## 3 Results
### Robust Synthesis: \(\mu\)-Controller
The first mechanical load in each of the simulations that came after the disturbance was 10 N at the free end; K(s) is the controller, \(\Delta\) is the uncertainty plan (Equation (12)), B (Figure 1a) is the smart structure, and x and y are from Equations (7) and (15). Figure 2 shows the \(\mu\)-values of the computed controller's \(\mu\)-values. The majority of the frequencies indicate that the controller is reliable [8; 9; 10; 11; 12]. A higher \(\mu\)-value indicates better system performance and robustness. One way to create a \(\mu\)-controller is to use the D-K iteration process. As previously mentioned, this is an approximation process that provides boundaries for the \(\mu\)-value. Figure 3 illustrates a comparison of \(\mu\)-controller performance at the free end, which serves as an indicator of the overall effectiveness. It is clear that the \(\mu\)-controller surpasses the others in performance, albeit at the expense of an increased control effort. In Figure 3, the upper part shows the displacement of the free end of the beam. The blue line represents the displacement for the open-loop configuration of the beam, which means without control; the red line represents the displacement with the controller with \(\mu\)-analysis. The results were very good when vibration suppression was achieved. In Figure 3 (bottom window), we consider the control voltages for the free end of the beam with a maximum voltage of 50 Volt. Figure 4 (left window) confirms this result. This might be the result of numerical issues with the \(\mu\)-controller computation caused by the poor condition number of the plant. Another possible reason could be the high order of the \(\mu\)-controller. Further research is required in either event. A higher \(\mu\) value indicates a better robustness against uncertainties. The control system design using \(\mu\)-analysis involves optimizing the controller to maximize \(\mu\) and ensure robust stability and performance in the presence of uncertainties. Applying \(\mu\)-analysis to smart structures involves considering uncertainties in the structural parameters and control elements and designing controllers that maximize \(\mu\) to enhance robustness. Figure 4 illustrates the displacement of the beam's free end; the red line represents the displacement for the open loop of the beam, which means without control; the black line represents the displacement for the controller with \(\mu\)-analysis. The results were very good when vibration suppression was achieved. In the bottom-left window, we take the control voltages for the free end of the beam with a maximum voltage of 15 Volt (Figure 4 bottom-left). The previous results are with K\({}_{\text{o}}\) = 1.9 K: when K\({}_{\text{o}}\) = 0.1 K (this translates to a \(\pm\)90% deviation from the nominal value of the stiffness matrix K), the control voltages are 50 Volt (right down), and the displacement for the free end of the beam with and without control is shown in Figure 4 (upper-right window). Moreover, m\({}_{\text{p}}\) = \(\pm\)0.9 and k\({}_{\text{p}}\) = 0, which corresponds to a \(\pm\)90% variation from the nominal. Figure 2: The \(\mu\)-controller’s boundaries for m\({}_{\text{p}}\) = 0 and k\({}_{\text{p}}\) = \(\pm\)0.9. The displacement responses of this controller for the first mechanical input are shown in Figure 5. We take the results for the four nodes of the beam, and the results with the blue line are for the open loop, i.e., without control. The results with the green line are for \(\mu\)-analysis, and the results with the \(\mu\)-analysis are much better as we successfully rejected the oscillations. The previous results were in the state-space and frequency domains. The boundaries of the \(\mu\) values are shown in Figure 6. As can be seen, the system continues to function robustly and steadily because the upper limits of both values remain below 1 for all relevant frequencies. This finding is confirmed in Figure 7, where the applied voltage and free-end movement are displayed at the maximum degree of uncertainty.