FOC Workshop New-Xue

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Slide 1 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü 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 RoboticsFaculty of Information Sciences and

EngineeringNortheastern University

Shenyang 110004, P R China

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Computational Aspect of Fractional-Order Control Problems

Outlines and Motivations of Presentation

Computations in Fractional CalculusHow 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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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 ExamplesCriteria selection, design examples via Simulink

Implementation of the ControllersContinuous and Discrete

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Main Reference

Chapter 13 of the Monograph

Fractional-order Systems and Controls ---Fundamentals and ApplicationsBy 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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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 IOsAnalytical 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

where

Special cases

Derivatives

MATLAB function

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Code

Podlubny’s code mlf() embedded

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Examples to tryDraw curves

Code

Other functions

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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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MATLAB Implementation

Easy to program

Syntax

ExamplesOrginal function

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1.3 Closed-Form Solutions to Linear Fractional-Order Differential Equations

Mathematical FormulationFractional-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 SyntaxCode

Syntax

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Example

Fractional-order differential equation

with step input u(t)

MATLAB solutions

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1.4 Analytical Solutions to Linear Fractional-order Differential Equations

Important Laplace transform property

Special cases: Impulse input:

Step inputs:

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Partial fraction expansion of Commensurate-order Systems

Commensurate-order systems, base order

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 ToolboxSpecify 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 FOTFsStability Assessment of FOTFs Numerical 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 ExamplesExample1

Example 2

Example 3:

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2.2 Modelling of FOTF Systems

Series connection: G1*G2

Overload 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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Theoretical ResultsSeries connection

Parallel connection

Feedback Connection

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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 TFs

MATLAB function

Example: the previous

closed-loop system

For non-commensurate-order systems, works

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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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ExamplesClosed-loop model

Model with input

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2.5 Frequency Domain AnalysisExact evaluation of Overload functionsBode.mNyquist.mNichols.m

Via Examples

Slopes. Not integer times of 20dB/sec

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2.6 Norm Measures of FOTFsNorms2-norm Infinity norm

Overload functions

Examples

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3 Simulation Studies of Fractional-order Nonlinear Systems

Problems of Existing MethodsGrunwald-Letnikov definitions and others only app

lies 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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Filters for Approximating FO-DsFilter Approximations of FO-D’sContinued fraction approximationOustaloup’s filterModified Oustaloup’s filter

Simulink Modelling of NL-FO Systems Masking a Simulink block with the Oustaloup’s filte

r and othersSimulation of nonlinear frcational-order systems wit

h examplesValidation of simulation results

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3.1 Continued Fractions

Math form

For

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3.2 Oustaloup’s FilterIdea of Oustaloup’s Filter

Method

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MATLAB code

Syntax

Example

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3.3 Modified Oustaloup’s Filter

Method

Code

Syntax

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3.4 Simulink ModellingMask a Simulink block --- the key element

Possibly with a low-pass filter

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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 modelc10mfod2.mdl

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Example 3: fractional-order delay system

Rewrite

Simulink modelcxfdde1.mdl

Control loops can beestablished

With Simulink,complicated systemscan be studied.

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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 parametersThe frequency rangeThe order NThe filter, Oustaloup, modified, and others

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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 criteriaLQ optimization, artificial, no methods for Q and R ISE criterion, H2 minimization, Hinf, may be too conservativeFastest, most economical, and other

Criteria on integrals of error should 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-timeTested in an example

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4.2 Design Examples with MATLAB/Simulink

Plant model, time-varying

Simulink

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Optimum DesignEstablish 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 DesignController 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 ImplementationOustaloup’s filterModified Oustaloup’s filterOther implementations

Discrete ImplementationApproximations of FO OperatorsVia Step/Impulse Response Invariants

Frequency Domain Fitting

Sub-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

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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 controllerController, with QFT method


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

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