## 5 System Identification and Notch Filter Design
The basic equations can be written as follows for the exact values (Equation (50)) and for the noisy observations (Equation (51)):  
\[Y_{i}=H(\omega_{i},\ \mathbf{C})U_{i},\ i=1,2,\cdots,G \tag{51}\]  
\[Y_{mi}=H(\omega_{i},\ \mathbf{C})\cdot(U_{mi}-n_{ui})+n_{yi},\ i=1,2,\cdots,G \tag{52}\]  
The maximum-likelihood estimate of the unknown quantities is equivalent to the minimization of the following cost function:  
\[J(\mathbf{U},\mathbf{Y},\mathbf{C})=\sum_{i=1}^{\mathrm{G}}\left(\frac{(U_{ mi}-U_{i})(\overline{U_{mi}-U_{i}})}{2\sigma_{ui}{}^{2}}\right)+\sum_{i=1}^{ \mathrm{G}}\left(\frac{(Y_{mi}-Y_{i})(\overline{Y_{mi}-Y_{i}})}{2\sigma_{yi}{}^ {2}}\right), \tag{53}\]  
which is subject to the constraints \(Y_{i}=H(\omega_{i},\ \mathbf{C})U_{i}\), \(i=1,2,\cdots,G\) The upper bar means the complex conjugate, and \(\sigma_{ui}\) and \(\sigma_{yi}\) are the standard deviations of the input and output, respectively. Using this above methodology, a transfer function with 9 zeros and 11 poles has been identified, which has four major peaks up to around 500 Hz. These four peaks should be attenuated with proper control methods. Figure 15 compares the frequency response functions from the experiment and the system identification, and Equation (53) shows the formulation of the transfer function. This result seems to fit with the experimental results for up to 500 Hz. Figure 15 also shows the root locus for checking which poles are related to each peak. Some parts of the locus go over the right half plane, which indicates instability. \[G(s)=\frac{10^{6}s(s^{2}+140s+13530)(s^{2}+6s+18250)(s^{2}+10s+418800)(s^{2}+1 0s+5153000)}{(s+30)(s^{2}+2.4s+7738)(s^{2}+8s+34360)(s^{2}+28s+477700)(s^{2}+2 6s+3117000)(s^{2}+120s+11470000)}. \tag{54}\]  
With the identified poles and zeros of the system, notch filters are designed to reduce the amplitude of each peak. A stop band filter (BSF) or a notch filter refers to a band-elimination or blocking filter that exhibits sharp attenuation characteristics in only a specific frequency band. Peak amplitudes can be reduced by adding zeros near poles to maintain the DC gain. Since two zeros are added for each peak, two poles should also be added, which is similar to replacing the damping ratio at each resonance frequency. These are the resonance frequencies and damping ratios of each peak. The filters are derived by maintaining \(\omega_{n}\) and changing the damping ratio \(\zeta\) to 0.7. This is done to confirm that an actual solution exists and that enough control authority can be obtained from the actuators for the actual system. Table 3 shows the transfer functions of the notch filters based on Equation (54), in which the original damping ratio \(\zeta_{z}\) replaces the higher damping ratio \(\zeta_{p}\) at each resonance frequency so that the amplitudes at each peak can be attenuated. Figure 15: Comparison of: (**a**) frequency response functions; and (**b**) root locus. \[G_{c}(s)=\frac{s^{2}+2\zeta_{z}\omega_{n}s+{\omega_{n}}^{2}}{s^{2}+2\zeta_{p}\omega _{n}s+{\omega_{n}}^{2}} \tag{55}\]  
As shown in Figure 16, the amplitudes of the peaks are significantly reduced compared to the original system, and the root locus of the filtered system has been changed. The part in the right-hand plane is automatically removed by these filters with the added zeros near poles. The control results are also described in the time domain. Numerical simulations were conducted with impulse and step inputs to the system. Figures 17 and 18 show the impulse response and the step response of the structure. The notch filters show dramatically reduced vibration amplitudes compared to the case with no control. However, these filters will work on only the baseline structure, assuming that the identified model is accurate.