  The modulating functions method has been used for the identification of
linear and nonlinear systems. In this paper, we generalize this method to the
on-line identification of fractional order systems based on the
Riemann-Liouville fractional derivatives. First, a new fractional integration
by parts formula involving the fractional derivative of a modulating function
is given. Then, we apply this formula to a fractional order system, for which
the fractional derivatives of the input and the output can be transferred into
the ones of the modulating functions. By choosing a set of modulating
functions, a linear system of algebraic equations is obtained. Hence, the
unknown parameters of a fractional order system can be estimated by solving a
linear system. Using this method, we do not need any initial values which are
usually unknown and not equal to zero. Also we do not need to estimate the
fractional derivatives of noisy output. Moreover, it is shown that the proposed
estimators are robust against high frequency sinusoidal noises and the ones due
to a class of stochastic processes. Finally, the efficiency and the stability
of the proposed method is confirmed by some numerical simulations.
