Advances in fractional calculus: Control and signal processing applications

Fractional calculus is more than a three hundred-year-old concept way back during the time of de l'Hospital and Leibniz focusing on derivative and integrals having non-integer orders. Almost four decades ago, engineers and scientists began to venture into the field of fractional calculus by unfolding its applications where fractional differential equation models are valid. It has been found that fractional calculus indeed is becoming ubiquitous, seeing applications in many fields of sciences and engineering, from fractional diffusion equations and various biomedical applications, to signal processing and control engineering applications. A conclusion was then later proposed that fractional calculus is actually a generalization of integer-order calculus, being so powerful, it could overcome the advantages of its integer-order counterparts. This paper offers a comprehensive discussion on the applications of fractional calculus in the design and implementation of fractional-order systems in the form of electronic circuits which could be used for signal processing and control engineering applications. The article starts with the introduction to fractional calculus including some history and mathematical definitions. The second part of the article focuses on fractional-order differential equations and systems. Example circuit designs and implementation are then discussed which includes an elaboration of some papers related to this area. The final part of the article presents possible research topics in this area.