A dynamic load estimation method for nonlinear structures with unscented Kalman filter

Mechanical Systems and Signal Processing 101 (2018) 254–273

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A dynamic load estimation method for nonlinear structures with unscented Kalman filter L.N. Guo, Y. Ding ⇑, Z. Wang, G.S. Xu, B. Wu Key Lab of Structures Dynamic Behavior and Control of the Ministry of Education (Harbin Institute of Technology), Harbin 150090, China School of Civil Engineering, Harbin Institute of Technology, Harbin 150090, China

a r t i c l e

i n f o

Article history: Received 10 June 2017 Accepted 22 July 2017

Keywords: Force identification Unscented Kalman filter Nonlinear structures Isolated structure Real-time substructure test Shaking table test

a b s t r a c t A force estimation method is proposed for hysteretic nonlinear structures. The equation of motion for the nonlinear structure is represented in state space and the state variable is augmented by the unknown the time history of external force. Unscented Kalman filter (UKF) is improved for the force identification in state space considering the ill-condition characteristic in the computation of square roots for the covariance matrix. The proposed method is firstly validated by a numerical simulation study of a 3-storey nonlinear hysteretic frame excited by periodic force. Each storey is supposed to follow a nonlinear hysteretic model. The external force is identified and the measurement noise is considered in this case. Then a case of a seismically isolated building subjected to earthquake excitation and impact force is studied. The isolation layer performs nonlinearly during the earthquake excitation. Impact force between the seismically isolated structure and the retaining wall is estimated with the proposed method. Uncertainties such as measurement noise, model error in storey stiffness and unexpected environmental disturbances are considered. A real-time substructure testing of an isolated structure is conducted to verify the proposed method. In the experimental study, the linear main structure is taken as numerical substructure while the one of the isolations with additional mass is taken as the nonlinear physical substructure. The force applied by the actuator on the physical substructure is identified and compared with the measured value from the force transducer. The method proposed in this paper is also validated by shaking table test of a seismically isolated steel frame. The acceleration of the ground motion as the unknowns is identified by the proposed method. Results from both numerical simulation and experimental studies indicate that the UKF based force identification method can be used to identify external excitations effectively for the nonlinear structure with accurate results even with measurement noise, model error and environmental disturbances. Ó 2017 Published by Elsevier Ltd.

1. Introduction The exact information of external force of structures can contribute to structural system identification, condition assessment and maintenance. As a kind of inverse problem of structural dynamics, the force estimation is also an optimization process. In this process, the calculated structural response based on identified force is expected to be close enough to the

⇑ Corresponding author. E-mail address: [email protected] (Y. Ding). http://dx.doi.org/10.1016/j.ymssp.2017.07.047 0888-3270/Ó 2017 Published by Elsevier Ltd.

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measured response. Although advanced force transducers have been developed recently, the direct measurement of all the external force for structure will be infeasible due to the limited number of sensors required in engineering and lack of access to the position of the loads. Structures may perform nonlinearly during severe external excitation such as the excitations related to earthquake or strong wind [1,2]. Although the parameter identification is very important to the structural system it is difficult to initialize the parameter identification without the information of external force for nonlinear structures. The force estimation of nonlinear structural system would contribute effectively to the parameter identification. In other words, the accurate identification of the external force is one efficient way and first step to the system identification for nonlinear system. For the target nonlinear structures in this study with isolations or other energy dissipation components the main structure may have very few degrades in strength and stiffness during the moderate or even the severe earthquake but the exact knowledge of external excitation related to the earthquake definitely contribute to the methodology of the seismic design. Therefore, the proper external force identification methods for these kinds of nonlinear structures are of great importance in this condition. Currently, the investigations of dynamic load identification for linear structures is increasingly carried out while the identification of external force for nonlinear structure is still a difficult task due to the nonlinearity, model error and measurement noise. A lot of force estimation methods have been developed and extensive reviews of structural external force identification are provided [3–6]. The external excitation estimation methods can be classified into two categories of frequency domain methods and time domain methods. Frequency response function (FRF) as a powerful tool in frequency domain generally used in structural identification has also been applied to force estimation based on the inverse Fourier transformation [7]. Time domain methods are actively investigated currently as alternative tools for the force identification. Time history response can also be directly used as the measured information and the objective of the optimization process is to minimize the error between the measured and the predicted structural response with the time domain method. Examples of vehiclebridge interaction force, wind load and white noise excitation are concerned in the external force estimation [8–10]. This process can be formulated in state space and the time history of external force to be identified can be taken as unknowns. Regularization methods or some other optimization methods could be used to solve the state vector [11–14]. These methods in time domain are usually based on the finite element model of the structure which is often inaccurate due to the uncertainties in the property of structural material which affect the accuracy of inverse analysis [15]. Majority of the methods mentioned above consider the measurement noise but take the model of the structure as deterministic although some methods may be robust to the model errors [16]. The measurement noise and model error as stochastic process has been considered by state variable estimation in state space with Kalman filter [17]. Kalman filters have been applied in the state variable estimation and the external excitation estimation can be a separate process with recursive identification methods [18,19]. A large number of measurements are required and the separate identification process from the state space may cause the biased errors in the identification results based on the method mentioned above. Kalman filter as an effective tool for the online estimation has also been modified with a new state variable augmented by the discrete time history of the external force. The external force and the structural response, including displacement and velocity, are taken as the unknowns to be identified [20–23]. The measurements on structure may be limited but they can be re-constructed with the measured information and then be used for the load identification [20,24]. With the augmented Kalman filter, the force can be estimated on-line accurately. However, only a fraction of the system input identification methods was concerned with the force identification for nonlinear structural systems, which commonly exist in earthquake excitation [1]. More recently, methods of structural model updating and structural identification for nonlinear structures have also been developed with the extended Kalman filter. The extended Kalman filter is an effective means of parameter identification and external force estimation for the nonlinear structures and good results have been obtained [25–27]. It has been demonstrated that the identification based on EKF is effective for linear structures or structures with slight nonlinear property. Compared with EKF, the unscented Kalman filter (UKF) is an algorithm with higher accuracy for the nonlinear system identification [28–30]. In the identification process with UKF, the computation of Jacobin matrix is not necessary. UKF is more suitable to force identification of the civil structures with seismic energy dissipation component which performs with a strong nonlinear property during the severe external excitation such as earthquake, blast and strong wind [28–30]. The Monte Carlo methods including particle filters [1] and other ensemble Kalman filter can solve the inverse problem for the nonlinear systems with non-Gaussian posterior probability of the state [29,30]. These Monte Carlo methods can approximate the posterior probability of the state through the generation of a large number of samples. However, the Monte Carlo methods require a large number of samples, depending on the number of the factors in the state variable, making the identification process time consuming [29,30]. Until now, very few literature reports on the force identification methods with augmented state variable of force time history solved by UKF previously. As shown in Refs. [10,16,31] the external force estimation methods have been developed with orthogonal decomposition, regularization or least-square method in two-step identification procedure. This study will develop a new force identification method for nonlinear structure based on an improved UKF estimation tool to reduce the number of unknowns with the augmented state variable and efficient computation. This method is formulated recursively in state space with the augmented state variable by the discrete unknowns in the time history of external force. In this study it is supposed that the force identification is preceded by a system identification step. Therefore, the parameters of the structural system are assumed to be deterministically known constants. The position of the external excitation is also assumed to be determined while the time history is identified with the proposed method. A hysteretic nonlinear frame is firstly studied numerically with a single exci-

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tation to validate the proposed method. In this case measurement noise is considered. In the second case, a 14-storey seismically base isolated frame is studied. The nonlinear isolated structure is subjected to the earthquake excitation and impact force between the isolation layer and adjacent retaining wall. In this case the earthquake excitation is supposed to be recorded but the impact force is unknown. The impact force is identified based on the proposed method. For practical application purpose both of the process noise and the measurement noise are considered in this case and fairly accurate result is obtained. The proposed force identification method is also validated by real-time hybrid simulation [32,33]. A seismically isolated steel frame was investigated with real-time substructure test, which is also named as real-time hybrid test, to validate the proposed method which was conducted in the Hybrid Simulation Research Center of Harbin Institute of Technology. The superstructure is set as the numerical substructure and the isolator is taken as the physical specimen in the realtime substructure test. The interaction force between the superstructure and isolation layer is implemented by the actuator and identified by the proposed method. The results from numerical simulation studies and experimental verification indicate that the proposed method can identify the external force accurately even with the measurement noise and process noise. 2. Force identification in state space 2.1. The equations of motion for nonlinear system The equations of motion as used in Ref. [30] for nonlinear structural system with external force F can be written as:

€ðtÞ þ CxðtÞ _ Mx þ KzðtÞ ¼ LFðtÞ

ð1Þ

where M, C and K are the mass, damping and stiffness matrices of the structural system, respectively. L is the mapping € and x, _ respectively, are vectors of acceleration, velocity of the structural system. The vector matrix for the input forces. x T z ¼ ½ z1 z2    zndof  is the hysteretic component vector and ndof denotes the number of DOFs. The hysteretic component studied is represented with the generally used Bouc-Wen model as follows.

z_ i ¼ x_ i  bi jx_ i jjzi jni 1 zi  ci x_ i jzi jni

ð2Þ

where subscript i represent the ith element or storey. x_ i and zi are relative velocity and hysteretic component of the ith storey, and n, b and c are the nonlinear parameters of the Bouc-Wen model. When the nonlinear system is subjected to the earthquake excitation and uncertain external impact force between superstructure and retaining wall the equation of motion can be written as follows.

€ðtÞ þ CxðtÞ _ €g þ LF Mx þ KzðtÞ ¼ MGx

ð3Þ

€ g denotes the acceleration of the ground motion and G represents the location matrix for the earthquake effect. LF is where x €g is supposed to be measured but the the external force applied on structure during the earthquake excitation. In this study x external force F is unknown. The equation of motion can also be expressed in the state space as shown in Eq. (4).

_ €g ðtÞ; FðtÞÞ þ wðtÞ XðtÞ ¼ f ðXðtÞ; x

ð4Þ

_ where X denotes the state vector represented as XðtÞ ¼ ½ xðtÞ xðtÞ , w is the process noise assumed to follow a Gaussian distribution with zero mean and a covariance matrix of Q, and f is a nonlinear function of state vector X. The measurement equation can be written as Eq. (5).

€g ðtÞ; FðtÞÞ þ v ðtÞ yðtÞ ¼ hðXðtÞ; x

ð5Þ

where v is the measurement noise that also follows a Gaussian distribution with zero mean and a covariance matrix of R. The continuous state space equation can be expressed in discrete form as follows.

€g;k1 ; Fk1 Þ þ wk1 Xk ¼ FðXk1 ; x

ð6Þ

The function F can be obtained with the following integration.

€g;k ; Fk Þ ¼ FðXk1 ; x €g;k1 ; Fk1 Þ þ FðXk ; x

Z

ðkþ1ÞDt

kDt

€g;k1 ; Fk1 Þdt f ðXk1 ; x

ð7Þ

The integration can be calculated with numerical tools, such as fourth-order Runge-Kutta method [30,36]. The deterministic part of discrete system equation can be obtained with fourth-order Runge-Kutta method while the stochastic part in discrete system equation is directly taken as wk1 [34,35]. The influence of different treatment of stochastic differential equations on wk1 and the identification process will be further studied [37]. 2.2. The augmented state space equation of nonlinear system When the external force is known, the state variable including displacement and velocity as shown in Eq. (4) can be estimated recursively. However, the external force is always unknown and need to be estimated in practical problems. When the

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external force F is unknown, the discrete unknowns Fk in time history of external force can be represented based on Fk1 of the previous time step. The increment of the unknown external force is considered with an appropriate set of the covariance matrix of this process. The relationship between Fk and Fk1 is shown as follows.

Fk ¼ Fk1 þ gk1

ð8Þ

where gk1 is a part of process noise. This noise is assumed to have zero-mean and standard deviation S. For the force identification problem of nonlinear structural system, the state space equation can be rewritten as



Xk Fk





¼

€g;k1 ; Fk1 Þ FðXk1 ; x



Fk1



þ

wk1



ð9Þ

gk1

A new augment state space equation can be obtained as

€g;k1 Þ þ lk1 Zk ¼ GðZk1 ; x

ð10Þ

where the new state variable Zk is represented by ½Xk ; Fk T and the new process vector of l is denoted by ½wk1 ; gk1 T . G is the new nonlinear equation for the new state variable and process noise. The observation equation based on the new state variable can be written as

€g;k Þ þ v k yk ¼ HðZk ; x

ð11Þ

where H is the new nonlinear function for the measurement equation. 2.3. UKF and its improvement in force identification For the nonlinear dynamic structural system represented by the augmented nonlinear equation in state space as shown in Eq. (10) and observation equation as Eq. (11), UKF can be used for the state variable estimation. With UKF the identification ^ 0 for the initial state vector X0 and the calculation of covariance matrix P0 . The defprocess can be started with the guess X ^ 0 and P0 are provided as inition of X

^ 0 ¼ EðX0 Þ X

ð12Þ

^ 0 ÞðX0  X ^ 0 ÞT  P0 ¼ E½ðX0  X

^k1 constructed as: For the k th step, there are 2n + 1 of the deterministic sampling points, namely the sigma points v

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ k  ði þ kÞPk ; i ¼ 1; 2; . . . ; n ^ k; X ^ k þ ði þ kÞPk ; X ^k ¼ X v

ð13Þ

where n is the number of the factors in state vector and k is a gain parameter with a definition related to the dimension and ^ ^ . Matrix of P should the distribution of the state vector. In the identification process, P is calculated as P ¼ E½ðX  XÞðX  XÞ be symmetric positive definite which is required by the generally used Cholesky factorization in the calculation for the square root of matrix P, which is always ill-posed. In the calculation of the square root of P, singular value decomposition (SVD) is used instead of Cholesky decomposition to improve the robustness performance of UKF in this study. With SVD, the covariance matrix Pk can be represented as: T

Pk ¼ USVT

ð14Þ

where U is an m  m unitary matrix, the matrix S is an m  n diagonal matrix with nonnegative real numbers on the diagpffiffiffi onal, and V denotes the n  n unitary matrix. The square root of Pk can be represented with U S based on the symmetric assumption of the covariance matrix. The predicted sigma points of the k th step can be calculated as:

^k ¼ Fð^ vk1 ; uk1 ; wk1 Þ v

ð15Þ

^  and covariance P of the predicted where u denotes all the inputs of the nonlinear structural system. Then the mean X k k sigma points can be obtained considering the weighting matrix of each point as follows.

^ ¼ v ^k Wm X k

ð16Þ

^ k W½X ^ k T þ Q Pk ¼ X k1

ð17Þ

  W ¼ ðI  ½W m ; . . . . . . ; W m Þ  diag W 0c ; . . . . . . ; W 2n  ðI  ½W m ; . . . . . . ; W m ÞT c

ð18Þ

 W im ¼ W ic ¼

k=ðn þ kÞ; i ¼ 0 k=2ðn þ kÞ; i ¼ 1; . . . ; 2n

ð19Þ

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where Wm denote the weighing coefficients of mean and Wc is the weighing factor of covariance. The measurement updated step is implemented as following Eqs. (20)–(24).

  ^ k ; uk Yk ¼ h v

ð20Þ

yk ¼ Yk Wm

ð21Þ

1

T    T Kk ¼ vk W Yk Yk W Y k þ Rk

ð22Þ

 Xk ¼ Xk þ Kk yk  yk

ð23Þ

 

T Pk ¼ Pk  Kk Yk W Yk þ Rk KTk

ð24Þ

where yk is the measured response of structure, y k is the mean of the predict response. K is the gain of UKF which is similar to Kalman filter. This step of UKF is similar to Kalman filter estimation. But UKF involves the unscented transformation (UT) with deterministic sampling as Eq. (13) to solve the inverse problem for nonlinear structure. The deterministic sampling of sigma point in UT can be used to evaluate the mean and covariance of the variable to be identified. With the UKF, the sigma points are propagated with the nonlinear equation in state space as shown in Eq. (15) and the statistical characteristics of the propagated points the can be approximated. Therefore, there is no need to calculate the Jacobian matrix in each step with UKF compared to the EKF. With the UKF and its improvement introduced in this Section, the external force can be identified based on the measurement on nonlinear structural system. 2.4. Implementation procedure Step 1: Obtain the initial model of the target structure, including mass, damping matrices and determine the nonlinear model for the resisting force. Step 2: Conduct dynamic measurement on the structure. In the simulation studies, compute the responses of the structure under excitation as the ‘‘measured” responses. Step 3: Establish the state space equation for the purpose of force identification for the nonlinear system, which may be inaccurate with model errors. Step 4: Identify the state variable including structural response and external force with the proposed improved UKF algorithm as shown in Section 2.3 from Eqs. (12)–(24). 3. Numerical studies In this section two nonlinear structures are numerically studied to check the validity of the force identification method proposed in Section 2. First structure is a 3-storey hysteretic frame and the second structure is a 14-storey frame with base isolation layer. The specific first two scenarios for simulation studies are shown in Table 1. The last scenario mainly discusses the influence of the arrangement of the measurement on the identification results and is not listed in Table 1. In this simulation study, when there is noise in the ‘‘measured” response, the contaminated response is simulated by adding a normal random component to the unpolluted structural responses as

€m ¼ x € þ EP Nnoise rðx €Þ x

ð25Þ

where Ep is the percentage of the RMS noise, Nnoise is stochastic process following a standard normal distribution with zero €Þ is the standard deviation of the response without noise. In the simulation studies of mean and unit standard deviation, rðx the 14-storey seismically isolated frame, the noise in measured ground motion is considered, and the contaminated measured ground motion is also simulated as shown in Eq. (25). It is possibly to set the covariance of the force according to €max þ cx_ max þ kxmax which can be taken as simple approximation of the peak for the external excitation the summation of mx Table 1 Cases and scenarios for numerical studies of force identification for nonlinear structures. Force identification cases

Scenarios

Model error (%)

Measurement noise level (%)

Environmental base excitation

Case 1

Scenario 1 Scenario 2 Scenario 3

0 0 0

0 5 5

0 0 0

Case 2

Scenario Scenario Scenario Scenario

0 0 5 5

0 5 0 5

0 Yes 0 Yes

1 2 3 4

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and the square of this term can be used as the initial covariance for the force identification. The parameter of m,c and k can be obtained with the Guyan or other iterative dynamic reduction methods. 3.1. Case 1 – force identification for a 3-storey hysteretic frame A three-storey hysteresis nonlinear shear frame is firstly studied as shown in Fig. 1. Bouc-Wen model is employed to simulate the resisting force for each storey. The parameters for Bouc-Wen model of this 3-storey frame are taken as b = 4, c = 2 and n = 1.1. The stiffness and damping coefficient of each storey is taken as 40 kN/m and 0.32 kN/(m/s), respectively. The shear frame structure is subjected to the excitation on the top of the frame. The external excitation is set as FðtÞ ¼ F1 ðtÞ þ F2 ðtÞ, where F1 ðtÞ ¼ A  sinð4  p  tÞ and F2 ðtÞ ¼ 2  cosð3  p  tÞ. In the time history of excitation, ‘A’ is the time-variant amplitude of the excitation which is steadily increased from 3 kN to 9 kN in 2 s. The sampling rate of measurement is 1000 Hz in this case. The external force in this case is low frequency excitation. The initial guess of state variable including displacement, velocity and the external force are all set as zero. When the measurement noise is not considered, the identified time history of the external force is shown in Fig. 2(a). The comparison of the power spectrum of the original external force and the identified force is shown in Fig. 2(b). Considering the civil engineering structures sensitive to the low frequency excitation the components corresponding to the low frequency is shown only in Fig. 2(b). The time history and the spectrum of the identified external force and the real external force nearly overlap as shown in Fig. 2(a) and (b). The frequencies of the external force can be identified as 1.95 Hz and 1.47 Hz in Fig. 2(b) with peak picking. It is demonstrated from Fig. 2 that the external force identification result is very accurate. When the measurement noise level is considered as 5%, the time history of identified external force is shown in Fig. 3(a) and the power spectrum is shown in Fig. 3(b). There are some high frequency fluctuations in the peak of the time history of the identified external force as shown in Fig. 3(a). The low frequency power spectrum of the identified external force shown in Fig. 3(b) is accurate while the high frequency component exists in the identification result, which is consistent with the result shown in Fig. 3(a). In the third scenario, random excitation with zero mean is applied on the top of the structure. The standard deviation of the stochastic excitation is the same as the first two scenarios. The time history of the identified force and the frequency domain information are shown in Fig. 4(a) and (b) respectively. The identification results are shown in Fig. 5 considering the measurement noise. The identification results are still accurate in time domain and frequency domain for the random excitation identification even taking the noise into consideration. And the simulation study of this scenario indicates that the identification efficiency of the proposed method will not be limited by the version of external excitation. 3.2. Case 2 – impact force identification for a seismically isolated frame Seismically base isolated structures are increasingly used in the area with high density of earthquake disasters. The characteristics of the external excitation, such ground motion and the interaction between the retaining wall and isolation layer, will contribute to the seismically isolated building design effectively. It is should be noted that, there will be a very large displacement for the isolation layer while the width between the isolation layer and retaining wall is limited compared to the unknown displacement of the isolation layer during each earthquake. Therefore, there will be a risk that the isolation layer may have an interaction with the retaining wall. It is necessary to estimate this kind force to conduct the seismic performance evaluation for the isolated building. In this study, a 14-storey seismically base isolated frame as shown in Fig. 6 is studied to verify the proposed force identification method. In this case the base excitation is the El-Centro (1940, NS) earthquake ground motion. Impact force on the isolation layer of seismically isolation structure may exist during the earthquake, which is always caused by the interaction force between the superstructure and the retaining wall or the neighboring building due to the limited width of isolation gap. In this case the superstructure above the isolation layer is assumed to be linear but the base isolation layer performs

Fig. 1. Three-floor nonlinear shear frame.

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15 Identified value Real value

External excitation (kN)

10

5

0

-5

-10

-15

0

0.5

1

1.5

2

2.5

3

Time (s)

a) Time history 50

Power/frequency (dB/Hz)

Real force Identified force X: 1.953 Y: 19.08

0

X: 1.465 Y: 12.09

-50

-100 0

2

4

6

8

10

12

14

16

18

20

Frequency (Hz)

b) Power spectrum density Fig. 2. Identified periodic force for 3-storey frame without measurement noise.

nonlinear during the earthquake excitation. An impact force on the isolation layer is applied on the isolation layer when the displacement is largest in time history. The ground motion is supposed to be measured by the accelerometer while the impact force is taken as unknown time history to be identified based on the proposed force identification method. The stiffness of each storey is 114 kN/m and mass for each storey is set as115 ton. The mass of the isolation layer is 80 ton. Rayleigh damping assumption is used for the superstructure in the linear condition and the damping ratio of the first two modes are taken as 0.05. Bouc-Wen model is also used to simulate the performance of the isolation layer of this structure and the nonlinear parameters of the Bouc-Wen model are set as b = 2, c = 2 and n = 1.1. The sampling rate of measurement is 100 Hz. All of horizontal accelerations from the first storey to the 14th storey are taken as the ‘‘measured” responses in the first two scenarios. And the ‘‘accelerometer” position is discussed in the third scenario. Although some adaptive UKF may be available to the identification with acceleration measurement only, the displacement for the isolation layer is also measured for the stable identification considering the influence of the model error and noise. These measurements can be implemented in the practical engineering. In the first scenario, the model error and the measurement noise in the ground motion record and structural response are not considered. The identification result in time domain is shown in Fig. 7. The accuracy of this condition is similar to the first scenario in the first case study. The impact force can be accurately identified in this ideal condition and this result verifies the accuracy of the proposed method without measurement noise and process noise. In the second scenario, the measurement noise in both ground motion and structural responses are considered and the time domain identification result is shown in Fig. 8(a). Fluctuations are caused by the measurement noise and there are some errors in the peak of the identified impact force. The frequency domain comparison of the identified force and real force are shown in Fig. 8(b). Similar to the first case, the measurement noise may cause the unexpected fluctuations in both low frequency and high frequency component in fre-

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15 Identified value Real value

External excitation (kN)

10

5

0

-5

-10

-15

0

0.5

1

1.5

2

2.5

3

Time (s)

a) Time history 50 Identified value Real value

Power/frequency (dB/Hz)

X: 1.953 Y: 19.08

X: 1.465 Y: 12.09

0

-50

-100 0

2

4

6

8

10

12

14

16

18

20

Frequency (Hz)

b) Power spectrum density Fig. 3. Identified periodic force for 3-storey frame with 5% measurement noise.

quency domain. Although the level of the noise in this scenario is set as the same as Scenario 2 in the first Case, the influence of the measurement noise on the identification result in this scenario is larger than Scenario 2 in the first Case. This is because the impact force only contributes partially to the structural response in the case and the remaining of response is contributed by the earthquake excitation. Generally, the impact force cannot be identified when the measurement noise submerges the response contributed by the external impact force. Influence of model error of structure on identification result is studied in Scenario 3. There is 5% model error in the interstorey stiffness of the initial model used for the identification is added to the frame model and can be represented as follows.

k ¼ lðkÞ þ d  lðkÞ  Nnoise

ð26Þ

where k is the vector of storey-stiffness of the target structure, lðkÞ denotes the mean value, d is the coefficient of variation which is taken to be 0.05 in this study. Nnoise is a Gaussian random variable with zero mean and unit standard deviation. The force identification result in time domain is shown in Fig. 9. The time history of the identified external force is shown in Fig. 9 (a) and there are some low frequency errors in the identified force. The amplitude-frequency curve of the identified impact force is shown in Fig. 9(b). It is shown in Fig. 9(b) that the amplitude of low frequency is not as accurate as the high frequency amplitude of the identified impact. It is indicated that the model error would cause more errors in low frequency in the identification result for civil structures. This is also partially because civil structures are the low frequency systems and a few low frequency modes contribute to the majority of the response during the earthquake. In Scenario 4, the measurement noise and model error are considered in the impact force identification process. The results are shown in Fig. 10. Time domain and frequency domain information of the identified impact force are compared with the real impact force. The impact force is still fairly accurately identified. It is demonstrated from the identified result that the proposed method are robust to the process error and measurement error.

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15 Identified force Real force

Real force (kN)

10 5 0 -5 -10 -15 -20 0

0.02

0.04

0.06

0.08

0.1

Time (s)

a) Time history (first 0.1 second)

Power/frequency (dB/Hz)

50 Identified force Real force

0

-50

-100

0

5

10

15

20

25

30

35

40

45

50

Frequency (Hz)

b) Power spectrum density Fig. 4. Identified stochastic force for 3-storey frame without measurement noise.

When the measurement noise and the model error are considered, the covariance matrix P in Eq. (12) will be influenced by the noise and may lose the property of positive definite. The identification process with UKF based on Cholesky decomposition considering the model error or the measurement noise was commonly interrupted. The identification results in Cases 1 and 2 show that the identified values coincide with the true applied force or excitation efficiently with the estimation tool of SVD-UKF. It is illustrated that the SVD-UKF is with more robustness to model error or measurement noise in the calculation with covariance matrix. 3.3. Case 3 – force identification with incomplete measurement In the third case, the influence of the selected structural response observation on the identification accuracy is studied. The structures and external force are the same as the first and fourth scenarios in Case 2. The measurement of the displacement in the isolation layer is remained for the better identification accuracy. It is noted that the displacement measurement on isolated layer is always required in the structural health monitoring system for the isolated building or isolated large span structures. Therefore, the measurement of displacement in this study is consistent to the practical engineering. Then the incomplete measurements of the acceleration response will be used and studied for the force identification, which is different from the offline force identification for linear structures with the studies of the system Markov parameter matrix [38]. During the earthquake or impact excitation, only first a few modes contribute to the majority response of the buildings commonly in civil engineering while the first mode will occupy the main response of the building for isolated structure [39]. The first three mode shapes of the isolated frame in linear condition are shown in Fig. 11. For general cases of civil structures, avoiding the setting the accelerometer on the nodes of the first three modes will obtain the successful response information for the identification. Considering the first mode contribute to the majority response, it can be predicted that avoiding using the lower storey measurement may lead to successful identification in this study.

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15 Identified force Real force

Real force (kN)

10

5

0

-5

-10

-15

-20

0

0.02

0.04

0.06

0.08

0.1

Time (s)

a) Time history 50

Power/frequency (dB/Hz)

Identified force Real force

0

-50

-100

0

5

10

15

20

25

30

35

40

45

50

Frequency (Hz)

b) Power spectrum density Fig. 5. Identified stochastic force for 3-storey frame with 5% measurement noise.

Fig. 6. 14-floor nonlinear isolated shear frame.

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x 10

0.5 0

Identified value Real value

Impact force (kN)

-0.5 -1 -1.5 -2 -2.5 -3

0

5

10

15

20

25

30

Time (s) Fig. 7. Identification result for 14-storey structure without measurement noise and model error.

0.5

x 10

5

External force (N)

0 -0.5 -1 -1.5 -2 -2.5 -3

Identified force Real force

0

5

10

15

20

25

30

Time (s)

a) Time history 4000 Identified force True force

3500

Amplitude (N)

3000 2500 2000 1500 1000 500 0

50

100

150

200

250

300

350

400

Frequency (Hz)

b) Amplitude-frequency curve Fig. 8. Identified impact force for 14-storey isolated frame with 5% measurement noise.

According to the analysis above, the acceleration should be set to the position with larger response contributed by each mode. In the following study, four scenarios are investigated without considering the measurement noise to discuss the arrangement of the acceleration measurement. In the first scenario, the acceleration measurement of the first storey is used.

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0.5

x 10

5

Identified value Real value

0

Impact force (kN)

265

-0.5 -1 -1.5 -2 -2.5 -3

0

5

10

15

20

25

30

Time (s)

a) Time history Identified force True force

4000

Amplitude (N)

3500 3000 2500 2000 1500 1000 500 0

50

100

150

200

250

300

350

400

Frequency (Hz)

b) Amplitude-frequency curve Fig. 9. Identified impact force for 14-storey isolated frame with 5% model error.

In the second scenario, the acceleration on 14th storey is measured. In the third scenario, the accelerations from first storey to 6th storey are measured. In the last scenario, the accelerations on the 10th storey to the 14th storey are measured. All the identification results are shown in Fig. 12 compared to the ‘‘real” force. It is shown from the identification results of the first scenarios that the force cannot be identified with the only acceleration measurement on the first storey but can be fairly accurately identified with the only acceleration measurement on the 14th storey. In the third scenario, it is demonstrated from the figure that the force cannot be identified even with the mult-storey response measurement. However, when the measurements are arranged from 10th to 14th the force can be identified accurately although there are some fluctuations after the peak. It is can be concluded that when the measurement contain the information of the majority modes excited by the external force and the measured response contributed by each mode are relatively large, the force can be identified accurately. To estimate the force accurately, it is also suggested that the setting of the measurement should follow the observability and controllability conditions as introduced in Refs. [34,35]. It is should be noted that the nonlinear property of the structural system will also influence the controllability and obsevability, which is different from the linear system especially considering the severe nonlinear performance of the structural system. The study of controlability and obsevebility for different nonlinear models will be further studied but not included in this paper.

4. Laboratory validation 4.1. Real-time substructure test The proposed force identification method with UKF algorithm will be validated by the real-time substructure test (RTST) of a scaled base isolated structure which is conducted and introduced in this section.

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x 10

5

Impact force (kN)

0 -0.5 -1 -1.5 -2 Identified value Real value

-2.5 4

4.5

5

5.5

6

6.5

7

7.5

8

8.5

Time (s)

a) Time history

Identified force True force

4000

Amplitude (N)

3500 3000 2500 2000 1500 1000 500 0

50

100

150

200

250

300

350

400

Frequency (Hz)

b) Amplitude-frequency curve Fig. 10. Identified impact force for 14-storey isolated frame with 5% measurement noise and 5% model error.

Before the real-time substructure test, the nonlinear relationship between the resisting force and displacement is verified with a quasi-static test with 5 mm as displacement amplitude. The load applied in the quasi-static testing to the isolator is periodic with the mode of displacement control. The relationship of one loop between the resisting force and displacement is shown in Fig. 13. The measured force shown in Fig. 13 does not consist of the damping force due to the loading process with a very low speed. It is also demonstrated that the resisting force is a nonlinear function of the displacement. Though the nonlinear relationship between the force and displacement can be obtained from the quasi-static testing, the quasi-static testing only provided the periodic load and cannot take the effect of earthquake into consideration. The RTS test has a closer relationship to the practical condition of earthquake than the quasi-static testing because of the consideration of the ground motion. The force applied on the structure in the RTS test is calculated by equation of motion and can be accurately measured by force transducer. Compared to the shaking table testing, the applied force in RTS test can be measured directly for the validation. So, the RTS test is selected as an alternative way to validate the proposed force identification method for nonlinear structure with the comparison of the measured force applied by the actuator and the force identified by the proposed method in this study. The real-time substructure test was conducted in the Hybrid Simulation Research Center of Harbin Institute of Technology. The configuration of the test system employed to the validation is shown in Fig. 14. The real-time substructure system consists of test system provided by MTS Corporation, a dSpace DS1104 control board and the target physical substructure was fixed to the loading frame. In the real-time substructure test, the superstructure is simulated as numerical substructure. The mass of the superstructure is 2500 kg. The majority of the inertial force and the damping force of the superstructure are simulated on computer. The isolation with 125kg lumped mass on the top for the nonlinear structural identification purpose is taken as the physical substructure which is also named as experimental substructure. The interface force between the numerical substructure and physical substructure is applied by the actuator as shown in Fig. 14 and can be measured by

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267

14 12

Floor number

10 8 6 4 2 0 0

0.2

0.4

0.6

0.8

1

Mode shape

a) First mode 14 12

Floor number

10 X: 0.05159 Y: 8 X: -0.1344 Y: 7

8 6 4 2 0 -1

-0.5

0

0.5

1

0.5

1

Mode shape

b) Second mode 14 X: -0.1276 Y: 11

12

Floor number

10 8 6 X: -0.03985 Y: 4

4 2 0 -1

-0.5

0

Mode shape

c) Third mode Fig. 11. First three mode shapes of the isolated 14-storey frame in linear condition.

the force transducer. The measured force from force transducer as analog signal is sampled in MTS system. The sampled measured force then would be sent to dSpace to solve the equation of motion. The discrete equation of motion in ith time step solved in dSpace is shown as follows.

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x 10

5

0.5

Impact force (kN)

0 -0.5 -1 -1.5 Identified value of Identified value of Identified value of Identified value of Real value

-2 -2.5 5

5.5

6

6.5

Scenario 1 Scenario 2 Scenario 3 Scenario 4

7

7.5

Time (s) Fig. 12. Identified force considering the different arrangement of acceleration measurement.

€n;i þ Cx_ n;i þ Re;i ðx; €g _ zÞ ¼ MGx Mx

ð27Þ

where subscript n denotes the values from numerical substructure, subscript e means the response or effect from the experimental substructure and Re in Eq. (27) represents the resisting force from experimental substructure. Eq. (27) is solved by dSpace with central difference method in this study. After solving the equation of motion for the whole structure, the displacement command calculated by dSpace would be sent to MTS system and then applied on the specimen by actuator. The real-time substructure test will be performed in this procedure step by step. Considering the condition that the first mode accounts to the majority of the response for seismically isolated structures, the superstructure is taken as rigid body in this real-time substructure testing study to simplify the calculation in dSpace. In other words, the whole system is taken as a single-degree-freedom system. The damping force in the superstructure and partial of the inertia force are simulated numerically. In the identification process, the physical substructure with the lumped mass is the target structural system. The equation of motion for this target system can be expressed as Eq. (1) and the state space equation can be denoted by Eq. (9). The interaction force between the isolator and superstructure is provided by the resisting force Re, which is also the external force F of the target structure system and will be identified with the proposed method. To estimate the interaction force augmented SVD-UKF is used with the measurement of the displacement only. The identification result is shown in Fig. 15. The measured force denotes the force recorded by the force transducer on the actuator. Although there may be noise in the measurement it will be viewed as the ‘real force’ to verify the accuracy of the identified force. It is shown in this figure that the external force of the nonlinear base isolation is identified accurately compared to the measured force. The force identified as show in Fig. 15 consists of the hysteretic force and the damping force provided by the isolation. This is because the isolation is velocity-dependent component.

2000 1500

Force (N)

1000 500 0 -500 -1000 -1500 -6

-4

-2

0

2

4

6

Displacement (mm) Fig. 13. Nonlinear relationship between the resisting force and displacement in quasi-static test.

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Fig. 14. Configuration of the real-time substructure test system.

3000 Measured force Identified force

2000

Force (N)

1000

0

-1000

-2000

-3000 0

5

10

15

20

25

30

35

40

45

Time (s) Fig. 15. Resisting force identification of the isolator in real-time substructure test.

Table 2 Parameters of steel. E (GPa)

G (GPa)

t

Post-yielding stiffness ratio

Yielding stress (MPa)

Density (g/cm3)

206

79

0.31

0.02

215

7.85

It is noted that in this experimental study, the physical substructure is taken as the target structure in the force identification. So, the model error in the superstructure will not influence the identification result while the model error of the isolation layer may be involved. The Kalman filter is powerful enough to deal with the model error and measurement noise in this process, which is also accounting for the reason to select UKF as the estimation tool in this study. For the seismic isolation structure, the superstructure should keep linear during the moderate or severe earthquake, and this is the design object. Therefore, it is reasonable to suppose the superstructure as linear for base isolated structure during earthquake. 4.2. Shaking table test Shaking table test is the direct way to evaluate the seismic performance of the structural system which can conduct the scaled testing. In this part, the time history of the ground acceleration is taken as unknown in state variable and identified

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Fig. 16. Shake table test of a two-storey steel frame.

Fig. 17. Plan of the two-storey steel frame.

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(a) Connection plan

(b) Base Isolation

Fig. 18. The base isolation and connection plate.

m2

c2

k2

m1

c1

k1

Base isolation Fig. 19. Simplified model of the isolated steel frame.

Table 3 Comparison of natural frequencies of the frame model. Order

Experimental frame model

Numerical frame model

1

Modal frequency Damping ratio

4.06 3.05%

4.08 3.05%

2

Modal frequency Damping ratio

8.36 1.22%

8.42 1.22%

with the proposed method. A two-storey steel frame structure model with the scaled ratio of 1/6 is fabricated from grade Q235 steel with the properties shown in Table 2. The photograph of the steel frame is shown in Fig. 16 and the floor plan of the steel frame is shown in Fig. 17. The beams and columns are constructed with 40mm  60mm  2mm rectangular steel tubes. The unbraced frame is connected to the top of the isolations with bolts through the base plates of columns. Each column of the frame is welded at the bottom to a base plate which has four U20 mm bolt holes with the function of connecting to the base isolation as shown in Fig. 18. The four of the isolations are used and fixed to the surface of the shaking table. The acceleration of ground motion is estimated with the proposed method during the seismic excitation. The isolated steel frame model weighs 67 kg and there are 678 kg additional mass added on each floor level to simulate the inertia effect from floor mass. Each storey of the steel frame is 483.3 mm high. Considering the mass of the structure mainly simulated at each floor level the structure can be simplified into a 2-dimensional frame as Fig. 19 according the Code for seismic design of buildings of China. The first two orders of frequencies and damping ratios for the simplified frame model and the frame on shaking table are compared in Table 3. The modal frequencies and modal damping are very close to each other, which indicate that the numerical model used for the ground motion identification is fairly accurate. The sampling rate is 200 Hz and horizontal accelerations at the isolation layer and 3rd floors are measured for the excitation identification. In the experimental study, the earthquake excitation is taken as unknown input to be identified. The proposed external excitation identification method is applied. During the earthquake, the steel frame above the base isolation layer performs as linear structure while the isolation layer during the earthquake performs nonlinearly. The identified ground excitation in time domain is shown in Fig. 20. Though there are some errors between the identified and real input of the shake table, yet the acceleration time histories of the ground motion are very close. It is shown in the identification result that there is a time delay of the identified time history of the acceleration due to the model error which has been illustrated in the simulation study of this paper. This experimental result is also consistent to the simulation study results. The frequency domain

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0.15 Measured ground motion Identified ground motion

Ground motion (m/s 2)

0.1 0.05 0 -0.05 -0.1 -0.15 -0.2

0

2

4

6

8

10

Time (s) Fig. 20. Identification result in time domain of the acceleration.

-3

4

x 10

Measured ground motion Identified ground motion

3.5

Amplitude (m/s 2)

3 2.5 2 1.5 1 0.5 0

5

10

15

20

25

30

35

40

45

50

Frequency (Hz) Fig. 21. Identification result in frequency domain of the acceleration.

comparison of the identified acceleration and the measured one are shown in Fig. 21. It is noted that the error in band from 5 Hz to 20 Hz in frequency domain, which is a little large, was caused by the noise in the actuator for the shaking table. The property of the actuator of the shaking table is not considered in the experimental study and it can be viewed as the expected environmental noise. The measurement noise also performs the adverse influence in the identification results as shown in Fig. 21. Although the measurement noise in the acceleration is not very large, the measurement noise may cause the unexpected fluctuations in both low frequency and high frequency component in frequency domain which is similar to the analysis of numerical case studies. The identifiability conditions can guarantee that the measurements information reveal the characteristics of the external excitation. Considering the known position of the force, the positions of the measurements are selected randomly as long as the system controllability and observability condition is guaranteed in this study. To estimate the force accurately, it is suggested that the setting of the measurement should follow the observability, controllability, stability and uniqueness conditions as introduced in Refs. [34,35]. It is should be noted that the nonlinear model will also influence the controllability and observability which is different from the linear system. The study of controllability and observability for different nonlinear models will be further conducted.

5. Conclusions In this study a new force identification method based on an improved UKF estimation was proposed for nonlinear structures. Numerical studies with a 3-storey nonlinear shear frame and 14-storey seismically base isolated structure are conducted. The external force can be identified accurately even considering the polluted measurements, model error and

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unexpected environmental excitation. The proposed method is also validated by real-time substructure testing and shaking table testing of seismically isolated buildings. It is also shown from the experimental results that the proposed method can be efficiently used for the earthquake-related excitation estimation. It is also demonstrated by the results that measurement noise would influence the high frequency component in the identification result while the model error may cause unexpected effect in the low frequency component of the identified time history of the external load. Acknowledgements The work in this paper was supported by Projects 2016YFC0701106 of the National Key Research and Development Program of China, No. 51308160, No. 51161120360 and No. 51408157 of National Natural Science Foundation of China and Project No. LC201423 granted by Natural Science Foundation of Heilongjiang Province of China. References [1] B. Radhikaa, C.S. 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