## 1 Introduction  
Real-time hybrid simulation (RTHS) is a dynamic experimental method that deals with time-dependent specimens [1], and is an extension of hybrid simulation [2, 3] originating from the pseudo-dynamic test [4]. In RTHS, the emulated structure, which is typically large-scale, is divided into numerical and physical portions. The former is numerically simulated by a computer program, and is usually called a numerical substructure (NS), while the latter is physically tested in a laboratory and is called a physical substructure (PS) [5, 6]. Considering the cost and space requirement, RTHS has become an attractive alternative to the widely used quasi-static test and shaking table test to evaluate the performance of structures equipped with time-dependent members and/or devices [7, 8, 9]. A typical requirement of RTHS is that it must be executed in real time over a tiny time interval. However, owing to the computation, signal acquisition, and transmission processes, as well as the inherent dynamics of the actuator and PS (referred to as a control plant in this study), there are phase shifts--also known as time delays--and amplitude discrepancies between the desired and measured displacement, resulting in de-synchronization at the interface of the NS and PS. Using the energy-balance method, [PERSON], et al. [10] showed that the time delay is equivalent to negative damping artificially introduced into the testing system with a stiffness specimen. When this negative damping is greater than or equal to the system damping, it can give rise to test instability issues and further threaten the safety of the testing system and specimen. Hence, to safely and stably run the RTHS and to obtain reliable RTHS results, the time delay must be effectively compensated/controlled. To deal with the time delay, numerous delay compensation schemes have been proposed and applied to RTHS over the past 20 years. Based on the constant-delay assumption, polynomial extrapolation [11] and kinematic predictors [12] were proposed and then improved [13], among which the third-order polynomial extrapolation is being widely applied [11; 14]. Time delay resulting from the dynamics of control plant can be theoretically viewed as the phase lag divided by the corresponding frequency, and methods from control theory have thus been employed, such as the phase-lead network [15], pole assignment [16], feedforward control [17], inverse control [18], model-based control [19], state-based control [20; 21], \(H_{\infty}\) control [14; 22], and sliding mode control [23]. Owing to the dynamics/nonlinearity of the control plant and control-structure interaction [24], time delay often varies during the RTHS, resulting in the effectiveness of the above-mentioned delay compensation method reduced. To address the variant time delay issue, online delay estimation and updating strategies were proposed [25; 26]. Then, adaptive law was employed to combine with inverse control [27; 28; 29] and model-based control [30], where the model parameters of the control plant are estimated online. Recently, a general adaptive compensation method based on a discrete model of the control plant was proposed and validated [13; 31], where the compensation commands of actuators were generated using the desired displacements, measured displacements, and previous displacement commands, and the least squares method was used to estimate the system model parameters. Due to the effect of measurement noise and initial parameter values, the estimated parameters for the discrete system model changed significantly in [PERSON] et al. 's method [13], especially at the beginning of the simulation. Obviously, the use of parameter estimation methods for the delay compensation of RTHS is very demanding. Firstly, the parameter estimation method must be convergent to guarantee the estimated time delay or system model parameters that are approximate to the actual ones. Secondly, the parameter estimation method must be sensitive to capture variations of the time delay/system model parameter.