A complete review of the known differentiators and their time‐discretizations have been addressed in this study. To resolve the drawbacks of the explicit (forward Euler) discretization, which is commonly utilized in sliding‐mode‐based differentiators, implicit time discretization methods are proposed to handle the set‐valued functions. The proposed schemes are supported by some analytical results to show their crucial properties, for example, finite‐time convergence, exactness, invariant sliding‐surface, chattering elimination, insensitivity to the gains during the sliding‐phase, and the well‐posedness. The causal implementation of the proposed implicit schemes has been addressed clearly and supported by flowcharts. Semi‐implicit schemes are also presented to provide a compromise between the performance and the ease of implementation. Finally, an exhaustive comparison is made using numerical experiments among 25 different state‐of‐the‐art differentiators to evaluate the behaviors of the differentiators facing the noise and initial error. Realistic conditions are considered in the simulations to provide useful information based on practical conditions. General conclusions are that implicit discretizations can supersede explicit and semi‐implicit ones.

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This study presents a new chattering free full‐order terminal sliding‐mode controller (FOTSMC) for maximum power point tracking of photovoltaic cells. The proposed system consists of two loops, namely seeking loop and tracking loop. The seeking loop utilises an adaptive perturb and observe method for maximum power point (MPP) searching. Since the accuracy of the seeking loop depends on the tracking loop, a new FOTSMC is used in the tracking loop for tracking the MPP. The proposed control law is continuous, and therefore a chattering free system is achieved. Furthermore, since the derivatives of terms with fractional powers do not exist in the control law, the singularity problem, which exists in traditional terminal sliding‐mode controllers (TSMCs), is avoided. Stability and robustness of the proposed system is presented. In order to evaluate the results of the system, a traditional TSMC is provided. Afterwards, the performance of the control system is verified through simulation and experiment.

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