Globally convergent differentiators with variable gains

A new robust exact sliding mode (SM) based differentiator is proposed which provides for the fast global finite-time convergence of its outputs to the first n exact derivatives of its input f(t). The differentiator utilises the knowledge of a function L(t) providing the estimation |f ^{(n + 1)} ₀| ⩽ L(t), and satisfying |L|/L ⩽ M for a known bound M. The standard accuracy of the homogeneous SM-based differentiator is preserved in the presence of discrete sampling and noises in both f and L. The proposed discretisation scheme ensures the same accuracy in computer realisation.