Students of mathematics, sciences, and engineering encounter the differential operators d/dx, d2/dx2, etc., but probably few of them ponder over whether it is necessary for the order of differentiation to be an integer. Why not be a rational, fractional, irrational, or even a complex number? At the very beginning of integral and differential calculus, in a letter to L’Hôpital in 1695, Leibniz himself raised the question: “Can the meaning of derivatives with integer order be generalized to derivatives with non-integer orders?” L’Hôpital was somewhat curious about that question and replied by another question to Leibniz: “What if the order will be 1/2?” Leibniz in a letter dated September 30, 1695 replied: “It will lead to a paradox, from which one day useful consequences will be drawn.” The question raised by Leibniz for a non-integer-order derivative was an ongoing topic for more than 300 years, and now it is known as fractional calculus, a generalization of ordinary differentiation and integration to arbitrary (non-integer) order.

Before introducing fractional calculus and its applications to control in this book, it is important to remark that “fractional,” or “fractional-order,” are improperly used words. A more accurate term should be “non-integer-order,” since the order itself can be irrational as well. However, a tremendous amount of work in the literature use “fractional” more generally to refer to the same concept. For this reason, we are using the term “fractional” in this book.


Fractional Order Fractional Calculus Historical Overview Integer Order Laplace Domain 
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© Springer-Verlag London Limited 2010

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