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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
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7

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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
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7

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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
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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
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of 63 Computational Aspects of Fractional

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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
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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
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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
where
Special cases
Derivatives
MATLAB function
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Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Code
Podlubny’s code
mlf()
embedded
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Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Examples to try
Draw curves
Code
Other functions
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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
Other approximation methods, with
Others
Caputo's Derivatives, Riemann

Liouville’s, Cauchy’s
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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
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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
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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
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Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
MATLAB Code and Syntax
Code
Syntax
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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
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Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
1.4 Analytical Solutions to Linear
Fractional

order Differential Equations
Important Laplace transform property
Special cases:
Impulse input:
Step inputs:
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Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Partial fraction expansion of
Commensurate

order Systems
Commensurate

order systems, base order
Transfer function
After partial fraction expansion, step responses
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Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Example:
Partial fractional expansion
Step response, theoretical
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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
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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
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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)
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Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Object creation
Syntax
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Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Display function
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Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Model Entering Examples
Example1
Example 2
Example 3:
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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
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Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Theoretical Results
Series connection
Parallel connection
Feedback Connection
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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
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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
For non

commensurate

order systems, works
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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
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Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Examples
Closed

loop model
Model with input
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Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
2.5 Frequency Domain Analysis
Exact evaluation of
Overload functions
Bode.m
Nyquist.m
Nichols.m
Via Examples
Slopes. Not integer times of 20dB/sec
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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
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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 systems
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
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Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Filters for Approximating FO

Ds
Filter Approximations of FO

D’s
Continued fraction approximation
Oustaloup’s filter
Modified Oustaloup’s filter
Simulink Modelling of NL

FO Systems
Masking a Simulink block with the Oustaloup’s
filter and others
Simulation of nonlinear frcational

order systems
with examples
Validation of simulation results
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Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
3.1 Continued Fractions
Math form
For
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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
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Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
MATLAB Implementation
MATLAB code
Syntax
Example
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Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
3.3 Modified Oustaloup’s Filter
Method
Code
Syntax
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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
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Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Example 1: Linear model
Denote
Simulink
modelling
c10mfode1.mdl
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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
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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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Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
3.5 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
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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
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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
Criteria on integrals of error should be used
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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
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Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Selection of finite

time
Tested in an example
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Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
4.2 Design Examples with
MATLAB/Simulink
Plant model, time

varying
Simulink
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Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Optimum Design
Establish a MATLAB objective function
Design via optimization
Visualizing output curves in optimization
Allow nonlinear elements and complicated
systems, constrained optimizations possible
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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
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Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
MATLAB objective function
Optimal controller design
Optimal Controller found
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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
Approximations of FO Operators
Via Step/Impulse Response Invariants
Frequency Domain Fitting
Sub

Optimal Integer

Order Model Reduction
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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
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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
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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
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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
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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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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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Order Control Problems
Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Integer

order fitting model
Comparisons
Over a larger frequency interval
Compaisons
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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
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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
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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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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 researchers, for immediate
acquisition of plots and research results
Code available from
http://mechatronics.ece.usu.edu/foc/wcica2010tw/
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