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

Display 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

Model Entering Examples

Example1



Example 2



Example 3:

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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

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

Overload functions


Bode.m


Nyquist.m


Nichols.m

Via Examples



Slopes. Not integer times of 20dB/sec

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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

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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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

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

3.3 Modified Oustaloup’s Filter

Method


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

3.4 Simulink Modelling

Mask a Simulink block
---

the key element



Possibly with a low
-
pass filter

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

Example 1: Linear model



Denote



Simulink


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



Simulink model


c10mfod2.mdl

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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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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

MATLAB/Simulink based Optimal Controller
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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of 63 Computational Aspects of Fractional
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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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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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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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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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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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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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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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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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Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010

A complicated controller

Controller, with QFT method



MATLAB Implementation

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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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5.5 Rational Approximation to
Fractional
-
Order Transfer Functions

Original model


Fitting integer
-
order model



Fitting criterion


where

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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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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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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/