The fractional order calculus (FOC) is as old as the integer one although up to recently its application was exclusively in mathematics. Many real systems are better described with FOC differential equations as it is a well-suited tool to analyze problems of fractal dimension, with long-term “memory” and chaotic behavior. Those characteristics have attracted the engineers' interest in the latter years, and now it is a tool used in almost every area of science. This paper introduces the fundamentals of the FOC and some applications in systems' identification, control, mechatronics, and robotics, where it is a promissory research field.

Regenerative Braking in vehicles is projected as an intelligent and simple technique to improve the energy replenishment and autonomy in them, due to the fact that it is based on the concurrent process of deceleration. It is smart because takes advantage of the capability of the electric machine to behave as a generator. So, it is possible to recover the kinetic energy store in the rotating inertias by using regenerative braking during the deceleration and restore it into batteries. Some authors are focused on maximizing the energy regenerated, applying techniques as independent factors of the process, like storage elements, control of speed, acceleration, current, etc. In this work we exposes the regenerative braking process and its main parts in a holistic model, where discrete and continuous states and events are related, also linked into the dynamic behavior, bringing the possibility not only to have a general model that commands all the process, but also improve and monitor each stage as a whole. Thus our management architecture pretends to maximize the gain energy in every deceleration, and minimize the energy consumed.