Fixed-time convergent sliding-modes-based differentiators

• Published in: Communications in nonlinear science & numerical simulation 2022-01, Vol.104, p.106033, Article 106033
• Main Authors: Djennoune, Said, Bettayeb, Maamar, Al-Saggaf, Ubaid Muhsen
• Format: Article
• Language: English
• Subjects: Algorithms; Convergence; Differentiators; Errors; Estimating techniques; Fixed-time observers; Homogeneity; Initial conditions; Liapunov functions; Lyapunov functions; Mathematical functions; Modulating functions; Noise; Noise measurement; Sliding modes; Time dependence
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2021.106033

Conventional sliding-modes based differentiators make it possible to estimate successive derivatives of a given time-varying signal in finite-time and with exact convergence in noise free case. In general, the convergence time is an unbounded increasing function of initial estimation errors. Most already proposed solutions guarantee a convergence in a maximum time independent of initial conditions. In this paper, novel sliding mode differentiators with a prescribed convergence time are proposed. The convergence time can be chosen arbitrary whatever large initial estimation errors. The proposed key solution is based on a time-dependent transformation using modulating functions which make it possible to cancel the effect of initial conditions on the convergence time. New arbitrary order differentiators including the super-twisting algorithm based on modulating functions are introduced. Lyapunov functions and homogeneity tools are used to prove the convergence of the proposed first-order and arbitrary order differentiators, respectively. Robustness with respect to measurement noise is also addressed. •Development of innovative sliding-modes-based arbitrary order differentiators.•The predefined convergence time is chosen whatever initial estimation errors.•Time dependant transformation is used to annihilate initial estimation errors.•Convergence is based on Lyapunov theory and homogeneity properties.