This book presents an overview of fractional calculus and fractional order control systems. In particular, the author focuses on topics such as modeling of fractional order systems, solutions and control designs for linear and nonlinear systems. The author carefully wrote the introductary part to give a detailed history and the basic mathematical theories to understand the fractional calculus. Definitions and properties of required functions and transforms are clearly formulated and with necessary details. The book contains a good survey on the literature and provides many numerical problems.
Ten chapters form the book including an introduction, basic theories, solution derivations and control designs. Chapters 1 to 3 present a history about the factional calculus and its basic ideas about useful functions and transforms. The fractional derivatives in the sense of Caputo and Riemann-Liouville and their relationships, higher order integrals in fractional sense are clearly explained.
Chapter 4 is devoted to solutions of linear fractional order systems, here the author presents the closed-form solutions for initial value problems and some numerical algorithms like Taylor’s auxiliary algorithm and high-precision algorithms. Some results for irrational systems are presented in detail.
Approximation techniques for fractional operators using Carlon’s method, Matsuda-Fujii filter, some other filter techniques together with the integer-order approximations, higher and lower order approximations are explained in Chapter 5.
State space models and analysis of fractional order systems are presented in Chapters 6 and 7. Further, these chapters deal with a qualitative analysis of various types of fractional order systems including single input single output systems. In particular, the author also focuses on stability, controllability and observability results for linear and nonlinear cases.
Chapter 8 is devoted to numerical solutions of nonlinear fractional-order differential equations using some well-known numerical techniques.
The Chapters 9 and 10 are devoted to various control designs like PID, PD and fuzzy controls and frequency domain controllers for single and multivariable fractional order systems.
The book ends up with some necessary definitions of functions and transforms involving fractional and irrational operators, as well as some numerical toolbox features like MATLAB and Simulink.
Overall, the book will be useful for the one interested in the theoretical and numerical tools to design control systems.