## 3 Compound Controller Design
### Stability Analysis  
In order to analyze the stability of the closed-loop system with the proposed DTSMC, the following assumption condition about the discrete-time terminal sliding surface is required. **Assumption 3**. : _There is a positive constant \(\epsilon\) such that \(|\kappa_{1}\mathcal{C}\delta|\leq\epsilon=O(T^{2})\) and \(k\in N\)._  
During the stability analysis, the following proposition is needed. **Proposition 1**. : \(\epsilon\leq\kappa_{1}\xi_{2}\Phi^{\beta}\leq\Phi(1-\kappa_{1}\xi_{1})\)_._  
The proof of the proposition can be seen in the Appendix A. **Theorem 1**. : _Considering the state-space model (17) along with the discrete-time terminal sliding surface (22) and Assumption 3, if we employ the DTSM control law (29), the following sliding mode dynamics can be guaranteed._  
_(1) The discrete-time sliding mode dynamics from any initial state will enter the region \(\Delta\) within \(K^{*}\) steps, where_  
\[\Delta=\Bigg{\{}s(k):|s(k)|\leq\sigma=\psi(\beta)\cdot max\bigg{\{}\Big{(} \frac{e}{\kappa_{1}\xi_{2}}\Big{)}^{\frac{1}{\beta}},\Big{(}\frac{\kappa_{1} \xi_{2}}{1-\kappa_{1}\xi_{1}}\Big{)}^{\frac{1}{1-\beta}}\bigg{\}}\Bigg{\}} \tag{33}\]  
\[K^{*}=\lfloor k^{*}\rfloor+1,k^{*}=\frac{s^{2}(0)-\sigma^{2}}{\mu^{2}},\mu= \kappa_{1}\xi_{1}\sigma+\lfloor\psi^{\beta}(\beta)-1\rfloor e,\]  
\[\psi(\beta)=1+\beta^{\frac{\beta}{1-\beta}}-\beta^{\frac{1}{1-\beta}},\Phi= max\bigg{\{}\Big{(}\frac{e}{\kappa_{1}\xi_{2}}\Big{)}^{\frac{1}{\beta}},\Big{(} \frac{\kappa_{1}\xi_{2}}{1-\kappa_{1}\xi_{1}}\Big{)}^{\frac{1}{1-\beta}} \bigg{\}}.\]  
_where the operator \(\lfloor\cdot\rfloor\) gives the nearest integer that is less than or equal to the original value._  
_For any \(0<\beta<1\) and \(0\leq\Phi\leq 1\), the following relationships are met [35]:_  
\[1<\psi(\beta)<2,\psi(\beta)\Phi-[\psi(\beta)\Phi]^{\beta}+\psi(\beta)\geq 1 \tag{34}\]  
_(2) Once the sliding mode dynamics enter the region \(\Delta\), it will never escape in the subsequent time._  
**Proof of Theorem 1**. : (1) Define a discrete-time Lyapunov function \(V(k)=s^{2}(k)\), it can be obtained that  
\[ \begin{split}\Delta V(k)=& V(k+1)-V(k)\\ =& s^{2}(k+1)-s^{2}(k)\\ =&-[s(k)-s(k+1)][s(k)+s(k+1)]\\ =&-\Big{[}\kappa_{1}\xi_{1}s(k)+\kappa_{1}\xi_{2} \lceil s(k)\rfloor^{\beta}-\kappa_{1}\mathbf{C}\left[\mathbf{D}(k)-\hat{ \mathbf{D}}(k)\right]\Big{]}\\ &\times\Big{[}2s(k)-\kappa_{1}\xi_{1}s(k)-\kappa_{1}\xi_{2} \lceil s(k)\rfloor^{\beta}+\kappa_{1}\mathbf{C}\left[\mathbf{D}(k)-\hat{ \mathbf{D}}(k)\right]\Big{]}\end{split}  \tag{35}\]  
In this work, we consider the following two cases if \(s(k)\
otin\sigma\). _Case 1_: Consider \(s(k)>\sigma=\psi(\beta)\cdot\Phi>0\), we can obtain that  
\[ \begin{split} s(k)-s(k+1)&=\kappa_{1}\xi_{1}s(k)+ \kappa_{1}\xi_{2}\lceil s(k)\rfloor^{\beta}-\kappa_{1}\mathbf{C}\left[\mathbf{ D}(k)-\hat{\mathbf{D}}(k)\right]\\ &\geq\kappa_{1}\xi_{1}s(k)+\kappa_{1}\xi_{2}\psi^{\beta}(\beta) \Phi^{\beta}-\kappa_{1}\mathbf{C}\left[\mathbf{D}(k)-\hat{\mathbf{D}}(k) \right]\end{split}  \tag{36}\]  
By Proposition 1 and Assumption 3, Equation (36) can be deduced as  
\[ \begin{split} s(k)-s(k+1)&\geq\kappa_{1}\xi_{1}s(k)+ \kappa_{1}\xi_{2}\psi^{\beta}(\beta)\Phi^{\beta}-\left|\kappa_{1}\mathbf{C} \left[\mathbf{D}(k)-\hat{\mathbf{D}}(k)\right]\right|\\ &\geq\kappa_{1}\xi_{1}\sigma+\left\lceil\psi^{\beta}(\beta)-1 \right\rceil\epsilon:=\mu\end{split}  \tag{37}\]Referring to \(s(k)>0\), the one-step forward sliding function \(s(k+1)>0\), it can be obtained that  
\[s(k)>\kappa_{1}\xi_{1}s(k)+\kappa_{1}\xi_{2}\lceil s(k)\rfloor^{\beta}-\kappa_{1} \mathbf{C}\left[\mathbf{D}(k)-\hat{\mathbf{D}}(k)\right] \tag{38}\]  
then  
\[ \begin{split} s(k)+s(k+1)&=2s(k)-\kappa_{1}\xi_{1}s (k)-\kappa_{1}\xi_{2}\lceil s(k)\rfloor^{\beta}+\kappa_{1}\mathbf{C}\left[ \mathbf{D}(k)-\hat{\mathbf{D}}(k)\right]\\ &\geq\kappa_{1}\xi_{1}s(k)+\kappa_{1}\xi_{2}\psi^{\beta}(\beta) \Phi^{\beta}-\kappa_{1}\mathbf{C}\left[\mathbf{D}(k)-\hat{\mathbf{D}}(k)\right] \geq\mu\end{split}