4.1.1. Modeling Uncertainties

• Controller design

When designing the *H*∞ controller, the modeling uncertainties and measurement noise should be considered first, followed by the dynamics of the controlled system. Hence, it is recommended that the designed order of the weighting function is *W*T, *W*S, and *W*R.

To determine the modeling uncertainties, a series of linear numerical models of the nonlinear model were calculated. Then, the uncertainties were obtained, whose singular values are shown in Figure 12. Later, the weighting *W*T can be designed. It is seen in the figure that the modeling error was very small when the frequency was not very high; then, the modeling uncertainties increased with the frequency. Hence, the weighting *W*T should cover the uncertainties over all the frequency ranges. Through several trials, the expression for *W*T was given by

=

$$W\_T(s) = 2 \times 10^{-5}s^2 + 0.005s + 0.25$$

(14)

**Figure 12.** Modeling uncertainties and weighting function *W*T.

Subsequently, the dynamics of the controlled system were considered. It is expected that the controlled system should respond to the command quickly and without a steady-state error. Hence, the other two weightings were determined after several trials, which were

$$\mathcal{W}\_{\rm S}(s) = \frac{0.01s + 63}{s + 0.001}, \mathcal{W}\_{\rm R} = 0.1\tag{15}$$

Eventually, a feasible solution was reached, and the *H*∞ central controller was

$$K(s) = \frac{7,731,236.8628(s + 1.413)(s^2 + 164.3s + 3.704 \times 10^5)}{(s + 1.021 \times 10^6)\ (s + 1171)\ (s + 568.5)\ (s + 0.001)}\tag{16}$$

Afterward, the performance of the controlled system was examined through performance curves and the step response, which are given in Figure 13. A fast response speed is observed in Figure 13a, and there is no steady-state error. It can also be seen from Figure 13b that the complementary sensitivity function *T* was almost identical to 1 when the frequency was less than 10 Hz, indicating that the controlled system could track the command very well. It is seen in the figure that the curves of the sensitivity function *S* and complementary sensitivity function *T* were below the weighting functions *<sup>W</sup>*S<sup>−</sup><sup>1</sup> and *<sup>W</sup>*T<sup>−</sup>1, respectively, indicating that the selected weighting function could meet the robust performance [24]. Moreover, over the concerned frequency, the sensitivity function *S* was far less than 1, indicating that the controlled system exhibited a strong robustness considering the disturbance.

• Modeling errors

To investigate the robustness of the *H*∞ controller to modeling uncertainties, two cases were considered here. For the first case, the multiplicative uncertainties of 50% were considered, while for the second case, the controller designed employing a linear model was used for the nonlinear model. Step responses for the two cases are shown in Figure 14, and the response without any uncertainties is also given in the figure for convenience of comparison.

**Figure 13.** Performance of the controlled system. (**a**) Step response; (**b**) performance curve.

**Figure 14.** Step responses. (**a**) Case 1: multiplicative uncertainties; (**b**) case 2: nonlinear model.

It is seen in the figure that when the multiplicative uncertainties were considered, steady-state errors and overshoots were still not observed, while the settling time was less than the initial state. When the *H*∞ controller was used for the nonlinear model, it was found that there were obvious fluctuations, and overshoot and steady-state errors occurred. However, the overshoot was very small (less than 0.5%), and the steady-state error tended to zero. Hence, the *H*∞ controller exhibited strong robustness to modeling uncertainties.

4.1.2. Variation of the Specimen

• Stiffness

Two different stiffness coefficients were considered for the PS, 1.5 and 0.1 times the initial one, respectively. The time histories of the change ratio under the step response are given in Figure 15. It is seen from the figure when the stiffness coefficient varied, the steady-state error occurred. The change ratio for the stiffness decrease was smaller than that for the stiffness increase, and they were within the acceptable range. Hence, the *H*∞ controller was robust to the variation in stiffness.

**Figure 15.** Change ratio of step response for different stiffnesses. (**a**) *KE* = 0.1*KE*0; (**b**) *kE* = 1.5*KE*0.
