Fractional operators' synthesis and realization using electrical and Bond Graph approaches

The aim of this article is to present an overview of the fractional operators such as integrators and differentiators based on resistive, inductive and capacitive components. Four different configurations of RLC based circuits that may produce a fractional behavior will be introduced. First, we will present these operators in the electrical domain than generalize our results in the Bond Graph representation. The results of the Bond Graph approach will be presented in state-space matrices where the main factors of power used are the flow and the effort. After that, we will show the main steps required to synthesis and realize these controllers starting from the four high-level parameters required (low and high frequencies of the phase locking, the non-integer order of derivation and the gain at the unit frequency). The number of cells is also important because we can't get a phase locking without using a minimum value as this property doesn't appear for a reduced cells' size. An application for the hydropneumatic suspension will emphasize, in more details, the way to synthesize and realize fractional operators and the compare the behavior between the ideal fractional integrator and the implemented one.