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2015 | Book

Fractional Order Models Read chapter Read first chapter

Fractional Order Differentiation and Robust Control Design

CRONE, H-infinity and Motion Control

Authors: Jocelyn Sabatier, Patrick Lanusse, Pierre Melchior, Alain Oustaloup

Publisher: Springer Netherlands

Book Series : Intelligent Systems, Control and Automation: Science and Engineering

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This book provides an overview of the research done and results obtained during the last ten years in the fields of fractional systems control, fractional PI and PID control, robust and CRONE control, and fractional path planning and path tracking. Coverage features theoretical results, applications and exercises.

The book will be useful for post-graduate students who are looking to learn more on fractional systems and control. In addition, it will also appeal to researchers from other fields interested in increasing their knowledge in this area.

Table of Contents

Frontmatter
Chapter 1. Fractional Order Models
Abstract
In a letter to L’Hospital in 1695, Leibniz raised the possibility of generalizing the operation of differentiation to non-integer orders and L’Hospital asked what would be the result of half-differentiating a variable. Leibniz in an open letter dated September 30, 1695 replied “It leads to a paradox, from which 1 day useful consequences will be drawn”. It was only at the beginning of the nineteenth century that the generalization of differentiation to real or complex orders was formalized with the work of Liouville and Riemann (Liouville 1832; Riemann 1876; Letnikov 1868). This generalization is referred to in the literature as “fractional calculus” (Oldham and Spanier 1974) but also as “fractional differentiation” or sometimes “generalized differentiation”. In this book, the terms “fractional differentiation” or “fractional integration” are used to describe fractional order differential operators. This denomination is the most prevalent, although it is restrictive as it subsumes under a single term differentiation orders that can be integer, fractional, real and/or complex.
J. Sabatier, C. Farges, A. Oustaloup
Chapter 2. Fractional Order PID and First Generation CRONE Control System Design
Abstract
This chapter presents the design of controllers for Single Input/Single Output (SISO) systems, that is to say only one signal to control only one measured output. The fractional order controller is presented as a generalization of the common PID controller. Then, it is shown how the first generation of the CRONE methodology is able to design robust controllers for a class of gain-like perturbed systems.
P. Lanusse, J. Sabatier, A. Oustaloup
Chapter 3. Second and Third Generation CRONE Control-System Design
Abstract
This chapter presents the second generation of the CRONE methodology that extends the field of application of the first generation one. By replacing the real fractional order by a complex order, the third generation of the CRONE methodology is defined for any perturbed SISO system and is finally used to extend the CRONE approach to the design of robust controllers for Multi-Input/Multi-Output (MIMO) systems.
P. Lanusse, J. Sabatier, D. Nelson Gruel, A. Oustaloup
Chapter 4. H ∞ Control of Commensurate Fractional Order Models
Abstract
A commensurate fractional order model can be represented by a pseudo state space representation similar to the state space representation of an integer order model as shown in Chap. 1. This similarity can be used to extend H ∞ synthesis methods developed for integer order models represented by a state space representation to the case of fractional order models.
J. Sabatier, C. Farges, L. Fadiga
Chapter 5. Fractional Approaches in Path Tracking Design (or Motion Control): Prefiltering, Shaping, and Flatness
Abstract
Automatic control has a long history in engineering. At the end of the seventeenth century, Hooke introduced a system of balls rotating around an axis in which the velocity was proportional to the velocity of the windmill: the greater the ball velocity, the larger the gap from the axis activating the windmill sails in order to reduce the velocity. During the industrial revolution, Watt adapted a ball regulator for steam engines: the greater the ball velocity, the wider the opening of a valve that released the steam. By lowering the boiler pressure, the velocity could be reduced. The main problem was to maintain a constant speed despite load variations. In 1868, the physicist Maxwell published the first mathematical analysis explaining some of the behaviors observed on the regulators employed at that time. That was the beginning of several studies on stability to which the mathematicians Hurwitz and Routh were the main contributors. After theoretical and technological developments, it became possible to handle linear and nonlinear multivariable systems thanks to the important contributions of Bellman in dynamical programming (Brassard and Bratley 1996), Kalman (1960) in filtering and linear quadratic control and Pontryagin et al. (1962) in optimal control. Their contributions still feed research in control theory today.
P. Melchior, S. Victor
Metadata
Title
Fractional Order Differentiation and Robust Control Design
Authors
Jocelyn Sabatier
Patrick Lanusse
Pierre Melchior
Alain Oustaloup
Copyright Year
2015
Publisher
Springer Netherlands
Electronic ISBN
978-94-017-9807-5
Print ISBN
978-94-017-9806-8
DOI
https://doi.org/10.1007/978-94-017-9807-5