Performance Examination of Nonlinear Signal-Based Control in Shake Table Experiments with Sliding Structures

. Tis study examines nonlinear signal-based control (NSBC) in shake table experiments with sliding structures, which have an isolation efect during an earthquake. NSBC uses a nonlinear signal obtained from the outputs of a controlled system and its linear model under the same input. Owing to the presence of the linear model, NSBC controllers are described by transfer functions, even for controlling nonlinear systems. NSBC achieved excellent control of the shake table in experiments with a specimen having nonlinear characteristics such as yielding of structural components. A sliding structure placed on a shake table signifcantly jeopardises its control because the nonlinear severity of sliding is greater than yielding, and its compensation has not yet been fully developed. Terefore, this study introduces NSBC into shake table experiments with sliding structures along with its linear model design to enhance their robustness, utilising the analysis stability to evaluate the design. Numerical simulations with a shake table with a sliding structure with a friction coefcient of 0.22 demonstrate the excellent performance of NSBC in table acceleration control. However, inversion-based control (IBC), a basic compensation approach, shows its inefectiveness. In actual shake table experiments with a sliding structure with a friction coefcient of 0.2, NSBC with a reasonable linear model achieved excellent table acceleration control with almost 100% accuracy, whereas IBC was inefective. Tis study clarifes that NSBC can solve the problem of control degradation caused by a sliding structure placed on the table.


Introduction
In the feld of earthquake engineering, a shake table plays an important role in examining the seismic performance of building structures and infrastructure [1,2]. In a shake table experiment, a specimen placed on the table is excited by reproducing a predetermined signal, which is typically derived from seismic acceleration data recorded during an earthquake in the past or from artifcially generated data for experimental purposes. Te reproduction of the excitation is a matter of control, and its accuracy must be sufciently high to precisely examine the specimen under the anticipated situations or conditions. Shake tables are commonly controlled using a proportional-integral-derivative (PID) method applied to table displacement or similar methods such as three-variable control (TVC) [3,4]. Tese controllers are designed such that the table dynamics, including the controller, obtain the expected dynamic characteristics, such as second-or higher-order dynamics. When a heavy specimen is placed on the table, the table dynamics afected by the interaction signifcantly difer from those without the specimen [5,6]. To compensate for this infuence, an inversion of the dynamics, which is referred to as inversion-based control (IBC) in this study, is commonly employed as a feedforward controller acting on reference signals [3,4,7]. Tis approach is efective only for cases in which the specimen is a linear system whose characteristics are accurately refected in the feedfoward controller.
In shake table experiments, nonlinear characteristics within an actuation system and the structure to be examined degrade its control. Shake tables driven by hydraulic actuators exhibit nonlinearity derived from the hydraulic fuid, servo-valves, and friction of cylinders within the actuation system [8][9][10][11][12][13]. Shake table experiments are mainly conducted to observe seismic damage or failures in structures or nonlinear characteristics in earthquake protection devices [2]. Te appearance of nonlinear characteristics during shaking is essential. When this occurs in a heavy specimen placed on a table, the nonlinear characteristics degrade the control, leading to instability in the worst case. An iterative approach based on IBC [3,14] is occasionally employed to compensate for such nonlinear characteristics, but it is mainly limited to fxed characteristics without uncertainty. An efective approach based on feedback actions is required to accurately control shake tables, regardless of the type of nonlinear characteristics. It will be benefcial to its advanced technique, substructuring shake table experimentation, which is a prime subject actively studied in earthquake engineering [15][16][17][18][19][20][21].
To the control issue of shake table experiments, minimal control synthesis (MCS) [22,23], which is a type of model reference adaptive control [24], was intensively studied and applied in various shake tables, mainly focussing on nonlinear characteristics within the actuation system [25][26][27]. Te success of MCS has led to the class of adaptive control methods being a major tool for shake table experiments involving nonlinear characteristics [28][29][30][31][32]. However, adaptive control approaches still have challenging issues, particularly for systems with large unmodelled dynamics or rapid parameter variations. Tese issues have been addressed via enhancements with additional techniques (e.g., real-time parameter estimation [33] or a backstepping technique [34][35][36][37]).
A class of improvements in IBC is another research topic in shake table experiments with nonlinearity. Model-based control (MBC), which employs an error feedback action into IBC, has been actively investigated for shake table control [13,[38][39][40]. MBC with a linear quadratic Gaussian controller as its error feedback action has been applied to a shake table sustaining a specimen equipped with wooden braces that display nonlinear characteristics over its elastic range [39,41].
Nonlinear signal-based control (NSBC) [42][43][44][45], which has been specifcally established for nonlinear system control, is defned as the addition of a nonlinear signal feedback action into IBC. NSBC uses a nonlinear signal obtained from the outputs of a controlled system and its linear model under the same input. As NSBC relies on a linear model rather than a controlled system, its controllers are described by transfer functions, even for nonlinear system control. NSBC is advantageous because it requires only the output of a controlled part in the controlled system for its feedback action and does not require the outputs of other parts (e.g., structural responses in shake table experiments) for the action. Te efectiveness of NSBC has been demonstrated by dynamic substructuring experiments [46,47] and shake table experiments [44,48,49]. Te frst application of NSBC to shake table experiments [44] was performed using a single-axis shake table with a steel structure, whose weight was equal to the table weight. In the experiments, NSBC successfully realised an expected earthquake excitation with almost 100% accuracy, even when the structure on the table displayed severe nonlinear characteristics owing to the yielding of its structural components.
Regarding the severity of nonlinearity, sliding is one of the most severe phenomena observed in earthquake engineering. In addition, the occurrence of sliding has a positive impact on the seismic damage mitigation of structures [50][51][52], and modern sliding isolation systems [53][54][55] that intentionally utilise this aspect have become prominent earthquake protection devices in this feld [56]. Te excellence of sophisticated systems has led to the development of similar sliding systems based on primitive materials that are expected to exhibit an isolation efect during large earthquakes. Tese developments have been actively investigated along with studies on the friction of various materials, such as scraped tyres [57][58][59], sand [60], and common construction materials [61][62][63][64], which are typically tested in shake table experiments.
Te sliding of a specimen on a shake table afects the table control, and its infuence increases as the weight of the specimen increases [65]. When a heavy structure with a certain friction coefcient slides on a shake table, the sliding structure is shaken by an excitation that is completely diferent from the required one. A countermeasure to this control problem has not yet been fully developed, mainly because of the severities of the nonlinear characteristics caused by the friction and the interaction between the table and the sliding structure.
As NSBC can be a solution to this problem, this study numerically and experimentally examines the performance of NSBC in shake table experiments with a sliding structure with a friction coefcient of approximately 0.2. Tis study is an advancement of the preceding studies [44,65], and its novelty is summarised as follows.
(i) Tis study introduces a new linear model design where parameters in a linear model of a controlled system including a sliding structure are intentionally tuned to enhance the robustness of NSBC. In addition, this study introduces a method to approximately analyse the stability of NSBC for a shake table with a sliding structure. Te NSBC stability analysis based on the Nyquist stability criterion can be used to evaluate stability margins obtained from diferent linear models.
(ii) Tis study numerically and experimentally demonstrates the efectiveness of NSBC in shake table experiments with a sliding structure, which displays frictional behaviour whose nonlinear severity is greater than yielding of structural components and the friction behaviour of a sliding mass. Tis study clarifes that NSBC can solve the problem of control degradation caused by the sliding of the structure on the table.
Te remainder of this paper is organised as follows. Section 2 studies the infuence of a sliding structure on shake table control and describes the application of NSBC in shake table experiments with a sliding structure with N-storeys. 2 Structural Control and Health Monitoring Section 3 numerically examines the control performance of NSBC in shake table experiments with a single-degree-offreedom (SDOF) structure placed on a sliding mass, along  with its comparison with IBC. Section 4 introduces the  control performance of NSBC and IBC in actual shake table  experiments with a sliding structure consisting of a singlestorey steel structure and a sliding mass placed on a graphitelubricated sliding interface. Finally, Section 5 presents the conclusions of this study.  Figure 1(c), to investigate their interaction and infuence on shake table control.

Sliding Structures.
A mass placed on the ground, as shown in Figure 1(a), slides when an applied force in its sliding interface exceeds the maximum static friction of the sliding mass. When the force is caused by ground movement, which occurs during an earthquake, the slip and stick states of the mass are described by the following equation: where t is the time variable; m 1 is the sliding mass; y 1 , y g is the set of mass displacement and ground displacement in the absolute coordinate, respectively; x 1 (� y 1 − y g ) is the sliding displacement of the mass; sgn(a) � 1 (a > 0), { 0(a � 0), −1(a < 0)}; and F μv is the kinematic friction. When the kinematic friction F μv is equal to the maximum static friction F μs , it becomes F μv � F μs � μ max m 1 g, where μ max is the maximum static friction, and g is the gravitational acceleration.
Equation (1) rigorously distinguishes the states of stick and slip by _ x 1 (t) � 0 and _ x 1 (t) ≠ 0. Tis distinction is not practical for numerical simulations because of the strict requirement of zero velocity, which cannot be realised in computation. Terefore, this study employs the Karnopp model [66][67][68], which distinguishes between the two states by the limit velocity on the stick state, ε v . Subsequently, the friction for the stick and slip states in equation (1) can be uniformly expressed as follows: where χ(a) � 1 (a ≥ 0), 0 (a < 0) { }; and F 1 is the shear force applied to the sliding interface.
When a superstructure consisting of N-storeys is placed on the sliding mass, as shown in Figure 1(b), the equations of motion for both stick and slip states can be given as follows: , and f cki (� f ci + f ki ) is the resultant force of damping and restoring forces on the i th storey (i � 2, ..., N + 1). Note that F μv (t) � F μs � μ max ( N+1 i�1 m i )g when the kinematic friction F μv is equal to the maximum static friction F μs . In addition, when the superstructure is a linear system with timeinvariant damping and stifness coefcients on the i th storey, the resultant force becomes f cki (t)(� f ci (t) + f ki (t)) � c i ( _ y i (t) − _ y i−1 (t)) + k i (y i (t) − y i−1 (t)). According to equations (2) and (4), the equation of motion for the sliding interface with and without the superstructure can be uniformly described by the following equation: where s is the Laplace variable; y 0 is the table displacement; u is the control input signal, which has the dimension of displacement; and m 0b , ω 0b , ζ 0b , c 0b , k 0b is the set of table mass, circular frequency, its corresponding damping ratio, and equivalent damping and stifness coefcients, respectively, when no specimen is placed on the shake table.
According to equation (6), the bare shake table in the time domain can be described by the following expression: when a sliding mass sustaining an N-DOF superstructure is placed on the table as a specimen, as shown in Figure 1(c), its equations of motion are given by the following equations: where F ex (t) � m 1 € y 0 (t) + N+1 i�2 m i € y i (t); and m 0 , ω 0 , ζ 0 , c 0 , k 0 } is the set of the table mass, circular frequency, its corresponding damping ratio, and equivalent damping and stifness coefcients, respectively, when the specimen is placed on the table. Note that m 0 contains the bare table mass m 0b and the mass of some experimental rigs m 0α .
According to equation (8), the shear force F 1 , which is the friction force in the sliding interface, clearly disturbs the shake table. When the sliding structure has an interface with a near-zero friction coefcient, the friction force is negligible, and the sliding does not disturb the table control. However, when the force is not negligible owing to a high Sliding interface  friction coefcient, it disturbs the shake table control, resulting in control degradation. As F 1 in equation (8) is proportional to the mass of the sliding structure, the shake table control is considerably disturbed by the sliding of a heavy structure with a high friction coefcient. To properly excite a specimen on the table, the shake table must be accurately controlled, regardless of the properties of the specimen to be tested. Table Supporting a Sliding Structure. Tis study employed NSBC for shake table experiments with sliding structures and examined its control performance. Here, the principles of NSBC are briefy described and its controller design for table acceleration control is detailed. Table  Experiments. NSBC has the nonlinear signal feedback action as shown in Figure 2, as well as the feedforward controller derived from IBC, as shown in Figure 3. Te nonlinear signal enables us to measure the unmodelled dynamics within the controlled system by comparing it with its linear model as follows:

NSBC for Acceleration Control at Shake
where ∆G ∆τ (s) � (1 + (∆G(s)/G(s)))e − ∆τs − 1; σ is the nonlinear signal; y is the controlled output signal of the controlled system G; y is the output of the linear model G; ∆G(� G − G) denotes the unmodelled dynamics, which are typically derived from some nonlinear characteristics within G; and ∆τ(� τ − τ) is the diference between the pure time delay in the controlled system τ and its estimate τ. According to equation (9), the nonlinear signal at ∆τ � 0 becomes σ(s) � ∆G(s)e − τs u(s). Tis indicates that, when the estimate of the pure time delay is accurate, the unmodelled dynamics, which cannot be directly measured, can be indirectly measured in the form of the nonlinear signal. Note that the description of the controlled system in NSBC ( Figure 2) varies depending on the controlled output signal y: at displacement [acceleration] control, the controlled system is described by Using the nonlinear signal, the error signal can be obtained as follows: where e is the error signal; and r is the reference signal. As u in equation (10) is associated with the signals r, e, σ { }, the control input signal in NSBC is described as follows: where K r , K e , K σ is the set of controllers acting on signals r, e, σ { }, respectively. By substituting equation (11) into equation (10), the error signal can be rewritten as follows: Te NSBC controllers in equation (11) are designed to realise a zero error in equation (12) with τ � τ � 0; this condition indicates cancelling the infuence of the reference and nonlinear signals on the error signal. Ten, the general forms of NSBC controllers are described by the following equation:

Controlled system
Linear model Structural Control and Health Monitoring where F r , F e , F σ is the set of flters associated with the controllers K r , K e , K σ , respectively. In equation (13), K r is the feedforward controller based on the linear model transfer function G(s), and its sole use corresponds to IBC in Figure 3. K σ is the nonlinear signal feedback controller used to cope with the unmodelled dynamics (e.g., nonlinear characteristics) within the controlled system. K e is the error feedback controller optionally employed to forcibly minimise the error signal, which remains even after the activation of K r and K σ .
When NSBC was applied to shake table experiments with a steel structure [44], the controllers with K e (s) � 0 accurately realised an earthquake excitation, even though the structure displayed signifcantly severe nonlinear characteristics. In the experiments, the following control input signal was found to be suitable for the acceleration control of a shake table: where € σ(s) � € y 0 (s) − € y 0 (s); € r is the reference acceleration signal to be realised on the table; € y 0 is the shake table acceleration signal measured by a sensor, and € y 0 is the acceleration signal computationally calculated using G 0a and u. In equation (14), the signal u σ , which relies on the feedback signal € σ, drives to cancel the infuence of the nonlinear signal on the error signal e, resulting in a high control accuracy. When the linear model transfer function for the table acceleration and control input signal, G 0a (s), is given, it can be employed as G(s) � G 0a (s) in Figure 2. When the inversion of G 0a (s) is a stable and proper transfer function, the controllers in equation (14) become Te flter F σ basically limits the frequency components from being fed back, and its design afects the stability. A previous study [44] on the stability of NSBC demonstrated that the second-order bandpass Butterworth flter passing the range of 0.2-20.0 Hz could efectively maintain stability in shake table experiments with nonlinear characteristics. Tus, this study also employed the same bandpass flter as F σ . Tis bandpass range sufciently covers the frequency range of typical seismic acceleration records.

Stability of NSBC.
Te stability of NSBC depends on the feedback signals, e and σ, which are uniformly expressed by the following equation: where which is the closed-loop characteristic equation governing the stability.
According to equation (16), the error signal e has the following relation with the reference signal r: Equation (18) can be assessed for various controller designs, once the controlled system with nonlinear characteristics is equivalently or approximately described by a Laplace form.
Stability of NSBC with the controller designs can be assessed on the basis of equation (17). When the error feedback action is not activated, i.e., K e (s) � 0, which is the case in this study, it becomes

Structural Control and Health Monitoring
According to equation (19), the linear model G and flter F σ (s) can be used for maintaining stability and increasing stability margins, as they are factors that are fexibly designed in the equation. Table  Experiments with Sliding Structures. A linear model of the controlled system plays a fundamental role in NSBC. Tis study introduces a linear model design based on the stick state of the sliding structure because the structure remains within the state in a system identifcation test, which are commonly performed before the main experiments using earthquake excitations. Tis design is exemplifed for a shake table sustaining a sliding structure shown in Figure 1(c).

Linear Model and Controller Designs for Shake
Based on the stick state of the sliding structure, a linear model for the shake table and sliding structure can be described as follows: denotes the displacement, mass, damping coefcient, and stifness in the i th storey of the linear model, respectively; and y 0 , m 0 , c 0 , k 0 is the set denoting the displacement, mass, damping coefcient, and stifness associated with the shake table part in the linear model, respectively.
Tis linear model can be described by the set of transfer functions for the corresponding stories, which is described as follows: In equation (21), the transfer function G i/i−1 (s) contains G i+1/i (s), which also contains G i+2/i+1 (s), and this relation continues until i � N. Tus, the transfer function G 0/u (s) is obtained by calculating the transfer functions in the order of G N+1/N (s), G N/N−1 (s), · · · , G 1/0 (s), G 0/u (s). As G 0/u (s) denotes the relation between the table displacement and the control input signal, it can be expressed by G 0d (s) � G 0/u (s) and its relation with the table acceleration becomes G 0a (s) � s 2 G 0d (s). By employing G 0a (s) as G(s) in Figure 2, equation (14) can be used to determine the input signal for table acceleration control using the controller in equation (15).
Te design of the linear model is fairly fexible, allowing various settings for the linear model parameters: m i , c i , k i , (i � 1 · · · N + 1) and m 0 , c 0 , k 0 in this case. In fact, NSBC achieved perfect tracking control of multi-storey Structural Control and Health Monitoring nonlinear systems even by a linear model whose stifness coefcients are intentionally set to zero, despite the presence of stifness elements in the actual controlled system [45]. In addition, a linear model with a larger damping than the actual control system enhances the stability margin for nonlinear system control. Based on these features, a linear model for NSBC can be fexibly designed to enhance its performance.

Numerical Conditions.
A series of shake table experiments in the numerical simulation was performed using the setup in Figure 4(a), with the setting of a constant damping coefcient and a trilinear hysteretic spring shown in Figure 4(b) within the SDOF superstructure. An excitation employed for the simulation is the acceleration data shown in Figure 5, which was recorded by the Japan Meteorological Agency (JMA) during the Hyogo-Ken Nanbu (Kobe) earthquake in 1995. Tese data are used as the reference signal to be reproduced in the shake table experiments and referred to as the JMA Kobe motion in this study. Table and Sliding Structure. Te equations of motion for the table an sliding structure in Figure 4(a) are as follows:
As kinematic friction is generally afected by the sliding velocity [53,61,64,69], this study refects its characteristics in numerical simulations. Te kinematic friction in equation (22) can be expressed as follows: where μ min is the minimum friction coefcient during sliding and c is the intensity of the friction reduction. Note that equation (24) with c � 0.0 maintains the maximum static friction during the sliding state. Te detailed parameters employed in the simulation were based on a system identifcation test performed on an actual shake table sustaining a sliding steel structure, which is shown later in Section 4.1. Ten, the parameters of the shake table were set as follows: m 0 (� m 0b + m 0α � 200 +115 kg) � 315 kg, c 0 � 4.78 kNs/m, and k 0 � 71.06 kN/m. Te sliding structure was designed to have a sliding mass of m 1 � 225 kg and an SDOF superstructure with the following parameters: m 2 � 200 kg, c 2 � 120 Ns/m, and k 2 � 43.00 kN/m. Te trilinear hysteretic spring in equation (23) was modelled with Δ 21 � 0.03 m, Δ 22 � 1.5 kNs/mΔ 21 , r 21 � 0.5, and r 22 � 0.05. Te kinematic friction in equation (24) was modelled as μ max � 0.22, μ min � 0.15, and c � 20.0, according to a previous study on graphite lubrication of steel and mortar [64]. Te velocity limit for the Karnopp model used in equation (22) was set to ε v � 0.001, which is commonly used in this model [68]. Te pure time delay was set as τ � 4.0 ms, according to the identifcation test.
Prior to studying the interaction between a shake table and sliding structure, the structure with the aforementioned parameters was numerically placed on the ground and simulated with 80% JMA Kobe motion. Te response of the structure is shown in Figure 6, including the friction varying with the sliding velocity. By comparing with the structural response of the sliding mass rigidly fxed on the ground, as shown in Figure 6(c), it can be easily observed that the occurrence of sliding efectively mitigates the structural damage and contributes to maintaining the structure almost within the elastic range. Te numerical simulations for shake table experiments with a sliding structure are expected to produce identical or sufciently similar results.

Control Accuracy.
Tis study evaluated the control accuracy of a shake table experiment, based on the similarity of the realised table acceleration and its reference acceleration signal. Te similarities in the time and frequency domains were calculated by using the following eqaution: 8 Structural Control and Health Monitoring where {A r , A y0 } is the set of Fourier amplitude spectra of € r, € y 0 , respectively. S t directly measures the similarity of the two signals from the time history data, while S f is focussed on the amplitudes of the Fourier spectra in a limited frequency range. S f in this study is evaluated in the range of 0.1-20.0 Hz, which covers the frequency components in most earthquake excitations.

Numerical
Simulation. Te linear model required for NSBC was built based on the stick state of a sliding structure, and its equations of motion can be described by equation (20) with N � 1. Based on equation (21) with N � 1, the transfer functions for the linear model can be obtained as follows: By employing the linear model as G(s) � G 0a (s) in Figure 2, the NSBC input signal for Figure 4 is determined using equations (14) and (15). Te flter F σ was designed as a second-order Butterworth flter with a bandpass range of 0.2-20 Hz.
Te parameters in the linear model were designed using the initial parameters of the controlled system, m 0 , c 0 , k 0 , m 1 , m 2 , c 2 , k 2 � m 0 , c 0 , k 0 , m 1 , m 2 , c 2 , k 2 and τ � τ. Tis design is equivalent to the case where all initial parameters of the controlled system (i.e., the shake table and sliding structure) are known and refected in the NSBC controllers, and it is referred to as Linear Model 1 in this study. In the simulation, the unmodelled dynamics is the nonlinearity within the superstructure only, and the shake table does not have any nonlinear characteristics, although actual systems in practice do. Noise was added intentionally to the table acceleration and displacement to make the simulation more realistic.

Numerical Results of IBC.
Prior to examining NSBC, shake table experiments were computationally simulated with IBC, which was executed in the form of NSBC with K e (s) � K σ (s) � 0. Te simulations were performed for JMA Kobe motion with diferent amplitudes: {10%, 20%, 40%, 60%, 80%, and 100%}. Te results are summarised in Table 1. Te detailed results for 20% and 80% amplitudes are illustrated in Figures 7 and 8, respectively. Tese two amplitudes were selected to maintain the consistency with the experimental results shown later.
According to Table 1, the control accuracy for the 10% excitation is satisfactory, especially in S f . Tis is because a nonlinear characteristic does not appear under the excitation owing to the small amplitude, which is not sufciently large to induce sliding in the structure. As shown in Figure 7, the structure under 20% excitation clearly slid. Te sliding has degraded the table control of IBC, resulting in a signifcant reduction from the results of the 10% excitation, as shown in Table 1.
Te control accuracies decreased signifcantly at larger excitations, as shown in Table 1. At 80% excitation, the realised table acceleration signifcantly difered from the reference signal in time and frequency domains, as shown in Figure 8, resulting in extremely low accuracies: S f � 21.04% and S t � 16.95%. Tis result indicates that the expected experiment shown in Figure 6 is impossible using IBC. Te results of 100% excitation were even worse than those of the 80% excitation. Tese results clarify the inefectiveness of IBC for shake table experiments with a sliding structure.

Numerical Results of NSBC. Shake table experiments
based on NSBC with Linear Model 1 were computationally simulated for the JMA Kobe motion with the same amplitudes as those conducted with IBC. Te results are summarised in Table 2, and the detailed results of 20% and 80% excitations are illustrated in Figures 9 and 10, respectively.
As shown in Figure 9, NSBC has accurately realised 20% JMA Kobe motion, even though the structure has clearly slid on the table. Tis resulted in satisfactory accuracies: S f � 99.13% and S t � 95.73%, which are considerably better than those of IBC, which are shown in Table 1. As shown in Table 2, at even larger excitations, NSBC produced similar or better results than those at 20% excitation. At 80% excitation, the table acceleration realised by NSBC was satisfactorily similar to the reference signal in the time and frequency domains, as observed in Figure 10. In addition, the experiment realised  by NSBC was sufciently close to the expected experiment shown in Figure 6. Even at 100% excitation, NSBC did not exhibit control degradation, as observed in Table 2, confrming its efectiveness in shake table experiments with a sliding structure. Note that S t in Table 2 is improved at larger excitations because noise in the table acceleration becomes less infuential to the index as the amplitude increases.

Stability Margin and Its Improvement by Linear Model
Designs. NSBC with Linear Model 1 can efectively control a shake table sustaining a sliding structure, according to Section 3.2.2. However, stability is not guaranteed during the actual practice because the numerical simulations were performed without uncertainty within shake tables, which are typically observed during actual practices. To assess the stability and improve its margin, we introduce a stability analysis of NSBC for shake table control with a sliding structure and a linear model design in which linear model parameters are intentionally tuned to increase the stability margin. As a new design, Linear Model 1.5 is built by using G 0a (s) in equation (26) with the parameters, m 0 , c 0 , k 0 , Te stability analysis conducted in this study is based on table dynamics during two states (stick and slip) of the structure on the table. As the linear model is built based on the dynamics during the stick state, as shown in equation (26), the controlled system needs to be represented by the dynamics during the slip state. However, as sliding is a nonlinear phenomenon, it cannot be precisely described by a Laplace transform. Tus, by disregarding the infuence of the sliding structure and friction, this study describes the controlled system as follows:  Te modelling error dynamics ΔG, which is necessary for the stability analysis based on equation (19), can be approximated by the set of G(s) � G 0a (s) in equation (27) and G(s) � G 0a (s) in equation (26).
Stability of NSBC based on Linear Models 1 and 1.5 was analysed, and the results are illustrated in Figure 11 along with the relevant error. According to Figure 11(a), the stability margin of Linear Model 1.5 is greater than that of Linear Model 1, indicating the efectiveness of the design. In Figure 11(b), the error dynamics of Linear Model 1.5 are not signifcantly diferent from those of Linear Model 1.
Simulations with the numerical conditions identical to those described in Section 3.2 were performed with NSBC based on Linear Model 1.5, and the results are summarised in Table 3.

Structural Control and Health Monitoring 13
Linear Model 1.5 also achieved the satisfactory control accuracies, as shown in in Table 3, although the results are slightly lower than those of Linear Model 1, as shown in Table 2. Tis result indicates that Linear Model 1.5 attains a higher stability margin than Linear Model 1, achieving similar high control performance.       Figure 12(a), which has a size of 1.2 × 1.2 m and a weight of 200 kg. A set of mortar bases, shown in Figure 12(b), was placed on the table, and a sliding steel structure was placed on the mortar bases, as shown in Figure 12(c). Tis steel structure consisted of two masses connected by steel plates. Te frst mass corresponds to the sliding mass in Figure 4(a), and the second mass and plates correspond to the superstructure in   . Te steel plates demonstrated the fexibility of a building structure and functioned as an element that displayed nonlinear characteristics, similar to those shown in Figure 4(b). Te set of blocks shown in Figure 12(d) had an iron at its centre, which was placed on each corner of the sliding mass. Te fat area (20 × 20 mm) on the iron block, as shown in Figure 12(d), was attached to the bases, forming the sliding interface of the sliding structure. Tis interface was lubricated with graphite to achieve a friction coefcient of approximately 0.2. With this lubrication, 0.06 kg of graphite was scattered on the mortar base for each iron block, as shown in Figure 12(e). During the preparation of this experiment, the frst and second masses were measured as m 1 � 225 kg and m 2 � 200 kg, respectively, and the experimental rigs directly placed on the table were m 0α � 115 kg, resulting in m 0 � m 0b + m 0α � 315 kg. Based on the total weight of the sliding structure, m 1 + m 2 � 425 kg, and the contact pressure at the interface became 2.6 N/mm 2 .
Te shake table had a magnetostrictive displacement transducer and two servo accelerometers for control. Te sliding mass was equipped with four wire displacement transducers placed at its corners to measure its sliding displacement and two strain gauge accelerometers attached to the mass for its acceleration. Te superstructure was equipped with two wire displacement transducers placed on protection stands to measure its interstorey drift and two strain gauge accelerometers on the second mass.
System identifcation tests were performed with a bandlimited random excitation containing frequency components of 0.1-50 Hz, with two diferent amplitudes of 10.0 and 2.0 mm. Te dynamics of the table displacement and control input signal obtained for the two amplitude cases are illustrated in Figure 13.
Te dynamics in Figure 13(a), which were obtained with an amplitude of 10.0 mm, was modelled by the following expression: Based on m 0 , m 1 , m 2 � 315,225,200 { } kg and equation (26), the equivalent table parameters were found to be c 0 � 4.78 kNs/m and k 0 � 71.06 kN/m, and the superstructure was found to have the following parameters: c 2 � 120 Ns/m and k 2 � 43.00 kN/m.
In the system identifcation test with an amplitude of 2.0 mm, as shown in Figure 13(b), the obtained dynamics slightly difered from those obtained with an amplitude of 10.0 mm, indicating that the shake table had some nonlinear characteristics depending on its amplitude.

Experimental Results of IBC with Linear Model 1.
Shake table experiments with IBC were performed for two amplitudes of 10% and 20%. Te results are summarised in Table 4. Te results for the 20% excitation are illustrated in Figure 14. Te sliding interface had a friction coefcient of approximately 0.2, as observed in Figure 14(c).
At 10% excitation, the structure did not slide, and nonlinear characteristics that occurred in this experiment were limited. Te control accuracies in Table 4 are S f � 94.10% and S t � 81.82%, which were much lower than those of the numerical simulation in Section 3.2.1. Tis is attributed to the nonlinear characteristics of the shake table.
At 20% excitation, the structure clearly slid, as shown in Figure 14, and caused control degradation, resulting in S f � 79.43% and S t � 61.65%, which are again worse than those of the numerical simulations. It should be noted that experiments with larger excitations were not performed because their poor results were deduced from the numerical simulation and the experimental results of 10% and 20% excitations.

Experimental Results of NSBC with the Linear Model 1.
Shake table experiments with NSBC were scheduled to be performed using JMA Kobe motion with the two amplitudes of 10% and 20%. However, NSBC with Linear Model 1 became unstable in the experiment with 10% excitation, as shown in Figure 15. Tis can be attributed to the nonlinear characteristics of the  (15) with G(s) � G 0a (s) for the control input signal in equation (14). Te bandpass flter F σ (s) was the same as the one one used in Linear Model 1.
Shake table experiments based on NSBC using Linear Model 1.5 were conducted with JMA Kobe motion with the amplitudes: {10%, 20%, 40%, 60%, and 80%}. Te experimental results are summarised in Table 5, and the detailed results of the 20% and 80% excitations are illustrated in Figures 16 and 17, respectively. Note that the experiment with 100% excitation was not executed because the 80% excitation resulted in a large sliding displacement, which was near the limit of the allowable range of sliding.
NSBC based on Linear Model 1.5 maintained stability in all experiments with diferent amplitudes, unlike NSBC based on Linear Model 1, which became unstable at 10% excitation. Tis simply demonstrates the superiority of Linear Model 1.5 over Linear Model 1.
At 10% excitation, NSBC based on Linear Model 1.5 achieved the expected control with sufcient accuracies of S f � 99.45% and S t � 94.14%. During this experiment, the structure did not slide on the table owing to the small excitation. Although the 20% excitation clearly caused the structure to slide on the table, as shown in Figure 16, its table acceleration was sufciently close to the reference signal in both time and frequency domains. Te control accuracies were S f � 98.64% and S t � 91.96, which were remarkably better than IBC results for 20% excitation as shown in Table 4.
In the experiments with larger excitations, the control accuracies were similar to or better than those with smaller excitations, as shown in Table 5. At 80% excitation, the table acceleration was accurately controlled, as shown in Figures 17(a) and 17(b), with high accuracies of S f � 99.17% and S t � 95.22%, which are the best results among all the experiments conducted in this study. S t at this excitation was the highest in Table 5, because the infuence of the noise contaminating the table acceleration decreased as the excitation amplitude increased.
In addition to the accurate control of the shake table shown in Figures 17(a) and 17(b), the responses of the sliding structure in Figure 17(c)-17(e) reasonably match the numerical results of the sliding structure in Figure 6. Tis result also supports the reliability of the experiments conducted with NSBC based on Linear Model 1.5.

Conclusions
Tis study examined the performance of NSBC in shake table experiments with a sliding structure based on a graphite-lubricated interface. Its high performance was verifed through numerical simulations and actual experiments. Te results obtained in this study are summarised as follows: (i) As building a linear model of a controlled system is essential for NSBC, this study introduced its design for a shake table with a sliding structure. Te linear model is built based on the controlled system during the stick state of a sliding structure rather than the slip state. Te designs of the linear model and NSBC controllers were exemplifed by a shake table and a sliding structure, which consists of sliding mass and an N-DOF superstructure. In addition, this study introduced an approximate stability analysis of NSBC for such shake table experiments, in which the controlled system is represented by table dynamics during the slip state of the sliding structure. (ii) To enhance the robustness of NSBC against uncertainty within a shake table, this study additionally introduced a linear model design in which the model parameters corresponding to damping and stifness of the shake table are intentionally assumed to be larger than the identifed values. Its efectiveness was demonstrated by the stability analysis for two diferent models: Linear Model 1, built using the exact initial parameters of the controlled system, and Linear Model 1.5, built using 1.5 times the damping and stifness values corresponding to its shake table part. (iii) Numerical simulations were performed for shake table experiments with a sliding structure consisting of an SDOF superstructure and a sliding mass, and a sliding interface with a friction coefcient of 0.22. In the simulations, IBC, which is a basic compensation method for shake table control, produced satisfactory results only when the sliding structure remained in the stick state, and its control accuracy signifcantly deteriorated as the sliding amplitude increased. Contrarily, NSBC accurately realised an expected earthquake excitation with high accuracy: almost 100%, even when the structure on the table slid substantially. Linear Models 1 and 1.5 did not exhibit signifcant diferences in terms of control accuracies. (iv) In the experiments, a sliding structure consisting of a one-storey steel structure and a sliding mass was placed on a sliding interface lubricated with graphite, realising a friction coefcient of approximately 0.2. First, NSBC controllers were built using Linear Model 1 with the exact parameters of the controlled system obtained from a system identifcation test. However, a shake Te stability analysis of NSBC for shake table experiments involving the sliding phenomenon must be further elaborated for more accurate assessments. Furthermore, systematic designs of the linear model and NSBC controllers should be constructed. Tis study focused on examining the control performance of NSBC for shake table experiments with a single-storey sliding structure, based on the sliding interface having a friction coefcient of approximately 0.2. Its performance for diferent friction coefcients can be found in the study with a rigid sliding mass [65]. We will further investigate the control performance of NSBC in shake table experiments with sliding structures with multistorey superstructures. As applications of NSBC are currently limited to single-axis shake tables, we will investigate the applicability of NSBC to multi-axial shake tables.

Data Availability
Te data used to support the fndsing of this study are available from the corresponding author on reasonable request.