## 3 Compound Controller Design
### Design of the Novel DTSMC
Based on the trajectory tracking error, a discrete-time terminal sliding function is proposed as  
\[s(k)=\kappa_{1}e(k)+\kappa_{2}\lceil e(k-1)\rfloor^{\alpha} \tag{22}\]  
where \(\kappa_{1}>0\) and \(\kappa_{2}>0\) are positive constants. The notation \(\lceil e(k-1)\rfloor^{\alpha}\triangleq|e(k-1)|^{\alpha}sgn(e(k-1))\), \(\alpha\) denotes a ratio of two odd integers \(p\) and \(q\), where \(0<p<q\). The design of \(0<\alpha<1\) will produce a steeper slope of the sliding surface when the system states are close to the origin, and it results in a faster convergence speed [34]. In this work, the DTSMC is designed based on equivalent control, the following reaching law is used  
\[\Delta s(k)=s(k+1)-s(k)=0 \tag{23}\]  
The discrete-time terminal sliding function (22) can be deduced as follows:  
\[s(k)=\kappa_{1}e(k+1)+\kappa_{2}\left\lceil e(k)\right\rfloor^{\alpha} \tag{24}\]  
\[s(k)=\kappa_{1}\big{[}y(k+1)-y^{*}(k+1)\big{]}+\kappa_{2}\left\lceil e(k) \right\rfloor^{\alpha} \tag{25}\]\[s(k)=\kappa_{1}\big{[}\mathbf{C}(\mathbf{AX}(k)+\mathbf{R}u(k)+\mathbf{BU}(k)+ \mathbf{D}(k))-y^{*}(k+1)\big{]}+\kappa_{2}\left\lceil e(k)\right\rfloor^{\alpha} \tag{26}\]  
Based on the nominal model, the equivalent control input \(u_{eq}(k)\) can be derived as follows:  
\[u_{eq}(k)=(\mathbf{CR})^{-1}\left[y^{*}(k+1)+\kappa_{1}^{-1}s(k)-\kappa_{1}^{- 1}\kappa_{2}\big{[}e(k)\big{]}^{\alpha}-\mathbf{CAX}(k)-\mathbf{CBU}(k)- \mathbf{C\hat{D}}(k)\right] \tag{27}\]  
It must be pointed out that the equivalent control law \(u_{eq}(k)\) is derived based on the assumptions that the initial state is on the sliding surface and the estimation error \(\mathbf{\hat{D}}(k)=0\). However, in practice, the above assumptions are almost non-existent. Therefore, in order to deal with the problem, defining a terminal-sliding-model-type switching control law as  
\[u_{sw}(k)=-\big{(}\mathbf{CR})^{-1}\left[\xi_{1}s(k)+\xi_{2}\big{[}s(k)\big{]} ^{\beta}\right] \tag{28}\]  
where \(\xi_{1}>0\), \(\xi_{2}>0\), \(\big{\lceil}s(k)\big{\rfloor}^{\beta}\triangleq\big{\lvert}s(k)\big{\rvert}^{ \beta}sgn(s(k))\) with \(0<\beta<1\). Combining the equivalent control law (27) and the switching control law (28) can obtain the overall control law  
\[ \begin{split} u_{d}(k)=u_{eq}(k)+u_{sw}(k)=&( \mathbf{CR})^{-1}\Big{[}y^{*}(k+1)+\kappa_{1}^{-1}s(k)-\kappa_{1}^{-1}\kappa_{ 2}\big{[}e(k)\big{]}^{\alpha}-\mathbf{CAX}(k)\\ &-\mathbf{CBU}(k)-\mathbf{C\hat{D}}(k)-\xi_{1}s(k)-\xi_{2}\big{[} s(k)\big{]}^{\beta}\Big{]}\end{split}  \tag{29}\]  
Substituting Equations (29) into (21) yields  
\[e(k+1)=\mathbf{C}\left[\mathbf{D}(k)-\mathbf{\hat{D}}(k)\right]+\kappa_{1}^{- 1}s(k)-\kappa_{1}^{-1}\kappa_{2}\big{[}e(k)\big{]}^{\alpha}-\xi_{1}s(k)-\xi_ {2}\big{[}s(k)\big{]}^{\beta} \tag{30}\]  
Based on the proposed overall control law (29), the expression of \(s(k+1)\) becomes  
\[ \begin{split} s(k+1)&=\kappa_{1}e(k+1)+\kappa_{2} \big{[}e(k)\big{]}^{\alpha}\\ &=s(k)-\kappa_{1}\xi_{1}s(k)-\kappa_{1}\xi_{2}\big{[}s(k)\big{]} ^{\beta}+\kappa_{1}\mathbf{C}\left[\mathbf{D}(k)-\mathbf{\hat{D}}(k)\right] \end{split}  \tag{31}\]  
**Remark 2**. : _In order to alleviate the chattering phenomenon which may be induced by the \(sgn(\cdot)\), we employ the smoother saturation function \(sat(\cdot)\) to replace the \(sgn(\cdot)\)in this work,_  
\[sat(s(k))=\left\{ \begin{array}{ll}sgn(s(k)),&\quad if|s(k)|>\sigma\\ s(k)/\sigma,&\quad if|s(k)|\leq\sigma\end{array} \right. \tag{32}\]  
_where the parameter \(\sigma\) ensures the band width \(2\sigma\) of the sliding mode function \(s(k)\). In practice, the trial-and-error method is usually employed to obtain the parameter \(\sigma\) and make a compromise between the chattering phenomenon and the tracking accuracy._  
**Remark 3**. : _The switching control law \(u_{sw}(k)\) can drive the sliding mode dynamics into the DTSM domain with the band width of \(2\sigma\) and enhance the robustness feature in the presence of the estimation error \(\mathbf{D}(k)\) and other perturbations.