## 3 Results  
### Application in Smart Structures  
This section examines an eight-element cantilever smart structure (Figure 1a), in which four pairs of piezoelectric patches at the top and bottom surfaces of each beam element are symmetrically bonded. The measurements of several sections of this building are shown below, and Table 1 lists the beam specifications. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline L, for beam length: & 1.20 m \\ \hline W, for beam width: & 0.08 m \\ \hline h, for beam thickness: & 0.02 m \\ \hline \(\rho\), for beam density: & 1800 kg/m\({}^{3}\) \\ \hline E, for Young’s modulus of the beam: & \(1.5\times 10^{11}\) N/m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the smart structure. Figure 1: _Cont._  
Smart structures often involve the integration of sensors, actuators, and control systems to adapt to changing environmental conditions. The dynamical equation for smart structures can be complex and depend on the specific configuration and components used. In the next section, we examine the performance of an eight-element cantilever smart structure featuring four pairs of piezoelectric patches symmetrically affixed to both the top and bottom surfaces of each beam element. The section that follows lists the dimensions of each part of the intelligent structure. The following equations clarify the dynamic characterization of the system:  
\[\text{M}\tilde{q}(t)+\text{D}\dot{q}+\text{K}q(t)=fm(t)+fe(t) \tag{4}\]  
These equations capture the essence of how smart structures incorporate active and robust controls to adapt to dynamic loads or changing conditions. The challenge lies in designing effective control algorithms and systems that respond optimally to the state of the structure and external forces. Where \(K\) represents the global stiffness matrix, \(fm\) is the global external loading vector, and \(fe\) is the global control force vector produced by the electromechanical coupling effect, which is often determined by a feedback control system that uses sensor information to adjust the behavior of the structure in real-time. The overall goal is to optimize the performance or response of the structure under varying conditions. \(M\) represents the global mass matrix; \(D\) represents the viscous damping matrix. The mass \(M\) and stiffness \(K\) matrices were derived by assembling the local mass and stiffness matrices. The analysis was performed using [PERSON]'s theory, and we had two degrees of freedom at each node: vertical displacement and rotation. Damping \(D\) is a small percentage (0.0005) of the mass and stiffness matrices. The rotations \(\psi_{i}\) and vertical displacement \(w_{i}\) constitute independent variable \(q(t)\), i.e.,  
\[q(t)= \begin{bmatrix}w_{1}\\ \psi_{1}\\ \vdots\\ w_{n}\\ \psi_{n}\end{bmatrix}  \tag{5}\]  
where \(n\) denotes the number of finite elements used in the analysis. Vectors \(w\) and \(fm\) were positive upward. Figure 1: (**a**) Modeling of intelligent structure. (**b**) The block diagram in Matlab Simulink for the smart beam disturbance rejection. (**c**) The robust control design diagram of \(\mu\)-synthesis. To convert to a state-space control representation, allow (in the usual manner):  
\[x(t)= \begin{bmatrix}q(t)\\ \dot{q}(t)\end{bmatrix}  \tag{6}\]  
In addition, fe(t) can be defined as Bu(t) by expressing it as \(\mathrm{F}_{\mathrm{e}}^{*}\mathrm{u}\), where \(\mathrm{F}_{\mathrm{e}}^{*}\) (of size \(2\mathrm{n}\times\mathrm{n}\)) stands for the piezoelectric force for a unit established on the associated actuator [34; 35; 36], and u represents the voltages on the actuators [34; 35; 36]. Finally, \(d(t)=fm(t)\) denotes the disturbance vector. Then,  
\[