The reference structure used in this study is shown in Figure 1. It consists of a steel moment-resisting frame with 3 bays and 3 stories. The beams and columns are made from A36 and A992Fy50 steel, respectively. The beams are built-up sections while the columns are hot-rolled commercially available sections. A scaled El Centro historic record is used as the input ground motion to the system to generate the different responses. A scaling factor of 0.40 is selected to ensure that the lateral displacement of the frame is limited to maintain linear elastic behavior of all structural components.

To evaluate the performance of the RTHS algorithms, this benchmark uses a finite element (FE) model to capture the behavior and response of the reference structure. Figure 2 presents the schematic definition of the geometry and connectivity of the reference system. Each node has three DOFs: two translational DOF along the global x and y-axes; and one rotational DOF, θ around the z-axis, perpendicular to the xy-plane. The numbers in circles near the joints represent the node numeration and the numbers in rectangles close to the middle of beams and columns denote the element numeration. In Figure 2, each set of DOFs for any i-th node is represented by the triplet x i   y i   θ i .

The equation of motion for this reference system is given by: M ψ ¨ + C ψ ˙ + K ψ = − M Γ ⋅ x ¨ g , (1) where M, C, and K are the mass, damping and stiffness matrices of the reference structure. Γ is an influence vector that describes the inertial effects of the excitation on the masses on the system. In our case this is a column vector that contains a value of one for each mass that develops an inertial effect due to ground acceleration. The one-dimensional variable x ¨ g is the ground absolute acceleration and the vectors ψ ¨ , ψ ˙ , and ψ represent the absolute acceleration, relative velocity and relative displacement with respect to the ground for each degree of freedom considered in our analysis, respectively. Therefore, the number of DOFs in this FE model is 38 and they will be arranged in the displacement vector ψ according to Eq. (2). Any i-th element of the vector ψ will be represented by ψ i , where i = 1 . . . 38 , see Figure 3. ψ = x 1 … x 12 y 1 … y 12 θ 1 … θ 12 θ 14 θ 15 T . (2)

For instance, the DOFs of nodes 4 and 7 are represented by the vectors ψ 4 4,16,28 = x 4   y 4   θ 4 T and ψ 7 7,19,31 = x 7   y 7   θ 7 T , where the elements in brackets specify particular elements in an array and the subscript designates specific attributes of the array. In these examples, they refer to specific DOFs (element in brackets) of the vector ψ and their corresponding nodes (subscripts). The first 5 natural frequencies are 2.29 Hz, 12.74 Hz, 26.28 Hz, 26.53 Hz, and 29.91 Hz, with mass participation factors of up to 95%. The damping matrix is calculated here using the Rayleigh damping method with a damping ratio of 5% anchored to first and third modes. The damping matrix is assumed to be proportional to the sum of the mass and stiffness matrices of the reference system: C = α 1 M + α 2 K , (3) where α 1 and α 2 are constants used to generate the desired modal damping ratios for two chosen modes.

To conduct the RTHS, the reference model described in Section 2 must be partitioned into two subdomains: numerical and experimental. The partition chosen is shown in Figure 4. The portion in black (outermost structural elements) represents the numerical substructure, and the portion in red (central frame, simply supported) indicates the physical substructure, which is a steel moment resisting frame available in the IISL and it is assumed to be the less understood part of the entire structure.

In a partitioned system, these two substructures are separate, but connected to each other and synchronized through a feedback loop so that they share information at common interface nodes at every time step during execution. Figure 5 illustrates the main degrees of freedom of the interface nodes and the signals that transfer information between the numerical and physical substructures. Ideally, during every time interval, the numerical substructure is excited first, then the response of the interface nodes ψ ns 4 4,16,28 = x ns 4   y ns 4   θ ns 4 T and ψ ns 7 7,19,31 = x ns 7   y ns 7   θ ns 7 T , and their derivatives (velocity and acceleration) are imposed to the experimental substructure. Afterwards, the generated physical restoring forces, f es 4 = F es 4 x   F es 4 y   M es 4 T and f es 7 = F es 7 x   F es 7 y   M es 7 T , are measured and fed back to the numerical substructure. This configuration is usually denoted as ideal RTHS since no additional components such as hydraulic actuators or measured signals contaminated with high frequency noise are considered. Therefore, pure numerical models for the numerical and experimental substructures are used. Herein, the subscript ‘ns’ will be used for variables associated with the numerical substructure, and ‘es’ will be reserved for the experimental substructure variables. This partitioned analysis provides the realization of the most basic hybrid simulation which is helpful when designing and RTHS experiment. It allows one to study the stability of the partitioning chosen, assess the required forces to enforce the boundary conditions in the physical substructure (hence, to evaluate the actuator capacity to be used), define the signals from the physical domain and their structure necessary to close the loop with the specific type of numerical model, etc. For instance, this analysis facilitates the identification of the interface conditions to be enforced in both substructures regardless of whether an ideal RTHS is being executed or a physically actuated RTHS is being conducted. In this benchmark, these key interface conditions are represented by the vectors ψ ns 4 = x ns 4   y ns 4   θ ns 4 T , ψ ns 7 = x ns 7   y ns 7   θ ns 7 T , ψ es 4 = x es 4   y es 4   θ es 4 T , and ψ es 7 = x es 7   y es 7   θ es 7 T in Figure 5.

In an actual RTHS implementation, the experimental substructure is physically built and connected to the numerical substructure through a transfer system to ensure dynamic continuity and synchronized motion at common interface nodes during the experiment execution in real time. The inclusion of the transfer system in the experimental domain changes the dynamics of the experimental substructure, and a control approach must be included to compensate for these added dynamics and other effects. This benchmark includes and simulates these components and other relevant effects with identified models based on experimental data. This realistic computational implementation or virtual simulation is called vRTHS, which is the closest realization of an actual RTHS.

For RTHS execution, the reference model described in Section 2.1 must be partitioned into numerical and experimental substructures as described in Section 3.1. Assuming linear elastic behavior of the frame, the partitioned mass, damping and stiffness matrices can be written as the sum of numerical and experimental components: M ns + M es ψ ¨ + C ns + C es ψ ˙ + K ns + K es ψ = − M Γ ⋅ x ¨ g (4) where M = M ns + M es C = C ns + C es K = K ns + K es . (5)

Herein, the subscript ‘ns’ will be used for variables associated with the numerical substructure, and ‘es’ will be reserved for the experimental substructure variables. The ideal hybrid system with its active DOFs is shown in Figure 6. From this theoretical representation, a numerical substructure model for RTHS is obtained by defining an experimental substructure considering the DOFs ψ es i for i = 4, 7, 16, 19, 28, 31, 37, 38, that are shown in Figure 6. The corresponding matrices M es , C es , and K es are identified from static loading and free vibration tests, and the numerical substructure matrices are obtained by subtracting the identified experimental substructure matrices from the reference model, which was defined in Section 2. Then, from Eq. (4), the terms associated with the experimental substructure can be arranged into the right-hand side of equation of motion, yielding Eq. (6). M ns ψ ¨ + C ns ψ ˙ + K ns ψ = − M Γ ⋅ x ¨ g − M es ψ ¨ + C es ψ ˙ + K es ψ (6) f es = M es ψ ¨ + C es ψ ˙ + K es ψ . (7)

Here f es is the feedback force vector produced by the experimental substructure during the dynamic testing. In an ideal RTHS simulation, the feedback force may be obtained directly from Eq. (7). However, in physically actuated RTHS, it must be measured or estimated using measured data from the experimental domain. Finally, dividing the equation of motion by M ns , a mass-normalized form is obtained, see Eq. (8): ψ ¨ + M ns − 1 C ns ψ ˙ + M ns − 1 K ns ψ = − M ns − 1 M Γ ⋅ x ¨ g − M ns − 1 f es , (8) which, in this benchmark, is numerically integrated using the unconditionally stable constant-average-acceleration Newmark’s method with parameters α = 0.25 and δ = 0.5 (Bathe and Bathe, 2014). The time step is 0.976 ms since the goal is to run this RTHS Benchmark in real time with a sampling frequency of 1,024 Hz.

The physical specimen in the laboratory will be connected to the numerical substructure of the RTHS experiment. This frame was utilized in previous research (Gao, 2012; Gao et al., 2013) and as part of a past benchmark problem (Silva et al., 2020). Thus, the feasibility of the plant has already been demonstrated. Figure 7 shows the frame and its geometry, which is composed of a horizontal beam and two vertical columns made of structural steel A572 Grade 50. The boundary conditions at the base correspond to pinned connections and the column-beam joints are assumed rigid, transmitting axial, shear, and moment forces. The beam element is fabricated with a 50 mm × 6 mm plate (web) and two 38 mm × 6 mm plates (flanges), forming a custom-made I beam, and the columns are commercially available hot-rolled S3 x 5.7 sections.

In RTHS, additional hardware is needed to drive the experimental frame synchronously with the numerical substructure. The interface conditions discussed in Section 3.1 are enforced by the transfer system which in this case consists of two hydraulic actuators. Hereafter, the vertical DOFs along the global coordinate y will not be considered for convenience, and due to the negligible axial deformations in columns, vertical displacements in the nodes are small in comparison with the horizontal displacements. For instance, Figure 8A shows that for node 4 of the frame, the two associated DOFs of the numerical substructure ψ ns 4 4,28 = x ns 4   θ ns 4 T are imposed to node 4 of the experimental substructure (i.e., ψ es 4 4,28 = x es 4   θ es 4 T ). To represent this MDOF response in a more realistic fashion, these two interface boundary conditions are imposed by incorporating a multi-axial testing technique, which requires the use of multiple hydraulic actuators since each actuator provides translational motion only if used independently. Figure 8B illustrates that a minimum of two hydraulic actuators provide equivalent translational and rotational motion to node 4.

However, to use these two hydraulic actuators, a supplementary component between the frame and the actuator is designed and fabricated. This element attached to the physical frame is referred to herein as the coupler. The coupling of the linear stroke of both actuators through the coupler results in the translational and rotational motion of the coupler and subsequently the physical frame, as depicted in Figure 9.

The immediate effect of this setup is the complex internal coupling between the physical frame, coupler, and actuators. To understand these interactions, it is necessary to study each component individually.

Since RTHS requires a transfer system to drive the experimental substructure, the dynamics and response of the physical domain are affected by numerous well-known issues such as time delays, frequency-dependent time lags, measurement noise, control-structure interaction (CSI), servo-actuator dynamics, internal coupling in multi-actuator and multi-axial RTHS experiments, environmental and laboratory conditions at the time of the testing, etc. (Dyke et al., 1995; Phillips et al., 2013; Fermandois and Spencer, 2017; Nakata et al., 2023). These effects often play an important role in the accuracy and stability of RTHS, and if they are not considered, the quality of the RTHS can be substantially compromised.

In this regard, a properly designed and tuned control system is typically required to accommodate and compensate for these various issues if the goals of the test itself are to be achieved. The control system typically consists of three elements: 1) the plant to be controlled, which includes the dynamics of the system (structure) plus the transfer system (enforcer of the control action); 2) the sensing system, which comprises of all the required sensors to measure the responses of the plant; 3) a digitally implemented controller that takes the measured response(s) of the plant, estimate the necessary states if required, and generates a control action according to a specific control law. This element typically operates in closed loop, and includes one or more control layers for achieving the desired performance, and estimators for generating unmeasured or noisy states.

A block diagram of the key components described herein is presented in Figure 13A, where the signals and closed loops describe the maRTHS configuration and establish the physical or computational implementation of each component of this maRTHS. Figure 13B shows the experimental implementation of the control plant in the IISL.

Meeting the control objectives in this maRTHS scheme is a tracking problem. The main task of this benchmark is to design a control system (see Figure 13) such that the output of the control plant η _m (measured actuator displacements) tracks the target displacement vector η _ns (the response of the numerical substructure in actuator coordinates) and assesses the tracking performance and the overall RTHS performance. The control problem in Figure 13A is simplified in the closed-loop block diagram shown in Figure 14. Participants in this benchmark study will develop and implement their own controllers, following the details to be explained in Section 4. An example of designing and implementing a control scheme, based on a MIMO LQG approach is presented in Section 5.

An estimator is also necessary since the measured signals, η m , contain high frequency noise, and not all the states can be measured. Therefore, the state of the physical substructure must be estimated (the high frequency content is filtered out by the estimator). Then the estimated state η ^ m is used by the control law to realize a control input u, which is then sent to the control plant.

In Section 3.4, it was established that a target displacement vector, ψ ns 4 4,28 = x ns 4   θ ns 4 T , must be imposed to the physical substructure using two hydraulic actuators. Since hydraulic actuators provide translational motion only, a coordinate transformation between “frame coordinates” and “actuator coordinates” is required. Given that the control system must send command signals to the actuators (in actuator coordinates, i.e., linear displacements), the coordinate transformation functions are located between the numerical substructure and the control system, as shown in Figure 13A. Thus, η m = η m 1   η m 2 T is the multi-axial actuator displacement equivalent to the frame node target displacement vector ψ ns 4 4,28 . Then, the control system realizes the control input vector u = u 1   u 2 T , which are commanded to each hydraulic actuator to drive the physical substructure accordingly. Likewise, once the estimator computes the estimated actuator displacement vector η ^ m = η ^ m 1   η ^ m 2 T , these actuator coordinates are transformed back to frame coordinates ψ ns 4 4,28 .

The feedback signal required to satisfy equilibrium conditions at the interface node in the numerical substructure is the experimental force vector f m 4 = F m 4   M m 4 T , where F m 4 and M m 4 are the force and moment corresponding to the degree-of-freedom in the vector ψ ns 4 4,28 .

Some essential assumptions have been made to define the structure of the control system illustrated in Figure 13B. In principle, the target signal is ψ ns 4 4,28 , therefore, the controller should take measured translation and rotation ψ es 4 4,28 of the experimental frame joint so that a direct tracking error would be evaluated. Nonetheless, measuring these states directly at the joint is difficult. Therefore, considering the mechanical properties of the different components and their behavior, the following assumptions can be stated: first, the frame behavior is linear elastic, which guarantees small deformations and deflections. The coupler has a high stiffness and it can be assumed to be rigid. Likewise, the column-beam joint can be considered rigid (Castaneda, 2012). Finally, the coupler and joint are connected with four high-strength grade 5 bolts with a maximum tensile strength of 827.4 MPa (120 ksi), which is adequate for the range of forces required in this benchmark. These bolts connecting these two components will experience small deformations. Therefore, the strains experienced by the coupler and the column-beam joint are considered negligible, and it can be concluded that by measuring the displacement of the hydraulic actuators through LVDTs, the derived translational and rotational motion at node 4 of the experimental frame can be accurately obtained. In Section 3.7, performance indices J 2 and J 5 will be developed to quantify any errors produced by our set of assumptions.

The realization of this maRTHS problem requires the discussion of specific characteristics of its implementation and the definition of certain constraints to reproduce as close as possible actual laboratory conditions.

To assess the overall performance of the maRTHS, the quantitative evaluations consider: 1) tracking control performance (minimize error between target and measured displacements); and 2) global RTHS experiment performance (minimize the error between the reference structure response and the hybrid system response) are required. A set of 10 evaluation criteria is considered in this benchmark. The first six assess the tracking performance of the control system, and the remaining four compute the global performance of the RTHS. These criteria are computed after the RTHS (or vRTHS) is concluded. Table 1 summarizes the indices and briefly describes each criterion. Most of the criteria are evaluated at the interface node at the first story and some are evaluated at upper stories nodes 2 and 3, see Figure 6.

Due to the MDOF characteristic of this benchmark, each of the indices in Table 1 are vectors. For instance, the tracking control index J 2 = J 2,1 J 2,2 T has two components corresponding to the two hydraulic actuators motion Actuator 1 and Actuator 2, whereas the global performance index for upper stories J 8 = J 8,2 J 8,26 J 8,3 J 8,27 T has four components that represents the translational and rotational DOFs of the second story at node 2 (i.e., ψ 2 , ψ 26 ) and of the third story at node 3 (i.e., ψ 3 , ψ 27 ). See Figure 3 or 6 for the DOFs definition.

With reference to Figure 13A, it can be useful to recall the notation of the vector components that represent the different responses. The subscript ‘ns’ and ‘m’ stands for “numerical substructure” and “measured,” respectively; when this subscript is absent, the variable represents the reference response (e.g., Eq. (15)). A “hat” over a variable indicates an estimated (or filtered) value. The next subscript (after a comma) represents the position in a vector (e.g., a specific DOF in the relative displacement vector ψ of the reference frame or a specific actuator displacement in the vector η m ). The variable ‘k’ in brackets represents the discrete time sequence and ‘N’ is the number of samples in the time series.

This vRTHS tool is implemented using scripts and block models in MATLAB/Simulink R2019b, respectively. This virtual implementation is as close as possible to RTHS. The companion package includes a Starting_Guideline.pdf file that explains how to work with the code. Figure 17 shows the basic organization of the code package. The top-level folder has three files only:

• A guideline document: Starting_Guideline.pdf

Additional folders contain experimental data for identification of the plant, input data, ground motions, finite element model of the reference structure, and necessary scripts such as functions for defining and loading different components of the vmaRTHS. The main script main_vmaRTHS.m is in charge of initialization, loading models and control system, running the vmaRTHS, assessment of the results, and it is the only script that needs to be executed to run this tool. The files the participants must modify are the script S3_Controller.m and the corresponding control block in the Simulink model.

Figure 18 shows the execution flow of the principal files (scripts and block models) involved in this implementation as well as their related formulations. The user defines the control system and has the choice, based upon the specific features of the control scheme to be used, of developing its own nominal plant model with the experimental data for identification that is available or use the nominal plant provided as an example in this benchmark. With control design in mind, this tool can also be executed offline.

The control plant defined in Section 3.5 has two inputs u 1   u 2 T and two outputs η m , 1   η m , 2 T representing the control inputs and the hydraulic actuator displacements, respectively. According to Figures 8, 9, subscript 1 represents the bottom actuator and 2 represents the top actuator. Therefore, a compact 2 × 2 matrix description of the control plant is convenient. Eq. (19) shows a mathematical representation of the control plant in terms of a transfer function matrix. H = H 11 H 12 H 21 H 22 , (19) where H i j is the transfer function from input j to output i. The diagonal terms describe the direct relationship between the input and output of a specific actuator when this is commanded, whereas the off-diagonal terms provide the internal coupling behavior of one actuator when the other is commanded.

Since the frame contains component that are all part of one dynamic system, its poles should be common in Eq. 19 and the remaining poles of the system will depend on the model of the transfer system. Therefore, Eq. (19) can be written as H = n u m 11 s d e n 11 s n u m 12 s d e n 12 s n u m 21 s d e n 21 s n u m 22 s d e n 22 s ⋅ n u m es s d e n es s , (20) where n u m i j s / d e n i j s represents the numerator and denominator of the transfer function H i j and n u m es s / d e n es s characterizes the poles and zeros of the frame (experimental substructure).

The use of two hydraulic actuators to enforce translation and rotation requires a coordinate transformation between the degrees of freedom of the numerical substructure and the two actuator displacements. Four assumptions are made to develop this relation: 1) the coupler deformations are negligible. The analysis presented in Section 3.4 demonstrates that the maximum strains in the coupler justify this assumption; 2) the vertical motion of the nodes can be neglected. The axial deformation of the columns in the physical substructure is negligible due to their high axial stiffness, and the axial forces in the columns are small since the motion of the frame is mainly horizontal; 3) the rotations of the column-beam joints are small because the behavior of the frame in this benchmark is limited to linear elastic; 4) the connection of the coupler to the column-beam joint provided by the high-strength bolts is rigid, hence the deformations are negligible.

Therefore, the coupler can be considered as a rigid body, the boundary conditions of the coupler at the column-beam joint allows two degrees of freedom, and the kinematics of the actuators can be described entirely by the horizontal components of their motion. Figure 19A shows that the model of the coupler is defined by the rigid triangle AOB. The vertex O f is located at the intersection of the beam and column axes, and vertices A and B represent the location where the hydraulic actuators are attached.

Figure 19B illustrates the motion of the coupler (initial position in blue and final position in green) and the corresponding attached actuators trajectories when the frame moves laterally. The actuators axial displacements can be obtained by adding the effect of the horizontal displacement of the coupler c (translational DOF of node 4, ψ ns 4,4 , imposed on the physical frame) and the horizontal components of the vectors a and b when the coupler rotates the angle ψ ns 4,28 . For instance, considering the rotation effect only, the initial position of the top actuator can be represented by the vector p o , 2 and its final position by the vector p f , 2 = p o , 2 + b . Thus, the actuator axial displacements can be written as: η m , i = c + p f , i − p o , i , i = 1,2 (21) and its magnitude can be approximated by its horizontal component: η m , i = ψ m , 4 + p ⋅ cos α i + ψ m , 28 − cos α i .   i = 1,2 (22)

Where p = p o , i = p f , i since the coupler is assumed to be rigid. The inverse relation is: ψ m , 28 = sin − 1 η m , 1 − η m , 2 2 p ⁡ sin α 2 (23) and ψ m , 4 = η m , i − p ⋅ cos α i + ψ m , 28 − cos α i . (24)

Actual uncertainties in the plant such as imprecision in the size of elements, material properties, parameters, etc. and a simplified representation of the plant’s dynamics yield to model imprecision. Therefore, it is realistic to incorporate uncertainties into the control plant used in this benchmark so that the proposed control approach is tested realistically as well via virtual RTHS. H = 2165.2   s + 120 s + 90 s + 2.65 s + 40   4.5 e 6   s + 3 s + 3.5 s + 18.5 349.95   s + 120 s + 90 s + 2.65 s + 40   2165.2   s + 5 s + 3.5 s + 18.5 ⋅ n u m es s d e n es s , (25)

This benchmark problem considers uncertainties or model inaccuracies in the control plant by defining random variations in the transfer function matrix of a nominal plant model that has the form of Eq. (20). Random variations in the poles and zeros of the nominal plant model generate these differences by introducing changes from a standard normal distribution sampling process, which create a family of frequency response functions (FRF) where any FRF member represents a potential control plant. Table 2 presents the mean and standard deviation for each of these poles and zeros of the nominal plant model that can be described by Eqs 19, 20. In this benchmark, and commonly in practice, the nominal plant model is an identified plant model that is provided in the companion package tool. n u m es s d e n es s = 1 s + 50 + 63 j s + 50 − 63 j . (26)

Figure 20 shows a set of FRFs generated using the parameters of Table 2 that captures the uncertainty in modeling the control plant. To simulate an actual RTHS experiment, the proposed controller is designed considering an identified plant model (the nominal plant model in this benchmark). Then, a virtual RTHS is conducted where the designed controller tracks a control plant model randomly selected from the FRF family shown in Figure 20. To guarantee robust performance and stability of the controller, at least 20 virtual RTHS should be executed, each one with a randomly selected control plant model. The companion package tool implements these procedures and facilitates the processing of performance metrics data.

This benchmark includes a companion package that helps to implement the control system using vRTHS.

1.Models:

All files will be available on the MECHS website: https://mechs.designsafe-ci.org/

The participants are asked to produce the following to address the benchmark problem:

The high degree of internal coupling in this maRTHS sample demands a systematic procedure to analyze the control plant to obtain the necessary information for system identification. Experimental data was obtained using four energy levels of BLWN signals to the control plant. A first test was conducted using a 0–100 Hz BLWN signal input to the bottom actuator while the other was set to zero displacement and the displacement of both actuators were measured. This set of inputs and outputs was used to compute the experimental transfer functions of the plant, which is the first column of Eq. (19). Likewise, in another test the same signal was used as input to the top actuator while sending a zero to the bottom actuator to generate the second column of the experimental transfer function matrix.

To provide a basic but meaningful nominal model of the control plant, the experimental frame is considered as a 1-DOF second order system with a complex conjugate pair of poles. Even though the hydraulic actuators are of the same model from the manufacturer, they have slightly different behavior in an experimental setup and these dominate each of the columns of the system transfer functions. Thus, a different set of poles is identified for each actuator. The number and type of poles for each system are assumed according to a parametric model previously investigated. The models of the servo-valve and hydraulic dynamics of each actuator are assumed to be represented by first order differential equations (Maghareh et al., 2018). Consequently, in this sample, each actuator is modeled using two real poles. The bandwidth used for this identification process is 40 Hz since the frequencies of interest such as the input signal (ground motion record) and useful natural frequencies of the structure are well below this limit (see Section 2.2). Then, a model with two real poles is fitted to the experimental FRF, see Figure 21. Eqs 25 and 26 describe the identified model of the control plant.

Here, n u m es s / d e n es s is the transfer function that represents the behavior of the experimental frame. This system is then transformed to state-space form as z ˙ = A ⋅ z + B ⋅ u   η m = C ⋅ z + D ⋅ u , (27) where the command displacement vector u is the input to the system; z contains the states of the identified control plant; A, B, C, and D are typical constant matrices in state-space description; and the measured actuator displacement vector, η m , is the output vector.

The RTHS scheme presented in Figure 13B shows the feedback force being measured directly from the control plant, which is typical in RTHS experiments. Despite having load cells available while executing this maRTHS sample in the IISL, it would not be correct to use these measured forces directly. These measurements contain very large inertial forces associated with the coupler, which is not part of the original structural system (the whole frame). Thus, the force that is developed only by the frame must be estimated.

This sample shows a practical approach that uses the FE model of the experimental substructure discussed in Section 3.3. The estimated frame response ψ ^ m is differentiated twice to obtain its corresponding velocity and acceleration and, similar to Eq. (7), the feedback force would be given by Eq. (31). This implementation is shown in Figure 22A. f ^ m = M es ⋅ ψ ¨ ^ m + C es ⋅ ψ ˙ ^ m + K es ⋅ ψ ^ m . (31)

The code for this implementation is included in the companion package. Participants can use the identified model developed in this section, which is also available in the companion package, or choose another model based on a preferred methodology.

Among the vast variety of control methodologies feasible for RTHS, this sample is based on an optimal control strategy that is not intended to be competitive. This approach is selected here because it has acceptable performance and at the same time is simple enough to focus on the important features of this maRTHS while overcoming the control requirements and challenges from a hybrid simulation perspective. An LQG control scheme is chosen due to its versatility in introducing uncertainty in state-space form as added noise.

The LQG controller consists of two components in closed loop: (1) a deterministic LQR which assumes full state feedback, and (2) a Kalman filter that estimates the required states to be fed back to the LQR control law. The strategy for including the target signal η ns is to incorporate the error ε = η ns − η m by augmenting and additional state z ˙ e r r = ε , and find the optimal gain for the augmented plant, Eq. (32). The trade-off between performance and control effort is defined by selecting appropriate ratios for the weighting matrices Q and R, to minimize the cost function established by Eq. (34). Finally, one solves for the gains required to drive the error state to zero. Thus, the control law, u, is computed with Eq. (35). The MATLAB function ‘lqr’ with matrices A a , B a , Q, and R, provides the gains K p and K ε to realize the control input u. z ˙ a = A a z a + B a u + H η ns , (32) where: z a = z z e r r , A a = A 0 − C 0 , B a = B 0 , H = 0 1 (33) J = ∫ 0 ∞ z a T ⋅ Q ⋅ z a + u T ⋅ R ⋅ u (34) u = − K p − K ε z ε (35)

However, in this maRTHS experiment the states, z, are not available, i.e., the only measurements available are the measured displacements of the actuators, η m . Therefore, estimation of states is necessary to feed them to the control law (Eq. (35)). A Kalman filter is used to overcome this limitation by providing estimated states z ^ and filtered response η ^ m so that the control system is implemented as shown in Figure 22B. Considering that the process noise and measurement noise covariance matrices are additive with known distributions, Eq. (27) can be written as z ˙ = A ⋅ z + B ⋅ u + w k   η m = C ⋅ z + D ⋅ u + v k (36) where the distribution of the process noise is assumed to be w k ∼ N 0 , Q , and the distribution of the measurement noise is assumed to be v k ∼ N 0 , R .

The final values for Q, R and the Kalman estimator terms can be found in the companion package.

The following figures and table present a qualitative and quantitative assessment of the performance of the sample LQG control system on the maRTHS based on experimental results. The input to the reference and hybrid system for this example is the El Centro earthquake historic record with a scaling factor of 0.40. See Section 4.5 for additional ground motion records. Figures 23–25 shows qualitatively the tracking and global RTHS performance of the interface node (node 4 in Figure 5), while Table 3 presents the metrics defined by the performance indices (see Section 3.7) which includes not only the interface node information, but also additional nodes at the upper stories of the frame for a more comprehensive evaluation of the RTHS.

Figure 23 shows the tracking performance by comparing the measured actuator displacements ( η m ) and the target actuator displacements ( η ns ) computed from the numerical substructure node displacements. The results show a NRMS error of 23.8% for actuator 1 (bottom) and 13.2% for actuator 2 (top). These results are in agreement with the mechanics explained in Section 3.6: Actuator 1 is counteracted by the deformed frame effect. Conversely, Actuator 2 is “helped” by the same frame effect. From a control perspective, the effort required by actuator 1 to drive the motion of the frame node is greater. This particular behavior of the plant requires the selection of larger Q/R ratios for actuator 1 in the sample LQG controller. If the estimated measured actuator displacements ( η ^ m ) are considered, the NRMS tracking errors are 3.2% and 13% for Actuator 1 and Actuator 2, respectively. This result shows the benefits of estimators to enhance the tracking control performance. Despite that the actuator displacements provides direct measurements for tracking assessment, a more realistic evaluation of tracking performance is achieved by considering frame coordinates, i.e., the frame node motion. Figure 24 illustrates a comparison between the transformed (estimated) measured displacement vector ψ ^ m and the target displacement vector ψ ns at the interface node. The NRMS error for the translational and rotational DOF are 8.1% and 27.8%, respectively. The increased errors result from the assumptions described in Section 4.3, specifically, the effectiveness of the connection of the coupler to the joint frame, which demonstrates the challenges and limitation in enforcing rotational boundary conditions with the experimental setup.

Figure 25 provides a comparison between the reference response and the hybrid system (global performance) at the interface node. The NRMS error of the RTHS for the translational DOF at node 4 is 12.2% and for the rotational DOF at the same node is 26.2%. The fact that these errors are comparable to the errors based on the numerical substructure target signals (Figure 24) reveals that the partition is adequate even though the control approach is basic in this sample. Table 3 complements the global performance evaluation of the RTHS by providing the numerical values for the indices defined in Section 3.7 based on the average of three consecutive experiments.