The problem addressed in this paper is the online differentiation of a signal/function that possesses a continuous but not necessarily differentiable derivative. In the realm of (integer) high-order sliding modes, a continuous differentiator provides the exact estimation of the derivative f˙(t), of f(t), by assuming the boundedness of its second-order derivative, f¨(t), but it has been pointed out that if f˙(t) is casted as a Hölder function, then f˙(t) is continuous but not necessarily differentiable, and as a consequence, the existence of f¨(t) is not guaranteed, but even in such a case, the derivative of f(t) can be exactly estimated by means of a continuous fractional sliding mode-based differentiator. Then, the properties of fractional sliding modes, as exact differentiators, are studied. The novelty of the proposed differentiator is twofold: (i) it is continuous, and (ii) it provides the finite-time exact estimation of f˙(t), even if f¨(t) does not exist. A numerical study is discussed to show the reliability of the proposed scheme.

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