respectively.An extended state vector is defined as(2)Zt=xtT,x˙tT,θtTT.Then, one can obtain(3)dZtdt=x˙tM−1−Fx˙t,xt,θ+φufutθ˙t=uZt,fut,t+wt,where wt is the process noise vector with zero mean and a covariance matrix Q(t). If the time-independent structural parameters are considered, θ˙t in equation (3) will be zero.Here, limited acceleration and displacement responses are measured and fused into the observation equation. The observation equation at thek-th time instant can be discretized as(4)y¯k=ykayks=φax¨k+vkaφsxk+vks=−φaM−1Fx˙k,xk,θk−φfufku+vkaφsxk+vks,where yka and yks are acceleration and displacement measurements, respectively, φa and φs are associated with the placement of accelerometers and displacement transducers, respectively, and vka and vks are the corresponding measurement noise vectors with zero mean and covariance matrices Rka and Rks, respectively.A modified version of the observation equation can be derived with the aid of projection matrix [45],(5)Φ¯y¯k=Φ¯h¯Zk+Φ¯v¯k,where Φ¯=Φ00I;Φ=In−DDTD−1DT;D=−φaM−1φu;v¯k=vkavks;(6)h¯Zk=hZkφsxk;hZk=−φaM−1Fx˙k,xk,θk.Based on the aforementioned state equation and modified observation equation, the EKF-UI approach can be performed as follows, including priori estimation, posteriori estimation, and load identification. More details can be found in Ref. [45].Priori estimation:(7)Z^k+1k=Z^kk+∫k∆tk+1∆tuZ^kk,f^ku,k∆tdt,(8)Pk+1k=A1PkkA1T+A2RkaA2T+∆t2Qk,where ∆t is the time interval; Z^kk and f^ku are the estimates of Zk and fku, respectively, and Z^k+1k and Pk+1k are the priori state estimate and the corresponding error covariance matrix, respectively;(9)A1=I+∆tUkk−∆t0I−ΦHkk0;A2=0−∆tI−Φ0;Hk+1k=∂hZk+1∂Zk+1Zk+1=Z^k+1k;Ukk=∂uZk,fku,tk∂ZkZk=Z^kk,fku=f^ku.Posteriori estimation:(10)Z^k+1k+1=Z^k+1k+Gk+1Φ¯y¯k+1−Φ¯h¯Z^k+1k,(11)Pk+1k+1=I−Gk+1Φ¯H¯k+1kPk+1kI−Gk+1Φ¯H¯k+1kT+Gk+1Φ¯R¯k+1Φ¯TGk+1T,(12)Gk+1=Pk+1kH¯k+1kTΦ¯TΦ¯H¯k+1kPk+1kH¯k+1kT+R¯k+1Φ¯T−1,where Z^k+1k+1 and Pk+1k+1 are the posteriori state estimate and the corresponding error covariance matrix, respectively, Gk+1 is the gain matrix, R¯k+1 is the covariance matrix of measurements noise vector v¯k+1, and H¯k+1k=∂h¯Zk+1/∂Zk+1Zk+1=Z^k+1k.Load identification:(13)f^k+1u=DTD−1DThZ^k+1k+1−yk+1a.


## 2.2. Stage 2: Tracking Capability Improvement

As mentioned before, the EKF-UI approach introduced in the previous subsection is only suitable for time-invariant system and unable to identify time-variant parameters. In order to capture time-varying parameters, the following procedures are developed for the improvement of tracking capability.Based on the identified results shown in equations (10) and (13), the structural acceleration can be calculated according to equation (1).(14)x¨^k+1=−M−1Fx˙^k+1,x^k+1,θ^k+1−φuf^k+1u,where x˙^k+1 and x^k+1 are the estimates of velocity and displacement obtained from equation (10), respectively. This calculated acceleration shown in equation (14) can be divided into two parts. The first part named as x¨^k+1a is associated with the DOFs where the accelerometers are installed, whereas the other part named as x¨^k+1u corresponds to the unmeasured DOFs.By assumingx¨^k+1u as “fictitious measurements,” the observation equation shown in equation (4) can be updated as(15)y˜k+1=y˜k+1ayk+1s+v˜k+1avk+1s,where y˜k+1a=x¨^k+1uyk+1a and v˜k+1a is the noise vector of y˜k+1a with zero mean and covariance matrices R˜k+1a.Similar to equation (5), a modified observation equation can be found as(16)Φ˜y˜k+1=Φ˜h˜Zk+1+Φ˜v˜k+1,where Φ˜=Φ′00I;Φ′=In−D˜D˜TD˜−1D˜T;D˜=−M−1φu;h˜Zk+1=h′Zk+1φsxk+1;(17)h′Zk+1=−M−1Fx˙k+1,xk+1,θk+1v˜k+1=v˜k+1avk+1s.As can be seen from equations (15) and (16), all the acceleration responses including the measured and reconstructed ones are considered in the observation equation. The usage of these reconstructed responses is helpful to assure convergence speed and stability.For the purpose of tracking time-variant parameters, a forgetting factor matrixΛk+1 is introduced into the priori estimation error covariance matrix. The aim of using the forgetting factor is to place more importance on recent errors than older ones. As stated by Yang et al. [17], if the forgetting factor is set to be large, it has less tracking capability but is robust to noise. On the contrary, if a small value is adopted, it has better tracking capability, but is vulnerable to noise pollution. Λk+1 is a (2n + m)×(2n + m) diagonal matrix. The first 2n diagonal elements associated with displacement and velocity are assumed to be one. The last m diagonal elements corresponding to the unknown parameters are set to be close to one.Then, equation (8) can be updated as(18)P˜k+1k=Λk+1A˜1P˜kkA˜1TΛk+1T+A˜2R˜kaA˜2T+∆t2Qk,where P˜k+1k is the updated priori estimation error covariance matrix.(19)A˜1=I+∆tUkk−∆t0I−Φ′Hkk′0;A˜2=0−∆tI−Φ′0;Hk+1k′=∂h′Zk+1∂Zk+1Zk+1=Z˜k+1k.According to equation (8), the updated prior state estimate Z˜k+1k can be obtained as(20)Z˜k+1k=Z˜kk+∫k∆tk+1∆tuZ˜kk,f˜ku,k∆tdt,where f˜ku means the updated value of the identified load at the k-th time step.Based on equation (16), the updated posteriori state estimate Z˜k+1k+1 can be found as(21)Z˜k+1k+1=Z˜k+1k+G˜k+1Φ˜y˜k+1−Φ˜h˜Z˜k+1k,where G˜k+1 is the updated gain matrix.Similarly, the updated posteriori estimation error covarianceP˜k+1k+1 can be obtained basing on equation (11),(22)P˜k+1k+1=I−G˜k+1Φ˜H˜k+1kΛk+1P˜k+1kΛk+1TI−G˜k+1Φ˜H˜k+1kT+G˜k+1Φ˜R˜k+1Φ˜TG˜k+1T,where R˜k+1 is the covariance matrix of noise vector v˜k+1; H˜k+1k=∂h˜Zk+1/∂Zk+1Zk+1=Z˜k+1k.The updated gain matrix can be found as(23)G˜k+1=Λk+1P˜k+1kΛk+1TH˜k+1kTΦ˜TΦ˜H˜k+1kΛk+1P˜k+1kΛk+1TH˜k+1kT+R˜k+1Φ˜T−1.Finally, the identified force can be updated as(24)f˜k+1u=D˜TD˜−1D˜Th′Z˜k+1k+1−y˜k+1a.More details on the derivation of equations (16)–(24) can be found in the Appendix. Although the procedures introduced above make the EKF-UI approach possess tracking capability, the limitation of using a constant forgetting factor matrix may be especially evident for identifying abrupt changes in structural parameters. Therefore, in order to ensure the variations of structural parameters that can be efficiently captured, a covariance resetting technique is further employed.As stated by other researchers [19, 46, 47], abrupt changes in parameters introduce immediate discontinuities in structural acceleration responses. To clearly display these discontinuities in acceleration signals, the square errors between the measured accelerations and those reconstructed ones are calculated and considered an index.(25)∆i,k+1=yi,k+1a−x¨^i,k+1a2,where ∆i,k+1 is the square error of the acceleration at the i-th DOF; yi,k+1a is the measured acceleration at the i-th DOF; x¨^i,k+1a is the reconstructed acceleration as mentioned in equation (14).A jump increase of∆i,k+1 indicates the onset of parameter variations such as stiffness degradation. As this damage time instant is detected, a covariance resetting technique is used immediately. The posteriori estimate error covariance matrix is reset as follows:(26)P¯=diag01×2n,P112n+1,2n+1,…,P112n+m,2n+m.where P¯ is called as reset error covariance matrix; and P1|1 is the initial value of error covariance matrix which is given at the beginning. It can be seen that P¯ is a (2n + m)×(2n + m) diagonal matrix. The first 2n diagonal elements are set to be zero to reduce the oscillation of the estimated responses, and the last m diagonal elements are equal to the corresponding initial value of P1|1. As the usage of P¯ at the damage time instant, the error covariance of structural parameters is reset to a larger value which means more powerful estimation is conducted at this time. This makes the time-variant parameters can be captured more efficiently.The flowchart of the proposed approach is plotted in Figure1 for ease of understanding.Figure 1
Flowchart of the proposed two-stage EKF-FFM-UI approach.


## 3. Numerical Investigation

To demonstrate the effectiveness of the proposed approach, a six-story shear-type building structure without/with an MR damper and a fixed beam structure are considered as numerical examples in this section. For practical considerations, the measurements are simulated by the corresponding theoretical ones superimposed with noise of 5% noise-to-signal ratio in root mean square.


### 3.1. A Six-Story Building Structure

A six-story shear-type building structure is considered as the first numerical example. The mass and stiffness of each story are 100 kg and 120 kN/m, respectively. The Rayleigh damping model, i.e.C = α·M+β·K, is used. The proportional coefficients are α = 0.3739 and β = 1.8227 × 10−3. To mimic structural damage, the stiffness parameters of the 1st story is reduced abruptly from 120 kN/m to 72 kN/m at t = 4s. The concentrated random excitation is assumed to be applied on the top floor, and thus the force influence matrix can be expressed as φu = (0 0 0 0 0 1)T. By using the state space method, the structural responses can be calculated. The time interval in this example is 0.001 s. Here, the acceleration responses of the 1st, 3rd, 5th, and 6th floors as well as the displacement response of the 6th floor are assumed to be measured and used for the identification.The quantities to be identified include six stiffness parameters, two damping coefficients, and the external excitation. The structural parameters are initially assumed to be 50% of the actual ones. The posteriori estimate error covariance matrix is set to beP11=diagI1×12,101×614,101×210 at the beginning. The first 12 and the last 8 diagonal elements of forgetting factor matrix (FFM) are set to be 1 and 1.001, respectively. The reset error covariance matrix used for the enhancement of tracking capability is set as P¯=diag01×12,101×614,101×210. The proper assignment of process and measurement noise covariance matrices (i.e., Q and R in this study) plays an important role in ensuring the reliability and stability of recursive KF-based methods. As stated by Eftekhar Azam et al. [28], the Q matrix reflects the confidence put in the utilized structural model. A large value is usually selected if the model uncertainty is large. The R matrix reveals the confidence put in the observations. The larger value means the more measurement noise is considered. Research on proper adjustment of Q and R matrices has been conducted by some scholars using different techniques, such as Bayesian probabilistic algorithm [48, 49] and mathematical regularization techniques [28, 50]. However, the optimal determination of Q and R matrices is not the main purpose of this study. The constant values of these two matrices in this paper are selected based on other available literature [38, 51, 52]. Here, the Q and R matrices are set to be 10−4 × I and I, respectively, where I denotes identity matrix.The structural parameters can be identified by using the proposed approach as given in Table1. Clearly, the identified results are quite close to the actual ones. To investigate the tracking capability, the square errors defined in equation (21) are calculated and shown in Figure 2. It is obvious that a significant increase of ∆1 can be observed at t = 4 s. This means the structural damage is happening at this time. Only the time histories of the identified stiffness of the 1st and 6th stories are plotted in Figure 3 due to the limitation of the paper length. Similar results can be found for the remaining stories. Moreover, for ease of understanding, two additional cases are considered in this example for demonstrating the necessity of using FFM and P¯ matrix. In case 1, only FFM is used for the online identification. In this case, the error covariance matrix will not be reset by equation (22) even though the damage instant is detected. Instead, only P¯ matrix rather than FFM is used in case 2. It is clear from Figure 3(a) that neither of these two cases can track the abrupt change of k1. The proposed approach, however, is capable of accurately and stably capturing this sudden decrease.Table 1
The identified results of the building structure.
ParametersActualIdentifiedError (%)k1 (kN/m)120⟶72119.84⟶71.99−0.13⟶−0.01k2 (kN/m)120120.440.36k3 (kN/m)120120.130.11k4 (kN/m)120120.070.06k5 (kN/m)120119.56−0.37k6 (kN/m)120120.290.24α0.37390.3618−3.24β × 10−31.82271.86192.15Figure 2
Time histories of square errors in the 1st example. (a) Time history of square error of ∆1. (b) Time history of square error of ∆3.
(a)(b)Figure 3
The identified stiffness of building structure. (a) Stiffness of the 1st story. (b) Stiffness of the 6th story.
(a)(b)Besides identifying time-variant parameters, the unknown excitation can be identified as well. As shown in Figure4(a), the time history of the identified load is plotted by a solid line, whereas the actual one is given by a dashed line. The time segment from 4 s to 6 s is further given in Figure 4(b) for clarity of comparison. Obviously, the identified input is in good agreement with the actual one.Figure 4
The identified excitation applied to building structure. (a) The entire time history. (b) The segment from 4 s to 6 s.
(a)(b)


### 3.2. Six-Story Building Structure with MR Damper

The aim of this section is to investigate the performance of the proposed approach for the identification of nonlinear structure. The six-story building structure introduced in Section3.1 is employed herein. To mimic structural nonlinearity, an MR damper described by the Bingham model is assumed to be installed on the 1st story. The restoring force provided by the MR damper is expressed as follows:(27)FMR=fcsgn∆x˙i+cb∆x˙i+fb,where ∆x˙i is the interstory velocity and defined as ∆x˙i=x˙i−x˙i−1 and fc, cb, and fb are the model coefficients and set to be fc = 1 N, cb = 1000 N.s/m, and fb = 0.In this section, multiple damages are considered. The stiffness parameters of the 1st and 2nd stories are reduced by 50% att = 5 s. A random force is applied to the top floor, and the corresponding structural responses are computed by the Runge–Kutta method. The acceleration responses of the 1st, 3rd, 5th, and 6th floors and displacement response of the 6th floor are employed for identification. The sampling frequency is 1000 Hz. The parameters are initially set to be 50% of the actual ones. The first 12 and the last 10 diagonal elements of the FFM are set to be 1 and 1.002, respectively. The reset error covariance matrix is P¯=diag01×12,101×612,101×22,104,109.The parameters can be identified by using the proposed approach as listed in Table2. Satisfactory identification results can be observed even though structural nonlinearities are involved. The square errors used for detecting damage instant are also calculated as plotted in Figure 5. Due to paper length limitations, only the square errors of the 1st and 3rd stories are given as examples. Apparently, the damage instant is at t = 5s. The time histories of the identified time-varying stiffness are shown in Figure 6. Similarly, two additional cases, i.e., FFM only and P¯ matrix only, are considered for comparison. It can be found that only using FFM or P¯ matrix is unable to efficiently track time-varying parameters. The proposed approach with the combination of FFM and P¯ matrix is capable of capturing time-varying parameters with good accuracy. Also, the external excitation can be simultaneously identified as shown in Figure 7. It is obvious that the identified force is close to the actual one.Table 2
The identified results of the building structure with MR damper.
ParameterActualIdentifiedError (%)k1 (kN/m)120⟶60120.13⟶60.740.11⟶1.23k2 (kN/m)120⟶60120.73⟶60.500.61⟶0.83k3 (kN/m)120119.68−0.27k4 (kN/m)120120.330.27k5 (kN/m)120119.72−0.61k6 (kN/m)120119.41−0.49fc (N)10.9678−3.22cb (N·s/m)1000979.24−2.08α0.37390.3574−4.43β × 10−31.82271.85411.72Figure 5
Time histories of square errors in the 2nd example. (a) Time history of square error of ∆1. (b) Time history of square error of ∆3.
(a)(b)Figure 6
The identified stiffness of building structure with MR damper. (a) Stiffness of the 1st story. (b) Stiffness of the 2nd story.
(a)(b)Figure 7
The identified excitation applied to nonlinear structure. (a) The entire time history. (b) The segment from 5 s to 7 s.
(a)(b)


### 3.3. A Fixed Beam Structure

In this section, a fixed beam composed of eight elements is further considered, as shown in Figure8. The physical parameters of each element are given as follows: the area of the cross-section A = 2.25 × 10−2 m2, the material density ρ = 7850 kg/m3, modulus of elasticity E = 2 × 108 Pa, and the length l = 1.5 m. The Rayleigh damping model is adopted, and proportional coefficients are α = 0.0473 and β = 1.4866 × 10−2. The damage scenario considered herein is an abrupt stiffness degradation of three elements. We assume that the stiffness parameters of the 1st, 2nd, and 8th elements are decreased sharply by 50%, 50%, and 40%, respectively, at the time instant t = 5 s.Figure 8
The fixed beam structure.As shown in Figure8, there are nine nodes, and each node has two DOFs except node 1 and node 9. A random excitation is vertically applied at node 4, and the structural responses are computed numerically. Since the rotational responses are quite difficult to be measured, only the acceleration of nodes 2∼8 and the displacement of node 3 in the vertical direction are used for identification. The parameters are initially set to be 50% of the actual ones. The first 28 and the last 10 diagonal elements of FFM are 1 and 1.001, respectively. The P¯ matrix is set as diag01×28,101×814,101×26.The identified results are given in Table3. The square errors between the measurements and the corresponding estimated ones are plotted in Figure 9. Similarly, only ∆1 and ∆3 are given in Figure 9 as examples. The time histories of some identified stiffness are plotted in Figure 10. Two additional cases are discussed. It can be seen from these results that the proposed approach is able to satisfactorily identify the time-varying parameters of beam-type structures. Moreover, the unknown force applied to the beam can be simultaneously identified. It can be found from Figure 11 the identified force has a good agreement with the actual one.Table 3
The identified results of the beam structure.
ParameterActualIdentifiedError (%)k1 (N/m)5625⟶2812.55347.0⟶2827.7−4.94⟶2.14k2 (N/m)5625⟶2812.55564.0⟶2810.5−1.08⟶−0.07k3 (N/m)56255630.90.10k4 (N/m)56255583.7−0.73k5 (N/m)56255761.62.43k6 (N/m)56255618.6−0.11k7 (N/m)56255680.40.98k8 (N/m)5625⟶33755769.2⟶3314.72.56⟶−1.79α0.04730.0458−3.16β × 10−21.48661.48730.04Figure 9
Time histories of square errors in the 3rd example. (a) Time history of square error of ∆1. (b) Time history of square error of ∆3.
(a)(b)Figure 10
The identified stiffness of the beam structure. (a) Stiffness of the 1st element. (b) Stiffness of the 2nd element. (c) Stiffness of the 7th element. (d) Stiffness of the 8th element.
(a)(b)(c)(d)Figure 11
The identified excitation applied to the fixed beam. (a) The entire time history. (b) The segment from 5 s to 7 s.
(a)(b)


## 3.1. A Six-Story Building Structure

A six-story shear-type building structure is considered as the first numerical example. The mass and stiffness of each story are 100 kg and 120 kN/m, respectively. The Rayleigh damping model, i.e.C = α·M+β·K, is used. The proportional coefficients are α = 0.3739 and β = 1.8227 × 10−3. To mimic structural damage, the stiffness parameters of the 1st story is reduced abruptly from 120 kN/m to 72 kN/m at t = 4s. The concentrated random excitation is assumed to be applied on the top floor, and thus the force influence matrix can be expressed as φu = (0 0 0 0 0 1)T. By using the state space method, the structural responses can be calculated. The time interval in this example is 0.001 s. Here, the acceleration responses of the 1st, 3rd, 5th, and 6th floors as well as the displacement response of the 6th floor are assumed to be measured and used for the identification.The quantities to be identified include six stiffness parameters, two damping coefficients, and the external excitation. The structural parameters are initially assumed to be 50% of the actual ones. The posteriori estimate error covariance matrix is set to beP11=diagI1×12,101×614,101×210 at the beginning. The first 12 and the last 8 diagonal elements of forgetting factor matrix (FFM) are set to be 1 and 1.001, respectively. The reset error covariance matrix used for the enhancement of tracking capability is set as P¯=diag01×12,101×614,101×210. The proper assignment of process and measurement noise covariance matrices (i.e., Q and R in this study) plays an important role in ensuring the reliability and stability of recursive KF-based methods. As stated by Eftekhar Azam et al. [28], the Q matrix reflects the confidence put in the utilized structural model. A large value is usually selected if the model uncertainty is large. The R matrix reveals the confidence put in the observations. The larger value means the more measurement noise is considered. Research on proper adjustment of Q and R matrices has been conducted by some scholars using different techniques, such as Bayesian probabilistic algorithm [48, 49] and mathematical regularization techniques [28, 50]. However, the optimal determination of Q and R matrices is not the main purpose of this study. The constant values of these two matrices in this paper are selected based on other available literature [38, 51, 52]. Here, the Q and R matrices are set to be 10−4 × I and I, respectively, where I denotes identity matrix.The structural parameters can be identified by using the proposed approach as given in Table1. Clearly, the identified results are quite close to the actual ones. To investigate the tracking capability, the square errors defined in equation (21) are calculated and shown in Figure 2. It is obvious that a significant increase of ∆1 can be observed at t = 4 s. This means the structural damage is happening at this time. Only the time histories of the identified stiffness of the 1st and 6th stories are plotted in Figure 3 due to the limitation of the paper length. Similar results can be found for the remaining stories. Moreover, for ease of understanding, two additional cases are considered in this example for demonstrating the necessity of using FFM and P¯ matrix. In case 1, only FFM is used for the online identification. In this case, the error covariance matrix will not be reset by equation (22) even though the damage instant is detected. Instead, only P¯ matrix rather than FFM is used in case 2. It is clear from Figure 3(a) that neither of these two cases can track the abrupt change of k1. The proposed approach, however, is capable of accurately and stably capturing this sudden decrease.Table 1
The identified results of the building structure.
ParametersActualIdentifiedError (%)k1 (kN/m)120⟶72119.84⟶71.99−0.13⟶−0.01k2 (kN/m)120120.440.36k3 (kN/m)120120.130.11k4 (kN/m)120120.070.06k5 (kN/m)120119.56−0.37k6 (kN/m)120120.290.24α0.37390.3618−3.24β × 10−31.82271.86192.15Figure 2
Time histories of square errors in the 1st example. (a) Time history of square error of ∆1. (b) Time history of square error of ∆3.
(a)(b)Figure 3
The identified stiffness of building structure. (a) Stiffness of the 1st story. (b) Stiffness of the 6th story.
(a)(b)Besides identifying time-variant parameters, the unknown excitation can be identified as well. As shown in Figure4(a), the time history of the identified load is plotted by a solid line, whereas the actual one is given by a dashed line. The time segment from 4 s to 6 s is further given in Figure 4(b) for clarity of comparison. Obviously, the identified input is in good agreement with the actual one.Figure 4
The identified excitation applied to building structure. (a) The entire time history. (b) The segment from 4 s to 6 s.
(a)(b)


## 3.2. Six-Story Building Structure with MR Damper

The aim of this section is to investigate the performance of the proposed approach for the identification of nonlinear structure. The six-story building structure introduced in Section3.1 is employed herein. To mimic structural nonlinearity, an MR damper described by the Bingham model is assumed to be installed on the 1st story. The restoring force provided by the MR damper is expressed as follows:(27)FMR=fcsgn∆x˙i+cb∆x˙i+fb,where ∆x˙i is the interstory velocity and defined as ∆x˙i=x˙i−x˙i−1 and fc, cb, and fb are the model coefficients and set to be fc = 1 N, cb = 1000 N.s/m, and fb = 0.In this section, multiple damages are considered. The stiffness parameters of the 1st and 2nd stories are reduced by 50% att = 5 s. A random force is applied to the top floor, and the corresponding structural responses are computed by the Runge–Kutta method. The acceleration responses of the 1st, 3rd, 5th, and 6th floors and displacement response of the 6th floor are employed for identification. The sampling frequency is 1000 Hz. The parameters are initially set to be 50% of the actual ones. The first 12 and the last 10 diagonal elements of the FFM are set to be 1 and 1.002, respectively. The reset error covariance matrix is P¯=diag01×12,101×612,101×22,104,109.The parameters can be identified by using the proposed approach as listed in Table2. Satisfactory identification results can be observed even though structural nonlinearities are involved. The square errors used for detecting damage instant are also calculated as plotted in Figure 5. Due to paper length limitations, only the square errors of the 1st and 3rd stories are given as examples. Apparently, the damage instant is at t = 5s. The time histories of the identified time-varying stiffness are shown in Figure 6. Similarly, two additional cases, i.e., FFM only and P¯ matrix only, are considered for comparison. It can be found that only using FFM or P¯ matrix is unable to efficiently track time-varying parameters. The proposed approach with the combination of FFM and P¯ matrix is capable of capturing time-varying parameters with good accuracy. Also, the external excitation can be simultaneously identified as shown in Figure 7. It is obvious that the identified force is close to the actual one.Table 2
The identified results of the building structure with MR damper.
ParameterActualIdentifiedError (%)k1 (kN/m)120⟶60120.13⟶60.740.11⟶1.23k2 (kN/m)120⟶60120.73⟶60.500.61⟶0.83k3 (kN/m)120119.68−0.27k4 (kN/m)120120.330.27k5 (kN/m)120119.72−0.61k6 (kN/m)120119.41−0.49fc (N)10.9678−3.22cb (N·s/m)1000979.24−2.08α0.37390.3574−4.43β × 10−31.82271.85411.72Figure 5
Time histories of square errors in the 2nd example. (a) Time history of square error of ∆1. (b) Time history of square error of ∆3.
(a)(b)Figure 6
The identified stiffness of building structure with MR damper. (a) Stiffness of the 1st story. (b) Stiffness of the 2nd story.
(a)(b)Figure 7
The identified excitation applied to nonlinear structure. (a) The entire time history. (b) The segment from 5 s to 7 s.
(a)(b)


## 3.3. A Fixed Beam Structure

In this section, a fixed beam composed of eight elements is further considered, as shown in Figure8. The physical parameters of each element are given as follows: the area of the cross-section A = 2.25 × 10−2 m2, the material density ρ = 7850 kg/m3, modulus of elasticity E = 2 × 108 Pa, and the length l = 1.5 m. The Rayleigh damping model is adopted, and proportional coefficients are α = 0.0473 and β = 1.4866 × 10−2. The damage scenario considered herein is an abrupt stiffness degradation of three elements. We assume that the stiffness parameters of the 1st, 2nd, and 8th elements are decreased sharply by 50%, 50%, and 40%, respectively, at the time instant t = 5 s.Figure 8
The fixed beam structure.As shown in Figure8, there are nine nodes, and each node has two DOFs except node 1 and node 9. A random excitation is vertically applied at node 4, and the structural responses are computed numerically. Since the rotational responses are quite difficult to be measured, only the acceleration of nodes 2∼8 and the displacement of node 3 in the vertical direction are used for identification. The parameters are initially set to be 50% of the actual ones. The first 28 and the last 10 diagonal elements of FFM are 1 and 1.001, respectively. The P¯ matrix is set as diag01×28,101×814,101×26.The identified results are given in Table3. The square errors between the measurements and the corresponding estimated ones are plotted in Figure 9. Similarly, only ∆1 and ∆3 are given in Figure 9 as examples. The time histories of some identified stiffness are plotted in Figure 10. Two additional cases are discussed. It can be seen from these results that the proposed approach is able to satisfactorily identify the time-varying parameters of beam-type structures. Moreover, the unknown force applied to the beam can be simultaneously identified. It can be found from Figure 11 the identified force has a good agreement with the actual one.Table 3
The identified results of the beam structure.
ParameterActualIdentifiedError (%)k1 (N/m)5625⟶2812.55347.0⟶2827.7−4.94⟶2.14k2 (N/m)5625⟶2812.55564.0⟶2810.5−1.08⟶−0.07k3 (N/m)56255630.90.10k4 (N/m)56255583.7−0.73k5 (N/m)56255761.62.43k6 (N/m)56255618.6−0.11k7 (N/m)56255680.40.98k8 (N/m)5625⟶33755769.2⟶3314.72.56⟶−1.79α0.04730.0458−3.16β × 10−21.48661.48730.04Figure 9
Time histories of square errors in the 3rd example. (a) Time history of square error of ∆1. (b) Time history of square error of ∆3.
(a)(b)Figure 10
The identified stiffness of the beam structure. (a) Stiffness of the 1st element. (b) Stiffness of the 2nd element. (c) Stiffness of the 7th element. (d) Stiffness of the 8th element.
(a)(b)(c)(d)Figure 11
The identified excitation applied to the fixed beam. (a) The entire time history. (b) The segment from 5 s to 7 s.
(a)(b)


## 4. Experimental Validation

In order to validate the effectiveness of the proposed approach for real structures, a small-scale, six-story shear building model, as shown in Figure12, is designed and built in the lab. The structure consists of aluminum columns and plates assembled firmly by bolted joints. The size of each plate is 400 mm × 400 mm, and the thickness is 10 mm. The cross-section of each column is 20 mm × 4 mm. The height of each floor is 210 mm except for the top floor. To conveniently install a lock-up system to mimic an abrupt change in structural stiffness, the height of the top floor is designed at 410 mm. An additional iron column of 400 mm × 50 mm × 1 mm fixed on the top floor plate and a powerful electromagnet device fixed on the 5th floor plate are used to construct the lock-up system. When direct current (DC) is injected into the electromagnet device by a DC power supply, a large magnetic attraction force is generated. In this case, the iron column is locked by the electromagnet device, as shown in Figure 12, providing additional stiffness to the structural system. If the DC becomes zero, the iron column is immediately released by the electromagnet device, indicating a sudden decrease in stiffness on the 6th story. In other words, structural damage is simulated by such a lock-up system in this experiment. The “lock” state indicates additional stiffness is provided, whereas the “unlock” state indicates additional stiffness is zero.Figure 12
The six-story building model in the lab.The six-story shear building model in this section can be viewed as a 6-DOFs lumped mass structural system. The weight of plates, columns, and bolts is measured and equally distributed in each story. Notably, the length of the columns of the 6th story is longer than other stories due to a lock-up system being installed on this story. Since the heavy electromagnet device is fixed to the 5th story plate, as shown in Figure12, the concentrated mass of the 5th DOF is relatively larger than other DOFs. The stiffness of each story is determined by static tests, as schematically shown in Figure 13. Based on the weight of the calibrated mass block and the deformation of the building model recorded by displacement transducers, the structural stiffness can be determined. The mass distribution and stiffness of each story are summarized in Table 4.Figure 13
Schematic diagram of static test.Table 4
The basic information of the building model.
Floor numberMass distribution (kg)Stiffness (kN/m)Lock stateUnlock stateThe 1st story5.3554.8954.89The 2nd story5.3456.4456.44The 3rd story5.3653.9753.97The 4th story5.3360.6860.68The 5th story11.3956.6056.60The 6th story6.478.856.15The Rayleigh damping assumption is used for the consideration of damping effect. Moreover, in order to effectively identify structural frequencies, a sine sweeping-frequency excitation ranging from 0 Hz to 50 Hz is applied to the 5th floor of the model. Based on the frequency response functions (FRFs), the natural frequencies of the model are identified. The natural frequencies are 3.24 Hz, 6.95 Hz, 12.74 Hz, 20.36 Hz, 27.44 Hz, and 31.71 Hz in the lock state, and 3.14 Hz, 6.07 Hz, 12.67 Hz, 20.32 Hz, 27.43 Hz, and 31.67 Hz in the unlock state. For clarity, only the FRF of the 1st floor is shown in Figure14.Figure 14
FRF of the 1st floor.Dynamic tests are conducted on this six-story building model to validate the effectiveness of the proposed approach. A random force measured by a force transducer is applied to the 5th floor by an exciter. The displacement response of the 5th floor and the acceleration responses of the 1st, 3rd, 5th, and 6th floors are measured at a sampling frequency of 1000 Hz. The unknown quantities include six stiffness coefficients, two damping coefficients and the random force. The structural parameters are initially assumed to be 50% of the actual ones. The first 12 and the last 8 diagonal elements of FFM are set to be 1 and 1.001, respectively. TheP¯ matrix is defined as diag01×12,101×610,101×21 herein.The structural parameters can be identified by the proposed approach, as shown in Table5. The maximum error is 4.72%, which means the proposed approach can be employed for identifying time-varying parameters of real structure with acceptable accuracy. The square errors associated with the acceleration measurements are displayed in Figure 15. The jump increases can be observed at t = 8.13 s indicating the significant changes of structural parameters at this time. Figure 16 gives the time histories of the identified stiffness parameters. Only the stiffness parameters of 1st and 6th story are plotted due to the limitation of paper length. It can be found that the proposed approach is capable of efficiently capturing abrupt changes in structural parameters. The external excitation is identified and compared with the measured one, as plotted in Figure 17. Obviously, the identified force is close to the measured one.Table 5
The identified stiffness of the real building model.
ParametersActualIdentifiedError (%)k1 (kN/m)54.8954.37−0.95k2 (kN/m)56.4457.922.61k3 (kN/m)53.9754.641.25k4 (kN/m)60.6858.78−3.14k5 (kN/m)56.6058.162.75k6 (kN/m)8.85⟶6.158.51⟶5.86−3.84⟶−4.72Figure 15
Time histories of squares errors in the experiment. (a) Time history of square error of ∆1. (b) Time history of square error of ∆6.
(a)(b)Figure 16
The identified stiffness of the building model in the lab. (a) Stiffness of the 1st story. (b) Stiffness of the 6th story.
(a)(b)Figure 17
The identified excitation applied to the building model in the lab.


## 5. Conclusion

Online identification of time-varying parameters of linear and nonlinear structures under unknown loads is an important but challenging task. In this paper, a two-stage EKF-FFM-UI approach is proposed for simultaneously identifying time-variant parameters and external loads. In stage 1, an EKF-UI approach previously proposed by the authors is briefly introduced. At this stage, the parameters are assumed to be time-invariant. The aim of stage 2 is to improve this EKF-UI approach for the purpose of possessing tracking capability. First, based on the reconstructed acceleration, an updated observation equation is derived. Then, a forgetting factor matrix is introduced, and the formulas for the estimation of priori states, posteriori states, and unknown inputs are updated accordingly. Finally, in order to efficiently capture abrupt changes in structural parameters, the square errors associated with acceleration measurements are defined. The jump increases of these square errors indicate the occurrence of parameter variations. A covariance resetting technique is immediately used at this time instant to enhance the tracking capability. One of the merits of the proposed approach is that it is performed solely based on the EKF principle without any optimization procedures. Therefore, it is more straightforward and computationally efficient for tracking abrupt damages in real time. The effectiveness of the proposed approach is first validated via linear and nonlinear numerical examples. Experimental tests are also conducted to investigate the performance of the proposed approach for real structures. Results show that abrupt structural damages can be effectively captured and accurately identified. The unknown excitations can also be estimated at the same time. The proposed approach provides a promising way for real-time parameter identification and structural evaluation.



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*Source: 8379183-2023-02-21.xml*