03 FOC Workshop New Xue 3

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Slide 1 of 63 Computational Aspects of Fractional-Order Control ProblemsDingy Xue for WCICA 2010, Jinan, P R China, 07/2010

Tutorial Workshop on

Fractional Order Dynamic

Systems and ControlsWCICA2010, 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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Slide 2 of 63 Computational Aspects of Fractional-Order Control Problems

Dingy Xue for WCICA 2010, Jinan, P R China, 07/2010

Computational Aspect of

Fractional-Order Control ProblemsOutlines 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, isthere a similar way to solve fractional-order control

problems. Class based programming in MATLAB


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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 adoptedOptimum Controller Design for Fractional-

Order Systems through Examples

Criteria selection, design examples via SimulinkImplementation of the Controllers

Continuous and Discrete


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Main Reference

Chapter 13 of the MonographFractional-order Systems and Controls ---

Fundamentals and Applications

By Concepcion Alicia Monje, YangQuan Chen,Blas Manuel Vinagre, Dingyu Xue,

Vicente Feliu

Springer-Verlag, London, July, 2010Implementation part is from Chapter 12 of the book


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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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1.1 Evaluation of Mittag-Leffler Functions

Importance of Mittag-Leffler functionsAs important as exponential functions in IOs

Analytical solutions of FO-ODEs

DefinitionsML in one parameter

ML in two parameters

Special cases


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Mittag-Leffler Functions in more pars

Definitions

withDerivatives

MATLAB function


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Code

Podlubnys code mlf() embedded


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Examples to tryDraw curves

Code

Other functions


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Friday, 2010-7- 2,16:05:56 Slide 10 of 63 Computational Aspects of Fractional-Order Control Problems


1.2 Evaluations of Fractional-order

Derivatives

Definitions:

Grnwald-Letnikov's Definition

Others

Caputo's Derivatives, Riemann-Liouvilles, Cauchys


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MATLAB Implementation

Easy to program

Syntax

Examples

Orginal function


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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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From G-L definition

And

The closed-form solution can be obtained


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MATLAB Code and Syntax

Code

Syntax


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Example

Fractional-order differential equation

with step input u(t)

MATLAB solutions


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Partial fraction expansion of

Commensurate-order Systems

Definition

Transfer function

After partial fraction expansion, step responses


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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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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 sectionDesign of a FOTF Object

Modeling Using FOTFs

Stability Assessment of FOTFsNumerical Time Domain Analysis

Frequency Domain Analysis


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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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Object creation

Syntax


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Display function


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Model Entering Examples

Example1

Example 2

Example 3:


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2.2 Modelling of FOTF Systems

Series connection: G1*G2Overload functions are needed for mtimes.m

Similarly other functions can be writtenplus.m, feedback.m, uminus.m, mrdivide.m

simple.m, mpower.m, inv.m, minus.m


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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 TFsMATLAB function

Example: the previous

closed-loop system


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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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Examples

Closed-loop model

Model with input


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2.5 Frequency Domain Analysis

Overload functionsBode.m

Nyquist.m

Nichols.mVia Examples

Slopes. Not integer times of 20dB/sec


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2.6 Norm Measures of FOTFs

Norms

2-norm

Infinity norm

Overload functions

Examples


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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 diagrambased approach is needed.

A Simulink block is needed for FO-D


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3.2 Oustaloups Filter

Idea of Oustaloups Filter

Method


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3.3 Modified Oustaloups Filter

Method

Code

Syntax


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3.4 Simulink Modelling

Mask a Simulink block --- the key element

Possibly with a low-pass filter

E l 1 Li d l


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Example 1: Linear model

Denote

Simulink

modelling

c10mfode1.mdl


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Example 2: Nonlinear system

Rewrite the equation

Simulink model

c10mfod2.mdl


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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 Oustaloups filter parameters

The frequency range

The orderN

The filter, Oustaloup, modified, and others


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4.1 Optimal Criterion Selections

What kind of control can be regarded asoptimal? Time domain optimization is going

to be used in the presentation.

Other types of criteriaLQ 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


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Why Finite-Time ITAETwo 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

MATLAB/Simulink

Plant model, time-varying

Simulink


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Optimum Design

Establish a MATLAB objective function

Design via 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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5 Implementation of FO Controllers

Continuous ImplementationOustaloups filter

Modified Oustaloups filter

Other implementations

Discrete Implementation

Via Step/Impulse Response Invariants

Frequency Domain FittingSub-Optimal Integer-Order Model Reduction


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Continuous Implementations

As Discussed Earlier

Approximation to Fractional-order operators

(differentiators/integrator) only. Suitable for

FO-PID type of controllers

Functions to use

Discrete-Time Implementations


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Discrete-Time Implementations

FIR Filter, s work

Again for fraction-order operators

Also possible, Tustins approximation


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Step/Impulse Response Invariants

Approximation Models

The following functions can be used,

Dr Yangquan Chens work

Example

Discrete Time Approximation to


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Discrete-Time Approximation to

MATLAB solutions, due to Dr Chens code

Example

Rewrite asMATLAB solutions

5 3 F R Fi i f


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5.3 Frequency Response Fitting of

Fractional-Order Controllers

Criterion

MATLAB Function

Example

A i


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A complicated controller

Controller, with QFT method

MATLAB Implementation


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Integer-order fitting model

Comparisons

5 5 R ti l A i ti t


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5.5 Rational Approximation to

Fractional-Order Transfer Functions

Original model

Fitting integer-order model

Fitting criterion

where

M d l Fi i Al i h


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Model Fitting Algorithm

1. Select an initial reduced model2. Evaluate an error

3. Use an optimization (i.e., Powell's algorithm)

to iterate one step for a better estimatedmodel

4. Set , go to Step (2) until an

optimal reduced model is obtained5. Extract the delay from , if any

MATLAB F ti I l t ti


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MATLAB Function Implementation

Function call

Example

Finding full-order approximation

Reduction

C l di R k


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l d f l f l d l bl

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/
http://mechatronics.ece.usu.edu/foc/wcica2010tw/http://mechatronics.ece.usu.edu/foc/wcica2010tw/

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