The infinite state approach: Origin and necessity

The objective of this paper is to demonstrate that the Infinite State Approach, used for fractional order system modeling, initialization and transient prediction is the generalization of integer order system theory. The main feature of this classical theory is the integer order integrator, according to Lord Kelvin’s principle. So, fractional order system theory has to be based on the fractional order integrator, characterized by an infinite dimension frequency distributed state. As a consequence Fractional Differential Systems or Equations generalize Ordinary Differential Equation properties with an infinite dimension state vector. Moreover, Caputo and Riemann–Liouville fractional derivatives, analyzed through their associated fractional integrators, are no longer convenient tools for the analysis of fractional systems.