A Novel Model-Based Adaptive Feedforward-Feedback Control Method for Real-Time Hybrid Simulation considering Additive Error Model

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Introduction
1.1.Background and Motivation.Real-time hybrid simulation (RTHS) [1] is a cost-efective and promising testing technique and has been extensively adopted for structural performance evaluation with rate-dependent behavior under dynamic excitation [2,3].In RTHS, the emulated structure is typically divided into two parts: the structural portion with a clear constitutive model is modeled numerically by a computer software, which is generally regarded as the numerical substructure (NS), and the remaining member exhibiting stronger nonlinearities is loaded physically and termed as physical substructure (PS).Nowadays, some new progresses have been made in RTHS considering structural or external excitation uncertainties [4][5][6][7].In RTHS, the key issue is that displacement compatibility and force equilibrium on the boundary between NS and PS are enforced in real time [8][9][10].
Owing to the intrinsic dynamic of the transfer system (e.g., the servo-hydraulic actuator or shaking table), inevitable amplitude and phase errors exist between its accepted commanded and the actual response displacements.Moreover, the interaction between the actuator and PS [11], the data transition between the analog and digital signals, and the noise in the experimental process can further increase the displacement desynchronization between the NS and PS.In addition, the abovementioned issues will lead to the desynchronization representing time-varying characteristics.Generally, this phenomenon is called time delay, which will reduce RTHS's accuracy and even cause instability problems [12][13][14].Hence, the negative efect of time delay ought to be eliminated as much as possible.
In recent decades, various adaptive compensation/control methods have been proposed or improved to suppress RTHS's variable time delay.Te classical control theory-based adaptive control schemes have been developed and widely used.For instance, an adaptive inverse compensation method was developed by Chen et al. [15] and its parameters were estimated and updated by the displacement tracking indicator.Chae et al. [16] advanced an adaptive time series compensator, designed by the input-output displacement time series relationship of the test system, and the compensator parameters are continuously updated by the least-squares method.Chen et al. [17] developed an adaptive model-based tracking controller using a parameter projection algorithm, and the stability was ensured by Routh's stability criteria.By combining model-based feedforward control and robust calibration of adaptive gains, Gálmez and Fermandois [18] proposed a robust adaptive model-based compensation framework.Najaf and Spencer [19] introduced a reference model and an adaptive law into the model-based feedforward-feedback control scheme to obtain high-precision RTHS results.Strano and Terzo [20] devised an adaptive control scheme, in which the control law was established by an inverse frst-order transfer function and then updated online by the extended Kalman flter.Combining the frequency domain-based error indicators and the lead-lag controller, Mirza Hessabi et al. [21] designed a tracking error adaptive control method that exhibited the characteristics of high computational efciency.Under the framework of feedforward and feedback control, Ning et al. [22] built a new adaptive control strategy using the discrete-time model of the physical testing system.
In contrast to the classical ones, the modern control theory-based adaptive control strategies have drawn much attention and proven to be more suitable for dealing with nonlinear testing systems.To improve the robustness and accuracy of the linear-quadratic-Gaussian (LQG) controller, Zhou et al. [23] developed a novel robust LQG controller using a loop transfer recovery procedure, and its tracking ability was validated by the benchmark problem of RTHS.Te backstepping method was introduced by Ouyang et al. [24] to deal with variable time delay, and an adaptive control law was designed by Lyapunov stability analysis.To accomplish high performance and robust RTHS, Li et al. [25] established an adaptive sliding mode control method by combining the adaptive and robust control strategies.Aiming at enhancing RTHS's fdelity, Tsokanas et al. [4] introduced the adaptive strategy into the model predict control and designed a novel tracking controller.
Recently, adaptive strategies combining diferent control theories have been proposed and exhibited high control accuracy.For instance, Wang et al. [26] designed a two-stage adaptive control strategy, where the frst stage was polynomial extrapolation and the second stage was a discrete-inverse model-based adaptive delay compensator.Tao and Mercan [27] advanced an adaptive compensation scheme, in which the tracking error-based adaptive compensator was combined with a proportional-integralderivative controller.Xu et al. [28] developed an adaptive combined compensation strategy using a sliding mode compensator and an improved adaptive forward prediction method to suppress time delay in RTHS.Combining the passivity control with an adaptive feedforward flter, Peiris et al. [29] developed a combined control method, with the advantages of low cost and the lower dependency on the prior knowledge about the test system.
Although the adaptive compensation methods have represented exceptional performance in literature, three main factors still afect their abilities.First, the adaptive method's performance depends profoundly on the initial setting of the model parameter.It requires explicit prior knowledge of the experimental testing system in general [17].Second, the parameter estimation method plays a critical role in most adaptive compensation methods.A parameter estimation method with excellent ability ought to guarantee convergence, low dependency on the initial parameter setting, and high sensitivity to displacement tracking errors [30].Tird, the robustness is difcult to guarantee, especially for the complex nonlinear testing system [31].

Scope.
In the feld of robust control theory, the controller is generally established by using the nominal model of the control plant and its additive or multiplicative uncertainties, so as to ensure that the designed controller exhibits brilliant tracking capability and strong robustness in RHTS.On this basis, this study divides the control plant into a nominal model and an additive error model and establishes a novel modelbased adaptive feedforward-feedback control method to solve the challenges in adaptive control methods.Te proposed method comprises three parts: (1) a feedforward controller and (2) a discrete parametric adaptive controller to which a (3) feedback controller is added.In particular, the feedforward controller is designed by the control plant's inverse nominal model and its parameters are constant, and it can suppress the primary time delay and amplitude discrepancy.Te adaptive controller is determined by a discretized additive error model, which can minimize the residual time delay.Te feedback controller is introduced to enhance the robustness and further improve the control accuracy.
Te main contents of this study are as follows.Te background, the motivation, and the scope are introduced in Section 1. Te formula of model-based adaptive feedforward-feedback control considering the additive error model will be introduced in Section 2. Ten, Section 3 presents numerical validations, including the emulated structure and a nonlinear control plant, compensation strategies, tracking performance assessment, and parametric study.Subsequently, the experimental validations for the proposed method are demonstrated in Section 4, followed by the conclusions in Section 5.

Formula of Model-Based Adaptive Feedforward-Feedback Control Method
In this study, the control plant is composed of the transfer system and the PS.Its exact analytical model is often challenging to obtain, seriously afecting the design of the tracking controller.In robust control theory, the controller is generally designed based on the nominal model and the additive or multiplicative uncertainty of the control plant, and the robust controller has achieved excellent tracking performance [32].Based on this, a novel adaptive model-based control scheme is conceived in this section.
2. .As shown in Figure 1(a), the measured displacement d m can match excellently the desired displacement d in this case.
However, the existence of uncertainties in PS and/or the transfer system can signifcantly weaken the efect of the feedforward controller.An additional control quantity d a should be introduced to achieve outstanding tracking performance, as shown in Figure 1 Defning the tracking error e as then substituting (1)-( 3) into (4), the tracking error e can be further expressed as In RTHS, the tracking error e is always expected to be 0, and then, the additional control quantity can be obtained, namely, However, the additional control quantity shown in ( 6) is usually very difcult to use due to the existence of G ∆ .It is necessary to simplify (6).Let G ∆ + G 0 � TG 0 , then (6) can be expressed as where T is a frequency-dependent transfer function.Te magnitude of T is almost 1 at the low frequency range and then gradually increases with the increase of frequency.In the feld of earthquake engineering, the frequency of the structure and/or the earthquake excitation is generally very low, and one can always assume that the value of T is approximately 1, namely, Tus, the model-based feedforward controller can be reached, as shown in Figure 1(c).

Real-Time Error Model
Estimation.Typically, it is diffcult to achieve the additive error model G ∆ in (8), because its amplitude is very small, especially in the low frequency range, say lower than 10 Hz, while, in the moderate frequency range, its amplitude increases gradually.Te frequency response of the additive error model G ∆ exhibits diferential or higher-order diferential features, namely, the order of the numerator is higher than the denominator in the transfer function model.To facilitate practical use, this study adopts the discrete-time model of the additive error, and d e k at the k th time step can be expressed as where x is the additive error model parameter and q denotes the additive error model parameter number.Because of the uncertainties in the control plant, the dynamic characteristics of the additive error model cannot be completely described by (9) with the given parameters.To address this issue, the error and commanded displacements are utilized to online estimate the additive error model G ∆ , which is denoted as  G ∆ .It is assumed that the dynamics of the model error change slowly; thus, one can always treat the parameters in (9) as constants within a certain time span.Consequently, the problem of determining parameters in (9) belongs to a certain optimization or flter problem.While the least-squares method family is widely used, they are timeconsuming concerning the calculation.Hence, considering the outstanding estimation performance, the Kalman flter (KF) algorithm is adopted in this study.
Te state and measurement equations for estimating the additive error model can be expressed as where , and v k is unknown zero mean white measurement noise with known covariance R.

Structural Control and Health Monitoring
Te recursive estimation for the model parameters x k+1 can be expressed as in which K k is the KF gain that is calculated by where P is the error covariance matrix of the state estimate, which is calculated by in which I is a unit matrix with respect to the dimension of the state vector x.Tus, the additional control quantity can be written as where  x denotes the estimate of the additive error model parameter and r � −G −1 0 G −1 0 d.

Feedback Controller.
A feedback controller is introduced to further enhance the robustness and improve the tracking performance.Te proportional-integral (PI) controller is adopted in this study because of its facilitation, which can be expressed as where d FB is the displacement generated by the PI controller, K P and K I are the proportional and integral gains in the PI controller, respectively, and ξ denotes the diference between the desired displacement d and the measured displacement d m that has been compensated for by the feedforward controller.
Tus, the formula of the model-based adaptive feedforward-feedback control method has been achieved, whose block diagram is shown in Figure 2. It is seen that the inverse model G −1 0 is served as the feedforward controller to eliminate the primary time delay in the control plant, and the adaptive controller is designed utilizing the additive error model to eliminate the residual time delay by online estimating the adaptive model parameters.Furthermore, the PI controller can increase the robustness and obtain a superior control performance.

Numerical Validation
Te efectiveness and tracking performance for the proposed control strategy were accomplished and validated by two kinds of numerical simulations, including tracking performance assessment under prescribed displacements and a series of virtual RTHSs (vRTHSs).Considering that the testing system and PS will sufer diferent degrees of nonlinearity in an actual application, a nonlinear control plant, which comprises a nonlinear actuator model and a nonlinear specimen, is employed in this section.

Overview of the Simulation.
As shown in Figure 3, the emulated structure is a steel frame structure of three stories, and a magnetorheological (MR) damper is installed between the frst foor and the ground foor.Te steel frame structure is selected as the NS, while the MR damper is taken as the PS.
Only the horizontal degrees of freedom (DOF) are considered for the NS.It is assumed that the mass is lumped at the foor levels and the mass for each foor is 2.05 × 10 4 kg.Te lateral stifness for each story is 3.773 × 10 7 N/m, and the calculated natural frequencies for each story are 3.04 Hz, 8.51 Hz, and 12.30 Hz, respectively.Te structural damping matrix is calculated using the Rayleigh damping model, where the frst two modal damping ratios are assumed to be 2%.
In the numerical simulations, the nonlinear actuator system model provided by Zhao et al. [33,34] is employed to realize the displacement boundary coordination and force equilibrium between the two substructures, and its block diagram is revealed in Figure 4.As shown in the fgure, the  Structural Control and Health Monitoring nonlinear actuator model is mainly composed of an actuator and its servovalve, with the natural velocity feedback considered.Te model can account for the efects of nonlinear factors in RHTS, including the fow property, response delay of the servovalve, and the interaction between the PS and the actuator.In addition, the Bouc-Wen model developed by Weber et al. [35] is used to represent the numerical model for the MR damper (Figure 4).Te model is given by where E v is the evolutionary variable and ς(i) and λ(i) are functions related to the input current i in the damper.Te parameters and description of the servohydraulic actuator system and MR damper are shown in Tables 1 and 2, respectively.

Control Strategy.
As presented previously, the proposed adaptive control method incorporates a model-based feedforward-feedback strategy and a nominal model of the control plant should be identifed frst.Preliminary simulation with a duration of 60 s is performed using the uncontrolled control plant as stated previously, in which a 0.1∼10 Hz swept signal owing an amplitude of 5 mm is served as the commanded displacement.To capture more information from the control plant and to increase the reliability of the feedforward controller, a second-order transfer function without zeros is adopted to describe the dynamic of the nominal model, which could be expressed as where s represents the Laplace operator and the corresponding feedforward controller can be given in continuous time form in the time domain, namely, where t and u denote the input and output of the feedforward controller, respectively; the subscript "2" indicates that the inverse controller is second order.A four-parameter diference equation is used to describe the additive error model and design the adaptive controller, which can be expressed as Given the low nonlinearity of the control plant at the initial stage of the simulation, the model error between the nominal model and the control plant can be neglected.Hence, x 0 can be expressed as [0, 0, 0, 0] T , and P 0 is represented by a fourth-order unit matrix.As for R, its value is 1 × 10 −4 in this study.
Once the parameters are estimated, the additional control quantity d a generated by the adaptive controller can be given by

Structural Control and Health Monitoring
As for the feedback controller, namely, the PI controller, K P and K I gains are set to 2 and 1, respectively.Tey were previously determined by the sinusoidal signal excited test, and the frequency and amplitude are 2 Hz and 5 mm, and the duration is 60 s.
Tis study also investigates the control strategy where the proposed method's feedback controller is removed, termed additive error model-based adaptive controller (AEM-AC).In addition, the inverse model (IM) control method is considered to demonstrate the necessity of introducing and estimating the error model online, which is given in (18).
Figure 4: Nonlinear control plant model [33,34].For the qualitative aspect, three indicators, namely, the calculated time delay J 1 , the normalized roots mean square of the tracking error J 2 , and the peak tracking error J 3 , are used to evaluate the control performance.Tey are defned by in which ω is the number of displacement data points for d and d m .
To assess the global performance of the virtual RTHS, six more indicators are employed, which are calculated by where l denotes the l th foor of the emulated structure, J l+3 and J l+6 are the roots mean square error and the peak tracking error of the displacement for the l th foor, respectively, and D R and D N are the displacements measured from each story of the reference structure and numerical substructure, respectively.Tese nine evaluation indicators can also be found in Silva et al. [36].show that there are evident amplitude and phase errors between the desired and measured displacements of the IM and AEM-AC methods for each prescribed displacement.Tough the amplitude errors of IM and AEM-AC methods are comparable, the phase errors of AEM-AC are smaller than those of IM, indicating the necessity of estimating the additive error model parameters online.Te proposed method's measured displacement is almost identical to the desired displacements, illustrating its superior tracking ability.Comparing the results of the proposed and the AEM-AC methods demonstrates that the feedback controller is benefcial to improving the tracking performance.

Tracking Performance Assessment under
Te J 1 ∼J 3 values for diferent control methods under three earthquake displacement signals are plotted in Figure 7. Te J 1 ∼J 3 values of AEM-AC are smaller than those of IM, implying that on the basis of the inverse model, adding to the additive error model and estimating its model parameters online can improve the control accuracy.When AEM-AC is compared with the proposed method, the J 1 and J 2 values of these two methods are nearly identical to each other.However, the J 3 values of the proposed method are smaller than those of AEM-AC, demonstrating that the feedback controller is attributed to remedy the displacement discrepancies at the peak further.Te results indicate that the proposed controller exhibits strong robustness and brilliant tracking accuracy compared with the AEM-AC and IM methods.
Figure 8 presents the estimated time histories (ETHs) of the additive error model parameters under three earthquake displacement signals.Te estimated error model parameters are broadly identical to each other.Te estimated parameters start to update at the beginning of the simulation and then approach their corresponding constants at about the 5 th second.Te variation range of the estimated parameter for each earthquake displacement signal is relatively small.It could be attributed to the fact that the primary time delay is mainly suppressed by the feedforward controller, resulting in the smaller error displacement d e and then the smaller estimated parameters.It could also be concluded from Figure 8 that the KF parameter estimation method can adjust the estimated parameters timely to drive the measured displacement tracking the desired displacement.Although the error displacements are tiny, the KF is sensitive enough to estimate the almost true parameters for the additive error model with high accuracy.

Results of the vRTHS.
To better interpret the diferences between the three control methods mentioned above, 15 vRTHSs for each control method are carried out and the results are presented in this section.Tree diferent earthquake excitations are employed for each control method, namely, El Centro, Kobe, and Morgan Hill.Te PGAs are scaled to 0.2 g, 0.3 g, 0.4 g, 0.5 g, and 0.6 g for each earthquake excitation.
Te DTHs for three control methods under El Centro excitation with a PGA of 0.4 g are shown in Figure 9, with the enlarged views at zero and peak displacement at the same time provided.Figure 9(a) reveals a better agreement between the measured and desired displacements for diferent control strategies, whereas Figures 9(b) and 9(c) show notable diferences in the amplitude and phase for IM and AEM-AC.Te phase diference of AEM-AC is smaller than that of IM, while the amplitude discrepancy of AEM-AC is Structural Control and Health Monitoring     Structural Control and Health Monitoring higher than that of IM. Figure 9(d) shows an excellent agreement between displacements d and d m , demonstrating the superior performance of the proposed method.
To roundly compare the compensated ability of diferent control methods, Figure 10 gives the desired-measured (D-M) displacement plots under the considered three earthquake excitations with a PGA of 0.4 g.Here, the enlarged views of displacements d and d m in the range of [−1, 1] are also given.As shown in the fgure, obvious hysteresis loops can be observed from the results of the IM method, while the AEM-AC method only exhibits slight hysteresis loops under three excitations.For the proposed method, D-M displacement plots are the thinnest, and the slope is almost equal to 1, implying an excellent tracking ability.Tis is consistent with the conclusions drawn from Figure 9.
Te evaluation indicator values of the vRTHS are analyzed and plotted in Figure 11.Te "Mean" and "STD" denote the mean and standard deviation of the evaluation indicator for the 15 vRTHSs of each control method, respectively.As shown in Figure 11, the "Mean" and "STD" of J 1 ∼J 3 values provided by IM are the largest among the three control methods, and they have been reduced signifcantly by AEM-AC, especially for J 1 , whose "Mean" and "STD" both reduces to 0 ms.Tis demonstrates that it is essential to consider the additive error model.Focusing on the proposed method, it could be found that the "Mean" and "STD" of the above evaluation indicators are almost the smallest, indicating its superior tracking performance.It also implies that the feedback controller can improve the robustness of the proposed method.Te proposed method has the smallest J 4 and J 7 values, either "Mean" or "STD."Tis indicates the proposed method exhibits an excellent ft between the reference and numerical results of the frst foor.For the global performance of the second and third foors, the evaluation indicator values, namely J 5 , J 6 , J 8 , and J 9 , are almost identical to each other, implying the superior accuracy and robustness of the two methods.In summary, the newly proposed model-based adaptive feedforward-feedback control method has supreme control performance in vRTHS.
Figure 12 describes the ETHs of the additive error model parameters for the proposed adaptive control method under three earthquake excitations with PGA scaled to 0.4 g.It shows that, the evolutions of the estimated parameters are almost identical to each other.In particular, the estimated parameters remain unchanged for the frst 5 s because the small displacements and the nonlinearities are very weak.Ten, the estimated parameters start to update, but they no longer converge to constant values.Te reason could be that the displacements here are greater than those in Section 3.3.1,and the control plant comes into a strong nonlinearity.Nevertheless, the variation range of the identifed parameters is relatively small for the three cases.

Parameter Study.
As described in Section 2, the control performance of the proposed method is mainly afected by the parameter estimation method, the inverse model order, and the additive error model parameter number.In order to investigate the infuence of parameter settings in the proposed controller and provide guidance on controller design, a series of simulations are carried out using the control plant described in Figure 4 with the prescribed displacement commands as shown in Figure 5.Note that the values of the calculated time delay J 1 are 0 ms in the simulations, so they are not presented in this section.

Initial Settings in KF.
As presented in Section 2.2, the initial settings, namely, the initial state vector x 0 , the initial estimate error covariance matrix P 0 , and the measurement noise covariance R, play an important role in the KF method.Hence, we investigate their infuences on the proposed method using the controller designed in Section  Structural Control and Health Monitoring 3.2.A parametric expression of the initial setting is adopted to represent their variation conveniently, which is defned as where B 0 denotes the original initial settings x 0 , P 0 , and R, while B f represents the variation of those parameters and Q f is the variable factor, whose value is collected in Table 3.A nonzero initial state vector is used in this section to fulfll the parameter analysis, which is calculated by the leastsquares method utilizing the commanded displacement d c and the error displacement d e .Te error displacement is determined by the diference between the second-order nominal model as shown in (17) and the nonlinear control plant.Tus, the initial state vector x 0 used in ( 23) can be expressed by when Q f � 0, the initial state vector calculated by ( 23) is identical to that used in Section 3.2.Figure 13 plots the J 2 and J 3 values with diferent initial parameters under three prescribed displacements.Te results in Figures 13(a) and 13(d) show that as Q f becomes larger, the J 2 values almost remain unchanged for each described displacement, and the J 3 values also hold the same constant value under the Dis.3 command.In contrast, the J 3 values under Dis.1 and Dis.2 remain unchanged frst and then become larger.Furthermore, the variation degree of J 3 under Dis.1 is higher than that under Dis.2.Te J 3 value under Dis.1 begins to increase sharply when the value of Q f is assigned to 14.At this time, the initial state vector is [-0.2492,0.2366, −0.0490, 0.0672] T , which is a relatively Structural Control and Health Monitoring large value based on the results in Figures 8 and 12.Nevertheless, the J 3 values are still less than 4.5%.Te results demonstrate that the initial state vector x 0 has a limited infuence on the proposed method's tracking performance and verify the rationality of setting the initial state vector x 0 as a zero vector.Furthermore, the J 2 and J 3 values with J 2 (%) J 2 (%) Time (s) Time (s) Time (s)   13.It indicates that the initial estimation error covariance matrix P 0 and measurement noise covariance R have hardly any efect on the control ability of the proposed method.

Te Inverse Model Order.
To investigate the efect of the inverse model order on the control performance, four diferent inverse models, namely, frst-, second-, third-, and fourth-order, are considered in this section.Tey are determined by their corresponding nominal models by the system identifcation for the nonlinear control plant using the same swept signal described in Section 3.2.Te nominal and inverse models are listed in Table 4.Note that ( 19) is adopted to set up the additive error model.Te J 2 and J 3 values with diferent inverse models are shown in Figure 14.Tese values represent the mean of J 2 and J 3 under the three prescribed displacements.Te J 2 and J 3 values of the frst-order inverse model are both the greatest.Tis is because the frst-order nominal model is not accurate enough to represent the dynamic characteristics of the nonlinear control plant, leading to the poor control accuracy of its corresponding feedforward controller.Teoretically, the fourth-order inverse model is the most suitable one for designing the feedforward controller, but its J 3 value is higher than those of the second-and third-order inverse models, although the diferences in J 2 values of the three models are very small.Te reasons may be interpreted as follows.Te fourth-order inverse model can result in the lower error displacement d e , in which the measurement noise may account for a large proportion.Tus, inaccurate additive model parameters may be estimated by the KF, resulting in poor control accuracy.Although the higherorder model may be conducive to fully refecting the control plant's dynamic characteristics, it may adversely afect the control accuracy and complicate the control strategy design.Hence, the second-or third-order inverse model may be a candidate for the optimal model (Figure 14).Moreover, the frst-order inverse model can also exhibit superior control accuracy if the low nonlinear of the control plant or permissible testing error is present.17) and ( 18), respectively; (iii) the initial state vector x 0 is set to zero vector considering its limited efect on the control performance; (iv) the initial estimation error covariance matrix P 0 is set to a unit matrix, whose order is in accordance with the dimensions of x 0 ; and (v) the value of measurement noise covariance R is the same as the setting presented in Section 3.2.Similar to Figure 14, Figure 15 presents the mean values of J 2 and J 3 under three prescribed displacements with diferent additive error models.Te values of J 2 and J 3 decrease slowly with the increase of parameter numbers for the additive error model.On the contrary, the variations of the evaluation indicator are very limited, demonstrating that the proposed method is robust to the additive error model parameter number.Tis is because the primary time delay has been largely eliminated, and the error displacements are relatively small, whose dynamics can be well modeled by a fewer-parameter-described time-varying model.Tus, increasing parameter numbers in the additive error model cannot improve the control accuracy.Despite the fact that the 5-parameter additive error model has the best control accuracy, it is not conducive to the control strategy design in real applications, because the estimated parameters in the higher-order model are more sensitive to the disturbances, leading to the estimated parameters with large fuctuations and then poor control accuracy.Terefore, considering the high-frequency dynamics and unclear nonlinear of the testing system, the additive model with a 3-or 4-parameter is sufcient to obtain superior control accuracy.

Experimental Verification of the Proposed Method
4.1.Experimental Setup and Structure Parameters.In this section, actual RTHSs for demonstrating the feasibility and efectiveness of the proposed method were conducted on an energy dissipation structure at the Structural and Seismic Testing Center, Harbin Institute of Technology.As shown in Figure 16, the target structure was composed of a 4-story steel frame structure and a damper between the frst foor and the ground foor.Te frame structure was considered NS, and its dynamic model was simplifed to a story shear model that only considers the horizontal degree of freedom.Each story's mass and lateral stifness were assumed 20 × 10 3 kg and 5.9448 × 10 7 N/m, respectively.Te damping ratios for the frst two models were both set to 2%; thus, the calculated natural frequencies for each story were 3.01 Hz, 8.68 Hz, 13.29 Hz, and 16.31 Hz, respectively.Te experimental installation, consisting of an MTS servohydraulic system, a damper, and a dSPACE real-time controller, is adopted to verify the proposed adaptive control method and compare its capacity with existing compensation/control methods.Te MTS servoactuator can be capable of providing a maximum dynamic displacement of ±125 mm and a maximum force of 100 kN.As described in Figure 16, the damper was selected as the PS, which was in series with the actuator.Te dSPACE controller was used for signal transition and test data acquisition and monitoring during RTHSs.
Similarly, with the earthquake input setting in vRTHS, three seismic excitations were employed, and the PGAs were scaled to 0.2 g, 0.3 g, 0.4 g, 0.5 g, and 0.6 g for each controller.In the actual RTHSs, the central diference method was utilized to solve the structural dynamic equation of the target structure, and the integration step was 1/1024 s.

Controller Design.
In the actual RTHS, besides the three control methods mentioned above, this study took into account the most widely used polynomial extrapolation (PE) method [37] and the recently developed Kalman flter-based adaptive delay compensation (KF-ADC) method [30].Te swept signal used in Section 3.2 was employed as the commanded displacement to identify the nominal and inverse models and to obtain the initial parameter settings of the KF-ADC method by the least-squares method.
where θ 1 � 1.79 × 10 4 , θ 2 � 187.4,and θ 3 � 1.782 × 10 4 .Considering the discreteness of the testing system, the IM method can be represented by the discrete-time form using the forward diference method.Tus, the corresponding inverse controller can be expressed as where ∆t denotes the time interval between the adjacent displacement data points.In this study, the adjacent time interval was determined by every fve displacement points, namely, ∆t � (5/1024) s, and this will reduce the adverse efects of measurement noise and ensure the smoothness of the displacement curve.

Te Proposed and the AEM-AC Methods.
Te inverse model in the proposed and AEM-AC methods was the same as (26).For ease of use, the nominal model used to generate the error displacements was represented by a discrete transfer function.
where z denotes the transformation operator.
In addition, this study adopted the 4-parameter additive error model to set up the adaptive controller, whose parameter settings were the same as those used in Section 3.2.
For the proposed method, the optimal PI gain was previously determined by a series of tests.Tese tests were all conducted for a duration of 60 s, adopting a 2 Hz sinusoidal signal owning 5 mm amplitude as the displacement input of the control plant.A total of 10 PI combinations were considered (Table 5), and the J 1 ∼J 3 values are shown in Figure 17.Te results show that the PI gain has a limited efect on the control performance.Combination 7, K P � 1.0 and K I � 0.1, is set to be the optimal PI gain for the proposed method because the J 3 value is the smallest in this case.

PE Method.
Te PE method is one of the most widely used methods for the time delay problem in RTHS because of its simple design and better compensation efect.Tis method utilizes three parameters to predict the displacement command of the next time step: the control plant's sampling frequency, the time delay, and the polynomial coefcients.Considering the infuence of polynomial coefcients and the degree of nonlinearity in the control plant, this study used the 3-order PE method, which is expressed as where η � τ De /∆t and τ De is the calculated time delay.Te displacement signal used in the PI gain determination was used to estimate the time delay ofine, which was about 11.7 ms.

KF-ADC Method.
In the KF-ADC method, the adaptive time-varying parametric compensator was developed by the discrete-time inverse model of the control plant and its coefcients were estimated online by the KF using d c and d m at each time step.In this study, the 4parameter KF-ADC method was considered, which can be expressed as where  θ 1 ∼  θ 4 are the estimated model parameters at each time step.Teir initial values were 2.4456, −2.7503, 2.1531, and −1.3700, respectively, which were calculated by the least-squares method.Te initial estimation error covariance matrix was set to a 4-order diagonal matrix, whose diagonal elements are 2.4456 2 , (−2.7503) 2 , 2.1531 2 , and (−1.3700) 2 .Te measurement noise covariance was the same as that used in Section 3.2.For the IM and PE methods, there are signifcant diferences in the agreement of both the phase and amplitude between the measured and desired displacements, indicating that the two methods exhibit poor displacement synchronization.Focusing on the enlarged view at around zero displacements, it can be concluded that the phase shift of the PE method is lower than that of the IM method, indicating that the PE method has a better delay compensation ability.In addition, it can be seen that the KF-ADC method can signifcantly reduce both the phase shift and amplitude discrepancy and remarkably improve the synchronization between the measured and desired displacements.Tese facts demonstrate that the time delay is variable and can be efectively addressed by the KF-ADC method.Despite all this, there are still obvious displacement diferences.Compared with PE, IM, and KF-ADC methods, the displacement d m is in better agreement with the displacement d for the proposed and AEM-AC methods, indicating that the additional control quantity generated by the additive error model is benefcial for improving the tracking ability.
Te mean values of J 1 ∼J 3 for diferent control methods under three earthquake displacement signals are plotted in Figure 19.It is seen that the IM method has the largest values of J 1 ∼J 3 and exhibits the worst compensation capability in all compared control methods.Te values of J 1 ∼J 3 for the PE method are improved slightly compared to those of the IM method.Tough the J 3 values of PE and KF-ADC are roughly the same, the J 1 and J 2 values of the latter are signifcantly reduced, demonstrating the greater time delay compensation accuracy of the KF-ADC method.Te J 1 values of the AEM-AC and proposed methods are both 1 ms, but the J 2 and J 3 values of the proposed method are smaller.In general, error indicators of the proposed method are the smallest, implying that considering the modeling errors can signifcantly improve the tracking ability and the feedback control is conducive to promoting the control accuracy.

Swept Signal.
Figure 20 provides the absolute tracking error (ATE), i.e., the absolute error between displacements d and d m at each time step.Tis study also calculated and plotted the maximum and mean values of the tracking errors, denoted by "Max" and "Mean," respectively.Figure 20 demonstrates that the distribution ranges of the ATE for the proposed and the AEM-AC methods are smaller than those for IM, PE, and KF-ADC methods.Te tracking errors for IM, PE, and KF-ADC methods rise as time goes on, while for the proposed and the AEM-AC methods, the tracking errors increase gradually at the frst 20 s and then decrease progressively and reach the lowest values at around 85 s.As shown in Figures 20(a)∼20(c), both the "Max" and "Mean" of the IM method are the greatest, followed by the PE method.In contrast to the above two methods, the KF-ADC method shows a better control ability.However, the tracking error for the KF-ADC method has a spike point of about 0.2 mm at 0 s (marked by a green point in Figure 20(c)).Figures 20(d) and 20(e) show that the "Max" and "Mean" of the tracking error are identical in the proposed and the AEM-AC methods and are both signifcantly lower compared with the other three methods.Te results demonstrate that the proposed and the AEM-AC methods exhibit excellent control performance.
Te obtained J 1 ∼J 3 values with diferent methods are plotted in Figure 21.It is seen that the IM method performs the worst, followed by the PE and KF-ADC methods.Te J 1 ∼J 3 values are nearly identical for the proposed and the  18 Structural Control and Health Monitoring AEM-AC methods, and they are the smallest in all the comparative methods.In particular, the calculated time delay J 1 values in these methods are reduced to 0 ms. Figure 22 gives the ETHs of the additive error model parameters for the proposed method.Te error model parameters are updated immediately as soon as the test begins and has a fast change speed in the frst 20 s.Afterward, the estimated parameters gradually change.Although the estimated parameters do not converge to constant values, they fuctuate within a very limited range.Te main reason for this fact is that, under the swept signal, the control plant was negatively afected by diferent nonlinearities and the increasing frequency of the signal, resulting in a signifcant variation in time delay.To track the commanded displacement, the control plant parameters must be adjusted correspondingly.

Results of RTHS.
Figure 23 plots the DTHs of the considered methods under El Centro excitation with a PGA of 0.4 g, where the enlarged views of the displacement around zero and peak displacement at the same time are also provided.Te measured displacements with diferent control methods match the desired displacements very well on the whole, as shown in Figure 23(a).However, there are profound diferences among diferent control methods if focusing on the enlarged view.As depicted in Figures 23(b)∼ 23(d), there are very large amplitude discrepancies and phased shifts for IM and PE methods, while the KF-ADC methods efectively improve these.However, poor agreements between the desired and measured displacements also exist at peak displacement for the KF-ADC method.As presented in Figures 23(e) and 23(f ), amplitude discrepancies or phase shifts can hardly be recognized from the DTHs of the proposed and the AEM-AC methods.Furthermore, the proposed method yields a perfect agreement between the measured and desired displacements, regardless of zero or peak displacement.Te results indicate that the model-based adaptive feedforward-feedback control method has the best displacement ft and exhibits excellent control accuracy in RTHS among the considered control methods.
Figure 24 plots the hysteresis relationships of the damper in the RTHS under three seismic excitations with PGAs scaled to 0.4 g.It is seen in the fgure that the force increases with the growth of displacement in the positive direction, whereas the force measured from the damper is almost zero in the negative displacement range, demonstrating the damper sufers strong nonlinearity.In particular, the hysteresis relationship curves under El Centro and Morgan Hill are fatter than those under Kobe, demonstrating that the damper in RTHS sufered a stronger nonlinearity under the former earthquake excitations.
Te D-M displacement plots of diferent control methods under the considered three excitations with a PGA of 0.4 g are shown in Figures 25(a)∼25(c), respectively.As presented in the fgure, hysteresis loops can be detected for the IM and PE methods, while this phenomenon is efectively improved for the KF-ADC and AEM-AC methods.However, displacements d and d m cannot still perfectly anastomose in the D-M displacement plots for the KF-ADC method.For the proposed method, its D-M displacement plot is the thinnest, which is approximated as a straight line with a slope of 1.
Figure 26 plots the J 1 ∼J 3 values of the considered control methods in RTHS.In the fgure, the implications of "Mean" and "STD" are the same as the descriptions in Figure 11.Te results in Figure 26(a) show that the "Mean" value of J 1 ∼J 3 for the IM method is the greatest, followed by the PE method, and the proposed method is the smallest, indicating the perfect tracking performance of the proposed method and the necessity of adaptive control strategy.Compared with the IM method, the "Mean" values of J 1 ∼J 3 for the AEM-AC method shrank by more than 50%, indicating that the tracking performance can be improved by considering the additive error of the control plant.Moreover, the "STD" values of J 1 for the fve methods are shown to be nearly identical to each other in Figure 26(b), while the proposed method has the lowest "STD" value for J 2 and J 3 .It demonstrates that the proposed method exhibits stronger robustness.
A comparison of the three adaptive control schemes shows that the KF-ADC method has the worst control accuracy, whereas the proposed and the AEM-AC methods can more efectively solve the variable time delay problem in RTHS.Te reason may be attributed to the following facts.Te KF-ADC method cannot deal with the relatively highfrequency dynamics of the system, resulting in the accumulation of test errors, and the unreasonable initial parameter settings can further reduce its control accuracy.For the proposed and the AEM-AC methods, the feedforward controller largely eliminates the primary time delay, and the time-variant additive error model is conducive to  Furthermore, the proposed method performs the best, indicating the feedback controller contributes to improving the tracking accuracy.Figure 27 describes the ETHs of the additive error model parameters for the proposed method under three earthquake excitations with a PGA of 0.4 g.Te evolutions of the estimated parameters are similar to each other for the considered three excitations.Te estimated parameters are almost unchanged at the beginning because the control plant's nonlinearity is very weak.Subsequently, they are updated rapidly in the following 3 s and almost converge to their corresponding constant.Furthermore, the variation ranges for the error model parameters are relatively small for all excitations, namely, from −0.2 to 0.2.Te results in Figure 26 also imply that the parameter estimation method

Conclusions
Tis study proposed a novel adaptive model-based feedforward-feedback control method to deal with the variable time delay in RTHS.Compared with most existing adaptive control methods, it can reduce the dependence on the parameter estimation method as well as the initial parameters of the adaptive control law.Te feasibility and efectiveness were examined by the virtual simulations and actual experimental validations, respectively.Te major conclusions are as follows.
(1) Te formula of the proposed adaptive control method is presented in detail.In this control method, the primary time delay is suppressed by a feedforward controller based on the nominal inverse model of the control plant.Structural Control and Health Monitoring

1 .
(b).In Figure1, dc denotes the commanded displacements, d e is the error displacement produced from the additive error model G ∆ , d 0 denotes the displacement of the nominal model G 0 , and d FF represents the displacement generated by the inverse controller G 0 −Te form of the additional control quantity is derived as follows.Te displacements d c , d m , and d FF can be expressed as

Figure 2 :Figure 3 :
Figure 2: Block diagram of the proposed method.
Prescribed Displacement Command.Tree earthquake displacement signals, named Dis.1,Dis.2, and Dis.3, respectively, are considered in this study, whose time histories are shown in Figure5.Tey are obtained from the frst-foor displacement responses of the Benchmark structures[36] excited by the El Centro, Kobe, and Morgan Hill seismic excitations (their peak ground accelerations (PGA) are all 0.4 g), respectively.

Figure 6
plots the displacement time histories (DTHs) of diferent control methods under three earthquake displacement signals, along with the enlarged views.It is seen from Figures 6(a), 6(c), and 6(e), and d m greatly matches d on the whole, demonstrating the efectiveness of the three control strategies.Te results in Figures 6(b), 6(d), and 6(f )

Figure 9 :
Figure 9: DTHs for three control methods.(a) Global view; (b) enlarged view of the IM method; (c) enlarged view of the AEM-AC method; (d) enlarged view of the proposed method.

Figure 23 :
Figure 23: DTHs of comparative methods.(a) Global view; (b) enlarged view for the IM method; (c) enlarged view for the PE method; (d) enlarged view for the KF-ADC method; (e) enlarged view for the AEM-AC method; (f ) enlarged view for the proposed method.

22
Structural Control and Health Monitoring adopting the Kalman flter can identify the additive error model of the control plant quickly, demonstrating excellent parameter estimation ability.
1. Model-Based Feedforward Controller.In the proposed method, the control plant's analytical model is split into a nominal model G 0 and a corresponding additive error model G ∆ , as shown in Figure1(a).When G 0 can commendably catch the control plant's dynamic characteristics, namely, G ∆ infnitely approaches 0, an excellent feedforward controller can be derived by the inverse of the nominal model, G 0 −1

Table 2 :
[35]meter of the MR damper model[35].Simulation Results and Analysis.Te simulation results are evaluated from both qualitative and quantitative viewpoints.

Table 3 :
Q f settings for initial parameters.
Model Parameter Number.Tis section investigates the infuence of the additive error model parameter number.Te simulation settings are as follows:

Table 4 :
Inverse and nominal models with diferent orders.

Table 5 :
PI gain parameter settings for the proposed method in RTHS.
To further eliminate the residual time delay, an adaptive controller represented by a diference equation is designed by an additive error model.Finally, a feedback controller is introduced to improve its robustness.(2) Numerous simulations of RTHS are carried out to verify this method's efectiveness, utilizing a steel frame structure equipped with a damper.Te results reveal that the proposed method has a brilliant tracking ability and can efectively improve the accuracy of RTHS.It is necessary to incorporate the additive error model in the proposed method.(3) Parametric studies are carried out to investigate the critical factors of the proposed method.Results reveal that the initial parameter settings in KF and the additive error model parameter numbers can barely afect the tracking accuracy.Te high-order inverse model is unable to enhance the tracking ability of the adaptive control method efectively.Te second-or third-order inverse model is recommended to obtain stable and good control performance.(4) Virtual and actual RTHSs are conducted to validate the performance of the control method.Adaptive control strategies are more accurate than the IM and PE methods, and the proposed control method's errors are smaller than those of the AEM-AC and KF-ADC methods on the whole.Moreover, its standard deviations are the smallest, indicating superb tracking performance and strong robustness in all comparative methods.