Substantial, tempered, and shifted fractional derivatives: Three faces of a tetrahedron

The substantial, tempered, and shifted fractional derivatives, useful in medium range systems, are reviewed and highlighted in a unified framework. Their historical evolution is described and their properties studied. Moreover, they are characterized and assessed under the light of the strict sense criterion for the definition of derivatives. The relationship between these derivatives and several formulations developed in fractional calculus are explored. Additionally, the interpretations in the time and frequency domains are included. In the scope of the framework, new tempered linear systems and transfer functions are introduced. As particular applications, several systems for compensation and control purposes are revisited.