## 3 System Identification
The system identification toolbox (Matlab software, version 7.0, Mathworks, Natick, MA, USA) was used for system identification. Based on Equation (5), the identified transfer function using the chirp signal is expressed as Equation (7). For comparison, the identified transform function depicted as Equation (2) is also given in Equation (8). The modal parameters for the two identified models are detailed in Table 4. Meanwhile, Bode diagram of the two identified transfer functions in the frequency range of interests (0.1-1000 rad/s) are plotted in Figure 5.  
Figure 4: Measured frequency response of the system. \[G_{c}(s)=\sum_{i=1}^{m}\frac{k_{i}}{s^{2}+2\zeta_{i}\omega_{i}s+\omega_{i}^{2}}+D \tag{2}\]0 Time response of the chirp excitation and measured output signal: (**a**) chirp excitation signal; and (**b**) sensor output signal. \[G_{c}(s)=\sum_{i=1}^{m}\frac{k_{i}}{s^{2}+2\zeta_{i}\omega_{i}s+\omega_{i}^{2}}+D \tag{2}\]1  
\[G_{c}(s)=\sum_{i=1}^{m}\frac{k_{i}}{s^{2}+2\zeta_{i}\omega_{i}s+\omega_{i}^{2}}+D \tag{2}\]2  
It can be observed from Table 4 that the first two resonance frequencies of the identified models are 1.48 and 9.06 Hz, which are nearly the same as the results obtained from the measured frequency response in \[G_{c}(s)=\sum_{i=1}^{m}\frac{k_{i}}{s^{2}+2\zeta_{i}\omega_{i}s+\omega_{i}^{2}}+D \tag{2}\]3 Meanwhile, the two identified models have nearly the same damping ratios for the first two modes. Bode diagrams of the two identified models have similar characteristics nearby the resonance frequencies, as shown in \[G_{c}(s)=\sum_{i=1}^{m}\frac{k_{i}}{s^{2}+2\zeta_{i}\omega_{i}s+\omega_{i}^{2}}+D \tag{2}\]4 The values of the zeros and poles of the two models are listed in Table 5 based on the ascending order of the image parts. Their locations in the two models are also given in \[G_{c}(s)=\sum_{i=1}^{m}\frac{k_{i}}{s^{2}+2\zeta_{i}\omega_{i}s+\omega_{i}^{2}}+D \tag{2}\]5  
It is worth noting that the locations of the poles of the two identified models coincide highly, while the locations of the zeros of the two identified models are totally different, which is because the locations of the zeros are distorted due to the omission of the higher modes, and different system models lead to different relocation of the zeros as depicted before. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{2}{c}{**Natural Frequencies**} & \multicolumn{2}{c}{**Damping Ratio**} \\ \cline{2-5}  & \(f_{1}\) & \(f_{2}\) & \(\xi_{1}\) & \(\xi_{2}\) \\ \hline \(G_{c}(s)\) & 1.48 & 9.06 & 0.004 & 0.004 \\ \(G(s)\) & 1.48 & 9.06 & 0.004 & 0.004 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of modal parameters for the identified models. Figure 5: Bode diagrams of the identified models. System responses of the two identified models excited by the same chirp signal are illustrated in Figure 7a,b, both in the time and frequency domains. Besides, the measured system response of the smart structures is also given for comparison. As seen from these figures, system responses of the two identified models are both in good agreement with the experimental response. An excellent match between the identified models and experimental results can be noticed nearby the two resonance frequencies. However, at the beginning period of the chirp excitation (\(t<1.5\) s), there are some differences between the model \(G_{c}\)(s) and the measured system responses. It is noticed that system responses of the identified model \(G\)(s) show a better agreement with the experimental responses than that of the model \(G_{c}\)(s). As a result, the identified model \(G\)(s) is accurate both in the time and frequency domains. It should be pointed out that the proposed structure behaves like a collocated system, as shown in Table 4 and Figure 6 [28]. This is because the distance between the strain gauges and the actuators is small, compared to the beam length (the distance between the actuators and sensors is only 63.4 mm, while the length of the beam is 1 m).