Introduction

Abstract

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.