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Friday, 2010-7- 2, 16:05:56
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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Slide 2 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
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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Slide 3 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü 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 ExamplesCriteria selection, design examples via Simulink
Implementation of the ControllersContinuous and Discrete
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Slide 4 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
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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Slide 5 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü 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
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Slide 6 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
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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Slide 7 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Mittag-Leffler Functions in more pars
Definitions
with
Derivatives
MATLAB function
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Slide 8 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Code
Podlubny’s code mlf() embedded
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Slide 9 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Examples to tryDraw curves
Code
Other functions
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Slide 10 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
1.2 Evaluations of Fractional-order Derivatives
Definitions:Grünwald-Letnikov's Definition
Others Caputo's Derivatives, Riemann-Liouville’s, Cauchy’s
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Slide 11 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
MATLAB Implementation
Easy to program
Syntax
ExamplesOrginal function
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Slide 12 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
1.3 Closed-Form Solutions to Linear Fractional-Order Differential Equations
Mathematical FormulationFractional-order DEs
Denote
Original equation changed to
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Slide 13 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
From G-L definition
And
The closed-form solution can be obtained
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Slide 14 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
MATLAB Code and SyntaxCode
Syntax
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Slide 15 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Example
Fractional-order differential equation
with step input u(t)
MATLAB solutions
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Slide 16 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
1.4 Analytical Solutions to Linear Fractional-order Differential Equations
Laplace transform property
Special cases: Impulse input:
Step inputs:
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Slide 17 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Partial fraction expansion of Commensurate-order Systems
Definition
Transfer function
After partial fraction expansion, step responses
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Slide 18 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Example:
Partial fractional expansion
Step response, theoretical
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Slide 19 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü 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
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Slide 20 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
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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Slide 21 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü 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)
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Slide 22 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Object creation
Syntax
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Slide 23 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Display function
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Slide 24 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Model Entering ExamplesExample1
Example 2
Example 3:
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Slide 25 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü 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 writtenplus.m, feedback.m, uminus.m, mrdivide.m simple.m, mpower.m, inv.m, minus.m
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Slide 26 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Theoretical ResultsSeries connection
Parallel connection
Feedback Connection
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Slide 27 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Modelling Examples
Plant
Controller
Unity negative feedback connection
Closed-loop system
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Slide 28 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü 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
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Slide 29 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü 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
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Slide 30 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
ExamplesClosed-loop model
Model with input
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Slide 31 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
2.5 Frequency Domain Analysis
Overload functionsBode.mNyquist.mNichols.m
Via Examples
Slopes. Not integer times of 20dB/sec
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Slide 32 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
2.6 Norm Measures of FOTFsNorms2-norm Infinity norm
Overload functions
Examples
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Slide 33 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
3 Simulation Studies of Fractional-order Nonlinear Systems
Problems of Existing methodsGrunwald-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 diagram based approach is needed.
A Simulink block is needed for FO-D
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Slide 34 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Filters for Approximating FO-Ds
Continued fraction approximation
Oustaloup’s filter
Modified Oustaloup’s filter
Masking a Simulink block with the Oustaloup’s filter and others
Simulation of nonlinear frcational-order systems with examples
Validation of simulation results
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Slide 35 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
3.1 Continued Fractions
Math form
For s^0.5
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Slide 36 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
3.2 Oustaloup’s FilterIdea of Oustaloup’s Filter
Method
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Slide 37 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
MATLAB Implementation
MATLAB code
Syntax
Example
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Slide 38 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
3.3 Modified Oustaloup’s Filter
Method
Code
Syntax
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Slide 39 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
3.4 Simulink ModellingMask a Simulink block --- the key element
Possibly with a low-pass filter
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Slide 40 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Example 1: Linear model
Denote
Simulink modellingc10mfode1.mdl
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Slide 41 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Example 2: Nonlinear system
Rewrite the equation
Simulink modelc10mfod2.mdl
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Slide 42 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Example 3: fractional-order delay system
Rewrite
Simulink modelcxfdde1.mdl
Control loops can beestablished
With Simulink,complicated systemscan be studied.
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Slide 43 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
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 Oustaloup’s filter parametersThe frequency rangeThe order NThe filter, Oustaloup, modified, and others
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Slide 44 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü 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
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Slide 45 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü 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 criteriaLQ optimization, artificial, no methods for Q and R ISE criterion, H2 minimization, Hinf, may be too conservativeFastest, most economical, and other
Finite-time ITAE is to be used
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Slide 46 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Why Finite-Time ITAETwo criteria:
Which one
is better?
ITAE type of
criteria are
meaningful
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Slide 47 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Selection of finite-timeTested in an example
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Slide 48 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
4.2 Design Examples with MATLAB/Simulink
Plant model, time-varying
Simulink
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Slide 49 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Optimum Design
Establish a MATLAB objective function
Design via optimization
Allow nonlinear elements and complicated systems, constrained optimizations possible
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Slide 50 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
4.3 Optimal FO PID DesignController with 5 parameters
Design Example, Plant
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Slide 51 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
MATLAB objective function
Optimal controller design
Optimal Controller found
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Slide 52 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
5 Implementation of FO Controllers
Continuous ImplementationOustaloup’s filterModified Oustaloup’s filterOther implementations
Discrete ImplementationVia Step/Impulse Response Invariants
Frequency Domain Fitting
Sub-Optimal Integer-Order Model Reduction
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Slide 53 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü 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
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Slide 54 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü 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
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Slide 55 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü 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
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Slide 56 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü 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
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Slide 57 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
5.3 Frequency Response Fitting of Fractional-Order Controllers
Criterion
MATLAB Function
Example
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Slide 58 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
A complicated controllerController, with QFT method
MATLAB Implementation
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Slide 59 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
Integer-order fitting model
Comparisons
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Slide 60 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü 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
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Slide 61 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
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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Slide 62 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010
MATLAB Function Implementation Function call
Example
Finding full-order approximation
Reduction
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Slide 63 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü 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 users, for immediate acquisition of plots and research results
Code available fromhttp://mechatronics.ece.usu.edu/foc/wcica2010tw/
