The extended minimum variance unbiased estimation approach can be used for joint state/parameter/input estimation based on the measured structural responses. However, it is necessary to measure the structural displacement and acceleration responses at each story for the simultaneous identification of structural parameters and unknown wind load. A novel method of identifying structural state, parameters, and unknown wind load from incomplete measurements is proposed. The estimation is performed in a modal extended minimum variance unbiased manner, based on incomplete measurements of windinduced structural displacement and acceleration responses. The feasibility and accuracy of the proposed method are numerically validated by identifying the wind load and structural parameters on a tenstory shear building structure with incomplete measurements. The effects of crucial factors, including sampling duration and the number of measurements, are discussed. Furthermore, the practical application of the developed inverse method is evaluated based on wind tunnel testing results of a 234 m tall building structure. The results indicate that the structural state, parameters, and unknown wind load can be identified accurately using the proposed approach.
Wind load is one of the main loads during the design stage of tall buildings [
In recent years, a lot of force estimation methods have been developed [
To address the aforementioned issue, Hwang et al. proposed the Kalman filtering approach in modal domain to estimate modal loads on a structure using limited measured response [
To solve this problem, Wan et al. proposed a method called EGDF which is an extension of the unbiased minimum variance estimation for coupled state/input/parameter identification for nonlinear systems in state space [
A novel method of the modal extended unbiased minimum variance estimation for joint state/parameter/wind load estimation from incomplete measurements is proposed in this study. The proposed method is able to simultaneously estimate the wind loads and structural parameters without using complete acceleration measurements. Moreover, the data fusion of acceleration responses and interstory displacement responses is used to prevent the drifts of the identified displacements responses and wind loads. The content of this paper is organized as follows. In Section
The equation of motion of an
Based on modal coordinate transformation theory [
For proportionally damped system, the following can be obtained:
Equation (
In general, due to the limitation of the number and location of the sensors, a reduced order representation of equation (
The augmented state vector consists of the modal displacement, modal velocity, and the unknown structural parameters:
Denote that
Furthermore, the following is obtained [
By substituting
Only the partial windinduced displacement and acceleration responses are measured. The measurement vector is expressed as follows:
Using modal coordinate transformation theory, the measurement equation at the
Furthermore, the following is obtained:
Considering the measurement noise, the linearized measurement equation can be expressed as follows:
Furthermore, the following is obtained:
The changes in the eigenvalues and eigenvectors of the system due to changes in system parameters are used to calculate matrix
Thus, the derivative of eigenvalue matrix
The sensitively of eigenvector matrix
The derivative of damping matrix
For the case
According to equation (
Then, the derivative of the damping matrix subject to modal parameter
The derivative of eigenvalue matrix
The sensitively of eigenvector matrix
The time update for the predicted state estimate
According to equations (
The covariance matrix
Defining the innovation
As
Therefore,
According to equation (
Generally,
To obtain the unbiased minimum variance estimation of modal wind load
Now the covariance
Therefore, the optimal
The error of the estimated modal wind load
According to equation (
Define the final form of the updated state estimate
Therefore,
Based on Equations (
To obtain the unbiased minimum variance estimation of state
By substituting equation (
Similarly, substituting equation (
Based on equations (
Now the estimated displacement response
To verify the feasibility and accuracy of the proposed method, a tenstory shear building structure under wind load is considered. The mass coefficient of each floor is
The fluctuating wind speed is numerically simulated based on the autoregressive model method. The power spectral density is Davenport spectral. The vertical wind profile is taken as the power profile with an exponent of
Simulated fluctuating wind speed on the (a) fifth floor and (b) tenth floor.
Comparison of the power spectral density between the simulated and Davenport spectral on the (a) fifth floor and (b) tenth floor.
In this section, the interstory displacement and acceleration taken as the “measurements” are calculated based on the Newmark
Number of measurements and corresponding locations.
Number of measurements  Location (floor) 

7  2, 3, 4, 5, 7, 9, 10 
Comparison of the structural displacement responses for time and frequency domains. Displacement vs time on the (a) fifth floor and (b) tenth floor. Frequency vs power spectrum on the (c) fifth floor and (d) tenth floor.
Comparison of the structural velocity responses for time and frequency domains. Time vs velocity on the (a) fifth floor and (b) tenth floor. Frequency vs power spectrum on the (c) fifth floor and (d) tenth floor.
Moreover, based on the proposed method, the unknown wind load is identified using incomplete measurements. The identified wind load and the relative errors between the identified wind load and the exact one in percentage on the fifth and tenth floors of the numerical model are plotted in Figure
Comparison of the wind load time histories. Wind load on the (a) fifth floor and (b) tenth floor. Wind load errors on the (c) fifth floor and (d) tenth floor.
Mean errors and RMS errors for wind load estimation.
Floor number  Mean error (%)  RMS error (%)  Floor number  Mean error (%)  RMS error (%) 

1  0.22  4.11  6  4.09  4.56 
2  3.88  4.17  7  3.68  3.82 
3  4.12  4.31  8  3.89  4.21 
4  3.01  3.49  9  3.21  3.30 
5  0.38  3.87  10  1.16  2.92 
Figure
Estimation results of unknown stiffness coefficients on the (a) fifth floor and (b) tenth floor.
Estimation results of unknown damping coefficients: damping coefficients (a)
Estimation values and errors of structural parameters.
Structural parameter  Estimated value  Error (%)  Structural parameter  Estimated value  Error (%) 


2.55 × 10^{8}  3.97 

2.55 × 10^{8}  4.08 

2.37 × 10^{8}  3.44 

2.55 × 10^{8}  4.19 

2.35 × 10^{9}  4.15 

2.53 × 10^{8}  3.37 

2.54 × 10^{8}  3.84 

2.34 × 10^{8}  4.30 

2.35 × 10^{8}  3.96 

5.81 × 10^{−1}  4.25 

2.35 × 10^{8}  3.99 

3.58 × 10^{−3}  4.24 
To investigate the stability of the proposed method, the effect of the sampling duration on wind load and structural parameter estimation is discussed. The number and location of measurements are the same as in Table
RMS errors of the estimated wind load under different sampling durations.
Floor number  RMS error (%)  

Duration: 1 s  Duration: 10 s  Duration: 30 s  Duration: 60 s  
1  7.77  6.33  4.52  4.11 
2  5.86  5.68  4.45  4.17 
3  5.83  5.20  4.45  4.31 
4  6.65  5.92  3.97  3.49 
5  14.23  7.68  4.03  3.87 
6  7.46  5.15  4.58  4.55 
7  7.30  5.74  3.92  3.82 
8  6.23  5.41  4.35  4.21 
9  7.55  6.54  3.34  3.30 
10  11.94  6.20  3.81  2.92 
Structural parameter estimation results under different sampling durations.
Structural parameter  Duration: 1 s  Duration: 10 s  Duration: 30 s  Duration: 60 s  

Estimated value  Error (%)  Estimated value  Error (%)  Estimated value  Error (%)  Estimated value  Error (%)  

2.59 × 10^{8}  5.86  2.57 × 10^{8}  4.98  2.55 × 10^{8}  4.12  2.55 × 10^{8}  3.97 

2.33 × 10^{8}  4.89  2.36 × 10^{8}  3.55  2.37 × 10^{8}  3.45  2.37 × 10^{8}  3.44 

2.36 × 10^{8}  3.76  2.37 × 10^{8}  3.39  2.35 × 10^{8}  4.20  2.35 × 10^{8}  4.15 

2.56 × 10^{8}  4.55  2.55 × 10^{8}  4.14  2.55 × 10^{8}  4.15  2.54 × 10^{8}  3.84 

2.59 × 10^{8}  5.61  2.57 × 10^{8}  4.84  2.35 × 10^{8}  4.01  2.35 × 10^{8}  3.96 

2.26 × 10^{8}  7.62  2.28 × 10^{8}  6.99  2.35 × 10^{8}  4.02  2.35 × 10^{8}  3.99 

2.56 × 10^{8}  4.41  2.55 × 10^{8}  4.08  2.55 × 10^{8}  4.09  2.55 × 10^{8}  4.08 

2.56 × 10^{8}  4.35  2.55 × 10^{8}  4.21  2.55 × 10^{8}  4.21  2.55 × 10^{8}  4.19 

2.55 × 10^{8}  4.06  2.53 × 10^{8}  3.36  2.53 × 10^{8}  3.40  2.53 × 10^{8}  3.37 

2.28 × 10^{8}  6.97  2.29 × 10^{8}  6.53  2.34 × 10^{8}  4.52  2.34 × 10^{8}  4.3 

0.740  32.78  0.575  3.28  0.581  4.66  0.581  4.25 

0.00440  29.33  0.00325  4.37  0.00348  3.49  0.00348  2.24 
This section discusses the influence of the number of measurements for joint state/parameter/wind load estimation. The considered set of measurements is four to nine, and each set of measurements includes interstory displacement and acceleration responses. Table
Number of measurements and corresponding locations.
Number of measurements  Location (floor) 

9  1, 2, 3, 5, 6, 7, 8, 9, 10 
8  1, 3, 4, 5, 6, 8, 9, 10 
7  2, 3, 4, 5, 7, 9, 10 
6  1, 2, 4, 6, 8, 10 
5  2, 4, 6, 8, 10 
4  1, 4, 6, 9 
The time histories of the estimated wind loads on the fifth and tenth floors under different numbers of measurements are plotted in Figure
Time histories of the estimated wind load for different numbers of measurements on the (a) fifth floor and (b) tenth floor.
RMS errors of wind load for different numbers of measurements.
Floor number  Nine measurements  Eight measurements  Seven measurements  Six measurements  Five measurements  Four measurements 

1  2.43  3.47  4.11  4.77  6.27  9.29 
2  2.4  3.06  4.17  4.71  5.14  10.6 
3  2.06  3.24  4.31  4.7  6.38  8.12 
4  2.34  3.01  3.49  4.38  6.09  10.78 
5  2.43  3.15  3.87  4.51  7.08  9.2 
6  2.98  3.42  4.56  5.33  5.59  8.23 
7  2.49  3.3  3.82  4.58  6.03  10.06 
8  2.24  3.45  4.21  4.77  5.77  9.2 
9  0.81  2.31  3.3  4.95  6.30  8.05 
10  1.1  2.42  2.92  3.48  5.94  9.25 
The time histories of the estimated errors of the fifth and tenth structural stiffness coefficients for different numbers of measurements are shown in Figure
Estimated stiffness coefficients for different numbers of measurements on the (a) fifth floor and (b) tenth floor.
Estimated damping coefficients for different numbers of measurements: damping coefficients (a)
Estimation of structural parameters under different numbers of measurements.
Structural parameter  Nine measurements  Eight measurements  Seven measurements  Six measurements  Five measurements  Four measurements  

Value (×10^{8})  Error (%)  Value (×10^{8})  Error (%)  Value (×10^{8})  Error (%)  Value (×10^{8})  Error (%)  Value (×10^{8})  Error (%)  Value (×10^{8})  Error (%)  

2.52  2.81  2.37  3.38  2.55  3.97  2.35  4.12  2.61  6.52  2.68  9.51 

2.39  2.56  2.52  2.88  2.37  3.44  2.55  4.02  2.60  6.28  2.68  9.48 

2.40  2.03  2.37  3.12  2.35  4.15  2.34  4.31  2.58  5.45  2.21  9.62 

2.40  1.97  2.53  3.24  2.54  3.84  2.30  5.98  2.61  6.44  2.22  9.26 

2.51  2.35  2.53  3.44  2.35  3.96  2.59  5.65  2.28  6.74  2.69  9.64 

2.51  2.26  2.38  2.76  2.35  3.99  2.57  5.07  2.29  6.39  2.70  10.08 

2.38  2.86  2.37  3.42  2.55  4.08  2.57  4.94  2.30  6.27  2.68  9.25 

2.39  2.28  2.52  2.98  2.55  4.19  2.32  5.43  2.62  6.74  2.71  10.46 

2.49  1.50  2.51  2.49  2.53  3.37  2.55  4.25  2.29  6.73  2.67  9.18 

2.50  2.12  2.53  3.40  2.34  4.30  2.58  5.43  2.29  6.62  2.69  9.61 

0.577  3.63  0.581  3.71  0.581  4.25  0.612  9.83  0.592  6.27  0.606  8.71 

0.00342  0.76  0.0035  2.8  0.00348  2.24  0.00358  5.33  0.0038  11.76  0.00384  13.01 
In order to verify the effectiveness of the proposed method in practical engineering application, a tall building with 58 floors is considered as an example in this study. The size of the building is 234 m (height) × 39 m (width) × 39 m (length). The synchronous multipressure scanning system wind tunnel test for simultaneous estimation of state, unknown structural parameters, and wind load was conducted in wind environment wind tunnel laboratory at Harbin Institute of Technology (Shenzhen), China. According to Chinese National Load Code (GB 500092012) [
Mean wind speed and turbulence intensity profiles: (a) mean wind speed; (b) turbulence intensity.
The model has the same length scale with that of wind field simulation, i.e., 1 : 300. The size of the model is 780 mm (height) × 130 mm (width) × 130 mm (length), as shown in Figure
Wind tunnel test model.
Coordinate system.
For building structures with large degreesoffreedom, it is complicated or even impossible to estimate the wind load and structural parameters strictly according to the prototype structure. Generally, a simplified equivalent model with less degreesoffreedom is needed in practical analysis. The equivalent model requires that the vibration responses of the two systems are same. This means that the workenergy between the prototype structure and the equivalent structure is equal. Based on the method proposed in [
Based on the mode decomposition method, the displacement can be represented as follows:
Table
Energy contribution ratio of the first five modal responses.
Number of modes  1  2  3  4  5 

Displacement  99.458  99.963  99.994  99.999  99.999 
Velocity  98.068  99.673  99.898  99.963  99.989 
Acceleration  85.276  94.096  96.825  98.396  99.369 
Equivalent mass and stiffness of the equivalent model.
Floor number  1  2  3  4  5 


3.62 × 10^{7}  2.38 × 10^{7}  2.30 × 10^{7}  2.11 × 10^{7}  1.16 × 10^{7} 

9.53 × 10^{8}  2.52 × 10^{8}  2.30 × 10^{8}  2.04 × 10^{8}  1.74 × 10^{8} 
Comparison of the first five natural frequencies between the prototype model and the equivalent model.
Modal order  1  2  3  4  5 

Real value  0.19  0.49  0.77  1.07  1.35 
Simplified model  0.19  0.51  0.76  0.90  1.00 
Error (%)  2.48  4.16  2.34  15.50  26.10 
Comparison of the mode shape: (a) the first mode shape; (b) the second mode shape.
The windinduced responses of the tall building cannot be measured directly from the pressure measurement on a rigid model in the wind tunnel test. Hence, windinduced responses including displacement, velocity, and acceleration responses are calculated based on the wind tunnel test results and the structural dynamic properties of the prototype building structure using the Newmark
Figures
Comparison of the structural displacement responses in direction
Comparison of the structural velocity responses in direction
Displayed in Figure
Comparison of the equivalent wind load time histories in direction
Mean errors and RMS errors for wind load estimation.
Floor number  12  24  36  48  58 

Mean value  4.95  4.59  −4.87  4.29  4.93 
RMS  1.63  3.85  5.27  5.55  5.89 
Estimation results of the equivalent stiffness coefficients.
Estimation results of damping coefficients.
Estimation values and errors of structural parameters.
Structural parameter  True value  Estimated value  Errors (%) 


9.53 × 10^{8}  9.02 × 10^{8}  5.40 

2.52 × 10^{8}  2.62 × 10^{8}  4.25 

2.30 × 10^{8}  2.20 × 10^{8}  4.41 

2.04 × 10^{8}  1.94 × 10^{8}  4.81 

1.74 × 10^{8}  1.66 × 10^{8}  4.52 

0.086  0.0898  3.80 

0.023  0.0234  2.11 
In this paper, a time domain joint state/parameter/wind load estimation method from incomplete measurements is proposed based on modal extended unbiased minimum variance estimation. The recursive procedure includes four parts: time update, modal wind load estimation, measurement update, and coordinate transformation. The measurement responses include the interstory displacement and acceleration responses on partial floors. The estimation quality of the proposed method is numerically validated by simultaneously identifying the structural state, parameters, and wind load of a tenstory shear building structure from incomplete measurements. The effects of crucial factors, including sampling duration and the number of measurements on the convergence and accuracy of the proposed method, are discussed. Moreover, a synchronous multipressure scanning system wind tunnel test on a 234 m tall building structure is used to demonstrate the proposed approach for real structure. A simplified equivalent model with five degreesoffreedom is derived for experimental validation. Results indicate that the proposed method shows much potential as an alternative way for simultaneously identify the unknown structural parameters, wind load, and windinduced responses from incomplete measurements.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this article.
This work was supported by the National Nature Science Foundation of China (Grant no. 51608153) and Shenzhen Knowledge Innovation Programme (Grant nos. JCYJ20170413105418298, JCYJ20180306171737796, and JCYJ20170811153857358).