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Research Papers

# An Exact Robust Differentiator Based on Continuous Fractional Sliding Modes

[+] Author and Article Information
Aldo Jonathan Muñoz-Vázquez

Mechatronic Engineering Polytechnic,
University of Victoria,
e-mail: amunozv@upv.edu.mx

Carlos Vázquez-Aguilera

Applied Physics and Electronics,
Umeå University,
Umeå 901 87, Sweden
e-mail: electroncvaitc@gmail.com

Vicente Parra-Vega

Center for Research,
Saltillo Coahuila 25900, Mexico
e-mail: vparra@cinvestav.mx

Anand Sánchez-Orta

Center for Research,
Saltillo Coahuila 25900, Mexico
e-mail: sanchez@cinvestav.mx

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received December 4, 2017; final manuscript received February 21, 2018; published online April 30, 2018. Assoc. Editor: Hashem Ashrafiuon.

J. Dyn. Sys., Meas., Control 140(9), 091018 (Apr 30, 2018) (5 pages) Paper No: DS-17-1604; doi: 10.1115/1.4039487 History: Received December 04, 2017; Revised February 21, 2018

## Abstract

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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Topics: Signals

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## Figures

Fig. 1

A signal with Hölder continuous but not IO differentiable derivative: (a) signal f(t) and (b) f˙(t)

Fig. 2

Comparison between the FO differentiator and the IO differentiator: (a) FO: Signal estimation x1, (b) IO: Signal estimation x1, (c) FO: Derivative estimation x2, (d) IO: Derivative estimation x2, (e) FO: Estimation error x̃1, and (f) IO: Estimation error x̃1

Fig. 3

Noisy case: Comparison between the FO differentiator and the IO differentiator: (a) FO: Signal estimation x1, (b) IO: Signal estimation x1, (c) FO: Derivative estimation x2, (d) IO: Derivative estimation x2, (e) FO: Estimation error x̃1, and (f) IO: Estimation error x̃1

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