## 2 Mathematical Methods and Experimental Details  
### Mathematical Model of the Motor of the GS  
As illustrated in Figure 1, the closed-loop control system of the GS was composed of a direct-current (DC) swing motor, a mirror, a rotor position detector (grating), and a servo driver. Position signals of the rotor were sampled by a grating position sensor and then transmitted to the controller after being processed by an NQC chip, together with the reference signals produced by a host computer. After being processed by the controller, the resulting digital voltage was converted to an analogue voltage using a digital-to-analogue converter (DAC), and the driving voltage on the motor of the GS was generated through the power-amplification circuit.  
The dynamic Equation of the motor of the GS under an open-loop control is [8]:  
\[E=K_{b}\frac{d\theta}{dt}+L\frac{di}{dt}+Ri(t) \tag{1}\]  
\[J_{r}\frac{d^{2}\theta}{dt^{2}}=K_{t}i(t)-B_{v}\frac{d\theta}{dt}-k_{r}(\theta -\phi) \tag{2}\]  
\[J_{m}\frac{d^{2}\phi}{dt^{2}}=k_{r}(\theta-\phi) \tag{3}\]  
where \(E\) and \(i\) refer to the armature voltage and current of the motor stator; \(R\) and \(L\) represent the resistance and inductance of the motor; \(J_{r}\) and \(J_{m}\) denote the inertia of the rotor and the mirror; \(\theta\) and \(\phi\) are the angular positions of the motor rotor and the mirror; \(K_{b}\), \(K_{t}\), \(B_{v}\) and \(k_{r}\) are the back electromotive force (BEF) constant, torque constant, damping coefficient, and constant of torsional rigidity, respectively.  
The angular position of the mirror is equivalent to that of the rotor due to the high rigidity of the contact between the motor and the mirror of the GS used in the experiment, therefore, Equations (2) and (3) can be rewritten as:  
\[J\frac{d^{2}\theta}{dt^{2}}=K_{t}i(t)-B_{v}\frac{d\theta}{dt} \tag{4}\]  
where \(J=J_{r}+J_{m}\).  
According to [14], by combining Equations (1) and (4), the open-loop transfer function of the motor is found to be:  
\[\frac{\theta}{E}=\frac{K_{t}}{LJs^{3}+(LB_{v}+RJ)s^{2}+(RB_{v}+K_{b}K_{t})s} \tag{5}\]  
According to [15], the linear dynamic model during the running of the motor is attained, as shown in Figure 2, where \(K_{u}\) is used to represent the amplification factor.  
Figure 1: Compositions of the closed-loop control system of the GS.  
Relative to the resistance \(R\), the inductance \(L\) is negligible, so the model shown in Figure 2 can be transformed into a state-space form:  
\[\left[\begin{array}{c}\dot{\theta}(t)\\ \dot{\omega}(t)\end{array}\right]=A\left[\begin{array}{c}\theta(t)\\ \omega(t)\end{array}\right]+Bu(t)+f(t) \tag{6}\]  
where \(A=\left[\begin{array}{cc}0&1\\ 0&P_{\omega}\end{array}\right]\), \(P_{\omega}=-\frac{B_{0}}{f}\), \(B=\left[\begin{array}{c}0\\ K_{\omega}\end{array}\right]\), and \(K_{\omega}=\frac{K_{t}K_{u}}{fR}\); \(f(t)\) represents the external disturbance.  
Equation (6) is discretized through a zero-order holding. In this way, the following expression of the discrete state space is obtained:  
\[\left[\begin{array}{c}\theta(k+1)\\ \omega(k+1)\end{array}\right]=\Psi\left[\begin{array}{c}\theta(k)\\ \omega(k)\end{array}\right]+\Gamma u(k)+f(k) \tag{7}\]  
where \(\Psi\in R^{2\times 2}\), \(\Gamma\in R^{2\times 1}\).