Fractional calculus is three centuries old, but interest in its applications is growing. This is a short book on fractional calculus with some useful mathematics and applications. It describes various definitions for fractional order derivatives and integrals, and various analytical and numerical methods for solving fractional order differential equations and fractional order integration. On the application side, it describes various fractional-calculus-based design techniques for controllers and specific application cases.

The first chapter describes four different types of fractional derivatives/integrals. Common properties such as linearity and the Leibniz rule follow. Each definition includes an evaluation, its relation to other definitions, and its area of application. The chapter also describes algorithms for numerical evaluation. The equations involving these definitions can be resolved using all these methods.

The second chapter is on techniques for designing controllers. The design techniques are classified into frequency based and states based. The major concerns are stability and associated transfer function. Proportional integral derivative (PID) controllers are discussed. After applying fractional order calculus, they are referred to as fractional order PID (FOPID) controllers. The chapter ends with some notes on optimization.

The third chapter covers control applications. The first application for controlling machine chatter has a nonlinear differential equation among its modeling equations. An adaptive fractional-order-based approach achieves better results than a single-order approach. A FOPID controller can also be used to control the nonlinear motions of a multilink robot. The chapter shows how FOPID controllers achieve stability. The chapter also presents numerical simulation results for trajectory controls. FOPID controllers are found to have superior performance. Controlling a rotary inverted pendulum (RIP) system is the third application discussed. The model of the application has a nonlinear subsystem. The simulation compares the integer-order approach with the fractional-order approach for the linear subsystem of a pendulum arm and the nonlinear subsystem of a rotary arm. For both, the fractional-order approach is shown to achieve better performance. The fourth application relates to internal combustion engine (ICE) speed control. The chapter details the model and shows better performance results for the fractional-order approach simulation.

This book is part of Morgan & Claypool’s “Synthesis Lectures on Mechanical Engineering” series. It is too short to be used as a textbook. Computing readers will appreciate chapter 1’s algorithms and chapter 3’s simulations. A strong foundation in calculus is required to understand the mathematics. The applications covered are in control systems. There is a growing interest in robotics, which makes the book more current.