## 1 Introduction
More advanced techniques, which are based on estimation and adaptation with regard to the unknown disturbance and system parameters, are discussed in [13, 14, 15, 16, 17]. There are many adaptive SMC approaches, which are based on artificial intelligence methods such as fuzzy logic and neural networks. Fuzzy logic and neural networks are often use for unknown parameter estimation or the controller structure or gains adaptation [18, 19]. Roughly all the proposed techniques alleviated the chattering but could not completely suppress it. Some of the adaptation techniques, through the adaptation phase, sacrifice the closed-loop dynamics, which guarantee the stability of the system. In some real-time applications, such trade-offs cannot be permissible. Regarding the aforementioned, the presented paper proposes a novel approach, which is capable of suppressing the chattering phenomenon completely for a higher-order plant with the super-twisted algorithm (STA). Theoretically, the STA has a smooth continuous output only when the STA is deployed with the first-order system; otherwise, a limited cycle (LC) exists. The LC results in the chattering of the controller output [20, 21]. The STA controller used most often is the used higher-order sliding mode controller (HSMC), where the discontinuous function appears in the first derivative of the sliding variable. Therefore, the STA controller output mainly regards the FSCM, except for the vicinity of the sliding manifold. The main advantage of the STA is the pure property of the FSCM, which ensures finite-time convergence and has a simple structure. Unlike the other HSMCs, the STA doesn't require the derivative of the sliding variable, which simplifies the implementation and computational burden in the real-time execution. The intention of the presented paper is to ensure the chattering-free output of the STA controller, regardless of the system order and unknown parasitic dynamic. The main idea to suppress the chattering phenomena lies in the adaptation of the STA coefficients, partially from the existing solutions given in [14, 15, 16]. The main difference in the proposed approach is that the STA parameters are only adapted in the prescribed region. The prescribed region presentsthe manifold vicinity of the sliding variable. The barrier function method (BFM) is employed for the adaptation mechanism of the STA coefficients. The barrier function (BF) adaptation method can be found in the recent scholars [22; 23; 24; 25]. The original idea of the BF adaptation law is suppression of unknown bounded disturbances and uncertainty [24]. The adaptation principle of BF can be applied to FSCM and HSMC algorithms as a feedback controller, as well as state estimators and differentiators [23; 24; 25]. In theory, the BF adaptation technique doesn't have the capability to suppress the chattering phenomenon of FSCM or HSMC controllers. Through the adaptation process, the BF can induce uncontrollably high gains, which can limit real-time implementation. The gain adaptation of BF is completely closed to the BF boundary [22; 23]. Theoretically, the BF gains can converge to infinity. The use of a larger gain in the SMC controller can improve the disturbance rejection efficiently, but, on the other hand, can deteriorate the smoothness of the controller output. In the presented research, the principal objective is not to deal with unknown disturbances, like in [23; 24; 25], but to improve STA operation sufficiently in the vicinity of the origin, with complete suppression of the chattering phenomena. In the presented approach, the entry boundary to the BF adaptation is prescribed. The entry boundary prescribes two modes of the STA controller operation. The first mode is the classic operation of STA, which is intended for system states far from the origin and larger disturbances values. The first mode of operation needs to ensure the stability of the known bounded disturbances, and to provide a proper time convergence of the sliding variable.