  In the paper we deal with linear fractional control problems with constant
delays in the state. Single-order systems with fractional derivative in Caputo
sense of orders between 0 and 1 are considered. The aim is to introduce a new
algorithm convenient for numerical approximation of a solution of the studied
problem. The method consists of the fractional differential transformation in
combination with general methods of steps. The original system is transformed
to a system of recurrence relations. Approximation of the solution is given in
the form of truncated fractional power series. The choice of order of the
fractional power series is discussed and the order is determined in relation to
the order of the system. An application on a two-dimensional fractional system
is shown. Exact solution is found for the first two intervals of the method of
steps. The result for Caputo derivative of order 1 coincides with the solution
of first-order system with classical derivative. We conclude that the algorithm
is applicable, efficient and gives reliable results.
