Adaptive Model Reference Sliding Mode Control of Structural Nonlinear Vibration

In this paper, an active mass damper (AMD) with adaptive control design is used to mitigate the vibrations of a multi-degree-offreedom (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. &e interstory shear restoring forces are simulated by the Bouc–Wen model. &is model could simulate the hinge effect of the yielding joints in steel structures or the performance of the hysteretic energy dissipation devices.&e 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.


Introduction
Structural control has received much attention in the research community during the last few decades [1][2][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][5][6][7][8][9][10][11][12][13].erefore, 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][15][16].erefore, 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.e UKF utilizes the unscented transform (UT) to estimate the propagated mean and covariance.e 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.e two methods were compared in the articles written in [17][18][19][20].It is a well-known procedure that has been applied to many real-time control systems.
ere are many studies of UKF combined with different control methods, such as fault-tolerant controls, model predictive controls [21][22][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.e Bouc-Wen model possesses two latent parameters.e 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][29][30][31][32][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][35][36][37][38][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][41][42][43][44][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.
e 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.e 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.e reference model used for MRSMC is the structure model with TMD. e efficiency of MRSMC-UKF is studied by simulation.
e numerical results demonstrate the effectiveness of this combined MRSMC-UKF method.

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.

State Estimation
Using a Traditional UKF.UKF addresses the nonlinear system with the state space form as follows: 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.e 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: where θ is the parameter vector of the system and x is the augmented state vector.

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 states x d and the latent states x l .e 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.is estimation is used as the observation of the next step: 2 Shock and Vibration where f 1 and f 2 represent the state update functions corresponding to x d and x l , respectively.x k and y k stand for the augmented state and observation vectors, respectively. x ++ lk is calculated by g 1 and is the estimation of the latent variables, used as observation for better estimating the state.g 1 is the expression of x l with respect to x d and u. g 1 is calculated with the force equilibrium equation x l � f 1 (x l , x d , u). w k−1 and v k are the augmented process and observation noise, with their covariance matrices being Q k−1 and R k , respectively.
Since the states of the system are continuous in time, when the time step is small, variance of a state is predictably small.is 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.
e detailed procedure of the modified UKF identification method is summarized in Appendix A.

Simulation Model and Control Design.
e 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.e system schematic diagram is shown in Figure 1.

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.
e displacement states of the controlled system are defined as x � x 1 x 2 x 3 x 4   T and y � y 1 y 2 y 3   T , where x 1 , x 2 , and x 3 are the displacements against the ground, x 4 is the displacement of the AMD mass relative to the third story, and y i is the interstory displacement.e parameter z i is a dimensionless hysteretic displacement.α, A, β, c, n, and D y are the parameters of the Bouc-Wen model, D y 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.e parameters m a , k a , and c a are the parameters of the AMD system.Setting a value to the ratio of the AMD mass to the main mass μ � m a /3m s , the parameter values of the AMD system can be solved from the optimum TMD parameters equation, as shown in the following equation [47]: e equilibrium equation for an AMD system excited by the ground acceleration a g can be expressed as e equilibrium equation of the main structure excited by the ground acceleration is where From the equation (6a), we can get the following equation: Shock and Vibration 3 Substituting this equation into equation ( 5), the equilibrium equation for the AMD system can be written as e 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: where M p ∈ R 4×4 and C p ∈ R 4×4 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, where H p is the force location vector and y is the interstory displacement of the model.

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 force U is equal to zero, the above AMD system changes into a TMD system.erefore, in this paper, the TMD control system is considered the reference model.e mathematical model of the reference model is defined as where , and the other parameters are defined as above.

MRSMC-UKF Control Law
In this paper, our goal is to reduce the third-story displacement and control the displacement of the AMD system.e signal error is defined as e � a(x 3 − x d3 ) + b(x 4 − x d4 ). a and b are displacement parameters in which proper values are obtained by the optimal parameter module in Simulink.
e control effect is to make the selected signal error e approach to zero gradually.x d3 and x d4 are the states that we want to track.In this simulation, we assume that x d3 � w 1 x m3 and x d4 � w 2 x m4 , where x m3 and x m4 are the states of the reference model and the w 1 and w 2 are coefficients.
e sliding mode surface is defined as where C c > 0. e Lyapunov function candidate can be defined as erefore, the active force U can be defined as 4

Shock and Vibration
where |a g | ≤ 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 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 C C > 0, in equation ( 12), we can derive that From LaSalle's theorem and the work of Xu and Özgüner [48], the active force U can realize the following equation: 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.

Numerical Simulation
MATLAB/SIMULINK is used for carrying out all simulations with a sampling frequency of 1000 Hz for a period of 100 s. e flow chart of the simulation is shown in Figure 2.
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 parameters k i , α, A, β, and c are unknown.e parameters n and D yi in the Bouc-Wen model have relatively little effect on the nonlinear behavior and are assumed to be known.e values of the structural parameters are shown in Table 1.
e El Centro earthquake with amplitude 490 Gal is employed as the seismic excitation.e initial states of the structure are set to zero, and the mass ratio μ 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 as C c � 2, a � 3, b � 0.1, w 1 � 0.5, and w 2 � 5.For the x 3 , the AMD control system based on MRSMC is better than the TMD system because of w 1 � 0.5.To improve the control of x 3 at the expense of magnifying the displacement of the AMD mass, we set w 2 to 5, and the magnified displacement is in our acceptable range.

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.e 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.
e calculation formula of the z state is expressed as It should be noted that all the unknown variables in the above equation used the value estimated in real time.Figures 3  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.e identified results are shown in Table 2. e state estimation results are shown in Figures 5-7, and the estimated

Shock and Vibration
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.

Control Results.
For chattering reduction, the sign(•) is replaced by arctan(•) in equation ( 15) since the arctan(•) function generates smooth control actions [37][38][39].e 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. e control results of the interstory displacement are shown in Figure 9. e states of AMD mass are shown in Figure 10.e active force generated by the actuator is shown in Figure 11, and the value of the sliding surface is shown in Figure 12.In Table 3, 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.

Conclusions
e 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.
e 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. e numerical model is a nonlinear 3-story frame structure with the AMD device on the top floor.e Bouc-Wen model is used to model the nonlinear restoring force of the structure.e 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.e 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.

Appendix
In this appendix the procedure of the modified UKF method is presented as follows: First, initialize the algorithm with where  □ represents the estimation of the corresponding variable, x 0 is the initial value of the states, x + 0 is the first estimation of x 0 , P 0 is the estimated covariance of x 0 ; by the same principle, P □ is the estimated covariance of x □ , and E(□) calculates the expectation of a random variable.
en, the algorithm starts.At the k-th step, Use x + k−1 , which is the estimation of x k−1 at the k − 1-th step, to formulate the sigma point vectors  With these updated sigma points, the first estimations of the mean and covariance of the states in step k are (A.5) Substituting the sigma points into the observation function gives From the sigma points of the observation vector lk  ,  x (i) lk is the value in  x (i)  k corresponding to the latent states.
e estimation of the observation and its covariance together with the cross covariance of x and y are computed as follows:

Figure 2 :
Figure 2: Flow chart of the simulation.

Figure 3 :Figure 4 :
Figure 3: Results of the Bouc-Wen parameter identification under different observations.(a) Only acceleration is input to UKF.(b) Both acceleration and state z are input to UKF.

Figure 5 :
Figure 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.

Figure 6 :
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.

Table 1 :
Structural parameters for simulation.

Table 2 :
Identified value and error of the unknown parameters.

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 controlFigure 12: Convergence of the sliding surface.Continue to the k + 1-th step. x + k is the states and parameters estimation in the k-th step, and P + k is its covariance.