Fitting of the initialization function of fractional order systems

The initial value problem of fractional order systems is studied further in this paper. Firstly, a new concept is put forward and named as aberration phenomenon, which reflects the complexity and the importance of the initial value problem. Then, the knowledge of the long memory property is reviewed for the purpose of revealing the nature of this phenomenon. As a result, it is involved with the so-called pre-initial process which describes the dynamic characteristic of fractional order systems before the initial state. Afterward, the definition of nonlinear time-varying initialization function is studied and the method of segmented linearization is proposed to fit the initialization function; meanwhile, other ideas on function fitting are provided as the follow-up study. Finally, several simulation examples are shown to illustrate the effectiveness of the proposed method.