5.1.2. Controller Design

Similar to that used in Section 4, the analytical model was established theoretically using the technical index of the servo-hydraulic actuator, which is provided by the manufacturer. The transfer function from *u* to *y* is

$$P = \frac{4\beta k\_0 A\_p}{\left(4\beta A\_p^2 + 4p A\_p^2 + V K\_c\right)s + 4\beta k k\_0 K\_E} \tag{18}$$

where the parameters are listed in Table 2. Hence, the numerical model is given by

$$P(s) = \frac{33.74}{s + 0.5739} \tag{19}$$


**Table 2.** Values of system parameters.

The mathematical model in between the input and output in the frequency domain is shown in Figure 19. It is seen in Figure 19a that within the frequency range of interest, the model was in good agreemen<sup>t</sup> with the test data, while for the phase, differences were observed when the frequency became large. However, the analytical model can reflect the major dynamics of the actual testing system.

**Figure 19.** Model validation. (**a**) Magnitude; (**b**) phase.

The weighting functions were determined using the recommended method given in Section 3. After several trials, they were selected as

$$\mathcal{W}\_{\text{S}} = \frac{0.05s + 62.83}{s + 0.01}, \mathcal{W}\_{\text{T}} = \frac{833.3333(s + 805) \text{ (s + 19.51)}}{(s + 2.09 \times 10^{4}) \text{ (s + 1.05 \times 10^{4})}}, \mathcal{W}\_{\text{R}} = 1 \times 10^{-6} \tag{20}$$

Eventually, a feasible solution was achieved and the controller was

$$K(s) = \frac{7.6251 \times 10^8 \text{ (s + 2.094 \times 10^4) \text{ (s + 1.047 \times 10^4) \text{ (s + 0.5739)}}}{(s + 2.812 \times 10^{10})(s + 0.01048)(s^2 + 1892s + 1.086 \times 10^2)}\tag{21}$$

It should be noted that the *H*∞ controller was designed in the continuous-time domain, while the dSPACE is a digital sampling system. Hence, the *H*∞ controller was converted to discrete form by the c2d command in MATLAB. The discrete method was Tustin, and the sampling time was identical to that in dSPACE, namely 1000 Hz.
