# Adaptive Model Reference Sliding Mode Control of Structural Nonlinear Vibration

Authors: Luyu Li; Naibang Wang; Han Qin
Journal: Shock and Vibration
 (2019)

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## Abstract

In this paper, an active mass damper (AMD) with adaptive control design is used to mitigate the vibrations of a multi-degree-of-freedom (MDOF) nonlinear structure under earthquake excitation. In the adaptive control design, a modified unscented Kalman filter (UKF) is developed to identify the unknown states and parameters adaptively. Based on the identified states and parameters, model reference sliding model control (MRSMC) is proposed for structural nonlinear vibration control. In the design of MRSMC, the structure with tuned mass damper (TMD) is used as a reference model. In the control process, the parameters and states needed to obtain the control forces are updated adaptively through UKF. A numerical example of a three-story shear-type model with an active mass damper (AMD) mounted on the top story is used to study the proposed controller. The interstory shear restoring forces are simulated by the Bouc–Wen model. This model could simulate the hinge effect of the yielding joints in steel structures or the performance of the hysteretic energy dissipation devices. The simulation results demonstrated that, with the help of the modified UKF method and the reference model, the vibration of the structure is effectively mitigated under the proposed MRSMC.


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## Body

## 1. Introduction

Structural control has received much attention in the research community during the last few decades [1–3]. Under strong earthquake excitations, a structure can experience nonlinear deformation that may cause damage or even collapse the structure. In steel structures, large deformation can induce yielding of the structure. In concrete structure, yielding of the structural components can also generate hysteretic forces. New energy dissipation devices, such as steel dampers, friction dampers, shape memory alloy dampers, pounding and impact dampers, and magnetorheological dampers, can also bring nonlinear forces into the structure [4–13]. Therefore, the study of structural vibration control considering the hysteretic effect is of great importance. To model the hysteretic effect, the Bouc–Wen model is widely used in civil engineering due to its ability to simulate various hysteretic behaviors [14–16]. Therefore, the nonlinearity of the structure is simulated using the Bouc–Wen model in this paper. For a nonlinear control design, the more the structural characteristics are known, the better the structure can be controlled. However, the model parameters of the structural nonlinearity are often unknown, and estimations of these parameters are necessary.Considering the estimation problem of nonlinear structures, the unscented Kalman filter (UKF) method has been used for parameter identification. The UKF utilizes the unscented transform (UT) to estimate the propagated mean and covariance. The unscented transform, which is the key to UKF, uses specially arranged points, which are called sigma points, to go through nonlinear transforms to estimate the updates of the mean and covariance. UKF was demonstrated to be more effective and accurate than the linearized counterpart of the Kalman filter, the extended Kalman filter. The two methods were compared in the articles written in [17–20]. It is a well-known procedure that has been applied to many real-time control systems. There are many studies of UKF combined with different control methods, such as fault-tolerant controls, model predictive controls [21–23], LQR controls [24], PID controls and its variants [25, 26], feedback linearization controls [27], and sliding mode controls [28]. However, when dealing with parameter identification of the systems possessing latent parameters, the results of UKF identification become less satisfactory. Latent parameters are those parameters that link indirectly with the observations. The Bouc–Wen model possesses two latent parameters. The states of the Bouc–Wen model are continuous. States change very little when using small integration step size. How to solve this problem for the Bouc–Wen model is important for the parameter identification and the controller design problems.Based on the estimation results, various control algorithms can be designed for structural nonlinear vibration. Regarding nonlinear control methods, the sliding mode control (SMC) shows its prominence in quick response, insensitivity to disturbances in the structure, and ease of use [28–33]. When specific purposes and performances are expected, model reference controls are often used. In vibration mitigation, a zero reference can induce large control forces, which is often unrealistic for the control actuator to realize. A structure with large damping was mostly used as the reference system [34–39].To realize an active control in civil structures, a commonly used control device is the active mass damper (AMD). An AMD generates control force through active motors and applies the force to structures by means of additional masses. Significant progress about using AMD to control structural vibration has been made in civil engineering [40–45]. Some studies are conducted to control structural nonlinear vibration using AMD. Li et al. [46] proposed a fuzzy logic control algorithm for structural nonlinear vibration control, which does not need the structure model. Incorporating the structural model into the controller design will benefit the control analysis and effect. However, fuzzy control is based on the fuzzy rule which is specified by the expert experience. Without a good mathematical model, fuzzy control may not have a good control effect especially for structural nonlinearities which are very complicated. In order to overcome this problem, a modified UKF is proposed in this paper to estimate unknown parameters and states. Based on this information, a reasonable controller can be proposed to control structural nonlinear vibration.In this paper, MRSMC is combined with UKF to solve the problem of vibration mitigation of a structure that contains nonlinearities. The UKF is used to identify the parameters and estimate the unknown structural states. To improve the performance of parameter identification, the hysteretic state of the Bouc–Wen model is calculated by substituting the identified values in the last step into the equilibrium equation. The obtained state is used as one of the observations in the current step to update the identification. Using the information estimated by UKF, MRSMC is used to determine the control law from these states and parameters. The reference model used for MRSMC is the structure model with TMD. The efficiency of MRSMC-UKF is studied by simulation. The numerical results demonstrate the effectiveness of this combined MRSMC-UKF method.


## 2. Modified UKF

To acquire all the states and parameters, an effective identification method is required. In this section, a modified UKF method is developed to effectively identify the unknown states and parameters in a hysteretic model. Traditional UKF used to identify the parameter is reviewed in the following subsection.


### 2.1. State Estimation Using a Traditional UKF

UKF addresses the nonlinear system with the state space form as follows:(1)mk=fmk−1,uk−1+w¯k−1,nk=Hmk,uk+v¯k,where f⋅ is the nonlinear state function, H⋅ is the observation function, m is the state vector of the system, u is the input vector of the system, and n is the observation vector. The parameters w¯k−1 and v¯k are, respectively, the process noise and the observation noise vectors, and they are assumed to be Gaussian.Generally, for parameter identification, the parameters are regarded as states, and then, the parameters are estimated together with the states:(2)x=mTθTT,where θ is the parameter vector of the system and x is the augmented state vector.


### 2.2. Modification of the UKF

With the uncertainty of the parameters and states, the unknown states and related parameters are often more difficult to identify using the traditional UKF. Fortunately, states in civil engineering are always continuous, and parameters are mostly varying slowly. It is, therefore, feasible to constrain the freedom of the parameter estimation along the time. In the modified UKF, states and parameters estimated in the last step are used to estimate the current states to offer the historical information as a reference by showing the consistency of the parameters and continuity of the states.In the modified method, the states are divided into two groups, the direct statesxd and the latent states xl. The latent states usually have their own evolving process, which has no observation variables. After each step, the latent states are estimated again as x^l++ by substituting other estimated states and parameters into the force equilibrium equation. This estimation is used as the observation of the next step:(3)xk=xlkxdkθk=f1xlk−1,xdk−1,uk−1f2xlk−1,xdk−1,uk−1θk−1+wk−1,yk=nkx^lk++=Hxlk−1,xdk−1,uk−1g1xdk−1,θk−1,uk−1+vk,where f1 and f2 represent the state update functions corresponding to xd and xl, respectively. xk and yk stand for the augmented state and observation vectors, respectively. x^lk++ is calculated by g1 and is the estimation of the latent variables, used as observation for better estimating the state. g1 is the expression of xl with respect to xd and u. g1 is calculated with the force equilibrium equation xl=f1xl,xd,u. wk−1 and vk are the augmented process and observation noise, with their covariance matrices being Qk−1 and Rk, respectively.Since the states of the system are continuous in time, when the time step is small, variance of a state is predictably small. This variance can be counted as the additive noise. In spite of a new uncertain observation being introduced, the latent variable is confined in a smaller range, which will largely increase the accuracy of the estimation as a whole. The detailed procedure of the modified UKF identification method is summarized in Appendix A.


## 2.1. State Estimation Using a Traditional UKF

UKF addresses the nonlinear system with the state space form as follows:(1)mk=fmk−1,uk−1+w¯k−1,nk=Hmk,uk+v¯k,where f⋅ is the nonlinear state function, H⋅ is the observation function, m is the state vector of the system, u is the input vector of the system, and n is the observation vector. The parameters w¯k−1 and v¯k are, respectively, the process noise and the observation noise vectors, and they are assumed to be Gaussian.Generally, for parameter identification, the parameters are regarded as states, and then, the parameters are estimated together with the states:(2)x=mTθTT,where θ is the parameter vector of the system and x is the augmented state vector.


## 2.2. Modification of the UKF

With the uncertainty of the parameters and states, the unknown states and related parameters are often more difficult to identify using the traditional UKF. Fortunately, states in civil engineering are always continuous, and parameters are mostly varying slowly. It is, therefore, feasible to constrain the freedom of the parameter estimation along the time. In the modified UKF, states and parameters estimated in the last step are used to estimate the current states to offer the historical information as a reference by showing the consistency of the parameters and continuity of the states.In the modified method, the states are divided into two groups, the direct statesxd and the latent states xl. The latent states usually have their own evolving process, which has no observation variables. After each step, the latent states are estimated again as x^l++ by substituting other estimated states and parameters into the force equilibrium equation. This estimation is used as the observation of the next step:(3)xk=xlkxdkθk=f1xlk−1,xdk−1,uk−1f2xlk−1,xdk−1,uk−1θk−1+wk−1,yk=nkx^lk++=Hxlk−1,xdk−1,uk−1g1xdk−1,θk−1,uk−1+vk,where f1 and f2 represent the state update functions corresponding to xd and xl, respectively. xk and yk stand for the augmented state and observation vectors, respectively. x^lk++ is calculated by g1 and is the estimation of the latent variables, used as observation for better estimating the state. g1 is the expression of xl with respect to xd and u. g1 is calculated with the force equilibrium equation xl=f1xl,xd,u. wk−1 and vk are the augmented process and observation noise, with their covariance matrices being Qk−1 and Rk, respectively.Since the states of the system are continuous in time, when the time step is small, variance of a state is predictably small. This variance can be counted as the additive noise. In spite of a new uncertain observation being introduced, the latent variable is confined in a smaller range, which will largely increase the accuracy of the estimation as a whole. The detailed procedure of the modified UKF identification method is summarized in Appendix A.


## 3. Model Reference Slide Mode Control (MRSMC)

### 3.1. Simulation Model and Control Design

The initial structural model is a 3-story shear frame structure, and the nonlinear behavior exists in the structure. In order to study the effect of control force in the nonlinear field, the Bouc–Wen model is used to model the nonlinear restoring force between stories and the control goal is to reduce the displacement of the third story relative to the ground. In this paper, the AMD control system is installed at the top of the structure. The system schematic diagram is shown in Figure1.Figure 1
Schematic diagram of AMD controlled system.


### 3.2. Model of the AMD Control System

In this work, the stiffness and damping elements of the AMD system are obtained by the design method of the optimum TMD parameters equation [47] and the active force of the AMD system is designed by the MRSMC control method.The displacement states of the controlled system are defined asx=x1x2x3x4T and y=y1y2y3T, where x1, x2, and x3 are the displacements against the ground, x4 is the displacement of the AMD mass relative to the third story, and yi is the interstory displacement. The parameter zi is a dimensionless hysteretic displacement. α, A, β, γ, n, and Dy are the parameters of the Bouc–Wen model, Dy is a parameter controls the magnitude of the hysteretic force of the Bouc–Wen model, and U is the active force generated by the actuator of the AMD. The parameters ma, ka, and ca are the parameters of the AMD system. Setting a value to the ratio of the AMD mass to the main mass μ=ma/3ms, the parameter values of the AMD system can be solved from the optimum TMD parameters equation, as shown in the following equation [47]:(4)βa=ωaω0=1−μ/21+μ,ζa=μ1−μ/441+μ1−μ/2.The equilibrium equation for an AMD system excited by the ground accelerationag can be expressed as(5)max¨3+x¨4+ag+kax4+cax˙4=U.The equilibrium equation of the main structure excited by the ground acceleration is(6a)Mx¨l+Cx˙l+Kxl+LKnz=−mseag−fc,(6b)z˙i=1DyiAiy˙i−βy˙izin−1−γiy˙izin,i=1,2,3,where xl=x1,x2,x3T, M=msI, C=c1+c2−c2−c2c2+c3−c3−c3c3, K=α1k1+α2k2−α2k2−α2k2α2k2+α3k3−α3k3−α3k3α3k3, L=1−11−11, Kn=1−α1k1Dy11−α2k2Dy21−α3k3Dy3, z=z1,z2,z3T, e=1,1,1T, and fc=0,0,U−kax4−cax˙4T. From the equation (6a), we can get the following equation:(7)msx¨3=c3x˙2−c3x˙3+α3k3x2−α3k3x3−1−α3k3Dy3z3−msag−U+kax4+cax˙4.Substituting this equation into equation (5), the equilibrium equation for the AMD system can be written as(8)max¨4=−mamsc3x˙2−c3x˙3+α3k3x2−α3k3x3−1−α3k3Dy3z3+1+mamsU−kax4−cax˙4.The governing equations of the AMD control system are expressed as equations (5), (6a), and (6b). Combining those two equations gives the motion equation of the AMD system as follows:(9)Mpx¨+Cpx˙+Fpx,z=Meag+HpU,Fpx,z=0K0−ka0mamsα3k3−mamsα3k31+mamskax+L00−mamsKnz,where Mp∈R4×4 and Cp∈R4×4 are the mass and damping matrices, respectively, while x, x˙, and x¨ represent the displacement, velocity, and acceleration vectors of the AMD control system, respectively.Specifically,(10)Mp=0M00000ma,Cp=0C0−ca0mamsc3−mamsc31+mamsca,Me=−ms−ms−ms0T,Hp=00−11+mamsT,y=LTx,where Hp is the force location vector and y is the interstory displacement of the model.


### 3.3. Reference Model

For vibration control, the sliding mode control can make the controlled states track the desired states. It is the best effect to let the controlled state approach zero; however, the AMD provides the control force mainly by the inertia of the mass. When the active forceU is equal to zero, the above AMD system changes into a TMD system. Therefore, in this paper, the TMD control system is considered the reference model. The mathematical model of the reference model is defined as(11)Mmx¨m+Cmx˙m+Fmxm,z=Mmag,z˙i=1DyiAiy˙mi−βy˙mizin−1−γiy˙mizin,i=1,2,3,where Mm=MP, Cm=Cp, Fm=Fp, Mm=Me, and the other parameters are defined as above.


## 3.1. Simulation Model and Control Design

The initial structural model is a 3-story shear frame structure, and the nonlinear behavior exists in the structure. In order to study the effect of control force in the nonlinear field, the Bouc–Wen model is used to model the nonlinear restoring force between stories and the control goal is to reduce the displacement of the third story relative to the ground. In this paper, the AMD control system is installed at the top of the structure. The system schematic diagram is shown in Figure1.Figure 1
Schematic diagram of AMD controlled system.


## 3.2. Model of the AMD Control System

In this work, the stiffness and damping elements of the AMD system are obtained by the design method of the optimum TMD parameters equation [47] and the active force of the AMD system is designed by the MRSMC control method.The displacement states of the controlled system are defined asx=x1x2x3x4T and y=y1y2y3T, where x1, x2, and x3 are the displacements against the ground, x4 is the displacement of the AMD mass relative to the third story, and yi is the interstory displacement. The parameter zi is a dimensionless hysteretic displacement. α, A, β, γ, n, and Dy are the parameters of the Bouc–Wen model, Dy is a parameter controls the magnitude of the hysteretic force of the Bouc–Wen model, and U is the active force generated by the actuator of the AMD. The parameters ma, ka, and ca are the parameters of the AMD system. Setting a value to the ratio of the AMD mass to the main mass μ=ma/3ms, the parameter values of the AMD system can be solved from the optimum TMD parameters equation, as shown in the following equation [47]:(4)βa=ωaω0=1−μ/21+μ,ζa=μ1−μ/441+μ1−μ/2.The equilibrium equation for an AMD system excited by the ground accelerationag can be expressed as(5)max¨3+x¨4+ag+kax4+cax˙4=U.The equilibrium equation of the main structure excited by the ground acceleration is(6a)Mx¨l+Cx˙l+Kxl+LKnz=−mseag−fc,(6b)z˙i=1DyiAiy˙i−βy˙izin−1−γiy˙izin,i=1,2,3,where xl=x1,x2,x3T, M=msI, C=c1+c2−c2−c2c2+c3−c3−c3c3, K=α1k1+α2k2−α2k2−α2k2α2k2+α3k3−α3k3−α3k3α3k3, L=1−11−11, Kn=1−α1k1Dy11−α2k2Dy21−α3k3Dy3, z=z1,z2,z3T, e=1,1,1T, and fc=0,0,U−kax4−cax˙4T. From the equation (6a), we can get the following equation:(7)msx¨3=c3x˙2−c3x˙3+α3k3x2−α3k3x3−1−α3k3Dy3z3−msag−U+kax4+cax˙4.Substituting this equation into equation (5), the equilibrium equation for the AMD system can be written as(8)max¨4=−mamsc3x˙2−c3x˙3+α3k3x2−α3k3x3−1−α3k3Dy3z3+1+mamsU−kax4−cax˙4.The governing equations of the AMD control system are expressed as equations (5), (6a), and (6b). Combining those two equations gives the motion equation of the AMD system as follows:(9)Mpx¨+Cpx˙+Fpx,z=Meag+HpU,Fpx,z=0K0−ka0mamsα3k3−mamsα3k31+mamskax+L00−mamsKnz,where Mp∈R4×4 and Cp∈R4×4 are the mass and damping matrices, respectively, while x, x˙, and x¨ represent the displacement, velocity, and acceleration vectors of the AMD control system, respectively.Specifically,(10)Mp=0M00000ma,Cp=0C0−ca0mamsc3−mamsc31+mamsca,Me=−ms−ms−ms0T,Hp=00−11+mamsT,y=LTx,where Hp is the force location vector and y is the interstory displacement of the model.


## 3.3. Reference Model

For vibration control, the sliding mode control can make the controlled states track the desired states. It is the best effect to let the controlled state approach zero; however, the AMD provides the control force mainly by the inertia of the mass. When the active forceU is equal to zero, the above AMD system changes into a TMD system. Therefore, in this paper, the TMD control system is considered the reference model. The mathematical model of the reference model is defined as(11)Mmx¨m+Cmx˙m+Fmxm,z=Mmag,z˙i=1DyiAiy˙mi−βy˙mizin−1−γiy˙mizin,i=1,2,3,where Mm=MP, Cm=Cp, Fm=Fp, Mm=Me, and the other parameters are defined as above.


## 4. MRSMC-UKF Control Law

In this paper, our goal is to reduce the third-story displacement and control the displacement of the AMD system. The signal error is defined ase=ax3−xd3+bx4−xd4. a and b are displacement parameters in which proper values are obtained by the optimal parameter module in Simulink. The control effect is to make the selected signal error e approach to zero gradually. xd3 and xd4 are the states that we want to track. In this simulation, we assume that xd3=w1xm3 and xd4=w2xm4, where xm3 and xm4 are the states of the reference model and the w1 and w2 are coefficients.The sliding mode surface is defined as(12)S=CCe+e˙,where Cc>0.The Lyapunov function candidate can be defined asV=1/2SmSS. Then,(13)V˙=SmsS˙=SmsCCe˙+ax¨3+bx¨4=SmsCce˙−aw1msx¨d3−bw2msx¨d4+a−bc3x˙2−c3x˙3+α3k3x2−α3k3x3−1−α3k3Dy3z3−amsag−a−1+13μbU−kax4−cax˙4.Therefore, the active forceU can be defined as(14)U=1a−1+1/3μbmsCce˙−aw1msx¨d3−bw2msx¨d4+a−bc3x˙2−c3x˙3+α3k3x2−α3k3x3−1−α3k3Dy3z3+amsD⋅signS+kax4+cax˙4,where ag≤D and D is an approximate rather than a strict upper bound value of an earthquake that is an external input. Substituting equation (14) into equation (13), the derivative of V can be rewritten as(15)V˙=S−a⋅ms⋅ag−a⋅ms⋅D⋅signS=−a⋅msag⋅S+D⋅signS≤0.If equation (15) is satisfied, the control law U designed by equation (14) can guarantee the controlled system is stable by the Lyapunov stability theory. Only when S=0 can we get V˙=0. Since CC>0, in equation (12), we can derive that(16)e⟶0.From LaSalle’s theorem and the work of Xu and Özgüner [48], the active force U can realize the following equation:(17)x3⟶xd3,x4⟶xd4.Regarding the control force (14), there is a sign function in the control force which will cause the chattering phenomenon [40, 49]. In this paper, an inverse tangent function is used to approximate the sign function.


## 5. Numerical Simulation

MATLAB/SIMULINK is used for carrying out all simulations with a sampling frequency of 1000 Hz for a period of 100 s. The flow chart of the simulation is shown in Figure2.Figure 2
Flow chart of the simulation.Firstly, the states of the structure and the parameters were updated in real time with the UKF. Secondly, the active force can be solved based on the estimated states and the identified parameters.In general, the acceleration of the structure is easy to measure; therefore, in the simulation, we assumed that only the acceleration state of the structure is known. To study the structure with uncertainties, we assumed that the parameterski,α,A,β,andγ are unknown. The parameters n and Dyi in the Bouc–Wen model have relatively little effect on the nonlinear behavior and are assumed to be known. The values of the structural parameters are shown in Table 1.Table 1
Structural parameters for simulation.
Story
m
s (t)
k
i (104 kN/m)
c
i (kN ∗ s/m)
D
y
i (cm)
n
1
345.6
9.315
545
1.9
5
2
7.605
445
1.7
3
6.165
359
1.5The El Centro earthquake with amplitude 490 Gal is employed as the seismic excitation. The initial states of the structure are set to zero, and the mass ratioμ of the AMD control system is set to 0.05. All the unknown parameters are assumed to be 0.6 times the actual value, respectively.In the simulation, the coefficients of the sliding mode surface are defined asCc=2, a=3, b=0.1, w1=0.5, and w2=5. For the x3, the AMD control system based on MRSMC is better than the TMD system because of w1=0.5. To improve the control of x3 at the expense of magnifying the displacement of the AMD mass, we set w2 to 5, and the magnified displacement is in our acceptable range.


### 5.1. State Estimation and Parameter Identification

Equations (6a) and (6b) shows that there is a little correlation between the acceleration states x¨ and the unknown Bouc–Wen parameters that are closely related to the values of state z. If the observations that are input to the UKF have little correlation with the unknown parameters, the identification effect will be poor. The state z is also needed to be given to the UKF as the observation; however, the z state is difficult to measure. In this paper, we propose a novel method. First, at time k, using the estimated values of the state and parameters, the calculated value of the active control force and the measured values of the external inputs and acceleration to calculate z by the dynamic equation of the AMD control system are shown in equation (6a). Second, the measured value of acceleration at time k and state z is input to the UKF as the observed value at time k to estimate the state and parameters at time k+ 1. The calculation formula of the z state is expressed as(18)z1e=11−α1k1Dy1−c1x˙1+cax˙4−α1k1x1+kax4−msx¨1+x¨2+x¨3+3ag−U,z2e=11−α2k2Dy2c2x˙1−c2x˙2+cax˙4+α2k2x1−α2k2x2+kax4−msx¨2+x¨3+2ag−U,z3e=11−α3k3Dy3c3x˙2−c3x˙3+cax˙4+α3k3x2−α3k3x3+kax4−msx¨3+ag−U.It should be noted that all the unknown variables in the above equation used the value estimated in real time. Figures3 and 4 show the parameter identification effect using only the acceleration as the observation input and using both acceleration and the state z as the observation input, respectively. The identified results are shown in Table 2. The state estimation results are shown in Figures 5–7, and the estimated states effectively converge the actual states. It is beneficial to obtain the control force when the UKF can identify the unknown parameters in a short time with a small error.Figure 3
Results of the Bouc–Wen parameter identification under different observations. (a) Only acceleration is input to UKF. (b) Both acceleration and statez are input to UKF.
(a)
(b)Figure 4
Results of the stiffness identification under different observations. (a) Only acceleration is input to UKF. (b) Both acceleration and statez are input to UKF.
(a)
(b)Table 2
Identified value and error of the unknown parameters.
Parameters
Actual value
Initial value
Identified by augmented observation
Identified by acceleration
Value
Error (%)
Value
Error (%)
k
1 (107)
9.315
5.5890
9.3537
0.42
9.2308
0.9
k
2 (107)
7.605
4.5630
7.6404
0.47
7.5410
0.84
k
3 (107)
6.165
3.6990
6.2121
0.76
6.1211
0.71
α
0.5
0.3
0.5070
1.41
0.5169
3.37
A
1
0.6
0.9906
0.94
1.0176
1.76
β
0.5
0.3
0.5190
3.80
0.3917
21.66
γ
0.5
0.3
0.5187
3.73
0.3893
22.13Figure 5
Comparison of structural displacement trajectory under MRSMC. (a) Displacement of the first story. (b) Displacement of the second story. (c) Displacement of the third story.
(a)
(b)
(c)Figure 6
Comparison of structural interstory displacement trajectory under MRSMC. (a) Interstory displacement of the first story. (b) Interstory displacement of the second story. (c) Interstory displacement of the third story.
(a)
(b)
(c)Figure 7
Dimensionless hysteretic displacement trajectory. (a)z1. (b) z2. (c) z3.
(a)
(b)
(c)


### 5.2. Control Results

For chattering reduction, thesign⋅ is replaced by arctan⋅ in equation (15) since the arctan⋅ function generates smooth control actions [37–39]. The control effect of AMD is compared with the response of the structure without control, and the TMD control system and the result of the control are shown in Figure 8. The control results of the interstory displacement are shown in Figure 9. The states of AMD mass are shown in Figure 10. The active force generated by the actuator is shown in Figure 11, and the value of the sliding surface is shown in Figure 12.Figure 8
Comparison of each story response for different control strategies. (a) The 1st story response. (b) The 2nd story response. (c) The 3rd story response.
(a)
(b)
(c)Figure 9
Comparison of each interstory displacement for different control strategies. (a) The 1st interstory displacement. (b) The 2nd interstory displacement. (c) The 3rd interstory displacement.
(a)
(b)
(c)Figure 10
Responses of AMD and TMD device.Figure 11
Applied active force.Figure 12
Convergence of the sliding surface.In Table3, different criterions are used to demonstrate the control effect of utilizing the method introduced in this paper. In terms of the maximum interstory drift, the control effect of the introduced AMD method is mostly far better than the TMD control effect in the control of the first two stories, though the control effect of these two methods in the third story is comparable. When the input potential of structure, which is expressed as the 2-norm of the interstory drift, is studied, the AMD control shows overwhelming advantage compared with the TMD control effect.Table 3
Control effect under different criteria.
Story
max
d
i
t
AMD control
max
d
i
t
TMD control
max
d
i
t
AMD control
max
d
i
t
No control
d
i
t
2
AMD control
d
i
t
2
TMD control
d
i
t
2
AMD control
d
i
t
2
No control
1
78.05%
68.78%
75.07%
62.56%
2
61.67%
50.13%
53.71%
44.36%
3
102.67%
83.28%
84.95%
70.38%


## 5.1. State Estimation and Parameter Identification

Equations (6a) and (6b) shows that there is a little correlation between the acceleration states x¨ and the unknown Bouc–Wen parameters that are closely related to the values of state z. If the observations that are input to the UKF have little correlation with the unknown parameters, the identification effect will be poor. The state z is also needed to be given to the UKF as the observation; however, the z state is difficult to measure. In this paper, we propose a novel method. First, at time k, using the estimated values of the state and parameters, the calculated value of the active control force and the measured values of the external inputs and acceleration to calculate z by the dynamic equation of the AMD control system are shown in equation (6a). Second, the measured value of acceleration at time k and state z is input to the UKF as the observed value at time k to estimate the state and parameters at time k+ 1. The calculation formula of the z state is expressed as(18)z1e=11−α1k1Dy1−c1x˙1+cax˙4−α1k1x1+kax4−msx¨1+x¨2+x¨3+3ag−U,z2e=11−α2k2Dy2c2x˙1−c2x˙2+cax˙4+α2k2x1−α2k2x2+kax4−msx¨2+x¨3+2ag−U,z3e=11−α3k3Dy3c3x˙2−c3x˙3+cax˙4+α3k3x2−α3k3x3+kax4−msx¨3+ag−U.It should be noted that all the unknown variables in the above equation used the value estimated in real time. Figures3 and 4 show the parameter identification effect using only the acceleration as the observation input and using both acceleration and the state z as the observation input, respectively. The identified results are shown in Table 2. The state estimation results are shown in Figures 5–7, and the estimated states effectively converge the actual states. It is beneficial to obtain the control force when the UKF can identify the unknown parameters in a short time with a small error.Figure 3
Results of the Bouc–Wen parameter identification under different observations. (a) Only acceleration is input to UKF. (b) Both acceleration and statez are input to UKF.
(a)
(b)Figure 4
Results of the stiffness identification under different observations. (a) Only acceleration is input to UKF. (b) Both acceleration and statez are input to UKF.
(a)
(b)Table 2
Identified value and error of the unknown parameters.
Parameters
Actual value
Initial value
Identified by augmented observation
Identified by acceleration
Value
Error (%)
Value
Error (%)
k
1 (107)
9.315
5.5890
9.3537
0.42
9.2308
0.9
k
2 (107)
7.605
4.5630
7.6404
0.47
7.5410
0.84
k
3 (107)
6.165
3.6990
6.2121
0.76
6.1211
0.71
α
0.5
0.3
0.5070
1.41
0.5169
3.37
A
1
0.6
0.9906
0.94
1.0176
1.76
β
0.5
0.3
0.5190
3.80
0.3917
21.66
γ
0.5
0.3
0.5187
3.73
0.3893
22.13Figure 5
Comparison of structural displacement trajectory under MRSMC. (a) Displacement of the first story. (b) Displacement of the second story. (c) Displacement of the third story.
(a)
(b)
(c)Figure 6
Comparison of structural interstory displacement trajectory under MRSMC. (a) Interstory displacement of the first story. (b) Interstory displacement of the second story. (c) Interstory displacement of the third story.
(a)
(b)
(c)Figure 7
Dimensionless hysteretic displacement trajectory. (a)z1. (b) z2. (c) z3.
(a)
(b)
(c)


## 5.2. Control Results

For chattering reduction, thesign⋅ is replaced by arctan⋅ in equation (15) since the arctan⋅ function generates smooth control actions [37–39]. The control effect of AMD is compared with the response of the structure without control, and the TMD control system and the result of the control are shown in Figure 8. The control results of the interstory displacement are shown in Figure 9. The states of AMD mass are shown in Figure 10. The active force generated by the actuator is shown in Figure 11, and the value of the sliding surface is shown in Figure 12.Figure 8
Comparison of each story response for different control strategies. (a) The 1st story response. (b) The 2nd story response. (c) The 3rd story response.
(a)
(b)
(c)Figure 9
Comparison of each interstory displacement for different control strategies. (a) The 1st interstory displacement. (b) The 2nd interstory displacement. (c) The 3rd interstory displacement.
(a)
(b)
(c)Figure 10
Responses of AMD and TMD device.Figure 11
Applied active force.Figure 12
Convergence of the sliding surface.In Table3, different criterions are used to demonstrate the control effect of utilizing the method introduced in this paper. In terms of the maximum interstory drift, the control effect of the introduced AMD method is mostly far better than the TMD control effect in the control of the first two stories, though the control effect of these two methods in the third story is comparable. When the input potential of structure, which is expressed as the 2-norm of the interstory drift, is studied, the AMD control shows overwhelming advantage compared with the TMD control effect.Table 3
Control effect under different criteria.
Story
max
d
i
t
AMD control
max
d
i
t
TMD control
max
d
i
t
AMD control
max
d
i
t
No control
d
i
t
2
AMD control
d
i
t
2
TMD control
d
i
t
2
AMD control
d
i
t
2
No control
1
78.05%
68.78%
75.07%
62.56%
2
61.67%
50.13%
53.71%
44.36%
3
102.67%
83.28%
84.95%
70.38%


## 6. Conclusions

The vibration mitigation of a nonlinear structure using an active mass damper with an adaptive control design is studied in this paper, and the adaptive force is provided by the model reference sliding model control (MRSMC) method. Due to the unknown parameters in the system, this paper proposed a novel control method by combining the unscented Kalman filter (UKF) with the MRSMC to form the integrated MRSMC-UKF. The UKF is used to identify the unknown parameters and estimate the structural states, and the MRSMC is used to determine the control law by these states and parameters based on the information estimated by UKF. The numerical model is a nonlinear 3-story frame structure with the AMD device on the top floor. The Bouc–Wen model is used to model the nonlinear restoring force of the structure. The stiffness of the simulated structure and the parameters of the Bouc–Wen model are assumed to be unknown, and these parameters are estimated in real time by using the proposed method based on the measured acceleration states. The control effect of AMD is compared with the responses of the structure without control and with the TMD control system. It turns out that the proposed MRSMC-UKF not only efficiently estimates the states and identifies the parameters but also effectively controls the structural nonlinear vibration. Based on the proposed method, a better active control device could be developed to suppress structural nonlinear vibration of the high-rise buildings under earthquake excitation. Moreover, other innovative active or semiactive control methods for structural nonlinear vibration may also be proposed since a good nonlinear mathematical model can be obtained by the modified UKF.



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*Source: 3612516-2019-04-28.xml*