Estimation of exact initial states of fractional order systems

The initial condition problem of fractional order systems is investigated in this paper. Firstly, a named aberration phenomenon is introduced, which reveals the nature of initial condition problem. And then, in order to interpret this odd phenomenon, definitions of the Riemann-Liouville derivative and Caputo derivative are revisited. As a result, the fractional order systems described by fractional differential equations and exact state-space models are linked much more closely, and it is found that relationship between these two kinds of models basically lies in the exact initial state distribution. The results also show the inborn defects of these two derivatives. Afterward, a practical method to estimate the exact initial states is studied naturally. At last, several simulation examples carefully illustrate the effectiveness of proposed method.