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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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tutorial workshop on fractional order dynamic systems and controls wcica 2010 jinan china
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 problems outlines and motivations of presentation
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
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
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
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
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
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

slide8
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
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
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
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
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

slide13
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
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
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
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
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

slide18
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

slide19
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
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

slide21

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

slide22
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

slide23
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
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
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
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
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
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
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
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
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
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
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
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
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
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 implementation37
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
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
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

slide40
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

slide41
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

slide42
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
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
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
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
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
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
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
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
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

slide51
MATLAB objective function
  • Optimal controller design
  • Optimal Controller found

Slide 51 of 63 Computational Aspects of Fractional-Order Control Problems

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

5 implementation of fo controllers
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
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
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
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
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
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
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

slide59
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
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
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
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
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