TUTORIAL WORKSHOP TITLE
Fractional Order Calculus
and Its Applications in Mechatronic System Controls
TUTORIAL WORKSHOP WEB:
http://mechatronics.ece.usu.edu/foc/ieee-icma06-tutorial/
WORKSHOP ORGANIZERS
(see below for bios)
Dingyu Xue－Northeastern Universtiy,
PRC
YangQuan
Chen – Utah State University,
USA
CONTACT INFORMATION: For more information about
the workshop, please contact:
Professor Dingyu
Xue
Institute of Robotics and Artificial
Intelligent
School of Information
Science and Engineering
Northeastern
Universtiy, Shenyang
110004, P R China
email: xuedingyu@ise.neu.edu.cn, T/F:
+86-24-23899982
or
YangQuan Chen, Ph.D, Acting
Director,
Center for Self-Organizing and
Intelligent Systems (CSOIS),
Assistant Professor, Dept. of Electrical
and Computer Engineering,
Utah
State University,
4120 Old Main Hill, Logan, UT 84322-4120,
USA
E: yqchen@ece.usu.edu, T: 1(435)797-0148; F: 1(435)797-3054;
WORKSHOP DURATION: Half day (4 hours)
LIST OF PRESENTERS
Dingyu Xue－Northeastern Universtiy, China
YangQuan Chen – Utah State University, USA
WORKSHOP ABSTRACT AND OBJECTIVES
The purpose
of this tutorial workshop is
to introduce the fractional calculus and its
applications in controller designs. Fractional order calculus, or integration
and differentiation of an arbitrary order or fractional order, is a new tools
that extends the descriptive power of the conventional calculus. The tools of
fractional calculus support mathematical models that in many cases more
accurately describe the dynamic response of actual systems in electrical,
mechanical, and automatic control applications etc. The theoretical and
practical interest of these fractional order operators is nowadays well
established, and its applicability to science and engineering can be considered
as emerging new topics. The need to digitally compute the fractional order
derivative and integral arises frequently in many fields especially in automatic control and digital signal processing. Fractional order proportional-integral-derivative
(PID) controllers are based on the fractional order calculus where the
derivative or integral can be of a non-integer order. Due to the extra tuning
knobs, it is expected that better control performance can be achieved if the
fractional order PID controller is used. Fractional calculus has much to offer
science and engineering by providing not only new mathematical tools, but more
importantly, its application suggests new insights into the system dynamics as
well as controls.
WORKSHOP AUDIENCE
The expected audience includes
engineers, scientists, postgraduate students, and academics. The workshop will
be self-contained so that it is suitable for systems and control researchers
and practitioners who may not be familiar with the concept of fractional order systems as well as to those with some initial
background in the field.
ORGANIZATIONAL
DETAILS
1. Attendees will be given hardcopies of the presentation
slides and will be provided electronic copies as well.
2. The organizers and presenters will use electronic
projection in PowerPoint or PDF format. We will
provide our own computers, but we require the use of an LCD projector.
3. We would prefer to not have more than 50 attendees.
WORKSHOP PROGRAM
The workshop
begins with an introduction to the theoretical foundation of FOS including fractional
calculus and its basic properties. It is now well known that many
physical phenomena can be modeled accurately and effectively using fractional
derivatives, whereas the integral derivative based models capture these
phenomena only approximately. It has been demonstrated that many fractional
derivatives based controls are far superior than integer order derivative based
control schemes. After this introduction, approaches
to digital computation of fractional order
derivative and integral will be presented. A new synthesis approximation method
is introduced to approximate fractional order differentiator operator 
in a given frequency
range of interest and realize frequency-band limited FOS. This new method can achieve
better approximation than the existent discretization methods in the whole frequency range. Next, it is shown how the building Simulink model
can be used to obtain numerical solution of fractional order calculus equation.
We will illustrate that this Simulink model makes it possible to resolve the
fractional order linear and nonlinear calculus equation.
Next we will begin to consider design and apply the FOC.
According to the desired gain margin and phase margin, the designed fractional
order PID controller can meet the stability robustness of the feedback control
loop. Designed FOC can be applied to the integer order system and fractional
order system. Then we apply the FOC to a basic control problem in automatic
control — position
servomechanism control system. The better robustness performances of FOC are
shown by the comparison of fractional order PID control’s responses with normal
PID control.
WORKSHOP SCHEDULE (June
ORGANIZERS’ and PRESENTERS’
BIOGRAPHIES
Dingyu Xue
(Organizer) received the B.S.
and M.S. degrees in automatic control engineering from Shenyang Polytechnic University
and Northeastern University, respectively. He received the D.Phil. in control engineering from Sussex University. He took up teaching posts in Northeastern University since 1993, and was appointed the
professor in the Faculty of Information Sciences and Engineering in Northeastern
University since 1997. He is the author
and coauthor of many monographs and textbooks from Tsinghua University Press,
mainly concentrate on computer aided control systems design, system simulation
and MATLAB based mathematical problem solutions.
YangQuan Chen (co-Organizer) is presently an
assistant professor of Electrical and Computer Engineering Department and the
Acting Director for CSOIS (Center for Self-Organizing and Intelligent Systems)
at Utah State University. He obtained his Ph.D. from Nanyang Tech. Univ. (NTU),
Singapore. Dr Chen has 12 US patents granted and 2 US patent applications
published, most related to iterative learning control and repetitive control.
He published over 160 academic papers, two textbooks, one research monograph,
and (co)authored over 50 industrial reports. He has been an Associate Editor in
the Conference Editorial Board of IEEE Control Systems Society since 2002 and
is a founding member of the ASME subcommittee of "Fractional
Dynamics" in 2003. He is a senior member of IEEE, a member of ASME and a
member of International Society for Information Fusion.
DETAILED WORKSHOP SYNOPSIS
Brief historic review of fractional calculus and overview
of its applications
There is an increasing interest in dynamic systems of
non-integer orders. Extending derivatives and integrals from integer orders to
non-integer orders has a firm and long standing theoretical foundation. For
example, Leibniz mentioned this concept in a letter to L’Hospital over three hundred years ago and the earliest
more or less systematic studies have been made in the beginning and middle of
the 19th century by Liouville, Riemann and Holmgren. In the literature, people
often use the term “fractional order calculus”, or “fractional order dynamic
system” where “fractional” actually means “non-integer”.
Among the literature, the definitions commonly used are
the Riemann-Liouville definition and the
Grünwald-Letnikov definition, which are given below
With such definitions, the fractional
order calculus are established and used in many fields. Control theory is one of the areas, where
fractional order calculus used widely. Many
originally considered lumped systems can be more exactly described by
fractional order systems. And in
controller design, fractional order controllers also exhibit their advantages. For instance, in fractional order PID
controllers, 
, two extra tuning knobs, l and m, make the controller more flexible and the behaviors of the controller
may superior to conventional controllers.
Computer
solutions to fractional calculus problems
The computation of the fractional order calculus is very
useful in the studies of controller design.
In the tutorial workshop, systematic methods and MATLAB based computer
routines will be given, where
l
Off-line
fractional order differentiators using Grünwald-Letnikov definition
l
On-line approximation of differentiators with improved
Oustaloup’s filters where
where
，
, 
l
Exact
solutions to fractional order linear differential equations
l
Block-diagram
based algorithms for fractional order nonlinear differential equations
l
Frequency
domain analysis to fractional order linear systems.
l
Realization
and discretization for fractional order systems
l
Properties of
fractional order systems
Applications of fractional order controllers design in
control systems
Fractional order PID controller is a very good example for process
control systems, whose mathematical model is given by
where two extra knobs are provided. It can be seen from the figure that, the
conventional PI controller, PD controller and PID controllers are special cases
of fractional order PID controllers. Flexibility
isare provided in the design of such controllers. Better performance can be expected in using
such controllers. It will be shown in
the next section that the fractional PID controller may have superior
properties to the integer order PID controllers. The controller design techniques given in
this part are:
l
Introduction to fractional order PID controllers
l
Discussions in performance index selections in controller
design.
l
ITAE criterion based optimum controller design technique.
l
An optimum order controller designer and its applications
in controller design.
Fractional
order controller versus integer order controllers: comparative studies through
benchmark problems
A well-established position
servomechanism is used as a benchmark problem in the comparisons of best integer order PI/PID controllers and the
best fractional order PI/PID controllers.
Comparisons will be made through different aspect. The control performance will be compared first,
and the robustness of the controllers to mechanical nonlinearities, the
robustness to elastic parameter changes, and so on, will be given.
The block diagram of the system to
be controller in given by
The
mathematical model of the system is given as
For this
benchmark model, the robustness is very high as demonstrated below
Last updated:
5/3/2006.