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

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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 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

• 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
• 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
• 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
• 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

• 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

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
• 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
• cxfdde1.mdl
• Control loops can be

established

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

• Plant model, time-varying

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

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
• 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

• 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

• 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