## 2 Model Identification of Ground-Based Laser Communication Servo Turntable
### System Model Identification  
For practical engineering application, system model identification utilized the mathematical method of medium-low frequency linear fitting and high-frequency two-order model fitting. The essence of this method is to consider the middle and low frequency bands of the system as an ideal linear model and then a polynomial mathematical model is adopted to fit the frequency characteristic curve of the middle and low frequency bands. For the high frequency band, the multi-order resonance frequency in the frequency characteristic curve is fitted by one or more two-order mathematical models. Therefore, the ideal mathematical model of the system can be expressed as follows:  
\[G(s)=G_{L}(s)G_{H_{1}}(s)G_{H_{2}}(s)\cdots G_{H_{n}}(s). \tag{2}\]  
In view of Equation (2), \(G_{L}(s)\) represents the transfer function model of ideal linear link in middle and low frequency band, which is composed of inertial link. \(G_{Hi}(s)\) (\(i\) = 1, 2,..., _n_) represents the transfer function model of the resonance link in high frequency band, which is composed of a two-order mathematical model in series, and its mathematical model is expressed as:  
\[G_{H}(s)=\frac{\frac{1}{\omega_{z}^{2}}s^{2}+2\frac{\zeta_{z}}{\omega_{z}}s+1}{ \frac{1}{\omega_{p}^{2}}s^{2}+2\frac{\zeta_{p}}{\omega_{p}}s+1}. \tag{3}\]  
The transfer function shown in Equation (3) includes a pair of conjugate complex zero and a pair of complex poles. The complex zeros correspond to systems of high frequency locked rotor frequency \(\omega_{z}\) and damping coefficient \(\zeta_{z}\). The double pole corresponds to the resonance frequency \(\omega_{p}\) and the damping coefficient \(\zeta_{p}\). According to the frequency characteristic curve of the pitch axis as shown in Figure 3, the linear transfer function of the middle and low frequency band system can be obtained by the polynomial fitting method as follows:  
\[G_{L}(s)=\frac{3.5}{0.264s+1}. \tag{4}\]  
Shown in the frequency characteristic curve, the third-order resonance mode can have the greatest influence on the high frequency band, and the other modes can be ignored. Therefore, only a second-order mathematical model is sufficient to fit the high-frequency characteristic curve. The high frequency mathematical model is expressed as:  
\[G_{H_{1}}(s)=\frac{\left(0.0095s\right)^{2}+0.00054s+1}{\left(0.0088s\right)^{ 2}+0.00066s+1}. \tag{5}\]  
In conclusion, the system model is expressed by the transfer function as:  
\[G(s)=G_{L}(s)\cdot G_{H_{1}}(s). \tag{6}\]  
The continuous time equation of state corresponds to the transfer function as shown in Equation (6) and can be expressed as:  
\[\left\{ \begin{array}{l}\dot{x}(t)=Ax(t)+Bu(t)\\ y(t)=Cx(t)\end{array} \right., \tag{7}\]where \(x(t)\in R^{n\times 1}\) is the state vector, \(y(t)\in R^{n\times 1}\) is the output vector, and \(A\in R^{n\times n}\), \(B\in R^{n\times n}\), and \(C\in R^{1\times n}\) are system model parameter matrices. Therefore, according to Equations (5) and (6), the parameter matrixes in Equation (7) can be specifically expressed as follows:  
\[A=10^{4}\times\left[ \begin{array}{cccc}-0.0012&-1.295&-4.89\\ 0.0001&0&0\\ 0&0.0001&0\\ 0.0002&0.0009&1.7123\\ \end{array} \right],B=\left[ \begin{array}{c}1\\ 0\\ 0\\ 0\\ \end{array} \right]\,.\]  
The characteristic of the open loop frequency domain of the mathematical model shown in Equation (7) has been tested. Given the data presented in Figure 3, the frequency characteristic fitting curve of model identification was constructed and is shown in Figure 4 and the model identification error frequency characteristic curve is presented in Figure 5.  
According to the comparison curve of frequency characteristics, the following conclusions can be drawn. The frequency is less than 2 Hz, and the motor is in the running start stage. There is an  
Figure 4: Frequency characteristic fitting curve of model identification. Figure 5: Model identification error frequency characteristic curve.