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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.

Advances in Robust Fractional Control will be a useful reference for the large number of academic researchers in fractional control, for their industrial counterparts and for graduate students who want to learn more about this subject.

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.
Fabrizio Padula, Antonio Visioli

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.
Fabrizio Padula, Antonio Visioli

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.
Fabrizio Padula, Antonio Visioli

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.
Fabrizio Padula, Antonio Visioli

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.
Fabrizio Padula, Antonio Visioli

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.
Fabrizio Padula, Antonio Visioli

Chapter 7. Control Design Based on Input–Output Inversion

Abstract
This chapter deals with the input–output system inversion of fractional-order minimum-phase scalar linear systems. Given an arbitrarily smooth output function, the corresponding input is computed explicitly in order to obtain a smooth transition of the system from a steady-state value to a new one in a predefined time interval. The minimum-time constrained transition problem is addressed. Given the predefined output function and set of constraints on the input and output signals and their derivatives, the existence of an optimal feasible input is proven under very mild conditions. The proposed methodology is conveniently used in the second part of the chapter for the synthesis of the feedforward action for a fractional control system in order to achieve a predefined process variable transition from a steady-state value to another. In particular, the feedforward action is implemented either as a signal to be added to the feedback control variable or as a command signal to be applied (instead of the typical step signal) to the closed-loop system. Finally, the approach is combined with the robust controller synthesis of Chap. 6 in order to minimize the worst-case settling time for family of systems subject to constraints on the control variable and on the maximum overshoot of the system output.
Fabrizio Padula, Antonio Visioli

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