# Novelty in Intelligent Controlled Oscillations in Smart Structures
## 2 Motion Formula of the Smart Structure
### Connection to the Issue of Beam Control  
The given problem involves taking the disturbance (d) and measurement noise (n) as inputs and producing displacement measurements (y) and controller voltages (u) as the outputs. The provided structural diagrams and equations were employed to simulate this particular issue involving a beam and were implemented using MATLAB v. 5.0. In the frequency domain, the objective is to determine the optimal transfer function N. To achieve this, it is beneficial to derive the input-output relationships for the initial model [16; 21]. \[ \begin{bmatrix}\text{u}\\ \text{e}\end{bmatrix} =\text{F}(\text{s}) \begin{bmatrix}\text{d}\\ \text{n}\end{bmatrix} \Rightarrow\text{z}=\text{F}(\text{s})\text{w}\]  
Figure 5: Smart structure. as illustrated in Figure 5.  
In Figure 6, K(s) is the controller, d is the disturbance, e is the error, n is the noise, u is the control voltage, and x is the state vector. Where the beam is explained by [31]  
\[\dot{\mathrm{x}}(\mathrm{t})=\mathrm{Ax}(\mathrm{t})+[\mathrm{B}\,\mathrm{G}] \begin{bmatrix}\mathrm{u}(\mathrm{t})\\ \mathrm{d}(\mathrm{t})\end{bmatrix} \]  
In the frequency domain, the transfer function H(s) is  
\[\mathrm{H}(\mathrm{s})=(\mathrm{sI}-\mathrm{A})^{-1} \tag{15}\]  
J is used to select the states that we aim to control, which may differ from y. In most studies, J is  
\[\mathrm{J}=\left[ \begin{array}{cccccccccccccccccccc}1&0&0&0&0&0&0&0&0&0&0&0&0&0 &0&0\\ 0&0&1&0&0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&1&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&1&0&0&0&0&0&0&0&0\end{array} \right] \tag{16}\]  
We commenced by gradually re-performing Figure 6.  
It is clear from Figure 7 that the disruption to the inaccuracy in the transfer function is  
\[\mathrm{T}_{\mathrm{de}}=\mathrm{J}\cdot(\mathrm{I}-\mathrm{HBKC})^{-1}\mathrm{ H}\cdot\mathrm{G} \tag{17}\]  
Equation (17) is a frequency domain equation, and an attempt was made to derive the transfer function Tde that relates the external disturbance to the error. Figure 6: Beam with noise output, error, disturbance input, and controller. Figure 7: Block diagram disturbance and errors. It is evident from Figure 8 that the noise-to-error transfer function is  
\[\text{T}_{\text{ne}}=\text{J}\cdot(\text{I}-\text{HBKC})^{-1}\text{HBK} \tag{18}\]  
Figure 9 illustrates the disturbance for controlling the transfer function:  
\[\text{T}_{\text{du}}=(\text{I}-\text{KCHB})^{-1}\text{KCH}\cdot\text{G} \tag{19}\]  
Figure 10 shows that the noise-to-control transfer function is  
\[\text{T}_{\text{nu}}=(\text{I}-\text{KCHB})^{-1}\text{K} \tag{20}\]  
When we combine these, we obtain  
\[\text{e}=\text{J}\cdot(\text{I}-\text{HBKC})^{-1}\text{H}\cdot\text{Gd}+\text {J}\cdot(\text{I}-\text{HBKC})^{-1}\text{HBKn} \tag{21}\]  
\[\text{u}=(\text{I}-\text{KCHB})^{-1}\text{KCH}\cdot\text{Gd}+(\text{I}-\text{ KCHB})^{-1}\text{Kn} \tag{22}\]  
or  
\[ \begin{bmatrix}\text{u}\\ \text{e}\end{bmatrix} = \begin{bmatrix}(\text{I}-\text{KCHB})^{-1}\text{KCHG}&( \text{I}-\text{KCHB})^{-1}\text{K}\\ \text{J}(\text{I}-\text{HBKC})^{-1}\text{HG}&\text{J}(\text{I}-\text{HBKC})^{- 1}\text{HBK}\end{bmatrix}\begin{bmatrix}\text{d}\\ \text{n}\end{bmatrix}  \tag{23}\]  
\[ \begin{bmatrix}\text{u}\\ \text{e}\end{bmatrix} = \begin{bmatrix}\text{Fu}_{\text{du}}&\text{Fu}_{\text{ nu}}\\ \text{F}_{\text{de}}&\text{F}_{\text{ne}}\end{bmatrix}\begin{bmatrix}\text{d}\\ \text{n}\end{bmatrix} \Rightarrow\text{z}=\text{F}(\text{s})\text{w} \tag{24}\]  
We proceeded by making the necessary weighting adjustments and redesigning Figure 6 to fit our particular issue. Figure 8: Noise and errors in block diagram form. Figure 10: Noise and control voltages presented in a block diagram form. Figure 9: Disturbance and control voltages presented in a block diagram form. Next, we created a new version of Figure 11, restructuring it into a two-port diagram. This new diagram should be formatted in a manner similar to that depicted in Figure 6. This comparison will help us better understand the differences and similarities between the two representations.