## 1 Introduction  
Many electrical devices such as motors, computers and household appliances use passive rectifiers for supplying energy to DC loads. The rectifying action usually injects lower order harmonics which increase the Total Harmonic Distortion (\(THD_{i}\)) of the current, reduces the Power Factor (PF) and worsens the energy quality of electrical grids [1, 2, 3]. Active rectifiers have become an attractive alternative for overcoming these problems [4, 5]. They are current-controlled rectifiers used to control the current in the AC side and provide a regulated DC voltage to load, their controllers are usually designed for keeping PF and \(THD_{i}\) within admissible ranges (PF \(>\) 0.9 and \(THD_{i}<\) 5%) according to IEEE Std. 519 and IEC/EN 61000-3-2 [6, 7]. Several active rectifiers with PFC-based on boost converter have been proposed to replace passive rectifiers [8, 9, 10]. Among these, SBBC is a promising topology since it standsout by reducing the number of diodes in the current path from source to load, decreasing conduction power losses and improving overall efficiency. In addition, SBBC topology has two clamped diodes that connect the source to the circuit ground, decreasing common mode noise and electromagnetic interference [8,11,12]. Due to the aforementioned reasons, the SBBC topology was selected as the topology under study of this paper. SBBC controllers are usually based on classic cascade PI or linear controllers which feature an external voltage loop and an inner current loop that use Pulse Width Modulation (PWM) for generating the required control signals and activating power switches. However, this kind of controllers present start-up overvoltages causing instability and affecting sensitive loads [10,13,14,15,16,17]. Particularly, [PERSON] et al. [17] made a comparative analysis for PFC of several high efficiency AC/DC boost topologies. They showed that linear controllers allow obtaining high PFs and low \(THD_{i}\)s; however, there are some issues to highlight: (1) DC voltage response has over peaks; (2) AC current stabilization time has a delay of several cycles in face of disturbances and load changes; and (3) the AC current waveform near zero crossing presents distortions [9,11,14]. In general terms, linear PI controllers are designed around an operating point which reduces the dynamic response in presence of large disturbances; furthermore, the dynamic performance of the controller is degraded when the operation region of the converter moves away from the equilibrium point used in the design [8,11,13]. In addition, large disturbances and extensive changes in the operation point supremely affect sensitive loads [18]. To improve the response of power systems, the use of non-linear controllers with fast response under disturbances and high working range is desirable. SMC is a non-linear control strategy that represents a good alternative for controlling variable structure systems, as SBECs. SMC improves the robustness against large and fast disturbances and reduces the sensitivity of the system to the variation of parameters. Furthermore, SMC deals with uncertainty in modeling parameters. Moreover, it can directly provide the switching signals of power switches by means of hysteresis modulation. Consequently, the dynamic response in closed loop is the fastest possible [19, 20, 21, 22, 23, 24]. The most relevant and recently published papers related to SBBC controllers, along with their main contributions and some drawbacks, are summarized hereafter. In [12,25], the authors proposed a hybrid PI-SMC for a SBBC that allows reducing the injection of DC current into the power network. They implemented hysteresis modulation; nonetheless, the hysteresis band amplitude is constant, obtaining a variable switching frequency which increases \(THD_{i}\). Their sliding surface function consists of reducing the error between voltage and current waveforms, guaranteeing AC current and voltage in phase.