**1. Introduction**

Real-time hybrid simulation (RTHS) [1], or the real-time substructure pseudo-dynamic test, is a cost-effective and versatile experimental technique to evaluate structural performance under dynamic excitations. It originated from the pseudo-dynamic test [2] first proposed by a Japanese researcher in the 1970s, which is known as hybrid simulation (HS) nowadays. HS takes advantage of numerical analysis and physical experiments, in which the emulated structure is divided into several substructures: the part that cannot be simulated exactly is experimentally tested in the laboratory, which is denoted as the physical substructure (PS), and the rest is simulated by a computer program, which is denoted as the numerical substructure (NS) [3,4]. In RTHS, boundary conditions between the two substructures are imposed on the PS by a transfer system, a servo-hydraulic actuator or shaking table, in a real-time manner. This allows RTHS the ability to test rate-dependent components, such as TMD, AMD, and MR dampers. In recent decades, much progress has been achieved [5–8].

However, due to the inherent nonlinear dynamics of the actuator–specimen system, the desired displacement cannot be realized at the end of the time integration step. This is often called the time delay, which will decrease the accuracy and may lead to RTHS instabilities [9]. Therefore, to carry out a successful RTHS, the detrimental effect of time delay must be mitigated. Horiuchi et al. [9] assumed a constant time delay and proposed a polynomial extrapolation method. Subsequently, more accurate strategies have been investigated to consider the variation in time delay [10–13].

**Citation:** Ning, X. Mixed Sensitivity-Based Robust *H*∞ Control Method for Real-Time Hybrid Simulation. *Symmetry* **2021**, *13*, 840. https://doi.org/10.3390/sym13050840

Academic Editor: Jan Awrejcewicz

Received: 16 April 2021 Accepted: 7 May 2021 Published: 10 May 2021

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Modern control theory was also used to deal with time delay, where the control plant includes the servo-hydraulic and the specimen. The inverse control technique was introduced by Chen and Ricles [14], where the servo-hydraulic actuator is modeled by a first-order transfer function. Carrion and Spencer [15] proposed a model-based control approach, where a low-pass filter is combined with the inverted actuator system plant. In this method, a higher-order control plant can be used. Ning et al. [16] proposed an adaptive feedforward control method, where the Kalman filter is used to estimate the adjustable parameters. Xu et al. proposed a frequency evaluation index-based compensation for RTHS [17]. A two-stage delay compensation method, combining the feedforward and polynomial extrapolation, was proposed by Wang et al. [18]. A polynomial-based feedforward prediction algorithm was combined with a robust linear-quadratic-gaussian controller that was proposed by Zhou et al. [19] to deal with the adverse effects of time delay. Ning et al. [20] proposed an adaptive feedforward and feedback control method based on a discrete control plant, of which the model order is not restrained.

However, there are differences between the control plant model and the actuator– specimen system. Hence, preliminary discussions have been made on the robustness of the control strategy to model uncertainties [21–23]. In this study, a mixed sensitivity-based *H*∞ control method was introduced to deal with the time delay and uncertainties in real-time hybrid simulation. The *H*∞ control theory is overviewed in Section 2. The selection of a performance weighting function is presented in Section 3, where the influence of the weighting function on the system dynamics is discussed. Subsequently, the proposed method is validated through numerical simulations and actual RTHS in Sections 4 and 5, respectively.
