Symposium on Fractional Signals and Systems

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Symposium on Fractional Signals and Systems

• Stability Preservation Analysis in Direct
  Discretization of Fractional Order Transfer
  Functions
• Mohammad Saleh Tavazoei
• Electrical Engineering Department, Sharif
  University of Technology
• (tavazoei_at_sharif.edu)

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Contents
Stability Preservation Analysis in Direct
Discretization of Fractional Order Transfer
Functions

• A Brief Introduction to Stability Preservation
  Problem and Previously Obtained Results
• Analysis of the Stability Preservation Problem in
  Direct Discretization of Fractional Order
  Transfer Functions
• Numerical Illustrations
• Conclusions

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Introduction
Stability Preservation Analysis in Direct
Discretization of Fractional Order Transfer
Functions

• At the present time, using the integer order
  approximations of fractional order operators in
  practical implementation of fractional order
  dynamics is unavoidable.
• The original transfer function and its
  approximation should be the same in the sense of
  stability.

Stability Preservation Problem
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Stability Criteria for Commensurate Order
Transfer Functions
Stability Preservation Analysis in Direct
Discretization of Fractional Order Transfer
Functions
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Stability Preservation Analysis for
Continuous-Time Approximations
Stability Preservation Analysis in Direct
Discretization of Fractional Order Transfer
Functions

• Original transfer function
• Original stability boundary
• It has been proved that when one uses the
  approximation
• the stability boundary
  converts to the curve
• in the
  complex plane.
• Ref Tavazoei M.S. and Haeri M., Rational
  Approximations in Simulation and Implementation
  of Fractional Order Dynamics A Descriptor System
  Approach, Automatica, In Press, 2009.

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Discretization of Fractional Order Transfer
Functions
Stability Preservation Analysis in Direct
Discretization of Fractional Order Transfer
Functions

• To find a discrete approximation for a fractional
  order transfer function, there are two different
  approaches
• Using Indirect Methods
• Using Direct Methods

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Discretization of Fractional Order Transfer
Functions
Stability Preservation Analysis in Direct
Discretization of Fractional Order Transfer
Functions

• Indirect Methods

Continuous-Time Fractional Order Transfer
Function
Continuous-Time Integer Order Approximation
Dicretization Method (ZOH, Tustin, )
Discrete-Time Integer Order Approximation
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Discretization of Fractional Order Transfer
Functions
Stability Preservation Analysis in Direct
Discretization of Fractional Order Transfer
Functions

• Direct Methods

is a discrete approximation for
fractional order operator
Continuous-Time Fractional Order Transfer
Function
Discrete-Time Integer Order Approximation
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Stability Preservation Analysis for Discrete-Time
Approximations
Stability Preservation Analysis in Direct
Discretization of Fractional Order Transfer
Functions
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Stability Preservation Analysis for Discrete-Time
Approximations
Stability Preservation Analysis in Direct
Discretization of Fractional Order Transfer
Functions
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Stability Preservation Analysis for Discrete-Time
Approximations
Stability Preservation Analysis in Direct
Discretization of Fractional Order Transfer
Functions
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Stability Preservation Analysis for Discrete-Time
Approximations
Stability Preservation Analysis in Direct
Discretization of Fractional Order Transfer
Functions
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Stability Preservation Analysis for Discrete-Time
Approximations
Stability Preservation Analysis in Direct
Discretization of Fractional Order Transfer
Functions
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Stability Preservation Analysis for Discrete-Time
Approximations
Stability Preservation Analysis in Direct
Discretization of Fractional Order Transfer
Functions
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Stability Preservation Analysis for Discrete-Time
Approximations
Stability Preservation Analysis in Direct
Discretization of Fractional Order Transfer
Functions
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Numerical Example
Stability Preservation Analysis in Direct
Discretization of Fractional Order Transfer
Functions
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Numerical Example
Stability Preservation Analysis in Direct
Discretization of Fractional Order Transfer
Functions
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Numerical Example
Stability Preservation Analysis in Direct
Discretization of Fractional Order Transfer
Functions
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Numerical Example
Stability Preservation Analysis in Direct
Discretization of Fractional Order Transfer
Functions
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Numerical Example
Stability Preservation Analysis in Direct
Discretization of Fractional Order Transfer
Functions
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Numerical Example
Stability Preservation Analysis in Direct
Discretization of Fractional Order Transfer
Functions
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Numerical Example
Stability Preservation Analysis in Direct
Discretization of Fractional Order Transfer
Functions
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Conclusions
Stability Preservation Analysis in Direct
Discretization of Fractional Order Transfer
Functions

• The stability boundary of direct methods for
  discretization of fractional order systems has
  been determined.
• This result helps us to apply an appropriate
  approximating filter in discretization of a
  fractional order transfer function. The discrete
  approximating filter should be chosen such that
  the stability or instability of the original
  system does not change in the procedure of
  discretization.

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The End
Stability Preservation Analysis in Direct
Discretization of Fractional Order Transfer
Functions
Thank You! Any Question?