## 3 Design and Simulation of a Velocity Loop Controller Based on Discrete SMC
### Proof of Robustness and Stability of SMC  
**Theorem 1**.: _The absolute value of the system's nonlinear function equation as shown in Equation (16) has an upper bound, and this upper bound is assumed as \(\omega\). The trajectories of the system from any initial state must arrive at the sliding mode surface driven by the proposed algorithm._  
Proof.: There must be an initial state such that the sign of the sliding mode switching functions s(0), s(1),... s(n) does not change, where n is a positive integer. The following proofs are discussed for two cases: s(0) \(<\) 0 and s(0) \(\geq\) 0.  
1. If s(k) \(\geq\) 0 (\(k=0,\,1,\,\ldots\,\,n\))  
Assuming that the system does not cross the sliding surface within the \(n\) step, recursive formulas can be obtained according to formula (15).  
\[\begin{array}{l}s(1)=(1-\lambda T)s(0)-(\kappa T\cdot\eta(0)+\omega(0))\\ s(2)=(1-\lambda T)^{2}s(0)-(1-\lambda T)(\kappa T\cdot\eta(0)-\omega(0))- \kappa T\cdot\eta(1)-\omega(1)\\ \vdots\\ s(n)=(1-\lambda T)^{n}s(0)-\sum\limits_{i=0}^{n-1}(1-\lambda T)^{n-1-i}( \kappa T\cdot\eta(i)-\omega(i))\end{array} \tag{19}\]  
There must beAssuming that the system trajectory reaches the sliding mode surface at time \(m\), then \(s(m)=0\), and according to Equations (19) and (20), the following equation can be obtained,  
\[s(m)=(1-\lambda T)^{m}s(0)-\delta\cdot\sum_{i=0}^{m-1}(1-\lambda T)^{m-1-i}=(1- \lambda T)^{m}s(0)-\delta\frac{1-(1-\lambda T)^{m}}{\lambda T} \tag{21}\]  
Therefore, the arrival time \(m\) can be expressed as:  
\[m=\log_{1-\lambda T}\frac{1}{\lambda T\cdot s(0)/\delta+1}. \tag{22}\]  
2. Similarly, if \(s(k)<0\) (\(k=0,1,\dots\ n\)), the moment that the system state reaches the sliding mode surface can be expressed as:  
\[m=\log_{1-\lambda T}\frac{1}{\lambda T\cdot s(0)/\delta+1}. \tag{23}\]  
Above all, driven by the sliding mode controller shown in formula (17), the state trajectory of the system from any initial position can reach the sliding mode surface in a limited time, and the arrival time is expressed as follows:  
\[m=\log_{1-\lambda T}\frac{1}{\lambda T\cdot\big{|}s(0)\big{|}/\delta+1}. \tag{24}\]  
Theorem 1 has been proved.  
**Theorem 2.**_Driven by the sliding mode controller, once the system state reaches the sliding mode surface, it will be stable in the QSMD and cannot escape, the control system is bounded. The QSMD can be expressed as:_  
\[\Phi=\big{\{}s(k)\big{|}\big{|}s(k)\big{|}\leq\Delta=\kappa T\cdot\eta+\omega \big{\}}, \tag{25}\]  
_where \(\omega\)/\(\kappa T\leq\eta<1\), \(\omega\) is the upper of the disturbance._  
**Proof.** The proof process can be divided into two cases: \(s(0)<0\) and \(s(k)\geq 0\) (\(k=0,\,1,\dots\ n\)).  
1. If \(s(k)\geq 0\) (\(k=0,1,\dots\ n\)), it can be obtained that  
\[\begin{split} s(k+1)=(1-\lambda T)s(k)-\kappa T\cdot\eta(k) \text{sgn}[s(k)]+\omega(k)\\ <s(k)-\kappa T\cdot\eta+\omega<s(k)\end{split} \tag{26}\]  
Therefore, when \(s(k)\geq 0\), the value of \(s(k)\) decreases successively. Assuming that the system state is in the QSMD at time \(n\), the system state at time \(n+1\) must be in the QSMD.  
2. If \(s(k)<0\) (\(k=0,\,1,\dots\ n\)), similar conclusions can be drawn that  
\[\begin{split} s(k+1)=(1-\lambda T)s(k)-\kappa T\cdot\eta(k) \text{sgn}[s(k)]+\omega(k)\\ >s(k)+\kappa T\cdot\eta+\omega>s(k)\end{split} \tag{27}\]  
Therefore, when \(s(k)<0\), the value of \(s(k)\) increases successively. When the system state is in the QSMD at time \(n\), the system state at time \(n+1\) must also be in the QSMD.  
Above all, \(s(k)\) deceases with time. Once the trajectory of the system reaches the sliding surface, the system state will stabilize in the QSMD. At this moment, the system is strongly robust to nonlinear disturbances. Therefore, the control algorithm is stable, and Theorem 2 is proved.  
**Remark 5.**_In view of [36; 44; 46; 47; 48; 49], the overestimation and underestimation problems may exist when the state is stable in the QSMD, we will study it in the future work._