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Substantial, tempered, and shifted fractional derivatives: Three faces of a tetrahedron
Author(s) -
Ortigueira Manuel D.,
Bengochea Gabriel,
Machado J. Tenreiro
Publication year - 2021
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.7343
Subject(s) - mathematics , fractional calculus , scope (computer science) , tetrahedron , range (aeronautics) , calculus (dental) , linear fractional transformation , pure mathematics , algebra over a field , control system , computer science , geometry , robust control , medicine , materials science , dentistry , electrical engineering , composite material , programming language , engineering
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.