Fractional calculus and its applications

2 Hour Tutorial Session at American Control Conference 2009

2009-02-03T11:07:00.000-08:00

2009 American Control Conference -- ACC2009
St. Louis, Missouri, USA
June 10 - 12, 2009

Tutorial Session Proposal Officially Accepted.

“Applied Fractional Calculus in Controls”
Web: http://mechatronics.ece.usu.edu/foc/

Organizer’s Contact:
Organizer:
YangQuan Chen, Ph.D, Associate Professor and Graduate Coordinator
Department of Electrical and Computer Engineering,
Director, Center for Self-Organizing and Intelligent Systems (CSOIS)
Utah State University, 4120 Old Main Hill, Logan, UT 84322-4120, USA
E: yqchen@ece.usu.edu or yqchen@ieee.org, T/F: 1(435)797-0148/3054;
W: http://www.csois.usu.edu or http://yangquan.chen.googlepages.com

1. Why Fractional Calculus
Why Fractional Calculus? Many real dynamic systems are better characterized using a non-integer order dynamic model based on fractional calculus or, differentiation or integration of non-integer order. Traditional calculus is based on integer order differentiation and integration. The concept of fractional calculus has tremendous potential to change the way we see, model, and control the nature around us. Denying fractional derivatives is like saying that zero, fractional, or irrational numbers do not exist.

2. Why Fractional Calculus in Controls
In the control side, clearly, for closed-loop control systems, there are four situations. They are 1)
IO (integer order) plant with IO controller; 2) IO plant with FO (fractional order) controller; 3) FO plant with IO controller and 4) FO plant with FO controller. From control engineering point of view, doing something better is the major concern. Existing evidences have confirmed that the best fractional order controller outperforms the best integer order controller. It has also been answered in the literature why to consider fractional order control even when integer (high) order control works comparatively well. Fractional order PID controller tuning has reached to a matured state of practical use. Since (integer-order) PID control dominates the industry, we believe FO-PID will gain increasing impact and wide acceptance. Furthermore, we also believe that based on some real world examples, fractional order control is ubiquitous when the dynamic system is of distributed parameter nature.

3. Organizer’s Related Credits (selected)
· Previously, Dr. Chen co-organized a one day tutorial at IEEE CDC2002 (Las Vegas)
and since then, the workshop notes and CDROM have been widely cited:
http://mechatronics.ece.usu.edu/foc/cdc02tw/
CONFIDENTIAL. Limited circulation. For review only.
Proposal submitted to 2009 American Control Conference.
Received September 19, 2008.
· Dr. Chen co-organized a half day tutorial on Fractional Order Control at IEEE Int
Conf. on Mechatronics and Automation (ICMA06) in 2006, Luoyang, China.
http://mechatronics.ece.usu.edu/foc/ieee-icma06-tutorial/
· Dr. Chen was the plenary lecturer for IFAC Workshop on Fractional Derivatives
and Applications (FDA) 2006, Porto, Portugal. His lecture title is "Ubiquitous
Fractional Order Controls?" slides at
http://mechatronics.ece.usu.edu/foc/fda06/01ifac-fda06-plenary-talk%235-chenutah.
ppt
· Dr. Chen was a plenary lecturer for IFAC Workshop on Fractional Derivatives
and Applications (FDA) 2008, Ankara, Turkey. Title: “Fractional Order Signal
Processing: Techniques, Applications and Urgency”
http://www.cankaya.edu.tr/fda08/lecturers.php
· Dr. Chen co-authored the first control textbook with a dedicated chapter on
Fractional Order Control,
o Dingyu Xue, YangQuan Chen* and Derek Atherton. “Linear Feedback
Control – Analysis and Design with Matlab”. SIAM Press, 2007, ISBN:
978-0-898716-38-2. (348 pages) Chapter-8: Fractional-order Controller -
An Introduction.
· Dr. Chen co-authored the first math+Matlab book with a dedicated section on
Fractional Calculus introducing systematically how to perform numerical
simulation
o Dingyu Xue* and YangQuan Chen. “Solving Advanced Applied
Mathematical Problems Using Matlab”. Taylor and Francis CRC Press.
2008 (448 pages in English, ISBN-13: 978-1420082500.)
· More credits can be found from http://fractionalcalculus.googlepages.com/

3. Confirmed contributions in this ACC Tutorial Session

(40 min.) Lead paper (Panel discussion)
Fractional Order Control – A Tutorial (12 pages)
YangQuan Chen, Utah State University, Logan, USA
Ivo Petras, Technical University of Kosice, Kosice, Slovakia
Dingyu Xue, Northeastern University, Shenyang, China

(20 min.) Tutorial Session Paper#1
Title:
FO-[PD]: Fractional-order [Proportional Derivative] Controller for Robust Motion Control
Systems: Tuning Procedure and Validation
Authors:
Ying Luo and YangQuan Chen
Center for Self-Organizing and Intelligent Systems (CSOIS)
Department of Electrical and Computer Engineering,
Utah State University, 4120 Old Main Hill, Logan, UT 84322-4120, USA
Abstract:
In this paper, a fractional-order [proportional derivative] (FO-[PD]) controller is proposed
for robust motion control systems. Focusing on a class of simplified models for motion
control systems, a practical and systematic tuning procedure has been developed for the
proposed FO-[PD] controller synthesis. The fairness issue in comparing with other
controllers such as the traditional integer order PID (IO-PID) controller and the fractional
order proportional derivative (FO-PD) controller has been for the first time addressed
under the same number of design parameters and the same specifications. Side-to-side
fair comparisons of the three controllers (i.e., IO-PID, FO-PD and FO-[PD]) via both
simulation and experimental tests have revealed some interesting facts: 1) IO-PID
designed may not always be stabilizing to achieve flat-phase specification while both FOPD
and FO-[PD] designed are always stabilizing; 2) Both FO-PD and FO-[PD] outperform
IO-PID designed in this paper; 3) FO-[PD] outperforms FO-PD more when the time
constant of the motion control system increases. Extensive validation tests on our realtime
experimental test-bench illustrate the same.
Keywords:
Fractional calculus, fractional order controller, motion control, robustness, controller
tuning

(20 min.) Tutorial Session Paper#2
Title:
Fractional Order Networked Control Systems and Random Delay Dynamics: A Hardware-
In-The-Loop Simulation Study
Authors:
Shayok Mukhopadhyay, Yiding Han and YangQuan Chen
Center for Self-Organizing and Intelligent Systems (CSOIS)
Department of Electrical and Computer Engineering,
Utah State University, 4120 Old Main Hill, Logan, UT 84322-4120, USA
Abstract:
In networked control systems (NCS), the spiky nature of the random delays hints us to
wonder if the “spikiness”, or we call “delay dynamics” is considered in the NCS controller
design, what benefits we can expect. It turns out that the “spikiness” of the network
induced random delays should be better characterized by the so-called $\alpha$-stable
processes, or processes with fractional lower-order statistics (FLOS) which is linked to
fractional calculus. Using a real world networked control system platform called CSOIS
SmartWheel, the effect of modeling the network delay dynamics using non-Gaussian
distributions, and compensating for such a delay in closed-loop systems using FOPI
(fractional order proportional and integral) controller has been experimentally studied.
The cases studied include the case when the delay compensated is exactly the same as the
actual delay. Other scenarios are the ones when the nature of the estimated delay is similar
to the actual delay, but the magnitude is slightly smaller. The effect of phase shifting
between the estimated and the original delay is also considered. Finally the order of the
fractional order proportional integral controller which gives least ITAE, ISE for a
particular distribution of the delay is presented. The conclusion is strikingly stimulating:
in NCS, when the random delay is spiky, we should consider to model the delay dynamics
using $\alpha$-stable distributions and using fractional order controller whose best
fractional order has shown to be related to the FLOS parameter $\alpha$ as
evidenced by our extensive experimental results on a real NCS platform.
Keywords:
Fractional order control, fractional lower-order statistics, $\alpha$-stable processes, spiky,
delay dynamics, networked control systems

(20 min.) Tutorial Session Paper#3
Title:
Impulse response-based numerical scheme for approximately solving fractional optimal
control problems
Abstract:
In this paper, we present a methodology for approximating a SISO linear fractional
transfer function with the purpose of solving fractional optimal control problems. Using
the analytical response of the system to an impulse signal, a linear time invariant model is
calibrated to match the dynamics of the fractional dynamics system. The calibration
method uses the singular value decomposition of a Hankel matrix to get the parameters.
The size of the Hankel matrix and the sampling of the analytical solution are optimized so
as to obtain the best approximation for a given desired dimension of the linear system. A
definition of the fractional optimal control problem is given in the sense of the Riemann-
Liouville fractional derivatives. The fractional problem is then reformulated into a finite
dimension optimal control one using the rational approximation. This allows to use
commercially available optimal control problem solvers, like RIOTS_95. One timeinvariant
example, a time-variant example and a free final time example from the
literature are considered to illustrate the effectiveness of the formulation.
Keywords:
Fractional calculus, fractional order optimal control, Hankel matrix, numerical methods.

(20 min.) Tutorial Session Paper#4
Title:
Robust path planning for mobile robot based on fractional attractive force
Authors:
Pierre MELCHIOR 1, Brahim METOUI 2, Slaheddine NAJAR 2, Mohamed Naceur
ABDELKRIM 2 and Alain OUSTALOUP 1
1. IMS (UMR 5218 CNRS, Université Bordeaux 1 - ENSEIRB - ENSCPB)
Département LAPS
351 cours de la Libération, Bât. A4 - F33405 TALENCE cedex, France
Phone: +33 (0) 540 006 607 - Fax: +33 (0) 540 006 644
Email: pierre.melchior@laps.ims-bordeaux.fr - URL: http://www.ims-bordeaux.fr
2. MACS (Unité de Recherche Modélisation, Analyse et Commande des Systèmes)
ENIG (Ecole Nationale d'Ingénieurs de Gabès)
rue Omar Ibn El Khattab - 6029 Gabès, Tunisia
Phone: +216 75 392 100 Fax: +216 75 392 190
Email: brahim.metoui@fsg.rnu.tn - URL: http://www.mes.tn/enig/index.htm
Abstract:
In path planning, potential fields introduce force constraints to ensure curvature
continuity of trajectories and thus to facilitate path-tracking design. In previous works, a
path planning design by fractional (or generalized) repulsive potential has been developed
to avoid fixed obstacles: danger level of each obstacle was characterized by the fractional
order of differentiation, and a fractional road was determined by taking into account
CONFIDENTIAL. Limited circulation. For review only.
Proposal submitted to 2009 American Control Conference.
Received September 19, 2008.
danger of each obstacle. If the obstacles are dynamic, the method was extended to obtain
trajectories by considering repulsive and attractive potentials taking into account position
and velocity of the robot with respect to obstacles.
Then, a new attractive force based on fractional potential was developed. The
advantage of the generalized normalized force is the possibility to control its variation. The
curve is continuously varying and depends only on one parameter, the non integer order of
the generalized attractive potential. But, in case of robot parameter variations, these 2
attractive forces do not allow to obtain robust path planning.
In this paper, a new fractional attractive force for robust path planning of mobile
robot in dynamic environment is presented. This method allows to obtain robust path
planning despite robot mass variations. Section 1 presents fractional calculus. Section 2
deals with the fractional attractive force definition. Section 3 presents the robustness
analysis. A comparison between a classical method and the proposed approach is
presented in Section 4. Finally a conclusion is given in section 5.
Keywords
Robotics, Mobile robot, Robust Path planning, Fractional potential, Attractive force,
Dynamic environment.

Third Symposium on Fractional Derivatives and Their Applications

2006-10-17T08:20:00.000-07:00

The third symposium on Fractional Derivatives and Their Applications will be held at Las Vegas, Nevada, September 4-7, 2007. To find the details about the symposium, go to
http://www.rpi.edu/~anderk5/IDETC2007/, and select SYMPOSIA from the left column.

I invite you to submit papers for this symposium, and attend the conference.

Om P Agrawal

FDA06 Plenary Lecture

2006-07-20T04:18:00.000-07:00

FDA06 Plenary Lecture -- July 20, 2006 -- Prof. S. G. Samko.
Auditorium E, 9:00.

FDA06 started

2006-07-19T07:29:00.000-07:00

2nd IFAC Workshop on Fractional Differentiation and its Applications

started in Porto, July 19, 2006.

MATLAB routine for evaluating the Mittag-Leffler function with two parameters.

2005-10-17T07:16:00.000-07:00

The Mittag-Leffler function with two parameters plays an important role and appears frequently in solutions of fractional differential equations (i.e. differential equations containing fractional derivatives).

The MATLAB routine for evaluating the Mittag-Leffler function with two parameters is now available at the MATLAB File Exchange:

http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=8738&objectType=FILE

The routine was written by Igor Podlubny and Martin Kacenak in 2001-2003.

Articles submitted to ASME 2005 conferences, Long Beach, CA, September 24-28, 2005.

2005-10-01T22:05:00.000-07:00

The 2005 ASME International Design Engineering Technical Conferences & Computers and Information In Engineering Conference was held at the Hyatt Regency in Long Beach, California from September 24-28, 2005.

The following articles on fractional derivatives and their applications have been submitted for the conference and presented in several dedicated sessions (for details, see the detailed conference program (PDF)):

DETC2005-84460: Control of Time-Delay Systems Using
Robust Fractional-Order Control and Robust Smith Predictor
Based Control.
Patrick Lanusse, LAPS - Bordeaux 1 University, Talence,
France, Alain Oustaloup, LAPS - UMR 5131 CNRS, Université
Bordeaux 1 - ENSEIRB, TALENCE Cedex, France

DETC2005-85182: LMI Tools for Stability Analysis of Fractional
Systems.
Mathieu Moze, Jocelyn Sabatier, LAPS - University Bordeaux
1, Talence, France, Alain Oustaloup, LAPS - UMR 5131
CNRS, Université Bordeaux 1 - ENSEIRB, TALENCE Cedex,
France

DETC2005-84819: Fractional Control of a Single-Link Flexible
Manipulator.
Vicente Feliu, Universidad de Castilla - La Mancha, Ciudad
Real, Spain, Blas Vinagre, University of Extremadura, Badajoz,
Spain, Concepción A. Monje, Universidad de Extremadura,
Badajoz, Spain

DETC2005-84744: Robust Controllability of Interval Fractional
Order Linear Time Invariant Systems.
YangQuan Chen, Hyosung Ahn, Utah State University, Logan,
UT, United States, Dingyu Xue, Northeastern University,
Shenyang, Liaoning, China

DETC2005-84344: Ziegler-Nichols Type Tuning Rules for
Fractional PID Controllers.
Duarte Valério, José Sá da Costa, Technical University of Lisbon
- IST, Lisboa, Portugal, Portugal

DETC2005-84864: Fractional Model of a Gastrocnemius Muscle
for Tetanus Pattern.
Laurent Sommacal, Pierre Melchior, LAPS - UMR 5131
CNRS, Université Bordeaux 1 - ENSEIRB, Talence, France,
Jean-Marie Cabelguen, INSERM EPI 9914, Bordeaux Cedex,
France, Alain Oustaloup, LAPS - UMR 5131 CNRS, Université
Bordeaux 1 - ENSEIRB, Talence Cedex, France, Auke
Ijspeert, EPFL, Swiss Federal Institute of Technology, LAUSANNE,
Switzerland

DETC2005-84784: Approximation and Identification of Fractional
Systems.
Amel Benchellal, Thierry Poinot, Jean-Claude Trigeassou,
Laboratoire d’Automatique et d’Informatique Insustrielle,
Poitiers, France

DETC2005-84743: Sub-Optimum H2 Rational Approximations
to Fractional Order Linear Systems.
Dingyu Xue, Northeastern University, Shenyang, Liaoning,
China, YangQuan Chen, Utah State University, Logan, UT,
United States

DETC2005-84796: High Performance Low Cost Implementation
of FPGA-Based Fractional-Order Operators.
X. Jiang, T. Hartley, Joan Carletta, The University of Akron,
Akron, OH, United States

DETC2005-84579: A Direct Approximation of Cole-Cole-Systems
for Time-Domain Analysis.
Markus S. Haschka, Volker Krebs, Universität Karlsruhe (TH),
Karlsruhe, Baden-Württemberg, Germany

DETC2005-84345: Damping in a Fractional Relaxor-Oscillator
Driven by a Harmonic Force.
B. N. Narahari Achar, John Hanneken, University of Memphis,
Memphis, TN, United States

DETC2005-85230: Synthesis of a Limited-Bandwidth Fractional
Differentiator Made in Hydropneumatic Technology.
Pascal Serrier, Xavier Moreau, LAPS / Université Bordeaux 1,
Talence Cedex, France, Alain OUSTALOUP, LAPS - UMR
5131 CNRS, Université Bordeaux 1 - ENSEIRB, Talence
Cedex, France

DETC2005-84336: Nonlinear Statical and Dynamical Models
of Fractional Derivative Viscoelastic Body.
Hiroshi Nasuno, Nobuyuki Shimizu, Iwaki Meisei University,
Iwaki, Japan

DETC2005-84452: Fractional Derivative Consideration on
Nonlinear Viscoelastic Dynamical Behavior under Statical
Pre-displacement.
Masataka Fukunaga, Nihon University, Sendai, Japan,
Nobuyuki Shimizu, Hiroshi Nasuno, Iwaki Meisei University,
Iwaki, Fukushima, Japan

DETC2005-85624: Flatness Control: Application to a Fractional
Thermal System.
Pierre Melchoir, Mikael Cugnet, Jocelyn Sabatier, Alain
Oustaloup, LAPS - UMR 5131 CNRS, Université Bordeaux 1 -
ENSEIRB, Talence Cedex, France

DETC2005-84952: Complex-order Distributions.
Tom Hartley, Jay L. Adams, University of Akron, Akron, OH,
United States, Carl Lorenzo, NASA Glenn Research Center,
Cleveland, OH, United States

DETC2005-84493: Numerical Scheme for the Solution of
Fractional Differential Equations of Order Greater Than 1.
Om Agrawal, Pankaj Kumar, Southern Illinois University at
Carbondale, Carbondale, IL, United States

DETC2005-84340: Solute Transport Simulated With the Fractional
Advective-Dispersive Equation.
Fernando San Jose Martinez, Politechnic University of
Madrid-ETSIA, Madrid, Madrid, Spain, Yakov A. Pachepsky,
USDA-ARS-BA-ANRI-EMSL, Beltsville, MD, United States,
Walter J. Rawls, USDA-BARC-ANRI-HRSL, Beltsville, MD,
United States

DETC2005-84914: On the Rarefied Gas Flow In Pipes.
Vladan D. Djordjevic, University of Belgrade, Faculty of
Mechanical Engineering, Belgrade 35, Serbia, Yugoslavia

DETC2005-84172: The Number of Real Zeros of the Single
Parameter Mittag-Leffler Function for Parameter Values
Between 1 and 2.
John Hanneken, David M. Vaught, B. N. Narahari Achar, University
of Memphis, Memphis, TN, United States

DETC2005-84348: Initialization Issues of the Caputo Fractional
Derivative.
B. N. Narahari Achar, University of Memphis, Memphis, TN,
United States, Carl Lorenzo, NASA Glenn Research Center,
Cleveland, OH, United States, Tom Hartley, University of
Akron, Akron, OH, United States

DETC2005-84601: The Fractional Hyperbolic Functions: With
Application to Fractional Differential Equations.
Carl Lorenzo, NASA Glenn Research Center, Cleveland, OH,
United States, Tom Hartley, University of Akron, Akron, OH,
United States

DETC2005-84951: Conjugated-order Differintegrals.
Tom Hartley, University of Akron, Akron, OH, United States,
Carl Lorenzo, NASA Glenn Research Center, Cleveland, OH,
United States, Jay L. Adams, The University of Akron, Akron,
OH, United States

DETC2005-85613: On Theory of Systems of Fractional Linear
Differential Equations.
Blanca Bonilla, Margarita Rivero, Juan J. Trujillo, Universidad
de La Laguna, La Laguna 38271. Tenerife, S/C de Tenerife,
Spain

DETC2005-84651: Frequency Domain Analysis of a Fractional
Derivative SDOF System.
Luigi Garibaldi, Politecnico di Torino, Torino, Italy, Silvio Sorrentino,
Sheffield University, Sheffield, United Kingdom

DETC2005-84862: A Fractional Calculus Perspective in Electromagnetics.
J. A. Tenreiro Machado, Isabel Jesus, Alexandra Galhano,
Institute of Engineering of Porto, Porto, Portugal

DETC2005-84266: Fractional Generalization of Ginzburg-Landau
and Nonlinear Schroedinger Equations.
George Zaslavsky, Courant Institute of Mathematical Sciences,
New York, NY, United States, Vasily Tarasov, Skobeltsyn
Institute of Nuclear Physics, Moscow State University,
Moscow, Russia

DETC2005-84390: About Lagrangian Formulation of Classical
Fields within Riemann-Liouville Fractional Derivatives.
Dumitru Baleanu, University of Cankaya, Ankara, Turkey, Sami
Muslih, International Center for Theoretical Physics, Trieste,
Italy

DETC2005-84057: Ergodicity Breaking in Fractional Diffusion
Processes (Presentation Only)
Eli Barkai, Dept. of Chem, Notre Dame, IN, United States

DETC2005-84647: The “Fractional” Kinetic Equations and
General Theory of Dielectric Relaxation.
Raoul Nigmatullin, Kazan State University, Kazan, Tatarstan
Republic, Russia

DETC2005-85299: Robustness of Boundary Control of Fractional
Wave Equations with Delayed Boundary Measurement
Using Fractional Order Controller and the Smith Predictor.
Jinsong Liang, Utah State University, Logan, UT, United
States, Weiwei Zhang, Michigan State University, East Lansing,
MI, United States, YangQuan Chen, Utah State University,
Logan, UT, United States, Igor Podlubny, Technical University
of Kosice, Kosice, Slovakia

DETC2005-85725: Experimental Identification of a Fractional
Derivative Linear Model for Viscoelastic Materials.
Giuseppe Catania, Silvio Sorrentino, University of Bologna,
Bologna, Italy

ASME sub-committee for fractional dynamics starts its blog

2005-10-01T20:27:00.000-07:00

The Fractional calculus and its applications blog started today in the framework of activities of the ASME sub-committee for fractional dynamics. The main purpose of this blog is promoting information exchange and collaboration in the field of fractional calculus and its applications.

It is supposed that, among others, information about conferences and other events, new books, special journal issues, individual articles, and preprints will be posted here.