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## Über dieses Buch

This monograph presents design methodologies for (robust) fractional control systems. It shows the reader how to take advantage of the superior flexibility of fractional control systems compared with integer-order systems in achieving more challenging control requirements. There is a high degree of current interest in fractional systems and fractional control arising from both academia and industry and readers from both milieux are catered to in the text. Different design approaches having in common a trade-off between robustness and performance of the control system are considered explicitly. The text generalizes methodologies, techniques and theoretical results that have been successfully applied in classical (integer) control to the fractional case.

The first part of Advances in Robust Fractional Control is the more industrially oriented. It focuses on the design of fractional controllers for integer processes. In particular, it considers fractional-order proportional-integral-derivative controllers, because integer-order PID regulators are, undoubtedly, the controllers most frequently adopted in industry.

The second part of the book deals with a more general approach to fractional control systems, extending techniques (such as H-infinity optimal control and optimal input‒output inversion based control) originally devised for classical integer-order control.

## Inhaltsverzeichnis

### Chapter 1. Introduction to Fractional Calculus

Abstract
In this chapter, fractional calculus is introduced. After a brief overview about this topic, some basic definitions and properties are presented. Moreover, some geometrical and physical interpretations of fractional operators are described, both in time and frequency domain.

### Chapter 2. Fractional Systems for Control

Abstract
This chapter is devoted to fractional linear time-invariant systems because they are the fundamental tool for fractional control. After a brief overview about fractional control, well-known properties such as stability, modes, and input-output properties for fractional linear time-invariant systems are discussed. Here, fractional systems are treated in external form because this is the form that will be used for the reminder of the book.

### Chapter 3. Fractional-Order Proportional-Integral-Derivative Controllers

Abstract
This chapter is devoted to the generalization to the fractional case of well-known proportional-integral-derivative controllers, namely fractional-order proportional-integral-derivative controllers. After an introduction about these controllers and their structure, much space is dedicated to the tuning strategies for these regulators. In particular, optimization-based tuning rules are presented for both asymptotically stable and non-asymptotically stable processes. Both set-point tracking and load disturbance rejection tasks are considered. Performance indexes are determined, in order to be used for the assessment of the controller performance.

### Chapter 4. FOPID Controller Additional Functionalities

Abstract
In this chapter some additional functionalities that FOPID controllers should posses for their use in industry are discussed. In particular, the problem of tuning the set-point weight for FOPID controllers is addressed in the first part, whereas the second part is devoted to the analysis of anti-windup strategies.

### Chapter 5. $$\fancyscript{H}_\infty$$ H ∞ Control of Fractional Systems

Abstract
In this chapter, the solution for the standard $$\fancyscript{H}_\infty$$ control problem for fractional linear time-invariant single-input-single-output systems is presented. The adopted approach consists of extending to the fractional case the procedure followed within the classical solution for the integer case. According to the classical route, the generalization to fractional systems of the standard Youla parametrization of all the stabilizing controllers is first considered. The $$\fancyscript{H}_\infty$$ optimal controller is then found among the class of stabilizing controllers, recasting the control problem into a model-matching one. The obtained results naturally extend well-established results to a fractional setting, including both the commensurate and incommensurate cases, thus providing a framework where $$\fancyscript{H}_\infty$$ design can be recast along the well-known and fruitful lines of the integer case.

### Chapter 6. $$\fancyscript{H}_\infty$$ H ∞ Optimization-Based FOPID Design

Abstract
In this chapter a fractional-orderFOPID controllers PID controller design based on the solution of a $$\fancyscript{H}_\infty$$ model-matching problem for fractional first-order-plus-dead-time processes is proposed. Starting from the analytical solution of the problem, a fractional-order PID suboptimal controller is obtained. Guidelines for the tuning of the controller parameters are given in order to address the robust stability issue and to obtain the required performance.