## 4 Barrier Function Based Adaptive Algorithm
The main property of the \(\overline{K}_{\text{BF}}(\sigma)\) is  
\[\lim_{\sigma\to\varepsilon}\overline{K}_{\text{BF}}(\sigma)=1. \tag{19}\]  
Figure 1: Barrier function \(K_{\text{BF}}(\sigma)\). The quasi Barrier Function is defined as  
\[\overline{K}_{BF}(\sigma)=\overline{L}\frac{sat_{\overline{\varepsilon}}(|\sigma|) }{(\varepsilon-sat_{\overline{\varepsilon}}|\sigma|)}, \tag{20}\]  
where the \(sat_{\varepsilon}\) is defined as  
\[sat_{\overline{\varepsilon}}(y)=\left\{ \begin{array}{ll}y&\mbox{for }\left|y \right|<\overline{\varepsilon}\\ \overline{\varepsilon}sign(y)&\mbox{for }\left|y\right|\geq\overline{ \varepsilon}\end{array} \right.. \tag{21}\]  
The QBF is presented in Figure 2.  
The adaptive STA controller with QFB adaptive coefficients (BSTA) is given as [23]  
\[ \begin{array}{l}u_{ASTA}=k_{1}\overline{K}_{BF}|\sigma|^{1/2}sign(\sigma)+v, \\ \dot{v}=k_{2}\overline{K}_{BF}^{2}sign(\sigma),\end{array}  \tag{22}\]  
The feedback system with BSTA is  
\[ \begin{array}{l}\dot{\sigma}=-k_{1}\overline{K}_{BF}(\sigma)|\sigma|^{1/2} sign(\sigma)+v+\delta_{2}.\\ \dot{v}=-k_{2}\overline{K}_{BF}^{2}(\sigma)sign(\sigma).\end{array}  \tag{23}\]  
For the stability proof of the BSTA adaptation procedure, we assumed that the sliding variable \(\sigma\) at the time \(\overline{t}\), \(\overline{t}\
eq 0\) is in the boundary of \(\varepsilon\), where is \(|\sigma|<\varepsilon\) and \(\overline{K}_{\rm BF}(\sigma)=K_{\rm BF}(\sigma)\). Let us consider the Lyapunov Barrier Function \(V_{\rm BF}(\sigma)\),  
\[V_{BF}(\sigma)=|\sigma|+k_{2}\varepsilon^{-1}\overline{L}\overline{K}_{BF}( \sigma)+\frac{1}{2}v^{2}. \tag{24}\]  
The time derivative of \(V_{\rm BF}(\sigma)\) is  
\[\begin{array}{lcl}\dot{V}_{BF}(\sigma)&=&\frac{\dot{\sigma}  
Defining  
\[\Delta=\left(1+\frac{k_{2}\overline{L}^{2}}{(\varepsilon-|\sigma|)^{2}}\right) \left(|\vec{v}|+\gamma\right),\ \Delta\geq 0, \tag{25}\]  
yields  
\[\dot{V}_{BF}(\sigma) \leq-\frac{k_{1}\overline{L}}{(\varepsilon-|\sigma|)}|\sigma^{3/2 }-\frac{k_{1}k_{2}\overline{L}^{3}}{(\varepsilon-|\sigma|)^{3}}|\sigma|^{3/2} -\frac{k_{2}\overline{L}^{2}}{(\varepsilon-|\sigma|)^{2}}|\sigma|^{2}|v|+\Delta,\] \[=-\frac{k_{1}k_{2}\overline{L}^{3}}{(\varepsilon-|\sigma|)^{3}}| \sigma|^{3/2}-\frac{k_{2}\overline{L}^{2}}{(\varepsilon-|\sigma|)^{2}}|\sigma| ^{2}|v|-\frac{1}{(\varepsilon-|\sigma|)}\left(k_{1}\overline{L}|\sigma|^{3/2} -(\varepsilon-|\sigma|)\Delta\right),\] \[=-\frac{k_{1}k_{2}\overline{L}^{3}}{(\varepsilon-|\sigma|)}| \sigma|^{3/2}-\frac{k_{2}\overline{L}^{2}}{(\varepsilon-|\sigma|)^{2}}|v|- \frac{\Delta}{(\varepsilon-|\sigma|)}\left(k_{1}\overline{L}|\sigma|^{3/2}+| \sigma|-\varepsilon\right).\]  
The upper bound of the Lyapunov function derivation is  
\[\dot{V}_{BF}(\sigma)\leq-\frac{k_{1}k_{2}\overline{L}^{3}}{(\varepsilon-| \sigma|)^{3}}|\sigma|^{3/2}-\frac{k_{2}\overline{L}^{2}}{(\varepsilon-| \sigma|)^{2}}|\sigma|^{2}|v|-\frac{\Delta}{(\varepsilon-|\sigma|)}F. \tag{26}\]  
where  
\[F=\frac{k_{1}\overline{L}}{\Delta}|\sigma|^{3/2}+|\sigma|-\varepsilon\geq 0. \tag{27}\]  
The convergence and stability of the BF adaptation approach are ensured if the \(F\) is a strict positive semidefinite function; \(F\geq 0\). From the expression Equations (25)-(27), it is obvious that the stability of the system is ensured if the sliding variable \(\sigma\) is smaller than the BF boundary \(\varepsilon\). If the variable \(\sigma\) is getting closer to the \(\varepsilon\), then the function \(\Delta\) converges to zero, and the expression Equation (27) holds if \(|\sigma|\leq\varepsilon\). The stability proof for BSTA is finished. In the presented paper, BF is used for chattering alleviation; therefore, the BF operation boundary \(\varepsilon\) is limited to the pre-selected value \(\overline{\varepsilon}\), Figure 2, where the chattering alleviation starts and holds, \(0<\overline{\varepsilon}<\varepsilon\).