## 3 Super-Twisted Positioning Controller Design  
The STA controller is the most applicable HSMC controller structure and has many positive features [14; 26; 27]. Like the robust FSCM structure, which has good rejection capability of the matched known bounded disturbances and uncertainties, the STA preserves this capability. The advantage of STA compared to the FSCM is that the nonlinear switching term is shifted behind the integrator, which smooths the controller output. Nevertheless, the HSMC- STA structure produced a continuous output and the continuity is strongly related to the order of the plant. The STA ensures continuous controller output where the relative degree of the system concerning the sliding function is one; otherwise, the limit cycle (LC) exists [8; 9; 12]. The LC can be effectively inspected with a describingfunction (DF) approach, where the Nyquist stability criteria are used [1]. For the system with a higher degree, some chattering alleviation technique can be used, but cannot be rejected. This topic will be discussed later in this article. The STA controller is also proclaimed as a nonlinear version of the well-known linear proportional integral (PI) controller, with nonlinear terms, which ensure a better tracking and disturbance rejection capability. Concerning the system order in Equation (6), and with the consideration that some unmodeled parasitic dynamics exist, the feedback system with STA will inevitably produce unwanted chattering phenomena at the system output. The system Equation (6) is used for the STA controller design. The STA controller structure is  
\[ \begin{array}{l}u_{STA}=k_{1}|\sigma|^{1/2}sign(\sigma)+v,\\ \dot{v}=k_{2}sign(\sigma),\end{array}  \tag{8}\]  
where the coefficients \(k_{1}\), \(k_{2}\) are the tuning parameters, and subject to the system stability and the feedback dynamic. The introduced sliding variable \(\sigma\) of STA Equation (8) is  
\[\sigma=e_{2}+\bar{w}e_{1}, \tag{9}\]  
where the first derivative is equal to  
\[\dot{\sigma}=\dot{e}_{2}+\bar{w}\dot{e}_{1}. \tag{10}\]  
The coefficient \(w\) is a tuning parameter and needs to hold \(w>0\). For a system Equation (6) with the selected sliding variable Equations (9) and (10), and STA control law Equation (8), the feedback system can be expressed as  
\[ \begin{array}{l}\dot{\sigma}=-gk_{1}|\sigma|^{1/2}sign(\sigma)+v+\delta_{2}.\\ \dot{v}=-gk_{2}sign(\sigma).\end{array}  \tag{11}\]  
where the perturbation term \(\delta_{2}\) is defined as  
\[\delta_{2}=-(f-w)e_{2}+\vec{d}. \tag{12}\]  
For the simplicity of the stability proof, we assume that the coefficient \(g=1\). For the stability assessment Equation (11), the given Lyapunov function \(V(\sigma)\) is used [6; 28]  
\[ \begin{array}{l}V(\sigma)=\frac{1}{2}v^{2}+2k_{2}|\sigma|+\frac{1}{2}\Big{(} k_{1}|\sigma|^{1/2}sign(\sigma)-v\Big{)}^{2},\\ V(\sigma)=\zeta^{T}P\zeta.\end{array}  \tag{13}\]  
The state variable \(\zeta\) and matrix \(P\) are  
\[ \begin{array}{l}\zeta^{T}=\Big{[}\begin{array}{cc}|\sigma|^{1/2}sign(\sigma )&v\end{array}\Big{]},\\ P=\frac{1}{2}\Big{[}\begin{array}{cc}4k_{2}+k_{1}^{2}&-k_{1}\\ -k_{1}&2\end{array}\Big{]}.\end{array} \]  
The time derivative of \(V(\sigma)\) is  
\[\dot{V}(\sigma)=\dot{\zeta}^{T}P\zeta+\zeta^{T}P\dot{\zeta},\]  
where the time derivative of \(\dot{\zeta}\) is equal to  
\[\zeta^{T}=|\sigma|^{-1/2}\Big{[} \begin{array}{cc}1\Big{(}-k_{1}|\sigma|^{1/ 2}sign(\sigma)+v+\delta_{2}\Big{)}&-k_{2}|\sigma|^{1/2}sign(\sigma)\end{array}  \Big{]}.\]  
The matrix form of the time derivative \(\dot{V}(\sigma)\) is  
\[\dot{V}(\sigma)=-\zeta^{T}Q\zeta+\delta_{2}\mathrm{q}\zeta,\]and matrices \(Q\) and \(q\) are defined as  
\[Q =|\sigma|^{-1/2}\left[ \begin{array}{cc}k_{1}k_{2}+\frac{k_{1}^{3}}{2}&- \frac{k_{1}^{2}}{2}\\ -\frac{k_{1}}{2}&\frac{k_{1}}{2}\end{array} \right],\] \[q =|\sigma|^{-1/2}\left[ \begin{array}{cc}\left(2k_{2}+\frac{k_{1}^ {2}}{2}\right)&-\frac{k_{1}}{2}\end{array}