## 3 Results
### Application in Smart Structures
\begin{split}\dot{x}(t)=\begin{bmatrix}0_{2n\times 2n}&I_{2n\times 2n}\\ -M^{-1}K&-M^{-1}K\end{bmatrix}x(t)+\begin{bmatrix}0_{2n\times n}\\ M^{-1}F_{\mathrm{e}}^{*}\end{bmatrix}u(t)+\begin{bmatrix}0_{2n\times 2n}\\ M^{-1}\end{bmatrix}d(t)\\ &=Ax(t)+Bu(t)+Gd(t)\\ &=Ax(t)+\begin{bmatrix}B&G\end{bmatrix}\begin{bmatrix}u(t)\\ d(t)\end{bmatrix}\\ &=Ax(t)+B\widetilde{u}(t)\end{split}  \tag{7}\]  
In this form, \(d\) is a \(2\mathrm{n}\times 1\) vector (where n is the number of nodes), and u is, at most, an \(\mathrm{n}\times 1\) vector (but it may be smaller). The units used were Newton (N), seconds (s), radians (rad), and meters (m). With a more detailed investigation of the stability in the frequency domain, our approach preserves the stability in the time domain [27]. \[\mathrm{y(t)=[x_{1}(t)\ x_{3}(t)\...\ x_{n-1}(t)]^{T}=Cx(t)} \tag{8}\]  
This state-space representation captures the first-order dynamics of the smart structure, where the state vector x(t) includes both displacement and velocity. Let us now use the following approach to the uncertainty in the M and K matrices:  
\[ \begin{split}\mathrm{K=K_{0}\ \left(I+k_{p}I_{2n\times 2n}\delta_{K} \right)}\\ \mathrm{M=M_{0}\ \left(I+m_{p}I_{2n\times 2n}\delta_{K}\right)} \end{split}  \tag{9}\]  
Furthermore, since D = 0.0005 (K + M), a suitable form for D is as follows:  
\[ \begin{split}\mathrm{D=0.0005}\big{[}\mathrm{K_{0}\big{(}I+k_{p} I_{2n\times 2n}\delta_{K}\big{)}+M_{0}\big{(}I+m_{p}I_{2n\times 2n}\delta_{M} \big{)}\big{]}=}\\ \mathrm{D_{0}+0.0005}\big{[}\mathrm{K_{0}k_{p}I_{2n\times 2n} \delta_{K}+M_{0}m_{p}I_{2n\times 2n}\delta_{M}\big{]}}\end{split}  \tag{10}\]  
However, given that, in general:  
\[\mathrm{D=\alpha K+\beta M}\]  
The structural damping matrix \(\mathrm{D}\) can be examined as a linear combination of mass and stiffness (Rayleigh damping). In this context, the values for \(\alpha\) and \(\beta\) were determined based on the first and second normal modes of vibration, with both \(\alpha\) and \(\beta\) set at 0.0005. D can be expressed in a manner similar to K and M, as follows:  
\[\mathrm{D=D_{0}\ (I+d_{p}I_{2n\times 2n}\delta_{D})} \tag{11}\]  
In the pertinent matrices, we incorporate uncertainty in the form of proportional deviation. Given the precision with which the length can be measured, this formulation of uncertainty aligns well with our context. It is more probable that uncertainty originates from factors other than the primary matrices. In this context, we assume that  
\[\left\|\Delta\right\|_{\infty}\overset{\mathrm{def}}{=}\left\| \begin{bmatrix} \mathrm{I_{n\times n}\delta_{K}}&0_{n\times n}\\ 0_{n\times n}&\mathrm{I_{n\times n}\delta_{M}}\end{bmatrix} \right\|_{\infty}<1 \tag{12}\]Hence, \(\mathrm{m_{p}}\) and \(\mathrm{k_{p}}\) are used to scale the value of the proportion, and the nominal values are represented by the zero subscript. (It is urged that, for matrix \(\mathrm{A_{n\times m}}\), the norm is determined via \(\|\mathrm{A}\|\)\(\infty\)\(=\)\(\max\limits_{1\leq j\leq m}\sum_{j=1}^{n}|\mathrm{a_{ij}}|\).) Taking these specifications into consideration, Equation (4) changes to,  
\[ \begin{split} M_{\mathrm{O}}\big{(}\mathrm{I}+m_{p}I_{2n\times 2 n}\delta_{M}\bar{q}(t)\big{)}+Ko\big{(}I+k_{p}I_{2n\times 2n}\delta_{K}q(t)\big{)}+ \Big{[}D+0.0005\Big{[}Kok_{p}I_{2\times 2}\delta_{K}+M_{\mathrm{O}}m_{p}I_{2 \times 2}\delta_{M}\Big{]}\hat{q}(t)+\\ f_{m}(t)+f_{e}(t)\\ \Rightarrow M_{\mathrm{O}}\bar{q}(t)+D_{\mathrm{O}}\hat{q}(t)+K_ {\mathrm{O}}q(t)=-\Big{[}M_{\mathrm{O}}m_{p}I_{2n\times 2n}\delta_{M}\bar{q}(t)+0.00 5\Big{[}Kok_{p}I_{2\times 2}\delta_{K}+M_{\mathrm{O}}m_{p}I_{2\times 2}\delta_{M} \Big{]}\hat{q}(t)+\\ K_{\mathrm{O}}k_{p}I_{2n\times 2n}\delta_{K}q(t)\Big{]}+f_{m}(t)+f_{e}(t) \\ \Rightarrow M_{\mathrm{O}}\bar{q}(t)+D_{\mathrm{O}}\hat{q}(t)+K_ {\mathrm{O}}q(t)=\widetilde{D}q_{u}(t)+f_{m}(t)+f_{e}(t)\end{split}  \tag{13}\]  
where:  
\[