**4. Numerical Validation**

In this section, a nonlinear model of the servo-hydraulic actuator [25] was employed to validate the effectiveness of the proposed *H*∞ control method used for RTHS.

Typically, the *H*∞ controller is designed by employing a nominal analytical model of the physical testing system, in which the PS is included. Hence, a linear model of the servo-hydraulic actuator, or nominal plant, is obtained from the nonlinear model for design convenience, which is given by

$$P(\mathbf{s}) = \frac{2\beta A\_p k\_0}{M\_\mathcal{E} V \mathbf{s}^3 + (\mathcal{C}\_\mathcal{E} V + 2\beta k M\_\mathcal{E} k\_0) \mathbf{s}^2 + (K\_\mathcal{E} V + 4\beta A\_p^2 + 2pA\_p^2 + 2k\beta \mathcal{C}\_\mathcal{E} k\_0) \mathbf{s} + 2k\beta K\_\mathcal{E} k\_0}. \tag{12}$$

where the symbols and their values are listed in Table 1. The damping coefficient *C*E is calculated by the damping ratio and natural frequency of the specimen. Then, the transfer function of the nominal model is

$$P(\text{s}) = \frac{7.748 \times 10^6}{\text{s}^3 + 165.7 \text{s}^2 + 3.706 \times 10^5 \text{s} + 5.235 \times 10^5}. \tag{13}$$


**Table 1.** Values of system parameters for simulation [25].

It should be noted that the units have been transformed into the international system of units.
