Vibration and Control in Structures under Single and Multiple Hazards
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ChaoJun Chen, ZuoHua Li, Jun Teng, WeiHua Hu, Ying Wang, "An ObserverBased Controller with a LMIBased Filter against WindInduced Motion for HighRise Buildings", Shock and Vibration, vol. 2017, Article ID 1427270, 18 pages, 2017. https://doi.org/10.1155/2017/1427270
An ObserverBased Controller with a LMIBased Filter against WindInduced Motion for HighRise Buildings
Abstract
Active mass damper (AMD) control system is proposed for highrise buildings to resist a strong wind. However, negative influence of noise in sensors impedes the application of AMD systems in practice. To reduce the adverse influence of noise on AMD systems, a Kalman filter and a linear matrix inequality (LMI) based filter are designed. Firstly, a tenyear return period fluctuating wind load is simulated by mixed autoregressivemoving average (MARMA) method, and its reliability is tested by wind speed power spectrum and correlation analysis. Secondly, a designed state observer with different filters uses windinduced acceleration responses of a highrise building as the feedback signal that includes noise to calculate control force in this paper. Finally, these methods are applied to a numerical example of a highrise building and an experiment of a single span fourstorey steel frame. Both numerical and experimental results are presented to verify that both Kalman filter and LMIbased filter can effectively suppress noise, but only the latter can guarantee the stability of AMD parameters.
1. Introduction
Active mass damper (AMD) is used to control the dynamic response of highly flexible buildings horizontally under environmental loadings such as strong wind [1–5]. Generally, a vector composition of displacement and velocity in the horizontal direction is used as a feedback signal for AMD control system [6, 7], but the whole displacements and velocities of each floor are too difficult to be measured directly. Therefore, a state observer design method is of great importance to the implementation of AMD control system in highrise structures. The references showed the state observers can solve the problem for linear uncertain systems [8–10] and nonlinear systems [11–15]. Compared with displacement and velocity, [16] shows that the acceleration signal is easier to be measured and control system based on acceleration feedback is more robust. Unfortunately, the problem in the design process of an observer is that accelerometers may lead to a large estimation error that is regarded as noise. Therefore, filters for noise have to be considered.
At present, such filter process is often based on Kalman filter. In [17], a Kalman filter technique was used to estimate effective signal to noise ratio (SNR) in wireless sensor network (WSN) systems. Based on a maximumlikelihood criterion, Kalman filter for discretetime systems was presented in [18]. In addition, an optimizationbased adaptive Kalman filtering method was proposed in [19]. Moreover, a hybrid Kalman filter was established to denoise fiber optic gyroscope (FOG) sensors signal for discretetime system in [20]. By unscented Kalman filter (UKF), extended Kalman filter (EKF), or particle filter (PF), the interacting multiple sensor filter (IMSF) had been presented in [21]. Similarly, based on filter and particle filter (PF), mixture Kalman filter (MKF) was built for conditionally linear dynamic systems in unknown nonGaussian noises by [22]. A robust cubature Kalman filter (CKF) was designed for multisensors discretetime systems with uncertain noise variances in [23]. Generally, Kalman filter, considering the disturbance as the observation input, can be used to estimate the system state by output data and is often applied in linear, discretetime and finite dimensional systems [24–27]. Normal Kalman filter cannot consider input excitation during state estimation. The state derivative of a general AMD control system includes the velocity and acceleration responses, which are closely related to the external excitation. As a result, it leads to a large estimation error when neglecting the influence of external excitation. Furthermore, since the Kalman filter is strongly dependent on the statistical properties of noise and the selected Kalman filter gain is not a global optimal solution, the problem of control forces and strokes that are oversized output in an AMD system with Kalman filter should be considered. Therefore, a new realtime filter with optimal Kalman filter gain that considers external excitation can be designed for highrise buildings based on linear matrix inequality (LMI) approach [28].
In this paper, a state observer design method based on structural acceleration is proposed for highrise buildings under strong wind firstly. For comparative analysis, a Kalman filter and a LMIbased filter that consider input excitation are presented to reduce the adverse influence of noise on AMD control systems. Specifically, based on variable substitution method [29, 30], the design problem of the LMIbased filter can be transformed into a group of nonlinear matrix inequalities, which can be turned into a group of convex and easily solved linear matrix inequalities. Finally, a numerical example of a highrise building and an experiment of a single span fourstorey steel frame are presented to verify the efficiency of the proposed filters. The result shows that only the control system with a LMIbased filter can guarantee the stability of the AMD parameters and effectively filter out noise.
2. An ObserverBased Controller with a Filter and Numerical Verification
2.1. An ObserverBased Controller Design
The force equilibrium of a multidegreeoffreedom (MDOF) system iswhere , , and are the mass, damping, and stiffness matrix of the system, respectively. is the control force. and are the location matrices of control force and strong wind, respectively. And , , and are the acceleration, velocity, and displacement of the system, respectively.
System state includes displacement and velocity. Then, (1) can be expressed into the statespace equation aswhere and are the control force and the input excitation, respectively. , , and are the state matrix, the excitation matrix, and the control matrix, and , , and are the state output matrix and the direct transmission matrix of excitation and control force, which can be expressed as
The control force of the system is
Substituting (4) into (2) leads towhere , , , and . A brief form of (5) is
The second equation of (6) can be written in the form of a partitioned matrix.where is a vector of displacement and velocity of the structure and its AMD and is a vector of acceleration, respectively. According to (7), the external excitation vector can be written as
Substituting (8) into (6) and (7) leads towhere , , , and .
Equation (9) can be written as
The state observer is
Substituting the second equation of (11) into the first equation leads towhere is the feedback gain of the observer. and can be used to estimate the estimated states of the structure and its AMD. is then used to calculate the control force.
2.2. The Simulation of WindInduced Motions of a HighRise Building
In this paper, a highrise building called KingKey Financial Center (KK100) shown in Figure 1(a) has a height of 441.8 m, and its slenderness ratio is 10.2. Its structural periods and frequencies are listed in Table 1. Moreover, the lumped mass method is used for establishing the mass matrix of KK100 whose total mass is 5.79 × 10^{5} tons. Its stiffness matrix that has taken into account structural flexural and shear deformations is built based on unitdisplacement method, and its structural damping ratio is 0.015. The first four natural mode shapes of KK100 along the minoraxis are given in Figure 2.

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Its AMD control system shown in Figures 1(b) and 1(c) includes two sets of synchronous AMD devices, which are located on both sides of the 91st floor, mainly used for the controlling windinduced motion along the minoraxis. The parameters of the control system are listed in Table 2.

KK100 is located in Caiwuwei Financial Center, Luohu District, Shenzhen, China. According to the Chinese loads code on buildings, the roughness category of the area is C and the basic wind pressure of tenyear return period is 0.45 kN/m^{2}. Based on Davenport spectrum, a fluctuating wind speed can be generated. The power spectral density of fluctuating wind speed is decreased as the following equation. where is the average wind speed at a height of 10 m above ground level and is the frequency of the fluctuating wind, respectively. is the coefficient related to ground roughness and can be expressed aswhere is the C category ground roughness exponent.
The spatial correlation of fluctuating wind in time domain is mainly related to transverse and vertical correlation and is represented by correlation function. In frequency domain, the coherence function is used to describe the spatial correlation. Compared with the vertical dimension of KK100, the lateral dimension is relatively small. Therefore, the vertical correlation of fluctuating wind load is only considered. The coherence coefficient of fluctuating wind pressure in vertical direction iswhere is the distance between two floors in vertical direction and m according to Chinese loads code on buildings.
Mixed autoregressivemoving average (MARMA) model [31] is proposed to simulate the stochastic process. A stochastic wind speed time series can be generated aswherewhere is the wind speed of the th random wind speed time series at time and is the order of autoregressive model. is a zero mean random number series that obeys normal distribution with a given covariance , respectively. The relationship between power spectral density and covariance satisfies WienerKhintchine approach that can be described as
can be obtained by (18), and is a regression coefficient matrix based on . Equation (16) can be separated by time , and the recursive matrix is expressed as
Discrete fluctuating wind speed vectors with a timeinterval can be derived from (19). In order to test the reliability of the simulation results, the Fourier transformation is applied to finish wind speed power spectrum and correlation tests shown in Figure 3. Figure 3(a) shows that spectrum analysis of the simulated fluctuating wind speed based on MARMA method is similar to Davenport spectrum in a wide frequency band ( Hz). It covers the natural frequency of highrise buildings. Figure 3(b) indicates the vertical correlation of two kinds of fluctuating wind speed time series is high goodnessoffit.
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Following these above steps, the fluctuating wind speed time series of each floor can be generated along the height of KK100. Timehistory curve of the fluctuating wind speed on 92nd floor (at 408.200 m above ground) is shown in Figure 4. As the fluctuating wind speed and structural information have been given, the simulated fluctuating wind load on each floor of KK100 can be calculated by (20). Timehistory curve of the tenyear return period simulated fluctuating wind load on 92nd floor is shown in Figure 5. The simulated wind load is only used for numerical analysis in the paper, and it cannot represent the realistic wind load of KK100.where is the fluctuating wind load at th floor and is the air density. is the average wind speed at th floor. is the fluctuating wind speed that is associated with height and time. and are the shape factor of a building and the area of windward side, respectively.
An observerbased controller shown in Figure 6(a) and an original controller shown in Figure 6(b) are designed to suppress the windinduced motions of KK100. The structural acceleration of the 87th floor under uncontrolled and controlled scenarios is shown in Figure 7, and AMD parameters of different systems are shown in Figure 8. Table 3 presents the control effects and values of AMD parameters. In this paper, control effect is quantified as the ratio between structural response reduction and the structural response without control, and AMD parameters include control force and stroke. From Figures 7 and 8, the original controller and the observerbased controller can obviously reduce the wind vibration response. The frequencies of KK100 in different vibration modes are obtained and nicely consistent with its theoretical values listed in Table 1. For example, its natural frequency shown in Figure 7(d) is 0.1399 Hz in line with its theoretical value (0.1398 Hz). Moreover, the maximum variations of the displacement and acceleration control effects between two different systems are only 0.0012% and 0.0712%, and the AMD parameters of the state observer increase by −7.6087 kN and 0.0001 m. In a word, the observerbased controller is used instead of the original controller, in order to overcome the difficulty in direct measurement of the state vector that includes both structural displacements and velocities in the horizontal direction of KK100.

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2.3. A Kalman Filter Design
When the external excitation is not taken into account, the statespace equation of a MDOF control system with noise can be described aswhere and are random process noise and measurement noise, respectively, and are assumed to be independent. Covariance matrices [18] of these Gaussian noises arewhere is the expectation value of .
The control force is
Substituting (23) into (21) leads to
A Kalman filter [32] for control systems can be constructed aswhere is the Kalman filter gain, is the optimal estimation of the state, and is the observation, respectively. According to [33], the Kalman filter gain can be written as where is the model state error covariance matrix and can be solved by the following Riccati equation
Note that the Kalman filter shown by (25) ignores the influence of external excitation during the system state estimation. However, the derivative of state vectors of a general AMD control system includes the velocity and acceleration responses, which are closely related to the external excitation. Therefore, it leads to a large estimation error when neglecting the influence of external excitation. In order to ensure that the Kalman filter can effectively correct the system state estimation values, the output of the filter includes displacement and velocity responses.
The rebuilt statespace equation of the control system is
In a linear system, the relationship between external excitation and state vector iswhere is an unknown transfer function matrix.
Substituting (29) into (28) leads to
Equation (30) is transformed into a discrete system.where is a timestep.
According to [34], a Kalman filter of discrete systems iswhere is the state transition matrix. and are the estimates of the state before and after correction. and are the estimates of model state error covariance matrix before and after correction.
Since the discrete filter shown (32) contains an uncertain coefficient matrix , it still cannot be used in practice. In general, is sufficiently short and ; thus
The discrete filter shown (32) can be written as
A Simulink block diagram of the rebuilding control system with a Kalman filter shown in Figure 9 is designed to filter noise. The state observer is depicted by the dashed box, and the symbol inside the solid box represents the Kalman filter.
In this paper, a measured acceleration signal in 87th floor has been collected by the health monitoring system of KK100. This signal includes noise and is processed by wavelet transformation to acquire the actual part of the structural acceleration. Thus, the characteristics of a special noise can be understood. Based on the measured and the actual acceleration signal, the estimate of the state can be obtained by the state observer, and the difference between the actual state and the estimated state is defined as the measurement noise of the system. Therefore, covariance matrix of the measurement noise can be solved. Additionally, since the output of the Kalman filter is the system state, is set up.
Based on the above obtained covariance matrices and , under a tenyear return period wind load, three systems can be established for KK100. System 1 does not contain noise, systems 2 and 3 include noise, and system 3 with a Kalman filter should be considered. The structural acceleration of different control systems is shown in Figure 10, and the corresponding control effects are listed in Table 4. Figure 10 indicates the control system with noise that does not take any measure is to diverge, and the acceleration control effects of different floors are negative. Compared with the control system without noise, the control system with filtering noise can also obviously reduce the acceleration response of the structure. From Table 4, the maximum variations of the control effects between System 1 and System 3 are only 0.8150% and 1.0505%, meaning the control effects are equivalent to the former. The fact can prove the effectiveness of the Kalman filter.

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According to (26) and (27), a Kalman filter gain of linear continuoustime systems should be solved by a model state error covariance matrix , covariance matrices of random process noise, and measurement noises and . Based on the above statistics, is used to calculate the Kalman gain of discrete systems; this process has been shown by (34). These above methods do not contain selection process for an optimal gain. Since the selected Kalman filter gain is not a global optimal solution, the observed control force and stroke of system 3 display in Figure 11 are diverging as time goes on. When the absolute value of the AMD stroke is maximum (−3.07 m), its AMD speed is up to (−0.90 m/s). Obviously, a realtime robust filter with optimal gain needs to be designed based on LMI approach to solve this problem.
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2.4. A LMIBased Filter Design
When the output contains displacement and velocity responses, the statespace equation of the Kalman filter can be written as
Subtracting the first equation in (28) from the first equation in (35), the residue equation is then defined aswhere is the state of the residue equation and .
Define
From (37), the residue equation is
is a given positive constant. In [35], if and only if there exists a symmetric positivedefinite matrix such that the following inequality holds, then the control system shown as (38) has a state feedback filter.
is a given positive constant. In [29], if and only if there exists symmetric positivedefinite matrices and such that the following inequalities hold, then the control system shown as (38) has a state feedback filter.
The first inequality of inequalities (40) can be satisfied by inequality (39). The variables , , , and are nonconvex and difficult to be solved due to the filter gain matrix coupling with the different matrices of , . Therefore, variable substitution method cannot be used to linearize these constraints. A public Lyapunov matrix can be found to handle the problem [29].
The optimization problems from inequalities (39) to (40) can be summarized as
Both sides of the first inequality of inequalities (42) are pre and postmultiplying , defining , and then the matrix inequalities (42) are
Substituting (37) into inequalities (43) leads to
Define ; then the constraint condition for solving the optimization problem (42) can be described as the following inequalities:
The optimal solutions of , , and are obtained through the solver MINCX of LMI toolbox of Matlab, respectively. Then the matrices of optimal feedback gain of the controller and filter are
The statespace equation of the LMIbased filter is
A LMIbased filter of discrete systems can be written as
A Simulink block diagram of the rebuilding control system with a LMIbased filter shown in Figure 12 is designed to consider the negative influence of noise, but it does not depend on statistical properties of noise. The symbol inside the solid box in Figure 12 represents the LMIbased filter. Its gain is different with the Kalman filter gain. In LMIbased filter, since the discretetime step is short, the actual state can be replaced by the estimated state [36].
The LMIbased filter shown as (48) is designed by ensuring that norm () of the input and output transfer function of the residue equation shown as (38) is minimum. norm () is taken as . A numerical example of KK100 is presented to verify the effectiveness of the LMIbased filter. Systems 1 and 2 are the same as Section 2.3. System 3 with a LMIbased filter has been established. The structural acceleration and the AMD parameters of different control systems are shown in Figures 13–15, and the corresponding control effects are listed in Table 5. Moreover, a shorter return period wind has a more significant effect on noise signal. Therefore, a oneyear return period simulated fluctuating wind load that acts on KK100 is also provided to illustrate the effectiveness of the proposed LMIbased filter.

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Figures 13 and 14 give a comparison of the system without noise and the system with filtering noise under tenyear and oneyear return period simulated fluctuating wind loads. The system with a LMIbased filter can obviously reduce the acceleration response of the structure. From Table 5, when tenyear return period wind load is considered, the maximum variations of the control effects between System 1 and System 3 are only 0.4264% and 0.5695%. This fact can prove the effectiveness of the LMIbased filter. Since the gain is a global optimal solution, the observed control force and stroke of system 3 displayed in Figure 15 are stable as time goes on. Obviously, compared with the Kalman filter, the system with a LMIbased filter can filter out noise in the feedback signals and guarantee the stability of the AMD parameters. As oneyear return period wind load is taken into account, the same results can be acquired.
3. Experimental Verification
This experimental system shown in Figure 16 consists of a fourstorey steel frame made of steel and an AMD control device installed on the fourth floor [7]. Specifically, the AMD system mainly consisted of a servo motor, servo controller, an EtherCAT bus system, a dSPACE with a type of DS1103, and a computer. The loading system is composed of a reducer, an inverter, and an eccentric mass. The measurement system utilizes GT02 force balance accelerometers and MicroEpsilon laser displacement sensors to measure the horizontal acceleration and displacement of the structure along the minoraxis. Acceleration signals are collected by a controller and used as the feedback signal to calculate the realtime control forces through a designed observer. An EtherCAT bus system can be used to transmit the forces to the servo motor. The displacement signals of the second, third, and fourth floors are used to verify the control effectiveness.
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Signal obtained from GT02 force balance accelerometer includes noise and is processed by wavelet transformation to acquire the actual part in this experiment. Based on the measured and the actual signal, the estimate of the state can be obtained by the state observer, and the difference between the actual state and the estimated state is defined as the measurement noise of the system. Thus, covariance matrix of the measurement noise can be solved. Additionally, since the output of the Kalman filter is assumed as the system state, so is set up. Then, the characteristics of noise in this accelerometer can be understood. A Kalman filter can be designed to the experimental system based on the above statistics. Meanwhile, a presented LMIbased filter is also designed to the system. A Simulink block diagram of the observerbased experimental control system with different filters shown in Figure 17 is established.
The structural responses and AMD parameters of different control systems with or without noise are shown in Figures 18 and 19. The duration of each scenario is 300 s, and the figures only give data in 30 s. Table 6 presents the corresponding control effects and the values of AMD parameters. The results show that AMD control system without a filter increases the structural response and play a negative role. The displacement and acceleration control effects of the system are all negative. Therefore, it is important to design filters to isolate noise. The control effects of the systems with two different filters are all obviously consistent with each other, and these systems can obviously restrain the structural responses. However, the LMIbased filter maintains the AMD parameters in an appropriate range compared with the Kalman filter, and the AMD parameters decrease by 4.1406 N and 0.0635 m.

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Because of the interaction between the AMD system and the structure and the coupling between the horizontal and vertical vibrations of the structure, the structural responses do not completely obey the sine law under a sinusoidal load. Since the acceleration control needs high frequency control force that stimulates the structural highorder modes and AMD device is placed on the fourth floor of the structure, the control effect of the third floor has an opposite highorder phase with the second and fourth floor and is significantly less than the control effects of other floors.
4. Conclusions
The paper mainly discusses how to reduce the adverse influence of noise on highrise buildings with AMD systems. A Kalman filter and a LMIbased filter are all presented for AMD systems with noise. Finally, a numerical example and an experiment are presented to verify the effectiveness of the proposed method. The main conclusions are as follows.
MARMA method is successfully used to simulate a fluctuating wind load, whose wind speed power spectrum is similar to Davenport spectrum in a suitable frequency band.
The original controller and the observerbased controller can obviously reduce the windinduced acceleration responses. And the observerbased control system has the same control effects and stable AMD parameters as the former.
A Kalman filter and a LMIbased filter are all designed in this paper. The control system with these above filters is consistent with the control system without noise, which indicates that these filters are effective.
However, the control system with a Kalman filter is strongly dependent on the statistical properties of noise, and its selected Kalman filter gain is not a global optimal solution. As a result, its AMD parameters lose stability.
Optimization of filter gain is accomplished based on LMI approach. Therefore, the control system with a LMIbased filter can guarantee the stability of its AMD parameters.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The research described in this paper was financially supported by the National Key Research and Development Program of China (Grant no. 2016YFC0701102), the National Natural Science Foundations of China (Grant nos. 51378007 and 51538003), and the Shenzhen Technology Innovation Programs (Grant nos. JSGG20150330103937411 and JCYJ20150625142543473).
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Copyright © 2017 ChaoJun Chen 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.