Shock and Vibration

Volume 2019, Article ID 4862983, 20 pages

https://doi.org/10.1155/2019/4862983

## Wind Load and Structural Parameters Estimation from Incomplete Measurements

Department of Civil and Environmental Engineering, Harbin Institute of Technology, Shenzhen 518055, China

Correspondence should be addressed to Kun Lin; nc.ude.tih@nuknil

Received 5 December 2018; Revised 28 February 2019; Accepted 20 March 2019; Published 17 April 2019

Academic Editor: Stefano Manzoni

Copyright © 2019 Huili Xue et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 wind-induced 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 ten-story 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.

#### 1. Introduction

Wind load is one of the main loads during the design stage of tall buildings [1, 2]. Studies show that the wind load on a tall building varies depending on the terrain condition, the shape of the building, and the surrounding buildings [3]. In most design processes, wind load is calculated according to the design code [4, 5]. However, it is difficult to calculate the time histories of wind load based on design code as the wind load design code is determined based on statistical information. Wind tunnel tests and computational fluid dynamic simulations are currently used to determine the time-varying wind load on a given structure [6, 7]. However, neither of the methods can exactly reproduce the incident turbulence and the characteristic of the surrounding buildings. Field measurement is regarded as the most accurate way of obtaining the time-varying wind load on tall buildings. However, due to the limitation of the wind load measurement technique, real-time measurement of wind load is difficult by field measurement. Compared to wind load measurement, the measurement of wind-induced displacement and acceleration responses is easier to achieve and more accurate.

In recent years, a lot of force estimation methods have been developed [8–10]. Law et al. identified wind load on a 50 m guyed mast from structural responses by regularization of the identification equation [11]. Lu and Law proposed a method for identifying unknown load using sensitivity measures of the dynamic response with respect to the input load [12]. Liu and Shepard developed a dynamic force identification approach based on the enhanced least squares approach in the frequency domain [13]. Ma et al. presented a Kalman filter method with a recursive estimator to determine the unknown excitation [14, 15]. The aforementioned approaches require that the unknown load has been generally point load acting at a specific DOF. However, wind loads acting on a building structure vary with space and time, and the methods in previous works cannot directly be used for wind load identification of building structures.

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 [16, 17]. Zhi et al. developed the Kalman filtering method and proper orthogonal decomposition technique to estimate the modal wind load on tall buildings [18, 19]. In 2007, Gillijns and Moor proposed an approach for joint input-state estimation in discrete-time dynamic system based on minimum variance unbiased solution [20]. Lourens et al. applied the method to estimate structural responses and unknown inputs in both numerical and experimental studies [21]. This method requires no assumption or prior knowledge about the unknown inputs, and it can be used for wind load identification in the physical domain [22]. Unfortunately, the aforementioned wind load estimation approaches assumed that the structural parameters are known a priori. However, for actual building structures, the structural parameters are generally determined based on the finite element model and it is difficult to calculate structural parameters exactly due to the material aging and damage in the structures.

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 [23]. Song developed the joint input-state estimation technique for joint input-state-parameter estimation based on the unscented minimum variance unbiased (UMVU) estimation method [24]. However, both EGDF and UMVU methods require that it is necessary to measure the acceleration responses at the locations where unknown inputs are applied, i.e., complete acceleration measurements at all DOFs are requested for simultaneous estimation of wind load and unknown structural parameters.

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 2, the proposed method of the modal extended unbiased minimum variance estimator is derived and the necessary mathematical proofs are given. In Section 3, a numerical validation by identifying wind loads and structural parameters from incomplete measurements on a ten-story shear building structure is addressed. Moreover, the effects of key factors, including sampling duration and the number of measurements, are discussed. In Section 4, a synchronous multipressure scanning system wind tunnel test on a 234 m tall building structure is carried out to validate the proposed method. In order to simplify the calculation, an equivalent model is carried out for experiment validation. Finally, in Section 5, the discussion and conclusion are summarized.

#### 2. Modal Extended Minimum Variance Unbiased Estimation

##### 2.1. System Model

The equation of motion of an *n*-degrees-of-freedom building structure can be written as follows:where , and are the structural mass, damping, and stiffness matrices, respectively; is the wind load vector; , , and are the wind-induced acceleration, velocity, and displacement vector, respectively.

Based on modal coordinate transformation theory [25], the structural displacement vector can be obtained as follows:where is the mass normalized modal shape matrix and is the modal displacement vector. By premultiplying and using equation (2), equation (1) can be written as follows:where is the modal mass matrix, is the modal damping matrix, is the modal stiffness matrix, and is the modal wind load vector.

For proportionally damped system, the following can be obtained:where is the *i*-th undamping natural frequency and is the *i*-th modal damping ratio.

Equation (3) can be rewritten as follows:

In general, due to the limitation of the number and location of the sensors, a reduced order representation of equation (6) is given:where , , and are the first *m* order modal acceleration, modal velocity, and modal displacement vector, respectively. and .

The augmented state vector consists of the modal displacement, modal velocity, and the unknown structural parameters:where denotes the unknown structural parameters vector. is the *i*-th structural stiffness coefficient. *α* and *β* are Rayleigh damping coefficients. Assuming that the unknown structural parameters are time invariant, the first-order differential equation of equation (8) can be obtained as follows:where is a nonlinear function consisting of the state vector, modal wind load vector, and time *t*.

Denote that with as the sampling interval and define and as the estimated values of and at time , respectively. Considering the process noise, the linearized expression of equation (9) can be expressed as follows:where is the process noise vector with a zero mean and covariance matrix .

Furthermore, the following is obtained [26]:

By substituting into the left side of equation (9), the state space equation can be obtained:

Only the partial wind-induced displacement and acceleration responses are measured. The measurement vector is expressed as follows:where denotes wind-induced displacement responses and denotes wind-induced acceleration responses.

Using modal coordinate transformation theory, the measurement equation at the *k*-th time step can be expressed as follows:where denotes the interstory displacements, is the mapping matrix associated with the DOFs of the measured displacement, and is the mapping matrix associated with the DOFs of the measured acceleration.

Furthermore, the following is obtained:

Considering the measurement noise, the linearized measurement equation can be expressed as follows:where is the measurement noise vector at the *k-*th time step with zero mean and covariance .

Furthermore, the following is obtained:where

##### 2.2. Eigenvalue and Eigenvector Sensitivity

The changes in the eigenvalues and eigenvectors of the system due to changes in system parameters are used to calculate matrix and . For the case , the structural stiffness coefficient and damping coefficient are chosen as the unknown parameters. The eigenvalue problem of proportionally damped systems can be solved according to [27] as follows:where is the *l*-th eigenvalue, is the *l*-th eigenvector, and is the *j*-th unknown parameter. Assuming that *m* modes are used for joint state-parameter-input estimation for an *n*-degrees-of-freedom, the eigenvector derivative can be calculated based on Wang’s method [28], as shown in the following equation:where

Thus, the derivative of eigenvalue matrix subject to structural parameter can then be obtained based on equation (19):

The sensitively of eigenvector matrix to the parameter can be given according to equation (20):

The derivative of damping matrix subject to structural parameter can then be calculated as follows:

For the case , the natural frequency and damping ratio are chosen as unknown parameters. The eigenvalue and eigenvector sensitivity can be calculated based on [29] as follows:

According to equation (4), the following equation can be obtained:in which is the *l*-th diagonal element in the matrix.

Then, the derivative of the damping matrix subject to modal parameter can then be calculated as follows:

The derivative of eigenvalue matrix subject to modal parameter can then be obtained based on equation (25):

The sensitively of eigenvector matrix to the modal parameter can be given:

##### 2.3. Modal Extended Minimum Variance Unbiased Estimation

###### 2.3.1. Time Update

The time update for the predicted state estimate at time can be calculated according to equation (12) as follows:

According to equations (10) and (31), the error of the predicted state estimate can be calculated as follows:where the coefficient matrices and . and are the estimation errors of state and modal wind load at time , respectively.

The covariance matrix related to the predicted state estimate can then be expressed as follows:where , , and .

###### 2.3.2. Modal Wind Load Estimation

Defining the innovation , according to equation (16), the following equation can be obtained:where the error is given by the following equation:

As is unbiased, it follows from equation (35) that , and consequently according to equation (34), can be obtained. Assume that the form of the estimated modal wind load is as follows:

Therefore, can be obtained. This indicates that the estimated modal wind load is unbiased if and only if satisfies .

According to equation (35), the covariance of the error can be obtained as follows:

Generally, , where *c* is a positive real number. This indicates that equation (34) does not satisfy homoscedasticity. Therefore, the estimator given in equation (36) is not the minimum variance estimation of modal wind load according to the Gauss–Markov theorem [30].

To obtain the unbiased minimum variance estimation of modal wind load , the optimal value of matrix in equation (36) should be determined. Assume that the covariance matrix in equation (37) is positive definite (i.e., ), an invertible matrix satisfying can be found. By premultiplying to equation (34), one can obtain the following equation:

Now the covariance , which satisfies homoscedasticity. Under the assumption that has full column rank, the unbiased minimum variance estimation of can then be obtained based on the Gauss–Markov theorem [30] as follows:

Therefore, the optimal is obtained:

The error of the estimated modal wind load can be given based on equations (36) and (39):

According to equation (41), the covariance matrix related to the estimated modal wind load is calculated as follows:

###### 2.3.3. Measurement Update

Define the final form of the updated state estimate , as follows:where is the gain matrix. The error of the updated state estimate can be calculated according to equations (12) and (43):

Therefore, , which indicates that is unbiased for all possible if and only if

Based on Equations (44) and (45), the covariance matrix related to the updated state estimate can be obtained as follows:

To obtain the unbiased minimum variance estimation of state , the optimal value of the gain matrix should be determined. Based on the Lagrange multipliers method [31], the optimal gain matrix can be calculated by minimizing the trace of under the unbiased condition shown in equation (45):where

By substituting equation (47) into (43), the unbiased minimum variance estimation of state can be calculated as follows:

Similarly, substituting equation (47) into (46), the covariance matrix related to can be expressed as follows:

Based on equations (41) and (44), the covariance matrices and can be obtained as follows:

Now the estimated displacement response , velocity response , structural parameters , and wind load at time can be calculated as follows:where , , and are the estimation of modal displacement response, modal velocity response, and structural parameters, respectively, which are obtained from the state estimate .

#### 3. Numerical Simulation

To verify the feasibility and accuracy of the proposed method, a ten-story shear building structure under wind load is considered. The mass coefficient of each floor is , and the stiffness coefficient of each floor is The damping is assumed to be Rayleigh damping which is calculated as with proportional coefficients of and . The corresponding damping ratio for the first two modes of vibration is approximate 5%.

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 and a reference height of according to the Chinese National Load Code [32]. The mean wind speed at the reference height is . Figure 1 shows the simulated fluctuating wind speed on the fifth and tenth floors. Figure 2 shows the comparison of the power spectral density between the simulated fluctuating wind speed and the Davenport spectral on the fifth and tenth floors. Figure 2 shows that the simulated power spectral density matches very well with the Davenport spectral. The wind load acting on the building structure is calculated according to [22]. The air density is assumed to be . The drag coefficient is set to 1.3, and the orthogonal exposed wind area of each floor is set to .