  The super-twisting differentiator, also known as the first-order robust exact
differentiator, is a well known sliding mode differentiator. In the absence of
measurement noise, it achieves exact reconstruction of the time derivative of a
function with bounded second derivative. This note proposes an upper bound for
its worst-case differentiation error in the presence of bounded measurement
noise, based on a novel Lipschitz continuous Lyapunov function. It is shown
that the bound can be made arbitrarily tight and never exceeds the true
worst-case differentiation error by more than a factor of two. A numerical
simulation illustrates the results and also demonstrates the
non-conservativeness of the proposed bound.
