## 2 Model Identification of Ground-Based Laser Communication Servo Turntable  
### System Frequency Domain Characteristic Test  
The azimuth or pitch axis control structure of ground-based laser communication servo turntable is shown in Figure 1. The system is composed of a velocity loop controller, a power amplifier, a torque motor, a circular grating encoder, and other parts.  
Traditionally, the sinusoidal sweep method is used to test the frequency domain characteristics of the system. As shown in Figure 1, point A indicates the application of the sinusoidal sweep signal, and the speed record at point B encoder feedback value. The input and output data can be used to obtain the system frequency characteristic curve, and the measured frequency domain characteristics include the power amplifier, motor, mechanical structure, and encoder components.  
The sinusoidal sweep signal is input at point A, and its digital sequence is expressed as:  
\[u(k)=A(k)\sin[2\pi\omega(k)]\] \[\omega(k)=f_{1}\Big{(}1+c(k)\cdot(k\cdot t)^{n}\Big{)},c(k)=\tfrac {f_{2}/f_{1}-1}{(n+1)T^{n}}\enspace, \tag{1}\]  
where \(f_{1}\) is the sweep signal starting frequency, \(f_{2}\) is the sweep signal termination frequency, \(t\) is the sampling time, \(n\) is the order of the sweep signal, and \(T\) is the sweep signal duration.  
Considering the pitch axis of ground-based laser communication servo turntable as an example, a frequency sweep experiment was carried out. During the experiment, the frequency range of sinusoidal sweep signal \([f_{1},f_{2}]\) was set to \([0.5,\,200]\), the signal amplitude \(A(t)\) was set to \(10^{\circ}\)/s, the  
Figure 1: Control structure of the pitch axis servo turntable system.  
sweep duration \(T\) was set to 25 s, and the signal sweep order \(n\) was set to the third to ensure sufficient low-frequency data. The sweep input signal and system response curve are shown in Figure 2.  
According to the input and output data, the open-loop transmission Bode diagram of the system can be obtained by discrete Fourier transform and power spectrum estimation, as shown in Figure 3. The following conclusions can be drawn explicitly: the turntable pitch axis tends to be linear in the low and middle frequency bands, but there are many nonlinear factors in the system because of the friction of the axis and other factors. There are multi-order resonance links in the high frequency band, and the third-order locked rotor frequency at 105 Hz and the resonant frequency at 113.3 Hz are the main frequencies that affect the system performance. The mechanical structure cannot achieve absolute rigid connection, because of torque imbalance and other nonlinear factors, but the numerical difference between the locked rotor frequency and the resonant frequency is small resulting in strong stiffness.  
Figure 3: Open-loop frequency characteristic curve of pitch axis velocity.  
Figure 2: Input and response signal curve of pitch axis open-loop test.  
**Remark 1**.: _It is worth noting that the inverse tangent function is adopted for phase calculation in this method. If the phase difference is less than 90\({}^{\circ}\), the angle value will jump, so \(\pm\) n180\({}^{\circ}\) (n = 1, 2... ) should be applied according to the phase angle data value._