## 5 System Identification and Notch Filter Design  
To design an appropriate controller, the structural dynamics of the system should be fully characterized. A frequency-domain system identification method was applied to derive the model of the system. Figure 12 shows the frequency response function of the structural model to be identified. The damping ratio of each peak can be estimated by the quadrature peak picking method. It is a classical method for determining the damping at resonance, and half power (\(-3\) dB) points of each resonance magnitude should be identified. For a particular mode, the damping ratio \(\zeta_{m}\) can be found from Equation (49). Figure 13 illustrates the concept of this method. \[\zeta_{m}=\frac{\omega_{b}-\omega_{a}}{2\omega_{d}},\ |H(\omega_{a})|=|H(\omega_{b})|= \frac{|H(\omega_{d})|}{\sqrt{2}} \tag{49}\]  
The accuracy of this method depends on the frequency resolution used for the measurement, since it determines how accurately the peak magnitude can be measured. For lightly damped structures, high-resolution analysis is required to measure the peak exactly. Consequently, repeated measurements at each resonance frequency are normally required to achieve sufficient accuracy. The obtained resonances are shown in Table 2.  
Figure 11: Frequency response function of beam structure: (**a**) two different configurations; and (**b**) comparison of frequency response function. Figure 12: Frequency response function of the structural model to be identified. A transfer function can be identified for the object structure based on the frequency response data. The least square estimation of the linear system parameters is presented as a complex-domain formulation [51]. Figure 14 shows a general model of the frequency-domain identification of linear systems. The system is represented by its transfer function \(H(s)\), where \(s=j\omega\) in the Laplace domain. \(H(s)\) has a rational form that is extended by a delay term \(e^{-j\omega t_{d}}\):  
\[H(s)=\frac{n_{m}s^{m}+n_{m-1}s^{m-1}+\cdots+n_{1}s+n_{0}}{d_{n}s^{n}+d_{n-1}s^{ n-1}+\cdots+d_{1}s+d_{0}}\cdot e^{-j\omega t_{d}} \tag{50}\]  
The excitation signal is represented by a Fourier series, with complex amplitudes \(U_{i}\) at angular frequencies \(\omega_{i}\). The response of the system is \(Y_{i}=H(\omega_{i})U_{i}\). Both the measured input and output complex amplitudes are corrupted by noise terms \(n_{u}\) and \(n_{y}\), respectively (errors in variables model), which are usually assumed to be Gaussian, uncorrelated between the input and output, and uncorrelated between different frequency points. Based on the above assumptions, a maximum-likelihood estimate of the system parameters is developed in real terms. The time-domain measurements are transformed into the frequency domain, and the calculated complex input and output amplitudes at selected angular frequencies \(\omega_{i},\;\;i=1,\cdots,G\) are \(U_{mi}\) and \(Y_{mi}\), respectively. The unknown parameters are the unknown coefficients of the numerator and the denominator of the transfer function (Equation (49)), which are collected in the vector \(\mathbf{C}\). Furthermore, \begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Peak Number** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline
**Frequency (Hz)** & 14.0 & 30.0 & 112.0 & 281.0 & 539.0 & 833.0 & 896.0 & 1330.0 \\
**Damping Ratio** & 0.0138 & 0.0222 & 0.0204 & 0.0074 & 0.0177 & 0.0058 & 0.0047 & 0.0053 \\ \hline \end{tabular}
\end{table} 
Table 2: Natural frequency and damping ratio of each peak. Figure 14: Frequency-domain identification of linear systems. Figure 13: Quadrature peak picking method. the exact complex input and output amplitudes \(\mathbf{U}\) and \(\mathbf{Y}\) are also unknown.