## 3 System Identification  
System identification can be used to experimentally extract the dynamic model of a system. For system identification, a mathematical model of the beam is necessary. Due to the spatially distributed nature of flexible structures, it is highly desirable to work with transfer functions, thus the vibration dynamics of a cantilever beam can be written as Equation (1)  
\[G(s)=k\sum_{n=1}^{\infty}\frac{1}{s^{2}+2\zeta_{n}\omega_{n}s+\omega_{n}^{2}} \tag{1}\]  
where \(k\) is the gain of the open loop, \(n\) is the mode number, \(\zeta_{n}\) and \(\omega_{n}\) are the damping ratio and natural frequency of the \(n\)th mode, respectively. However, it is impractical to identify the infinite number of vibration modes. In practice, only a certain bandwidth is of interest, and dynamic response of the flexible structures is often dominated by the vibrations of the lower modes. Therefore, some form of mode truncation for the higher modes is needed. A common reduced order model used by [PERSON] et al. (2015) and [PERSON] and [PERSON] (2017) is given as  
\[G_{c}(s)=\sum_{i=1}^{m}\frac{k_{i}}{s^{2}+2\zeta_{i}\omega_{i}s+\omega_{i}^{2}}+D \tag{2}\]  
where \(m\) is the number of modes kept in the model, \(k_{i}\) is the gain of the open loop for each mode, and \(D\) is a feedthrough term, which is added to compensate for the modal truncation effect. However, this neglect of higher modes still leads to the appearance of the direct energy transmission ([PERSON], 2017). Therefore, the poles of the truncated system, which represent the resonance frequencies, are natural property of the system and independent among different modes, but the locations of the zeros will be distorted due to the omission of the higher modes. Besides, the PZT actuators are located a distance of 63.4 mm from the strain sensors, these locations may lead to sort of time delay. Above all, the truncated model with a pure delay element is given by Equation (3)  
\[G(s)=k{e^{-\tau s}}\sum_{i=1}^{m}\frac{(s-a_{i})(s-b_{i})}{s^{2}+2\zeta_{i} \omega_{i}s+\omega_{i}^{2}} \tag{3}\]  
where \(a_{i}\) and \(b_{i}\) are the relocated zeros of the \(i\)th mode due to the omission of the higher modes. \(\tau\) is the delay time. Using Taylor series expansion, the pure delay element can be approximated as Equation (4)  
\[{e^{-\tau s}}=\left[\sum_{j=0}^{\infty}\frac{(\tau s)^{n}}{j!}\right]^{-1}= \left[1+\tau s+\dots\frac{(\tau s)^{n}}{j!}\dots\right]^{-1}\approx\frac{1}{ 1+\tau s} \tag{4}\]  
Then, the truncated model of the smart structure can be rewritten as Equation (5)  
\[G(s)=\frac{k}{1+\tau s}\sum_{i=1}^{m}\frac{(s-a_{i})(s-b_{i})}{s^{2}+2\zeta_{ i}\omega_{i}s+\omega_{i}^{2}} \tag{5}\]  
The pure delay element is approximated by a first-order inertia element as shown in Equation (5). Experiments were conducted to identify the parameters of the system model. A chirp signal with a frequency varies from 0.1 to 15 Hz was applied to the PZT actuators to have the chance of exciting the first two modes of the structure. The magnitude of the excitation signal was 5 V, and the identification experiment took 15 s. The sampling frequency was 200 Hz, and a low-pass filter was utilized with a cut-off frequency of 100 Hz. The time response signals for chirp excitation are plotted in Figure 3a,b. To find the transfer function of the system in the frequency domain, Fourier transform is performed on the chirp excitation signal \(u(t)\) and sensor output signal \(y(t)\). The frequency response function (FRF) of the open loop system is given by Equation (6)  
\[G(j\omega)=\frac{S_{uy}(j\omega)}{S_{uu}(j\omega)} \tag{6}\]  
where \(S_{uy}(j\omega)\) is the cross spectral density between signal \(u(t)\) and \(y(t)\), and \(S_{uu}(j\omega)\) is the auto spectral density of signal \(u(t)\). The measured frequency response of the system is plotted in Figure 4. It can be seen that the natural frequencies of the first two modes of the smart structure are 1.47 and 9.06 Hz, respectively.