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Dec 10, 2013 (4 years and 7 months ago)

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of 63 Computational Aspects of Fractional
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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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of 63 Computational Aspects of Fractional
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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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of 63 Computational Aspects of Fractional
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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

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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of 63 Computational Aspects of Fractional
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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
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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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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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of 63 Computational Aspects of Fractional
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Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010

Code

Podlubny’s code
mlf()

embedded

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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

Examples to try

Draw curves

Code

Other functions

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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

1.2 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
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Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010

MATLAB Implementation

Easy to program

Syntax

Examples

Orginal function

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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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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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of 63 Computational Aspects of Fractional
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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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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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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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Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010

Example:

Partial fractional expansion

Step response, theoretical

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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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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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of 63 Computational Aspects of Fractional
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Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010

Object creation

Syntax

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Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010

Display function

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Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010

Model Entering Examples

Example1

Example 2

Example 3:

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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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Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010

Theoretical Results

Series connection

Parallel connection

Feedback Connection

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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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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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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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Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010

Examples

Closed
-
loop model

Model with input

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Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010

2.5 Frequency Domain Analysis

Exact evaluation of

Bode.m

Nyquist.m

Nichols.m

Via Examples

Slopes. Not integer times of 20dB/sec

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Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010

2.6 Norm Measures of FOTFs

Norms

2
-
norm

Infinity norm

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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of 63 Computational Aspects of Fractional
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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

-
FO Systems

filter and others

Simulation of nonlinear frcational
-
order systems
with examples

Validation of simulation results

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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

3.1 Continued Fractions

Math form

For

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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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Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010

MATLAB Implementation

MATLAB code

Syntax

Example

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3.3 Modified Oustaloup’s Filter

Method

Code

Syntax

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Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010

---

the key element

Possibly with a low
-
pass filter

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Example 1: Linear model

Denote

modelling

c10mfode1.mdl

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Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010

Example 2: Nonlinear system

Rewrite the equation

c10mfod2.mdl

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Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010

Example 3: fractional
-
order delay system

Rewrite

cxfdde1.mdl

Control loops can be

established

complicated systems

can be studied.

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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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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

Design Procedures

Optimum Fractional
-
Order PID Controllers:
Parameter Setting via Optimization Through
An Example

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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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Why Finite
-
Time ITAE

Two criteria:

Which one

is better?

ITAE type of

criteria are

meaningful

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Selection of finite
-
time

Tested in an example

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4.2 Design Examples with

Plant model, time
-
varying

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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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4.3 Optimal FO PID Design

Controller with 5 parameters

Design Example, Plant

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MATLAB objective function

Optimal controller design

Optimal Controller found

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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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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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Discrete
-
Time Implementations

FIR Filter, ’s work

Again for fraction
-
order operators

Also possible, Tustin’s approximation

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Step/Impulse Response Invariants
Approximation Models

The following functions can be used,

Dr Yangquan Chen’s work

Example

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Discrete
-
Time Approximation to

MATLAB solutions, due to Dr Chen’s code

Example

Rewrite as

MATLAB solutions

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5.3 Frequency Response Fitting of
Fractional
-
Order Controllers

Criterion

MATLAB Function

Example

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A complicated controller

Controller, with QFT method

MATLAB Implementation

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Integer
-
order fitting model

Comparisons

Over a larger frequency interval

Compaisons

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5.5 Rational Approximation to
Fractional
-
Order Transfer Functions

Original model

Fitting integer
-
order model

Fitting criterion

where

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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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MATLAB Function Implementation

Function call

Example

Finding full
-
order approximation

Reduction

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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/