Epitome of the Fractional Calculus. Theory and its Applications in Automatic

Seria : Monografie t. 12 / Komitet Automatyki i Robotyki Polskiej Akademii Nauk 1640-8969This book is inspired by almost forty years' development in the Fractional Calculus and its applications in diff erent areas of science. The main task of this book is to acquaint Polish readers with a variety of applications of this mathematical tool including commonly known notions like derivatives and integrals. The book gives a comprehensive introduction to the Fractional Calculus. Chapter 1 contains Griinwald-Letnikov, Riemann-Liouville, Caputo and Nishimoto definitions of a non-integer order differ-integral. Some fundamental properties are presented. In Chapter 3, a one-sided Laplace transform of a fractional-order derivative and integral is discussed. Special attention is paid to an inverse Laplace transform. Series expansion -, Fourier series -, and Valsa methods of inverse Laplace transform of a strictly proper rational function of non-integer orders are presented. In Chapters 6 and 7 some applications of the Fractional Calculus in automatics are given. Fractional order dynamical system descriptions by a transfer function and state-space equations are introduced. Fundamental properties of a fractional-order integrator are discussed. Its transient and frequency characteristics are plotted. Oustaloup, series expansion, Roy, Charef and least square approximation methods of fractional-order integrator are compared. A relation between a state-space and a transfer function description is given. One state-space realisation of a fractional-order transfer function is also given. Michailov and Nyquist stability tests are applied to fractional-order dynamical systems. In Section 5 a closed-loop system with CRONE controller is introduced. One method of CRONE controller synthesis is given. Section 6 presents the properties of PID controller with fractional differentiation and integration orders. Transient and frequency characteristics of such a controller are presented. As one possible application of a fractional-order backward difference, a closed-loop system synthesis with an electromagnetic servo and a