Symposium on Fractional Signals

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Symposium on Fractional Signals Systems

• A Four Parameter Fractional Order Model Structure
  and its Use in Control System Design
• Mahsan Tavakoli-Kakhki
• Electrical Engineering Department, Sharif
  University of Technology
• (mah_tavakoli_at_ee.sharif.edu)

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Contents
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design

• Preliminary Concepts
• Introducing the proposed four-parameter
  fractional order model
• Parameter estimation of a four-parameter
  fractional order model
• Estimation of DC gain
• Estimation of fractional order
• Estimation of Parameter
• Estimation the value of dead time
• Estimation of Parameter
• Three strategies for estimating the parameters
• Simulation results
• Conclusion

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Preliminary Concepts
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design

• Riemann-Liouville Fractional Integral
• Riemann-Liouville Fractional Derivative
• where is the first integer which is not less
  than .
• The Laplace transform of the Riemann-Liouville
  based fractional
• derivative

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Preliminary Concepts
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design

• A class of fractional order transfer function
• if there exist a real number as the biggest
  common devisor of
• , and
  , this number is called as the
• commensurate order and the commensurate transfer
  function can
• be rewritten as

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Preliminary Concepts
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design

• The partial fraction expansion of a commensurate
  fractional order
• transfer function can be written in the following
  general form
• Impulse response of
• Step response of
• where ,

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Parameter Estimation of a Four-Parameter
Fractional Order Model
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design
Approximation of transfer functions with an
S-shaped unit step response
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Estimation of DC Gain K
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design

• Also this parameter can be determined by
  measuring
• the final value of the system output which
  has been
• scaled with the magnitude of the step input.

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Estimation of Fractional Order
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design

• In model , the parameter is
  considered equal to the
• commensurate order of the system transfer
  function or its
• state space model

• Equivalently, this parameter can be measured by
  paying
• attention to the asymptotic behavior of the
  system step
• response

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Estimation of Fractional Order
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design
Theorem In 2-parameter Mittag-Leffler function
when and is an arbitrary
complex number, the following expansion
holds where is an arbitrary integer
number.
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Estimation of Fractional Order
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design
According to the stated theorem, the asymptotic
behavior of the step response of the
four-parameter fractional order model can be well
approximated by function
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Estimation of Fractional Order
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design
In practical applications due to the existence of
measurement noise, it may not be easy to estimate
the value of fractional order by the mentioned
graphically method. In such cases the Least
Square method is beneficial.
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Estimation of Fractional Order
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design
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Estimation of Parameter T
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design
Taking logarithm from the both sides of the
four-parameter model
Therefore, when transfer function is available
and the fractional order is known beforehand by
considering the moment equalities
and parameter T can
be approximated by
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Estimation of Parameter T
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design

• Parameter T can also be computed utilizing the
  impulse
• response of the system.
• Based on the Laplace transform and fractional
  derivative
• definition

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Estimation of Dead Time L
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design
Since the tangent to the step response of the
four-parameter fractional order model possesses
the largest slope at time , the value of
the dead time in the fractional order model can
be estimated in the same graphical way as in the
three-parameter integer order model.
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Estimation of Expression
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design
1. Estimation of parameter T if L is known
beforehand
2. Estimation of parameter L if T is known
beforehand
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Estimation of Parameter
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design
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Estimation of Parameter
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design
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Estimation of Parameter
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design
Similarly, we can determine the percentage of the
unit step response of the presented
four-parameter transfer function
with respect to its final value at time
.
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Three Strategies for Estimating the Parameters
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design
Strategy One
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Three Strategies for Estimating the Parameters
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design
Strategy Two
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Three Strategies for Estimating the Parameters
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design
Strategy Three
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Simulation Results
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design
Example.1 Application of the obtained simple
model in designing an Internal Model Control
(IMC) system for a system modeled by
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Simulation Results
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design
Block diagram for IMC based closed loop system
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Simulation Results
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design
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Simulation Results
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design
Set point tracking and disturbance rejection by
applying three obtained controllers
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Simulation Results
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design
Example 2 In this example it is assumed that the
measurements of the unit step response of a
system are available. This system can be
approximated by a four-parameter model based on
the third strategy.
Unit step responses of the original system (Solid
Line) and the approximated model (Dashed Line)
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Conclusion
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design

• In this presentation
• A four parameter model was presented to
  characterize the
• dynamic response of a complex fractional
  order system
• possessing an S-shaped step response
• Three different strategies were proposed for
  estimating four
• parameters of the approximating model.

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Conclusion
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design

• We showed that the presented four-parameter
  model can be
• useful in design and tuning the parameters
  of some control
• systems such as IMC methodology.
• By an example the applicability of one of these
  strategies in
• finding an approximated four parameter model
  based on
• practical measurement data was shown.

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The End
A Four Parameter Fractional Order Model Structure
and its Use in Control System Design
Thank You! Any Question?