Fractional Order Systems: An Overview

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Fractional Order Systems An Overview S.Sen E
lectrical Engg. Department IIT Kharagpur
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• Topics
• 1.Real vs. integer number
• 2.Historical perspective
• 3.Fractional differentiation and Integration
• a. Grünwald-Letnikov Definition
• b. Riemann-Liouville Definition
• 4.Practical implications of a F.O.S

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5.Physical existence a. infinite line
transmission system b. diffusion
c. visco elasticity d. Flow
through a weir
6. Stability of a fractional order system
7. Application to control
a.
controller
b. Fractional state space
8. New areas of research
9. Conclusion
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Real vs. Integer number
Can there be derivative of order a (a real no.)?
?

• Difficult to conceive the physical
  interpretation, yet mathematically possible.

• The term fractional order is a misnomer, actually
  it means integration derivative of order (a
  real number).

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Historical Perspective
Leibnitz introduced the notation
. In a letter to LHospital in 1695 Leibnitz
raised the following question Can the meaning of
derivatives with integer order be generalized to
derivatives with non-integral orders?
The story goes that LHospital
was somewhat curious about that question and
replied by another question to Leibnitz. What if
n1/2? Leibnitz in a letter dated September 30,
1695 replied It will lead to a paradox, from
which one day useful consequences will be drawn.
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People who have worked on fractional calculus.

• Riemann
• Euler
• Fourrier
• Liouville
• Bode

For almost 300 years the investigation was mostly
mathematical. Only during last 30 years the
mathematical deductions found some physical
relevance.
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• Important Recent Events on Fractional Calculus
• 2002 IEEE CDC Tutorial Workshop on Fractional
  Calculus Applications on Automatic Control and
  Robotics (Las Vegas, USA).
• 2004 1st IFAC Workshop on Fractional
  Differentiation and its Applications (Bordeauax,
  France).
• 2006 2nd IFAC Workshop on Fractional
  Differentiation and its Applications (Porto,
  Portugal).
• (No. of papers presented in this workshop 88).
• 2008 3rd IFAC Workshop on Fractional
  Differentiation and its Applications (Ankara
  Turkey, Nov. 2008)

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Operators
gt0
Differentiation
1, 0
Integration
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Definition of fractional derivative.
Grünwald-Letnikov definition (generalization from
integer order).

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For integer Z
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For Fractional Differentiation (replacing n by
)
(t-a)Nh
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Definition of fractional integration.
Grünwald-Letnikov definition (generalization
from integer order).
(n-integer, a-fraction)
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Riemann-Liouville Definition
For integer n let
Taking Laplace Transformation
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Convolution
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Reimann Liouville definition of fractional
integration
How to define Differentiation? Define
n is an integer.
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Grünwald-Letnikov definition
Differentiation

Integration
Reimann Liouville Definition
Integration
Differentiation
,

• It can be shown that both the definitions are
  equivalent.

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Fractional order differential equation A typical
n-term fractional order linear differential
equation is given by

Where (j0,1,2,..,n) are constants and
(j0,1,2,..,n) are real numbers.

• Fractional order systems
• The physical systems those are governed by
  fractional order differential equations.
• Possibly so far we were approximating fractional
  order systems by integer order systems.

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• Different engineering approaches to work with
  fractional order systems
• Try to realize a fractional order system by an
  equivalent integer order system (normally of high
  order).
• Try to find out the devices those behave as
  fractional order systems.
• Try to model known physical systems by fractional
  order and have better understanding of their
  performances.
• Try to use knowledge of fractional calculus to
  design a system to achieve better performance.

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Implications of a fractional order
device Electrical Circuits
1. Resistance
2.Inductance
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3. Capacitance
-1ltalt1
4.Fractance
Laplace Transform
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A Constant Phase element (Fractance)
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Complex impedance plot
We cannot obtain this characteristics by using
finite no. of R-C combinations.
-20a db/decade
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R-F network.
Fractional differential equation
(0ltalt1)
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Realization
Frequency Characteristics of a fractional order
device
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Fractor developed by G.W. Bohannan
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Physical existence of fractional order systems

• Infinite Transmission line.
• Warburg Impedance
• Viscoelasticity
• Flow through weir

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Infinite Transmission line.
Equivalent impedance
and
when
(Fractional order system)
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Warburg Impedance
Equivalent circuit of impedance behavior of a
capacitive device immersed in a polarizable
medium (e.g. water). W Warburg impedance
Half order system. Diffusion of ions
through a porous medium also results in
fractional behaviour.
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Viscoelasticity
Fractional Kelvin-Voigt model
Kelvin-Voigt model
Fractional order model
Integer order model
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Flow through weir
Applying Bernoullis equation at a y
and
assumption
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For an elemental area
Flow through the elemental area
Flow rate is dependent on the shape in a
fractional way.
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Fractional order systems stability
Transfer function
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Stability of a system
Consider the T.F.
Apply transformation
Pole at
For stability Re(p)lt0,
In s-plane, pole is at
For stability,
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Stability region in s-plane
The stability region is not convex for lt1.
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State-space representation

• State space representation is not straight
  forward and normally leads to a high order
  system.

Example

• Find out the HCF of 1.4 and 0.6, i.e. 0.2 and
  the define the states as derivatives of order
  integer multiples of 0.2.
• Thus the number of states becomes 7.

. . . .
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State-space representation
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Controllability Observability
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Controller Design
Fractional PID Controller

• Advantage More maneuverability, so better
  performance expected.
• Challenges
• How to realize the fractional order controller?
• How to tune the controller?

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• Engineering Challenges
• Development of theory of Fractional order
  control.
• Applications of Fractional order control.
• Extend the horizon of control theory to
  fractional order
• Stability, optimal control, robust control,
  identification.
• 4. Fractional modeling of physical systems.
• Controlled production of fractional order
  devices.
• Development of suitable numerical algorithms for
  solving
• FOD equations.

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• New Areas of Research
• Chaos
• Oscillators
• Optimal control

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Fractional Order Chaos in Chuas Circuit
Basic Chuas Circuit
V-I Characteristics of Chuas diode
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Phase plane plot
Question What will happen if the capacitors are
replaced by fractors? -gt Fractional order chaos.
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Fractional order Wien bridge oscillator
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Optimal control
Euler-Lagrange equations (integer order system)
Find the extemum of the functional
Subject to
Euler Lagrange Equation

• Derivatives and integrals are all of integer
  orders.

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Euler-Lagrange equations (fractional order
systems)
Find the extremum of the functional
Subject to
Fractional Euler- Lagrange Equation

• Optimal control theory can be extended to
  fractional order systems.

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Conclusion

• Theory of fractional calculus evolved as a
  generalization of
• integer order calculus.
• Many physical systems exhibit fractional order
  behavior.
• A fractional order device can be realized either
  by
• (i) using a large no. of passive R, C
  components, or
• (ii) using fractance.
• Fractional order systems offer many challenging
  problems to researchers.

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The world is real! Let us accept it. Is it
Complex?
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Thank you.
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Fractional Derivatives Caputo Definition

• It can be shown that G-L definition and R-L
  definition
• are equivalent.
• Both the definition are generalization of
  integer order
• system and valid for an.

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Fractional Integration- Riemann-Liouville
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Graphical representation