1. Friday, 2010-7- 2,
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Slide 1 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Tutorial Workshop on
Fractional Order Dynamic
Systems and Controls
WCICA’2010, Jinan, China
Computational Aspect of Fractional-
Order Control Problems
Dingyu Xue
Institute of AI and Robotics
Faculty of Information Sciences and
Engineering
Northeastern University
Shenyang 110004, P R China

2. Friday, 2010-7- 2,
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Slide 2 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Computational Aspect of
Fractional-Order Control Problems
Outlines and Motivations of Presentation
Computations in Fractional Calculus
How to solve related problems with computers,
especially with MATLAB?
Linear Fractional-Order Transfer Functions
In Conventional Control: CST is widely used, is
there a similar way to solve fractional-order control
problems. Class based programming in MATLAB

3. Friday, 2010-7- 2,
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Slide 3 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Outlines and Motivations (contd)
Simulation Studies of Fractional-Order
Nonlinear Systems
How to solve problems in nonlinear systems? The
only feasible way is by simulation. Simulink based
programming methodology is adopted
Optimum Controller Design for Fractional-
Order Systems through Examples
Criteria selection, design examples via Simulink
Implementation of the Controllers
Continuous and Discrete

4. Friday, 2010-7- 2,
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Slide 4 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Main Reference
Chapter 13 of the Monograph
Fractional-order Systems and Controls
---Fundamentals and Applications
By Concepcion Alicia Monje, YangQuan Chen,
Blas Manuel Vinagre, Dingyu Xue,
Vicente Feliu
Springer-Verlag, London, July, 2010
Implementation part is from Chapter 12 of the book

5. Friday, 2010-7- 2,
16:05:56
Slide 5 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
1 Computations in Fractional Calculus
Evaluation of Mittag-Leffler functions
Evaluations of Fractional-order Derivatives
Closed-form Solutions to Linear Fractional-
order Differential Equations
Analytical Solutions to Linear Fractional-order
Differential Equations

6. Friday, 2010-7- 2,
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Slide 6 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
1.1 Evaluation of Mittag-Leffler Functions
Importance of Mittag-Leffler functions
As important as exponential functions in IOs
Analytical solutions of FO-ODEs
Definitions
ML in one parameter
ML in two parameters
Special cases

7. Friday, 2010-7- 2,
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Slide 7 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Mittag-Leffler Functions in more pars
Definitions
with
Derivatives
MATLAB function

8. Friday, 2010-7- 2,
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Slide 8 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Code
Podlubny’s code mlf() embedded

9. Friday, 2010-7- 2,
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Slide 9 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Examples to try
Draw curves
Code
 Other functions

10. Friday, 2010-7- 2,
16:05:56
Slide 10 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
1.2 Evaluations of Fractional-order
Derivatives
Definitions:
Grünwald-Letnikov's Definition
 Others
 Caputo's Derivatives, Riemann-Liouville’s, Cauchy’s

11. Friday, 2010-7- 2,
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Slide 11 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
MATLAB Implementation
Easy to program
Syntax
Examples
Orginal function

12. Friday, 2010-7- 2,
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Slide 12 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
1.3 Closed-Form Solutions to Linear
Fractional-Order Differential Equations
Mathematical Formulation
Fractional-order DEs
Denote
Original equation changed to

13. Friday, 2010-7- 2,
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Slide 13 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
From G-L definition
And
The closed-form solution can be obtained

14. Friday, 2010-7- 2,
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Slide 14 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
MATLAB Code and Syntax
Code
Syntax

15. Friday, 2010-7- 2,
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Slide 15 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Example
Fractional-order differential equation
with step input u(t)
MATLAB solutions

16. Friday, 2010-7- 2,
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Slide 16 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
1.4 Analytical Solutions to Linear
Fractional-order Differential Equations
Laplace transform property
Special cases:
 Impulse input:
Step inputs:

17. Friday, 2010-7- 2,
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Slide 17 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Partial fraction expansion of
Commensurate-order Systems
Definition
Transfer function
After partial fraction expansion, step responses

18. Friday, 2010-7- 2,
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Slide 18 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Example:
Partial fractional expansion
Step response, theoretical

19. Friday, 2010-7- 2,
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Slide 19 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Also works for the cases with multiple poles
For more complicated systems
Analytical solutions are too complicated

20. Friday, 2010-7- 2,
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Slide 20 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
2 Fractional-Order Transfer Functions
--- MATLAB Object Modelling
Motivated by the Control Systems Toolbox
Specify a system in one variable G,
use of * and +, and step(G), bode(G), convenient
Outlines in the section
Design of a FOTF Object
Modeling Using FOTFs
Stability Assessment of FOTFs
Numerical Time Domain Analysis
Frequency Domain Analysis

21. Friday, 2010-7- 2,
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Slide 21 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Fractional-Order Transfer Functions
Five parameters:
Possible to design a MATLAB object
Create a @fotf folder
Establish two essential functions
fotf.m (for creation), display.m (for display object)

22. Friday, 2010-7- 2,
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Slide 22 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Object creation
Syntax

23. Friday, 2010-7- 2,
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Slide 23 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Display function

24. Friday, 2010-7- 2,
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Slide 24 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Model Entering Examples
Example1
Example 2
Example 3:

25. Friday, 2010-7- 2,
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Slide 25 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
2.2 Modelling of FOTF Systems
Series connection: G1*G2
Overload functions are needed for mtimes.m
Similarly other functions can be written
plus.m, feedback.m, uminus.m, mrdivide.m
simple.m, mpower.m, inv.m, minus.m

26. Friday, 2010-7- 2,
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Slide 26 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Theoretical Results
Series connection
Parallel connection
Feedback Connection

27. Friday, 2010-7- 2,
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Slide 27 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Modelling Examples
Plant
Controller
Unity negative feedback connection
Closed-loop system

28. Friday, 2010-7- 2,
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Slide 28 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
2.3 Analysis of Fractional-Order Systems
Stability regions for commensurate-order TFs
MATLAB function
Example: the previous
closed-loop system

29. Friday, 2010-7- 2,
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Slide 29 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
2.4 Numerical Time Domain Analysis
Based on fode_sol function discussed earlier,
overload functions step and lsim are written
Step response
Time response to arbitrary inputs
No restrictions. Reliable numerical solutions
Validate the results

30. Friday, 2010-7- 2,
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Slide 30 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Examples
Closed-loop model
Model with input

31. Friday, 2010-7- 2,
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Slide 31 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
2.5 Frequency Domain Analysis
Overload functions
Bode.m
Nyquist.m
Nichols.m
Via Examples
Slopes. Not integer times of 20dB/sec

32. Friday, 2010-7- 2,
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Slide 32 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
2.6 Norm Measures of FOTFs
Norms
2-norm
Infinity norm
Overload functions
Examples

33. Friday, 2010-7- 2,
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Slide 33 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
3 Simulation Studies of Fractional-
order Nonlinear Systems
Problems of Existing methods
Grunwald-Letnikov definitions and others only
applies to the cases where input to a fractional-
order D/I is known
Step and lsim functions only works for FOTF
objects, not nonlinear systems
For nonlinear control systems, a block diagram
based approach is needed.
A Simulink block is needed for FO-D

34. Friday, 2010-7- 2,
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Slide 34 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Filters for Approximating FO-Ds
Continued fraction approximation
Oustaloup’s filter
Modified Oustaloup’s filter
Masking a Simulink block with the Oustaloup’s
filter and others
Simulation of nonlinear frcational-order
systems with examples
Validation of simulation results

35. Friday, 2010-7- 2,
16:05:56
Slide 35 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
3.1 Continued Fractions
Math form
For s^0.5

36. Friday, 2010-7- 2,
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Slide 36 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
3.2 Oustaloup’s Filter
Idea of Oustaloup’s Filter
Method

37. Friday, 2010-7- 2,
16:05:56
Slide 37 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
MATLAB Implementation
MATLAB code
Syntax
Example

38. Friday, 2010-7- 2,
16:05:56
Slide 38 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
3.3 Modified Oustaloup’s Filter
Method
Code
Syntax

39. Friday, 2010-7- 2,
16:05:56
Slide 39 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
3.4 Simulink Modelling
Mask a Simulink block --- the key element
Possibly with a low-pass filter

40. Friday, 2010-7- 2,
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Slide 40 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Example 1: Linear model
Denote
Simulink
modelling
c10mfode1.mdl

41. Friday, 2010-7- 2,
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Slide 41 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Example 2: Nonlinear system
Rewrite the equation
Simulink model
c10mfod2.mdl

42. Friday, 2010-7- 2,
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Slide 42 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Example 3: fractional-order delay system
Rewrite
Simulink model
cxfdde1.mdl
Control loops can be
established
With Simulink,
complicated systems
can be studied.

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Slide 43 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
3.6 Validations of Simulation Results
No analytical solution. Indirect methods:
Change parameters in equation solver, such as
RelTol, and see whether consistent results can
be obtained
Change simulation algorithms
Change Oustaloup’s filter parameters
The frequency range
The order N
The filter, Oustaloup, modified, and others

44. Friday, 2010-7- 2,
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Slide 44 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
4 Optimal Controller Design
What Criterion is Suitable for Addressing
Optimality of Servo Control Systems:
Criterion Selections
MATLAB/Simulink based Optimal Controller
Design Procedures
Optimum Fractional-Order PID Controllers:
Parameter Setting via Optimization Through
An Example

45. Friday, 2010-7- 2,
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Slide 45 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
4.1 Optimal Criterion Selections
What kind of control can be regarded as
optimal? Time domain optimization is going
to be used in the presentation.
Other types of criteria
LQ optimization, artificial, no methods for Q and R
ISE criterion, H2 minimization,
Hinf, may be too conservative
Fastest, most economical, and other
Finite-time ITAE is to be used

46. Friday, 2010-7- 2,
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Slide 46 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Why Finite-Time ITAE
Two criteria:
Which one
is better?
ITAE type of
criteria are
meaningful

47. Friday, 2010-7- 2,
16:05:56
Slide 47 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Selection of finite-time
Tested in an example

48. Friday, 2010-7- 2,
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Slide 48 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
4.2 Design Examples with
MATLAB/Simulink
Plant model, time-varying
Simulink

49. Friday, 2010-7- 2,
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Slide 49 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Optimum Design
Establish a MATLAB objective function
Design via optimization
Allow nonlinear elements and complicated
systems, constrained optimizations possible

50. Friday, 2010-7- 2,
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Slide 50 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
4.3 Optimal FO PID Design
Controller with 5 parameters
Design Example, Plant

51. Friday, 2010-7- 2,
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Slide 51 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
MATLAB objective function
Optimal controller design
Optimal Controller found

52. Friday, 2010-7- 2,
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Slide 52 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
5 Implementation of FO Controllers
Continuous Implementation
Oustaloup’s filter
Modified Oustaloup’s filter
Other implementations
Discrete Implementation
Via Step/Impulse Response Invariants
Frequency Domain Fitting
Sub-Optimal Integer-Order Model Reduction

53. Friday, 2010-7- 2,
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Slide 53 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Continuous Implementations
As Discussed Earlier
Approximation to Fractional-order operators
(differentiators/integrator) only. Suitable for
FO-PID type of controllers
Functions to use

54. Friday, 2010-7- 2,
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Slide 54 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Discrete-Time Implementations
FIR Filter, ’s work
Again for fraction-order operators
Also possible, Tustin’s approximation

55. Friday, 2010-7- 2,
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Slide 55 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Step/Impulse Response Invariants
Approximation Models
The following functions can be used,
Dr Yangquan Chen’s work
Example

56. Friday, 2010-7- 2,
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Slide 56 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Discrete-Time Approximation to
MATLAB solutions, due to Dr Chen’s code
Example
Rewrite as
MATLAB solutions

57. Friday, 2010-7- 2,
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Slide 57 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
5.3 Frequency Response Fitting of
Fractional-Order Controllers
Criterion
MATLAB Function
Example

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Slide 58 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
A complicated controller
Controller, with QFT method
MATLAB Implementation

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Slide 59 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Integer-order fitting model
Comparisons

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Slide 60 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
5.5 Rational Approximation to
Fractional-Order Transfer Functions
Original model
Fitting integer-order model
Fitting criterion
where

61. Friday, 2010-7- 2,
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Slide 61 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Model Fitting Algorithm
1. Select an initial reduced model
2. Evaluate an error
3. Use an optimization (i.e., Powell's algorithm)
to iterate one step for a better estimated
model
4. Set , go to Step (2) until an
optimal reduced model is obtained
5. Extract the delay from , if any

62. Friday, 2010-7- 2,
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Slide 62 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
MATLAB Function Implementation
Function call
Example
Finding full-order approximation
Reduction

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Slide 63 of 63 Computational Aspects of Fractional-Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Concluding Remarks
MATLAB code are prepared for fractional-
order systems, especially useful for beginners
Handy facilities can also be used by
experienced users, for immediate acquisition
of plots and research results
Code available from
http://mechatronics.ece.usu.edu/foc/wcica2010tw/