## 3 Design and Simulation of a Velocity Loop Controller Based on Discrete SMC
### Sliding Mode Simulation  
In this section, using the pitch axis system model of the ground-based laser communication servo turntable shown in Equation (11) as the simulation model, the stability of the proposed sliding mode algorithm was analyzed as follows. Assuming that the nonlinear disturbance of the system is \(\varepsilon(k)=[0;0;2.5\sin(2k\pi)+0.5]\), the initial state of the system is \([-1;-0.8;0]\), and the target state is [0;0;0].  
According to the discrete state equation of the system, the sliding mode coefficient shown in Equation (12) is a three-order matrix such that \(C_{e}=[C_{e1},C_{e2},1]\), the parameters should be chosen as [1,2,1] to satisfy lemma 1. The parameters of the sliding mode controller (17) were selected as: \(\lambda=5\), \(\delta\) = 0.15, \(\alpha=2\), \(\mu=10\), and \(\gamma=2\). To assess the proposed algorithm, it was compared to the classical exponential reaching law (28) with a \(O(T^{2})\) disturbance compensator and the chatter-reduced reaching law (29) with the \(O(T^{3})\) disturbance compensator described in Ref. [22]. The exponential function coefficients are the same, with values of \(\lambda_{c}\) = 5, \(\lambda_{[22]}\) = 5, and the variable structure function coefficients \(\kappa\), \(\kappa_{c}\), \(\kappa_{[22]}\) are determined by the disturbance estimation error in the next section. Other coefficients in the algorithm of Ref. [22] were set to \(\delta_{[22]}\) = 0.15, \(\mu_{[22]}\) = 10, and \(\gamma_{[22]}\) = 2.  
\[s(k+1)=(1-\lambda_{c}T)s(k)-\kappa_{c}T\cdot\mathrm{sgn}[s(k)]+C_{e}\varepsilon (k-1). \tag{28}\]  
\[s(k+1)=\Big{(}1-\lambda_{[22]}T\Big{)}\phi_{[22]}(k)\cdot s(k)-\frac{\kappa_{ [22]}T}{\phi_{[22]}(k)}\mathrm{sgn}[s(k)]+C_{e}[\varepsilon(k)-2\varepsilon(k -1)+\varepsilon(k-2)]\]  
\[\phi_{[22]}(k)=\delta_{[22]}+\Big{(}1-\delta_{[22]}\Big{)}e^{-\mu_{[22]}[s(k )]^{\gamma}[22]}\]  
The comparison curves of disturbance estimation errors between the three algorithms were determined and are shown in Figure 6. The maximum \(O(T^{3})\) amplitude was 0.01 for the algorithm disturbance estimation error curve for both the proposed algorithm and the Ref. [22] algorithm. This value is much smaller than that of the classical exponential reaching law algorithm, which exhibits a maximum \(O(T^{2})\) amplitude of 0.15. Therefore, the variable structure function coefficients \(\kappa\) and \(\kappa_{[22]}\) of both algorithms are set as 0.012, and the variable structure function coefficient \(\kappa_{c}\) of the classical algorithm is set as 0.17. According to Theorem 2, utilization of this parameter, larger than the maximum perturbation estimation, can ensure the stability of the sliding mode trajectory in the QSMD.  
Figure 7 shows the sliding mode trajectory contrast curves. Three reaching law algorithms can ensure that the system state arrives at the sliding mode surface in a limited time and is stable in the QSMD with a fixed width. Theorems 1 and 2 are proved here. Compared with the traditional exponential reaching law algorithm, the QSMD width of the proposed reaching law algorithm is narrower, \(\Delta_{p}\approx 0.038\), compared to the QSMD width of the traditional algorithm of \(\Delta_{c}\approx 0.31\). It should  
Figure 6: Disturbance estimation error contrast curve.  
be noted that a smaller QSMD width correlates to stronger robustness of the system to nonlinear disturbances. Compared with the chatter-reduced reaching law algorithm of Ref. [22], chattering in sliding mode can change to be continuous around the sliding surface, and the widths of both algorithms are almost equal. The proposed algorithm obviously eliminates sliding mode chattering and improves the stability of the system without reducing the robustness of the system.