Comparison of explicit integration algorithms for real-time hybrid simulation

Real-time hybrid simulation (RTHS) is a promising cyber-physical technique for the experimental evaluation of civil infrastructure systems subject to extreme dynamic loading. It allows for simulation of highly complicated civil infrastructure systems in a cost effective manner by partitioning them into numerical and physical substructures. And the coupling at the interface is achieved by enforcing equilibrium and compatibility at real-time. In RTHS, development of some systematic and random errors in the physical substructure are inevitable and can have substantial impacts on the quality of the simulation results. It has been shown that the reliability of RTHS results is highly dependent upon how successfully these sources of error in the physical substructure are mitigated. Some significant sources of error in RTHS are transfer system dynamics, e.g. dynamics of hydraulic actuator, and time delay, e.g. communication and computation delays. In this paper, we will show how a multi-degree of freedom (MDOF) RTHS system can be modeled with delay differential equations (DDE). Afterwards, a 4-story RTHS will be used as an illustrative numerical example to delineate how this DDE model can capture some important features of RTHS. Finally, the effects of transfer system dynamics on the quality of RTHS results will be shown for a 2-story structure and then the DDE model will be verified.

Real-time hybrid simulation combines experimental testing and numerical simulation by dividing a structural system into experimental and analytical substructures. Servohydraulic actuators are typically used in a real-time hybrid simulation to apply command displacements to the experimental substructure(s). Servohydraulic actuators may develop a time delay due to inherent actuator dynamics that results in a desynchronization between the measured restoring force(s) and the integration algorithm in a real-time hybrid simulation. Inaccuracy or even instability will occur in a hybrid simulation if actuator delay is not compensated properly. This paper presents an adaptive compensation method for actuator delay. An adaptive control law is developed using an error tracking indicator to adapt a compensation parameter used in the proposed compensation method. Laboratory tests involving large-scale real-time hybrid simulations of a single degree of freedom moment resisting frame with an elastomeric damper are conducted to experimentally demonstrate the effectiveness of the proposed adaptive compensation method. The actuator tracking capability is shown to be greatly improved and exceptional experimental results are still achieved when a good estimate of actuator delay is not available.

Gain-margin based discrete-continuous method for the stability analysis of real-time hybrid simulation systems

Implementation of real-time hybrid simulation using a large shake table with onboard actuators

For Real-time hybrid simulation (RTHS) to be stable and accurate, it is essential to address the time desynchronization issue between the numerical and physical substructures. Desynchronization is primarily caused by time delays, inherent dynamics of the control plant, system uncertainties, and noises. While existing adaptive compensators have shown effective tracking performance in single-input single-output (SISO) RTHS, their effectiveness in multi-input multi-output (MIMO) RTHS has not been fully demonstrated. MIMO-RTHS presents additional challenges due to its larger solution space, and significant dynamic coupling between actuators. To address these challenges, this study introduces an adaptive compensation framework for MIMO-RTHS. The proposed framework utilizes a control law based on the inverse dynamics of the control plant, incorporating real-time adaptive parameter updates through Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) methods. Both the transfer function (TF) and discrete-time state-space (SS) models of the plant are employed in distinct parameter estimation cases. The performance of the proposed compensation is validated through a multi-axial RTHS (maRTHS) benchmark problem. Extensive simulations on the maRTHS incorporating various earthquake inputs, sensor noise, and model uncertainties, demonstrated an excellent tracking performance and strong robustness across four parameter estimation cases (EKF-TF, UKF-TF, EKF-SS, and UKF-SS). The use of UKF with SS model (UKF-SS) achieved superior performance, effectively managing nonlinearities and noise without requiring low-pass filtering.

Impedance matching control design for the benchmark problem in real-time hybrid simulation

This paper presents a new family of explicit time integration algorithms with controllable numerical dissipation for structural dynamic problems by utilizing the discrete control theory. Firstly, the equilibrium equation of the implicit Yu-[Formula: see text] algorithm is adopted, and the recursive formulas of velocity and displacement for the explicit CR algorithm are used in the algorithms. Then, the transfer function and characteristic equation of the algorithms with integration coefficients are obtained by the [Formula: see text] transformation. Furthermore, their integration coefficients are derived according to the poles condition. It was indicated that the proposed algorithms possess the advantages of second-order accuracy, self-starting, and unconditional stability for linear systems and nonlinear systems with softening stiffness. The numerical dissipation of the algorithms is controlled by the spectral radius at infinity [Formula: see text]. It was also shown that the proposed algorithms have the same poles as the Yu-[Formula: see text] algorithm, and thus the same numerical properties. Compared with the implicit Yu-[Formula: see text] algorithm, the proposed algorithms are explicit in terms of both the displacement and velocity formulas. Finally, the effectiveness of the proposed algorithms in reducing the undesired participation of higher modes for solving the dynamic responses of linear and nonlinear systems has been demonstrated.

High performance compensation using an adaptive strategy for real-time hybrid simulation

Real-time hybrid simulations using large-scale magnetorheological (MR) dampers were conducted to evaluate the performance of various structural control strategies to control the seismic response of a three-story steel-frame building. Magnetorheological dampers were installed in the building to limit the story drift to less than 1.5% under the design-basis earthquake (DBE). The laboratory specimens, referred to as experimental substructures, were two individual MR dampers, with the remainder of the building modeled as a nonlinear analytical substructure. The experimental technique enables an ensemble of ground motions to be applied to the building, resulting in various levels of damage, without the need to repair the experimental substructures because the damage will be within the analytical substructure. Five different damper control algorithms, including passive and semiactive control algorithms, were selected. An ensemble of five ground motions scaled to the DBE was used for the real-time hybrid simulations to obtain statistical responses of the structure for each control. The real-time hybrid simulation results show that the MR dampers can control the drift, enabling the performance objective of 1.5% maximum story drift to be achieved. Although some semiactive controllers show better performance for a specific ground motion, the response statistics from the real-time hybrid simulations show that the overall performance of the semiactive control algorithms with the selected user-defined parameters is similar to that for the passive controller for the three-story building used in this study. A comparison of real-time hybrid simulation results with numerical simulation results using OpenSees was conducted to further gain insight into the performance of the damper control algorithms observed in the real-time hybrid simulations.

Real-Time Hybrid Simulation (RTHS) is a novel approach conceived to evaluate dynamic responses of structures with parts of a structure physically tested and the remainder parts numerically modelled. In RTHS, delay estimation is often a precondition of compensation; nonetheless, system delay may vary during testing. Consequently, it is sometimes necessary to measure delay online. Along these lines, this paper proposes an online delay estimation method using least-squares algorithm based on a simplified physical system model, i.e., a pure delay multiplied by a gain reflecting amplitude errors of physical system control. Advantages and disadvantages of different delay estimation methods based on this simplified model are firstly discussed. Subsequently, it introduces the least-squares algorithm in order to render the estimator based on Taylor series more practical yet effective. As a result, relevant parameter choice results to be quite easy. Finally in order to verify performance of the proposed method, numerical simulations and RTHS with a buckling-restrained brace specimen are carried out. Relevant results show that the proposed technique is endowed with good convergence speed and accuracy, even when measurement noises and amplitude errors of actuator control are present.

Integration algorithm is a key to ensure the better application of the dynamic constitutive model. As for the bounding surface model based on swell–shrink rules in this paper, an explicit integration algorithm based on sub-stepping and error control scheme has be described in detail. For being adaptable to the constitutive model, several modifications have been made. The modified explicit algorithm includes two procedures: the first procedure is the second-order forward modified Euler scheme with sub-stepping and error control; the second procedure is the stress correction. However, due to the shrink rule of the bounding surface, the second procedure is not needed at the unloading phase. Finally, several numerical simulations have been conducted for assessing the performance of the explicit algorithm, which demonstrates that the corresponding algorithm is stable and accurate, and can simulate the mechanical characteristics of the clay pretty well.

Adaptive model reference control method for real-time hybrid simulation

This article presents real-time hybrid simulation (RTHS) in a single-degree-of-freedom (SDOF) steel frame incorporated with tuned liquid column damper (TLCD). The SDOF steel frame is numerically simulated, and the TLCD alone is physically experimented on a shaking table. The delay-dependent stability of RTHS system for TLCD investigation is first assessed; and the delay-dependent accuracy is verified by comparing the responses obtained through the RTHS, the conventional shaking table test, and an analytical solution. Then, RTHSs are carried out to evaluate the effects of mass ratio, structural damping ratio, structural stiffness, and peak ground acceleration on the reduction effectiveness of STLCD. The nonlinear behavior of the STLCD is experimentally captured. Finally, the structural responses under STLCD and multiple TLCDs (MTLCD) control are compared. It is found that the performance of STLCD strongly depends on structural parameters and properties of earthquakes; both MTLCD and STLCD induce approximately the same response reductions, and the former can enhance the control performance in certain cases. These results presented here may contribute to improve the design and application of TLCDs in practical engineering.

Seismic performance assessment of a low-rise thin reinforced concrete wall building with unconnected fiber reinforced elastomeric isolators using real-time hybrid simulations

Magneto-rheological (MR) fluid dampers can be used to reduce the traffic induced vibration in highway bridges and protect critical structural components from fatigue. Experimental verification is needed to verify the applicability of the MR dampers for this purpose. Real-time hybrid simulation (RTHS), where the MR dampers are physically tested and dynamically linked to a numerical model of the highway bridge and truck traffic, provides an efficient and effective means to experimentally examine the efficacy of MR dampers for fatigue protection of highway bridges. In this paper a complex highway bridge model with 263 178 degrees-of-freedom under truck loading is tested using the proposed convolution integral (CI) method of RTHS for a semiactive structural control strategy employing two large-scale 200 kN MR dampers. The formation of RTHS using the CI method is first presented, followed by details of the various components in the RTHS and a description of the implementation of the CI method for this particular test. The experimental results confirm the practicability of the CI method for conducting RTHS of complex systems.