  In this paper we will present some alternative types of discretization
methods (discrete approximation) for the fractional-order (FO) differentiator
and their application to the FO dynamical system described by the FO
differential equation (FDE). With analytical solution and numerical solution by
power series expansion (PSE) method are compared two effective methods - the
Muir expansion of the Tustin operator and continued fraction expansion method
(CFE) with the Tustin operator and the Al-Alaoui operator. Except detailed
mathematical description presented are also simulation results. From the Bode
plots of the FO differentiator and FDE and from the solution in the time domain
we can see, that the CFE is a more effective method according to the PSE
method, but there are some restrictions for the choice of the time step. The
Muir expansion is almost unusable.
