FRACTIONAL CALCULUS APPLICATIONS IN AUTOMATIC CONTROL AND ROBOTICS

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FRACTIONAL CALCULUS
APPLICATIONS IN AUTOMATIC
CONTROL AND ROBOTICS
41st IEEE CDC2002 TUTORIAL WORKSHOP
Las Vegas, December 9 2002
Organizers:
Blas M. Vinagre
Blas M. Vinagre (University
of
Extremadura, Spain)
YangQuan Chen
YangQuan Chen
(Utah
State
University, USA)
PRESENTATION OUTLINE

TOPIC, PURPOSE AND STRUCTURE

WHY FRACTIONAL CALCULUS

WHAT’S FRACTIONAL CALCULUS

PIONEERING APPLICATIONS IN CONTROL

TUTORIAL WORKSHOP OVERVIEW

COMPLEMENTS

LECTURES AND SPEAKERS
TOPIC, PURPOSE AND
STRUCTURE
TOPIC AND PURPOSE

Idea of fractional operators is as old as the idea of the
integer order ones is.

The theoretical and practical interest of these operators is
nowadays well established, and its applicability to science
and engineering can be considered as emerging new
topics.

The fractional integro-differ
ential operators: Fractional
Calculus.

Give an overview of the fundamentals of Fractional
Calculus and its applications in Automatic Control and
Robotics.
TOPIC, PURPOSE AND
STRUCTURE
STRUCTURE

Mini-course covering:

Mathematical foundations of Fractional Calculus

Practical applications
in
Automatic
Control
and Robotics

Each topic presented by an expert on it

Software
tools will
be
presented used for:

Fractional systems identification

Fractional systems analysis

Fractional controllers design
WHY FRACTIONAL CALCULUS

The Procrustes Bed:
all people must fit the same bed:
if tall, cut the legs
if short, stretch the legs

The Purloined Letter:
same methods must be allways successfuly applicables:
if no solution is found,
there is no solution

A new paradigm:
a good opportunity for revisiting the origins
WHY FRACTIONAL CALCULUS

MODELING:

Time domain: is g(t) a linear combination of exponentials ?
Why not a function, just one, being exponentials, real or
complex, particular cases ?
WHY FRACTIONAL CALCULUS

MODELING:

Frequency domain: is H(jw)
a rational function of s
? Why
not a rational function of s0.4
in order to fit the slope with only
a factor ?
WHY FRACTIONAL CALCULUS

MODELING:

Weighted time: have all the past values the same weight
?
Or have different weights ?
WHY FRACTIONAL CALCULUS

CONTROL:

Basic control actions: why not to extended in a continuous
way?
WHY FRACTIONAL CALCULUS

CONTROL:

Reference systems for control: why not to use a different and
more robust (to gain and load changes) reference system for
design specifications ?
WHAT’S FRACTIONAL CALCULUS

TIME DOMAIN: operators defined by convolution
WHAT’S FRACTIONAL CALCULUS

LAPLACE AND FREQUENCY DOMAINS: fractional
(non integer) powers of Laplace or Fourier operators
PIONEERING APPLICATIONS IN
CONTROL

Bode (1945): Feedback amplifiers ideal open loop
transfer function (overshoot independent of gain):
fractional integrator

Tustin (1958): Control of massive objects: a constant
phase margin over a rather wide range of frequencies

Manabe (1961): Control with fractional integrator

Carlson and Halijak (1961): Control of a servo with
fractional integrator

Oustaloup (1981): Robust Control of Non Integer
Order (CRONE)
TUTORIAL WORKSHOP
OVERVIEW
PART I: FUNDAMENTALS

Historical introduction

Fractional calculus fundamentals

Fractional order systems and fractional order control actions

Analog and digital implementations of fractional order
operators

Software tools presentation I
TUTORIAL WORKSHOP
OVERVIEW
PART II: APPLICATIONS

Robus
t control

Other control applications:

Iterative learning control

Adaptive control

Control of distributed parameter systems

Robotics:

Robot manipulators

Mobile robots

Systems identification

Software tools presentation II
TUTORIAL WORKSHOP
OVERVIEW
HISTORICAL INTRODUCTION

L’Hôpital to Leibnitz: extension of meaning of dny/dxn, with n
fractional, irrational or complex.

Leibnitz to L’Hôpital: It will lead to a paradox, a paradox from
which one day useful consequences will be drawn.

For three centuries developed mainly as a pure theoretical
mathematical discipline (Riemann, Liouville, Abel, Lacroix,
Fourier, etc..).

Nowadays used in: electrochemistry, probability, mechanics,
control, robotics, signal processing, .....
TUTORIAL WORKSHOP
OVERVIEW
FRACTIONAL CALCULUS FUNDAMENTALS

The problem of definition of fractional order integro-
differential operators.

Different definitions: Riemann-Liouville, Weil, Caputo,
etc.

Laplace and Fourier transforms.

Fractional order differential equations: solutions and
numerical evaluation.
TUTORIAL WORKSHOP
OVERVIEW
FRACTIONAL ORDER SYSTEMS AND FRACTIONAL
ORDER CONTROL ACTIONS

Models and representations

Controlability and Obsevability

Stability

Time response

Bode's Ideal Transfer Function

Fractional Order Control Actions
TUTORIAL WORKSHOP
OVERVIEW
ANALOG AND DIGITAL IMPLEMENTATIONS OF
FRACTIONAL ORDER OPERATORS

The problem of implementation = the adavantage of
modelling: memory (definition by convolution)

The Procrustes Bed --
reverse

Analog implementations: circuits and devices
performing approximated fractional integrals and
derivatives

Digital implementations: fractional powers of
generating functions: selection of functions and
expansions: FIR vs. IIR digital filters
TUTORIAL WORKSHOP
OVERVIEW
ROBUST CONTROL

Robustness

CRONE control:

Commande Robuste Ordre Non Entier (Robust Control of
Non Integer Order)

First generation: Bode’s ideal transfer function

Second generation: Optimized loop shaping

Third generation: General complex fractional order

Application examples

CRONE suspension
TUTORIAL WORKSHOP
OVERVIEW
OTHER CONTROL APPLICATIONS

Pioneering works

Fractional PID’s

Iterative Learning Control

Adaptive Control

Optimal Control

Application examples
TUTORIAL WORKSHOP
OVERVIEW
ROBOTICS

Robot manipulators:

Trayectory control of robot manipulators

Fractional describing functions

Motion control:

Fractional motion control of cutting tables

Fractional path tracking of mobile robots
TUTORIAL WORKSHOP
OVERVIEW
SYSTEMS IDENTIFICATION

Systems identification using fractional Hammerstein
models

Systems identification using fractional state variable
filter

Time domain systems identification using modal
decomposition

Frequency domain system identification of
commensurate order systems: application to a
flexible structure
COMPLEMENTS

Software
tools presentations
:

CRONE Fractional Systems Toolbox for MATLAB

Time and frequency domain syst
em identification tool for
MATLAB

Tool for describing function and phase portrait analysis of
fractional systems with hard nonlinearities
•M
o
v
i
e
s
:

CRONE Team, University of Bordeaux

Center for
Intelligent and Self Organizing Systems

Documentation:

Lecture notes and presentation slides availables in:

CDROM

Web site: http://eii.unex.es/isa
LECTURES AND SPEAKERS
MORNING (I)

08:00 -
08:15 Workshop Presentation

Blas M. Vinagre, University of Extremadura, Spain

08:15 -
08:30 Historical Introduction

Lokenath Debnath, University
of Texas Pan-American, USA

08:30 -
09:30 Fractional Calculus Fundamentals

Lokenath Debnath, University
of Texas Pan-American, USA

09:30 -
10:15 Fractional Order Systems and Fractional Order
Control Actions

Blas M. Vinagre, University of Extremadura, Spain
LECTURES AND SPEAKERS
MORNING (II)

10:15 -
10:30
Break

10:30 -
11:30
Analog and
Digital
Imp
lementations of Fractional
Order Operators

J. A. Tenreiro Machado, Institure of Engineering of Porto, Portugal,
and Blas M. Vinagre, University of Extremadura, Spain

11:30 -
12:00 Fractional Calculus Tools I

12:00 -
13:00
Lunch
LECTURES AND SPEAKERS
AFTERNOON

13:00 -
14:00
Robust
Control

Patrick Lanusse, University of Bordeaux, France

14:00 -
15:00 Other Control Applications

YangQuan Chen, Utah State University, USA

15:00 -
16:00 Robotics

J. A.
Tenreiro
Machado,
Institure of Engineering of
Porto, Portugal,
and Pierre Melchior, University of Bordeaux, France

16:00 -
16:30 Systems Identification

Pierre Melchior, University of Bordeaux, France

16:30 -
17:00 Fractional Calculus Tools II