An Observer-Based Controller with a LMI-Based Filter against Wind-Induced Motion for High-Rise Buildings

Active mass damper (AMD) control system is proposed for high-rise buildings to resist a strong wind. However, negative influence of noise in sensors impedes the application of AMDsystems in practice.To reduce the adverse influence of noise onAMDsystems, a Kalman filter and a linearmatrix inequality(LMI-) based filter are designed. Firstly, a ten-year return period fluctuatingwind load is simulated bymixed autoregressive-moving average (MARMA)method, and its reliability is tested bywind speed power spectrum and correlation analysis. Secondly, a designed state observer with different filters uses wind-induced acceleration responses of a high-rise 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 high-rise building and an experiment of a single span four-storey steel frame. Both numerical and experimental results are presented to verify that both Kalman filter and LMI-based filter can effectively suppress noise, but only the latter can guarantee the stability of AMD parameters.


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][2][3][4][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 high-rise structures.The references showed the state observers can solve the problem for linear uncertain systems [8][9][10] and nonlinear systems [11][12][13][14][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 maximum-likelihood criterion, Kalman filter for discrete-time systems was presented in [18].In addition, an optimization-based adaptive Kalman filtering method was proposed in [19].Moreover, a hybrid Kalman filter was established to denoise fiber optic gyroscope (FOG) sensors signal for discrete-time 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 non-Gaussian 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, discrete-time and finite dimensional systems [24][25][26][27].Normal Kalman filter cannot consider input excitation during state estimation.The state derivative of a general AMD control system includes the 2 Shock and Vibration 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 real-time filter with optimal Kalman filter gain that considers external excitation can be designed for high-rise buildings based on linear matrix inequality (LMI) approach [28].
In this paper, a state observer design method based on structural acceleration is proposed for high-rise buildings under strong wind firstly.For comparative analysis, a Kalman filter and a LMI-based 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 LMI-based 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 high-rise building and an experiment of a single span four-storey steel frame are presented to verify the efficiency of the proposed filters.The result shows that only the control system with a LMI-based filter can guarantee the stability of the AMD parameters and effectively filter out noise.

An Observer-Based Controller with a Filter and Numerical Verification
where , , 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 Ẍ, Ẋ, and  are the acceleration, velocity, and displacement of the system, respectively.System state  includes displacement and velocity.Then, (1) can be expressed into the state-space equation as where  and  are the control force and the input excitation, respectively.,  1 , and  2 are the state matrix, the excitation matrix, and the control matrix, and ,  1 , and  2 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 to where  =  −  2 ,  =  1 ,  =  −  2 , and  =  1 .A brief form of ( 5) is The second equation of ( 6) can be written in the form of a partitioned matrix.
where  1 is a vector of displacement and velocity of the structure and its AMD and  2 is a vector of acceleration, respectively.According to (7), the external excitation vector can be written as Substituting ( 8) into ( 6) and ( 7) leads to where Equation ( 9) can be written as The state observer is Substituting the second equation of (11) into the first equation leads to where   is the feedback gain of the observer. 2 and  can be used to estimate the estimated states Ỹ1 of the structure and its AMD.Ỹ1 is then used to calculate the control force.

The Simulation of Wind-Induced Motions of a High-Rise
Building.In this paper, a high-rise 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 unit-displacement method, and its structural damping ratio is 0.015.The first four natural mode shapes of KK100 along the minor-axis are given in Figure 2. 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 wind-induced motion along the minor-axis.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 ten-year 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  (10) 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 as where  = 0.22 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 is where | 1 −  2 | is the distance between two floors in vertical direction and   = 60 m according to Chinese loads code on buildings.
Mixed autoregressive-moving average (MARMA) model [31] is proposed to simulate the stochastic process.A stochastic wind speed time series can be generated as where where   ( − Δ) is the wind speed of the th random wind speed time series at time ( − Δ) 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 Wiener-Khintchine approach that can be described as can be obtained by (18), and [Ψ  ] is a regression coefficient matrix based on   .Equation ( 16) can be separated by time Δ, and the recursive matrix is expressed as Discrete fluctuating wind speed vectors with a timeinterval Δ 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 ( ⩾ 10 −2 Hz).It covers the natural frequency of high-rise buildings.Figure 3(b) indicates the vertical correlation of two kinds of fluctuating wind speed time series is high goodness-of-fit.
Following these above steps, the fluctuating wind speed time series of each floor can be generated along the height of KK100.Time-history 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).Time-history curve of the ten-year 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 observer-based controller shown in Figure 6(a) and an original controller shown in Figure 6(b) are designed to suppress the wind-induced 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 observer-based 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 observer-based 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.

Shock and Vibration
Wind.mat Integrator Force.mat Wind.mat Force.mat

A Kalman Filter Design.
When the external excitation is not taken into account, the state-space equation of a MDOF control system with noise can be described as where  1 and  2 are random process noise and measurement noise, respectively, and are assumed to be independent.
Covariance matrices [18] of these Gaussian noises are where (⋅) 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 as where   is the Kalman filter gain, Ẑ is the optimal estimation of the state, and Ŷ 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 state-space equation of the control system is In a linear system, the relationship between external excitation  and state vector  is where  is an unknown transfer function matrix.Substituting ( 29) into (28) leads to Equation ( 30) is transformed into a discrete system.
According to [34], a Kalman filter of discrete systems is where Φ() is the state transition matrix.Z() and Ẑ() are the estimates of the state () before and after correction.P and P 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 lim 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 Ẑ is defined as the measurement noise  2 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 ten-year 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.
According to (26)    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 real-time robust filter with optimal gain needs to be designed based on LMI approach to solve this problem.

A LMI-Based Filter Design.
When the output contains displacement and velocity responses, the state-space 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 as where Δ() is the state of the residue equation and Δ() = () − Ẑ().

Define
From (37), the residue equation is is a given positive constant.In [35], if and only if there exists a symmetric positive-definite matrix  1 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 positive-definite matrices  2 and  such that the following inequalities hold, then the control system shown as (38) has a  2 state feedback filter.
The first inequality of inequalities (40) can be satisfied by inequality (39).The variables  1 ,  2 , , and   are nonconvex and difficult to be solved due to the filter gain matrix   coupling with the different matrices of  1 ,  2 .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

Shock and Vibration
Wind.mat Define  =   ; then the constraint condition for solving the optimization problem (42) can be described as the following inequalities: (45) 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 state-space equation of the LMI-based filter is A Simulink block diagram of the rebuilding control system with a LMI-based 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 LMI-based filter.Its gain   is different with the Kalman filter gain.In LMI-based filter, since the discrete-time step is short, the actual state ( + 1) can be replaced by the estimated state Ŷ() [36].
The LMI-based filter shown as (48) is designed by ensuring that  2 norm () of the input and output transfer function of the residue equation shown as (38) is minimum. ∞ norm () is taken as 1 × 10 −9 .A numerical example of KK100 is presented to verify the effectiveness of the LMI-based filter.Systems 1 and 2 are the same as Section 2.3.System 3 with a LMI-based 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 one-year return period simulated fluctuating wind load that acts on KK100 is also provided to illustrate the effectiveness of the proposed LMI-based filter.
Figures 13 and 14 give a comparison of the system without noise and the system with filtering noise under ten-year and one-year return period simulated fluctuating wind loads.The system with a LMI-based filter can obviously reduce the acceleration response of the structure.From Table 5, when ten-year 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 LMI-based 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 one-year  return period wind load is taken into account, the same results can be acquired.

Experimental Verification
This experimental system shown in Figure 16 consists of a four-storey 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 Micro-Epsilon laser displacement sensors to measure the horizontal acceleration and displacement of the structure along the minor-axis.Acceleration signals are collected by a controller and used as the feedback signal to calculate the real-time 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.
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  2 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 LMI-based filter is also designed to the system.A Simulink block diagram of the observer-based 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

Figure 1 :
Figure 1: KK100 and its AMD systems: (a) picture of the building; (b) locations of the AMD systems; (c) an AMD system.

Figure 3 :
Figure 3: Tests of the fluctuating wind speed time series on 92nd floor (408.200m): (a) wind speed power spectrum test; (b) correlation test.

Figure 4 :
Figure 4: A ten-year return period fluctuating wind speed time series on 92nd floor (408.200m).

Figure 9 :
Figure 9: Simulink block diagram of the system with a Kalman filter.

Figure 11 :
Figure 11: AMD parameters of the system with a Kalman filter: (a) AMD control forces, (b) AMD strokes, and (c) AMD speeds.

Figure 12 :
Figure 12: Simulink block diagram of the system with a LMI-based filter.

Figure 15 :
Figure 15: AMD parameters of the system with a LMI-based filter: (a) AMD control forces; (b) AMD strokes.

Figure 17 :Figure 18 :
Figure 17: Simulink block diagram of the experimental system.

Table 1 :
The periods and frequencies of KK100.

Table 2 :
Key parameters of the AMD system.

Table 3 :
Control effectiveness of structural responses.
(27)27), a Kalman filter gain of linear continuous-time 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

Table 4 :
Comparison of the acceleration control effects (%).

Table 5 :
Comparison of the acceleration control effects (%).

Table 6 :
Control effectiveness of structural responses.