## 4 Barrier Function Based Adaptive Algorithm  
The BF method in the feedback control system originated from the idea of preventing transgression in the system output constraints [22]. The BF methodology is further used as the SMC adaptation technique, where the gains of the controllers are adapted to the current value and the sliding variable. Many adaptation techniques for SMC have been proposed by recent scholars. The first proposed strategy uses constant gain increase until the sliding mode does not occur [29; 30]. This approach suffers from the unexpected disturbance growth, where the sliding mode can be lost for a certain time, until the growing gain again reaches the sliding mode behavior. The second approach overcomes this problem with use of the adaptations in terms of decreasing or increasing gain [14; 31]. This approach ensures the finite time convergence of the sliding variable to the neighborhood of zero, where the neighborhood size and convergence time depend on the disturbance [32]. The main disadvantage of the method is overestimated controller parameters, which produce an additional chattering effect. Adaptation laws also exist, which are based on the equivalent control approach as a disturbance estimator. Here, the low pass filter approximates the equivalent control strategy [15; 33]. The BF adaptation technique is used in the presented paper. Like the BF adaptation process presented in [24; 25; 32], this research used the BF method to suppress the chattering phenomena. The BF adaptation technique [24; 25] prescribes the closed boundary of the sliding variable. The sliding variable cannot leave the boundary. If the sliding variable is approaching the boundary, the gain of the controller converges to infinity. Also, the initial value of the sliding variable is assumed to be in the prescribed boundary; otherwise some additional adaptation algorithms are used, similar to [31], where the controller gain increases until the prescribed boundary of the BF is not reached. It needs to be mentioned that the original BF adaptation strategy is used for unknown disturbances, where the adapted controller coefficients are not overestimated, and no additional low pass filter is needed. In our case, the BF adaptation is used only in a certain boundary, otherwise the preselected STA coefficients remain unchanged. The BF strategy only adapts the controller coefficients near to the sliding manifold. The adapted STA coefficients are strictly bounded with preselected STA values, which is beneficial for real-time implementation. The benefits of the approach are its good disturbance rejection capability outside of the boundary and the fast time convergence of the STA coefficient in the prescribed BF region, which can suppress the chattering phenomena completely. The STA controller operates in two modes: outside the BF boundary and inside the BF boundary. The first mode is the normal operation of the STA, and the second mode is the BF adaptation procedure. Due to the limited operation of the BF in the original idea, we named this approach the quasi BF adaptation strategy. Before we describe the proposed adaptation strategy, the BF's definition needs to be presented. The BF is a positive, even function, depicted in Figure 1, defined on the interval \(\varepsilon>0\), where \(K_{\text{BF}}:\ x\in(-\varepsilon,\varepsilon)\to K_{\text{BF}}(x)\in[b, \infty)\) and \(K_{\text{BF}}\) is strictly increasing on [0,\(\varepsilon\)) and has a minimum at zero \(K_{\text{BF}}(0)=b\geq 0\). \[\lim_{x\to\varepsilon}K_{BF}(x)=+\infty. \tag{17}\]  
For the presented adaptation algorithm, the following BF is used,  
\[K_{\text{BF}}(\sigma)=\overline{L}\frac{|\sigma|}{\varepsilon-|\sigma|}. \tag{18}\]  
The course of the Barrier Function is presented in Figure 1.  
For the adaptation algorithm and chattering suppression, the quasi BF-QFB, \(\overline{K}_{\text{BF}}(\sigma)\) is proposed.