# Fractional calculus and its applications

## Tuesday, February 03, 2009

### 2 Hour Tutorial Session at American Control Conference 2009

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.