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Tutorial Workshop on Fractional-Order Dynamic Systems and Controls WCICA’2010, Jinan, China

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Tutorial Workshop onFractional-Order Dynamic Systems and ControlsWCICA’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

Slide 1 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 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, is there a similar way to solve fractional-order control problems. Class based programming in MATLAB

Slide 2 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

Slide 3 of 63 Computational Aspects of Fractional-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

Slide 4 of 63 Computational Aspects of Fractional-Order Control Problems

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

Slide 5 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

Slide 6 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

Slide 7 of 63 Computational Aspects of Fractional-Order Control Problems

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

Code

- Podlubny’s code mlf() embedded

Slide 8 of 63 Computational Aspects of Fractional-Order Control Problems

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

Examples to try

- Draw curves
- Code
- Other functions

Slide 9 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

Slide 10 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

Slide 11 of 63 Computational Aspects of Fractional-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

Slide 12 of 63 Computational Aspects of Fractional-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

Slide 13 of 63 Computational Aspects of Fractional-Order Control Problems

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

MATLAB Code and Syntax

- Code
- Syntax

Slide 14 of 63 Computational Aspects of Fractional-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

Slide 15 of 63 Computational Aspects of Fractional-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:

Slide 16 of 63 Computational Aspects of Fractional-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

Slide 17 of 63 Computational Aspects of Fractional-Order Control Problems

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

Example:

- Partial fractional expansion
- Step response, theoretical

Slide 18 of 63 Computational Aspects of Fractional-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

Slide 19 of 63 Computational Aspects of Fractional-Order Control Problems

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

Slide 20 of 63 Computational Aspects of Fractional-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)

Slide 21 of 63 Computational Aspects of Fractional-Order Control Problems

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

Object creation

- Syntax

Slide 22 of 63 Computational Aspects of Fractional-Order Control Problems

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

Display function

Slide 23 of 63 Computational Aspects of Fractional-Order Control Problems

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

Model Entering Examples

- Example1
- Example 2
- Example 3:

Slide 24 of 63 Computational Aspects of Fractional-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

Slide 25 of 63 Computational Aspects of Fractional-Order Control Problems

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

Theoretical Results

- Series connection
- Parallel connection
- Feedback Connection

Slide 26 of 63 Computational Aspects of Fractional-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

Slide 27 of 63 Computational Aspects of Fractional-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

Slide 28 of 63 Computational Aspects of Fractional-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

Slide 29 of 63 Computational Aspects of Fractional-Order Control Problems

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

Examples

- Closed-loop model
- Model with input

Slide 30 of 63 Computational Aspects of Fractional-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

Slide 31 of 63 Computational Aspects of Fractional-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

Slide 32 of 63 Computational Aspects of Fractional-Order Control Problems

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

Slide 33 of 63 Computational Aspects of Fractional-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

Slide 34 of 63 Computational Aspects of Fractional-Order Control Problems

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

3.1 Continued Fractions

- Math form
- For

Slide 35 of 63 Computational Aspects of Fractional-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

Slide 36 of 63 Computational Aspects of Fractional-Order Control Problems

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

MATLAB Implementation

- MATLAB code
- Syntax
- Example

Slide 37 of 63 Computational Aspects of Fractional-Order Control Problems

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

3.3 Modified Oustaloup’s Filter

- Method
- Code
- Syntax

Slide 38 of 63 Computational Aspects of Fractional-Order Control Problems

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

Slide 39 of 63 Computational Aspects of Fractional-Order Control Problems

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

Example 1: Linear model

- Denote
- Simulink

modelling

c10mfode1.mdl

Slide 40 of 63 Computational Aspects of Fractional-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

Slide 41 of 63 Computational Aspects of Fractional-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.

Slide 42 of 63 Computational Aspects of Fractional-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

Slide 43 of 63 Computational Aspects of Fractional-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

Slide 44 of 63 Computational Aspects of Fractional-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

Slide 45 of 63 Computational Aspects of Fractional-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

Slide 46 of 63 Computational Aspects of Fractional-Order Control Problems

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

Selection of finite-time

- Tested in an example

Slide 47 of 63 Computational Aspects of Fractional-Order Control Problems

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

4.2 Design Examples with MATLAB/Simulink

- Plant model, time-varying
- Simulink

Slide 48 of 63 Computational Aspects of Fractional-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

Slide 49 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

Slide 50 of 63 Computational Aspects of Fractional-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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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

Slide 52 of 63 Computational Aspects of Fractional-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

Slide 53 of 63 Computational Aspects of Fractional-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

Slide 54 of 63 Computational Aspects of Fractional-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

Slide 55 of 63 Computational Aspects of Fractional-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

Slide 56 of 63 Computational Aspects of Fractional-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

Slide 57 of 63 Computational Aspects of Fractional-Order Control Problems

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

A complicated controller

- Controller, with QFT method
- MATLAB Implementation

Slide 58 of 63 Computational Aspects of Fractional-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

Slide 59 of 63 Computational Aspects of Fractional-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

Slide 60 of 63 Computational Aspects of Fractional-Order Control Problems

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

Model Fitting Algorithm

- Select an initial reduced model
- Evaluate an error
- Use an optimization (i.e., Powell's algorithm) to iterate one step for a better estimated model
- Set , go to Step (2) until an optimal reduced model is obtained
- Extract the delay from , if any

Slide 61 of 63 Computational Aspects of Fractional-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

Slide 62 of 63 Computational Aspects of Fractional-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/

Slide 63 of 63 Computational Aspects of Fractional-Order Control Problems

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

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