###### Abstract  
To improve the experimental accuracy and stability of shaking table substructure testing (STST), an explicit central difference method (CDM) and a three-variable control method (TVCM) with velocity positive feedback (VPF) are proposed in this study. First, the explicit CDM is presented for obtaining an improved control accuracy of the boundary conditions between the numerical and experimental substructures of STST. Compared with the traditional CDM, the proposed method can provide explicit control targets for displacement, velocity, and acceleration. Furthermore, a TVCM-VPF is proposed to improve the control stability and accuracy for loading the explicit control targets of displacement, velocity, and acceleration. The effectiveness of the proposed methods is validated by experiments on a three-story frame structure with a tuned liquid damper loaded on an old shaking table originally designed with the traditional displacement control mode. The experimental results show that the proposed explicit CDM works well, and the response rate and control accuracy of the shaking table are significantly improved with the contribution of the TVCM-VPF compared with those of the traditional proportional integral derivative (PID) controller. This indicates the advantage of the proposed TVCM-VPF over the traditional PID for STST. A comparison between the traditional shaking table test and STST shows that when the latter is based on the TVCM-VPF, it exhibits an excellent performance in terms of the stability and accuracy of displacement and an acceptable performance in terms of the acceleration accuracy. shaking table substructure testing; velocity positive feedback control; three-variable control method; stability; accuracy; real-time hybrid simulation +
Footnote †: journal: Applied Soft  
system, such as a servo-hydraulic actuator or a shaking table, is employed to ensure the force equilibrium and deformation compatibility at the interface of the two parts. In the past three decades, owing to its outstanding advantages, RTHS has been extensively and systematically investigated [10; 11; 12; 13; 14; 15; 16]. Furthermore, it has been adopted in a range of applications for structural performance evaluation, including wind turbine structures [7], offshore platforms [17], steel structures [18; 19], and wood shear wall structures [20]. It is worthy to mention that RTHS is also useful for reinforced concrete [21; 22] and masonry structures owing to its unique features [23; 24]. Shaking table substructure testing (STST) is a branch of RTHS with a sophisticated shake table as the transfer system or auxiliary loading facility. Since 1994, when it was first proposed by [PERSON] et al. [25], STST has made significant progress in two aspects, namely, methodology theories of loading control and delay compensation, and application investigation in several fields. [PERSON] et al. [26] proposed a hybrid loading test method using a shake table and a hydraulic actuator to evaluate the real-time dynamic responses of structural systems under a strong seismic excitation; in this method, digital filters, namely, finite impulse response (FIR), are employed to compensate for the delays. [PERSON] [27; 28] proposed an STST with an inverse transfer function of the shaking table to accurately reproduce the interface accelerations and implemented it on a structure with a tuned liquid damper (TLD). A full-scale STST was proposed with a rubber-and-mass system that amplifies the table motion to reproduce large floor responses of high-rise buildings [29]. Open-loop inverse dynamics compensation via a simulation algorithm was implemented to generate commands using the model matching and Ho methods [29]. [PERSON] and [PERSON] [30] proposed an STST method where the lower portion of a structure was physically simulated, while the upper portion was numerically analyzed. A servo-hydraulic actuator in the displacement control mode was installed at the top of the ES to excite a mass.