## 3 Results
### Robust Synthesis: \(\mu\)-Controller  
\(\mu\)-Analysis, also known as mu-analysis, is a technique commonly used in control system design to analyze and address uncertainties in dynamic systems. This is particularly useful for understanding how uncertainties affect the robustness and stability of control systems. We applied \(\mu\)-analysis to our smart structures. In \(\mu\)-analysis, the goal is to assess the system performance in the presence of uncertainty by defining a performance metric denoted by \(\mu\). The uncertainty is bounded by identical constraints with a constant \(\mathrm{m_{p}}\) and \(\mathrm{k_{p}}\)[29; 37; 38; 39]. All the results were obtained using MATLAB Simulink (Figure 1b); moreover, in all simulations, noise as a percentage \(\pm\) 2% of the disturbance was obtained. The results are for the displacements and rotations of the smart beam nodes. In Figure 1b, a block diagram for smart beam disturbance rejection in MATLAB Simulink is presented. In the first result, a concentrated load of 10 N was applied to the edge of the support. Subsequently, a sinusoidal load with an oscillation part of 10 N and a period of 1 s was applied. Therefore, for the first mechanical load, we have 10 N at the edge of the cantilever, while, for the second mechanical load, we have a sinusoidal load in the 10 N range with a period of 1 s.  
This study uses advanced control techniques and finite element modeling with applications to smart materials. For this reason, because modeling requires many computational requirements, experimental results will be the subject of future research. m\({}_{\text{p}}\) = 0, k\({}_{\text{p}}\) = \(\pm\)0.9: This relates to a deviation of approximately 90% from the nominal value of stiffness matrix K (Equations (9) and (10)). As previously stated, the following commands were required to conduct this process in the MATLAB software platform (Mathworks, Natick, MA, USA) and are presented in Appendix A of the manuscript. This command generates a robust controller on the order of 56. This is a large value because of the operation of this algorithm. Despite being mentioned in the literature, this point has not been given enough emphasis, which is undoubtedly an oversight. To the best of our knowledge, lowering the order cannot be performed quickly without employing a laborious manual process. Figure 1c shows the \(\mu\)-synthesis robust control design framework. The white matrix for noise, control, error, and disturbance is for the control (u):  
\[W_{u}= \begin{bmatrix}\frac{7.856\times 10^{-6}s+0.005556}{s+100}&0&0&0\\ 0&\frac{7.856\times 10^{-6}s+0.005556}{s+100}&0&0\\ 0&0&\frac{7.856\times 10^{-6}s+0.005556}{s+100}&0\\ 0&0&0&\frac{5.556\times 10^{-6}s+0.005556}{s+100}\end{bmatrix}  \tag{16}\]  
For error e:  
\[W_{\text{e}=} \begin{bmatrix}\frac{22.723}{s+31.42}&0&0&0\\ 0&\frac{22.723}{s+31.42}&0&0\\ 0&0&\frac{22.723}{s+31.42}&0\\ 0&0&\frac{22.723}{s+31.42}&0\\ 0&0&0&\frac{22.723}{s+31.42}\end{bmatrix}  \tag{17}\]  
For disturbances d:  
\[W_{\text{d}=} \begin{bmatrix}20&0&0&0&0&0&0\\ 0&20&0&0&0&0&0\\ 0&0&20&0&0&0&0\\ 0&0&0&20&0&0&0\\ 0&0&0&20&0&0&0\\ 0&0&0&0&0&20&0\\ 0&0&0&0&0&0&20&0\\ 0&0&0&0&0&0&20\end{bmatrix}  \tag{18}\]  
For noise n:  
\[W_{n=} \begin{bmatrix}10^{-5}&0&0&0\\ 0&10^{-6}&0&0\\ 0&0&10^{-6}&0\\ 0&0&0&10^{-5}\end{bmatrix}  \tag{19}\]  
In the \(\mu\)-synthesis robust control design diagram, the primary focus is on enhancing the robustness of the system to uncertainties. The process involves evaluating the performance metric \(\mu\) by analyzing the closed-loop transfer function, considering both the system transfer function T(s) and uncertainty transfer function \(\Delta\)(s). The goal is to optimize the controller parameters to maximize \(\mu\), ensuring improved stability and performance in the presence of uncertainties.