## 4 Semi-Active Vibration Control of the Thin Plate Using the ERF Embedded ATVA
### Semi-Active Vibration Control Using the Interval Type-2 Fuzzy
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Shaker Amplitude (mV\({}_{\text{pk}}\))** & **50** & **100** & **200** & **50** & **100** & **200** \\ \hline \multicolumn{8}{c}{8 Hz Displacement (mV\({}_{\text{rms}}\))} & 10 Hz Displacement (mV\({}_{\text{rms}}\)) \\ \hline \multirow{4}{*}{E (kV/mm)} & 1 & 98 & 201 & 413 & 162 & 312 & 633 \\  & 1.5 & 82 & 181 & 363 & 148 & 267 & 515 \\  & 2 & 86 & 162 & 338 & 129 & 218 & 403 \\ \hline \multicolumn{8}{c}{13 Hz Displacement (mV\({}_{\text{rms}}\))} & 17 Hz Displacement (mV\({}_{\text{rms}}\)) \\ \hline \multirow{4}{*}{E (kV/mm)} & 1 & 17 & 35 & 68 & 98 & 194 & 378 \\  & 1.5 & 14 & 28 & 53 & 85 & 181 & 363 \\  & 2 & 8 & 13 & 27 & 72 & 155 & 352 \\ \hline \multicolumn{8}{c}{19 Hz Displacement (mV\({}_{\text{rms}}\))} & 21 Hz Displacement (mV\({}_{\text{rms}}\)) \\ \hline \multirow{4}{*}{E (kV/mm)} & 0 & 33 & 75 & 153 & 28 & 54 & 110 \\  & 0.5 & 36 & 77 & 158 & 25 & 52 & 106 \\  & 1 & 37 & 78 & 159 & 26 & 53 & 108 \\ \hline \multicolumn{8}{c}{23 Hz Displacement (mV\({}_{\text{rms}}\))} & 25 Hz Displacement (mV\({}_{\text{rms}}\)) \\ \hline \multirow{4}{*}{E (kV/mm)} & 0 & 21 & 41 & 80 & 16 & 34 & 69 \\  & 0.5 & 22 & 41 & 82 & 17 & 34 & 70 \\  & 1 & 22 & 41 & 81 & 16 & 34 & 71 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Frequency responses of this smart structure with respect to the different external electric field applied to ERF for the external vibration at 8–25 Hz. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Shaker Amplitude (mV\({}_{\text{pk}}\))** & **50** & **100** & **200** & **50** & **100** & **200** \\ \hline \multicolumn{8}{c}{74 Hz Displacement (mV\({}_{\text{rms}}\))} & 76 Hz Displacement (mV\({}_{\text{rms}}\)) \\ \hline \multirow{4}{*}{E (kV/mm)} & 1 & 4 & 7 & 13 & 9 & 20 & 39 \\  & 1.5 & **3** & **6** & **12** & **8** & **19** & **37** \\  & 2 & 5 & 8 & 14 & 10 & 20 & 37 \\ \hline \multicolumn{8}{c}{78 Hz Displacement (mV\({}_{\text{rms}}\))} & 80 Hz Displacement (mV\({}_{\text{rms}}\)) \\ \hline \multirow{4}{*}{E (kV/mm)} & 0 & 26 & 53 & 104 & **15** & **30** & 61 \\  & 0.5 & 18 & 33 & 68 & 15 & 30 & **60** \\  & 1 & 16 & 29 & 58 & 20 & 41 & 78 \\ \cline{1-1}  & 1.5 & **14** & **23** & **47** & 23 & 46 & 87 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Frequency responses of this smart structure with respect to the different external electric field applied to ERF for the external vibration at 74–80 Hz. Type-2 Fuzzy Controller Design for ATVA  
An interval type-2 Fuzzy Logic System (FLS) is studied for vibration control in the presence of system modeling uncertainties [17]. The type-2 fuzzy system has a similar structure with type-1 fuzzy, but the major difference is that the rule base has antecedents and consequents using type-2 fuzzy sets. First, the first mode of vibration is considered, a Gaussian function with a known standard deviation and a uniform weighting were used to represent a footprint of uncertainty as shaded in Figure 13. Because of using such a uniform weighting, it is usually called an interval type-2 fuzzy set (IT2 FS) [72]. For the IT2 FS, a type-reducer is needed to convert to a type-1 fuzzy set before defuzzification is carried out as shown in Figure 14. To obtain the output of the IT2 FS, we need two fuzzy rule tables; however, the same fuzzy rule table is used in this case as shown in Table 9. After obtaining the inference output, the type-reducer and defuzzifier are used to obtain the input-output relation and the two signals (\(D_{rms}\),\(f\)) were fed back to the proposed fuzzy controller. For type-2 fuzzy system, Figure 15 shows the inference operation to obtain the output by using the following equations. After obtaining the inference output, we used the type-reducer and defuzzifier to obtain the input-output relation as shown in Figure 16.  
\[\overline{f}=\prod_{i=1}^{N}\overline{u}_{\bar{A}}(x_{i}) \tag{7}\]  
\[\underline{f}=\prod_{i=1}^{N}\underline{u}_{\bar{A}}(x_{i}) \tag{8}\]  
where \(N\) is the number of the inputs.