70 N. Tsokanas and B. Stojadinovic Real-time hybrid simulation (RTHS) [4] has been developed in the past two decades enabling researches to run experiments in real-time. Albeit attractive, real-time hybrid simulation comes with numerous challenges since it requires all necessary procedures, such as experimental data acquisition and computational calculations, to be made in the range of milliseconds or even less [5]. Furthermore, boundary conditions need to be applied on the physical substructures in the same time intervals in order to obtain representative results, meaning that dynamic actuators performing on such time rates are needed. Hence quick hardware and software are crucial elements needed to perform RTHS. In cases that such equipment is not available or scaling has to be made, similitude laws should be taken into consideration when designing a RTHS or a fast hybrid simulation. Using similitude allows us to extend the notion of real-time hybrid simulation from strictly conducting simulations at the time scale of one to conducting simulations such that the rate-sensitive phenomena are correctly reproduced and to account for time scale distortion [6, 7]. Apart from time-sensitive phenomena, another critical issue in RTHS is time delays due to the kinematics of the actuator. Delays are inevitably introduced to the system due to the dynamics of the transfer system in combination with the timeconsuming procedure of data acquisition and interaction with the physical substructure [8–10]. These time delays have been shown to be equivalent to adding negative damping to the simulated structure [8]. As a result, instabilities in RTHS are caused when the negative damping is greater or equal to the numerical damping of the system. Therefore, time delays should be effectively compensated to guarantee robust performance of the hybrid model. Lately, control engineering theory was used to develop time delay compensation techniques to guarantee desirable performance of the interfaces between physical and numerical substructures [5, 11–13]. An additional crucial aspect in RTHS, and HS in general, is the fact that the parameters of the hybrid model whose dynamic response is being simulated, are often treated as deterministic. The values of these parameters are regularly determined through deliberate simplifications, ignoring the associated uncertainties. However, the effect of uncertainties may be significant. Stochastic hybrid simulation (SHS) is a significant extension of the state-of-the-art HS to address the dynamic response of uncertain structural systems under uncertain operating conditions and uncertain excitation. Under this concept, the parameters of the hybrid model are treated as random variables with known probability distributions. The results are probability distributions of the structural response quantities of interest. Robust control theory techniques are employed to deal with the presence of uncertainties [9, 11]. The arising question is to what extent does the closed-loop actuation control system at the interface between physical and numerical substructures affect the outcomes of SHS, while maintaining high fidelity of the simulation in the presence of uncertainties. Performing stochastic modelling and simulations requires adequate computational power and large dataset capacity since repeated testing of the stochastic hybrid model is essential. Surrogate modelling, also called meta-modelling, aims at decreasing the high costs of stochastic simulation by replacing the original computationally expensive model with simplified surrogates. Acquiring enough simulation data enables global sensitivity analysis for the simulated stochastic hybrid model. Sensitivity analysis aims at quantifying the relevant effects of the input random variables onto the variance of the stochastic hybrid model response considering the entire input space [14]. In this paper, model predictive control (MPC) will be implemented in the scope of the benchmark problem defined by Silva et al. [15] to evaluate the performance and robustness of a virtual RTSHS (vRTHS) and also a stochastic framework will be developed in order to conduct SHS. The outcomes of vRTHSs with the MPC and the proposed PID controller will be compared. Definition of the benchmark problem, the reference structure, and the SHS model parameters and their probability distributions is presented first. Then, the response of the benchmark stochastic prototype to random excitation is computed and treated as the reference model. Thereafter, the prototype is replaced by a virtual SHS model whose substructure interfaces are actuated in closed-loop control using MPC. Repeated real-time simulations are conducted to assess the performance and robustness of the tracking controller of the SHS model. Result post-processing is performed using uncertainty quantification techniques. Based on simulation data, three different surrogate models are developed: (1) based on polynomial chaos expansion (PCE), (2) Kriging, and (3) polynomial chaos kriging (PCK). Different meta-modelling techniques are used to cross-check and validate the results. Finally, global sensitivity analysis is performed using Sobol indices in order to identify the most sensitive stochastic input variables. 8.2 Problem Definition The reference model of the RTHS benchmark problem [15] is shown in Fig. 8.1. It is a three-story, two-bay, planar steel moment frame. A linear-time-invariant (LTI) numerical model with 3 lateral degrees-of-freedom (DOF) at the floor locations is provided for the reference structure. Its equation of motion (EOM) follows:
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