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  • 1. Friday, 2010-7- 2,16:05:56Slide 1 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Tutorial Workshop onFractional Order DynamicSystems and ControlsWCICA’2010, Jinan, ChinaComputational Aspect of Fractional-Order Control ProblemsDingyu XueInstitute of AI and RoboticsFaculty of Information Sciences andEngineeringNortheastern UniversityShenyang 110004, P R China
  • 2. Friday, 2010-7- 2,16:05:56Slide 2 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Computational Aspect ofFractional-Order Control ProblemsOutlines and Motivations of PresentationComputations 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, isthere a similar way to solve fractional-order controlproblems. Class based programming in MATLAB
  • 3. Friday, 2010-7- 2,16:05:56Slide 3 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Outlines and Motivations (contd)Simulation Studies of Fractional-OrderNonlinear SystemsHow to solve problems in nonlinear systems? Theonly feasible way is by simulation. Simulink basedprogramming methodology is adoptedOptimum Controller Design for Fractional-Order Systems through ExamplesCriteria selection, design examples via SimulinkImplementation of the ControllersContinuous and Discrete
  • 4. Friday, 2010-7- 2,16:05:56Slide 4 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Main ReferenceChapter 13 of the MonographFractional-order Systems and Controls---Fundamentals and ApplicationsBy Concepcion Alicia Monje, YangQuan Chen,Blas Manuel Vinagre, Dingyu Xue,Vicente FeliuSpringer-Verlag, London, July, 2010Implementation part is from Chapter 12 of the book
  • 5. Friday, 2010-7- 2,16:05:56Slide 5 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/20101 Computations in Fractional CalculusEvaluation of Mittag-Leffler functionsEvaluations of Fractional-order DerivativesClosed-form Solutions to Linear Fractional-order Differential EquationsAnalytical Solutions to Linear Fractional-orderDifferential Equations
  • 6. Friday, 2010-7- 2,16:05:56Slide 6 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/20101.1 Evaluation of Mittag-Leffler FunctionsImportance of Mittag-Leffler functionsAs important as exponential functions in IOsAnalytical solutions of FO-ODEsDefinitionsML in one parameterML in two parametersSpecial cases
  • 7. Friday, 2010-7- 2,16:05:56Slide 7 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Mittag-Leffler Functions in more parsDefinitionswithDerivativesMATLAB function
  • 8. Friday, 2010-7- 2,16:05:56Slide 8 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010CodePodlubny’s code mlf() embedded
  • 9. Friday, 2010-7- 2,16:05:56Slide 9 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Examples to tryDraw curvesCode Other functions
  • 10. Friday, 2010-7- 2,16:05:56Slide 10 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/20101.2 Evaluations of Fractional-orderDerivativesDefinitions:Grünwald-Letnikovs Definition Others Caputos Derivatives, Riemann-Liouville’s, Cauchy’s
  • 11. Friday, 2010-7- 2,16:05:56Slide 11 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010MATLAB ImplementationEasy to programSyntaxExamplesOrginal function
  • 12. Friday, 2010-7- 2,16:05:56Slide 12 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/20101.3 Closed-Form Solutions to LinearFractional-Order Differential EquationsMathematical FormulationFractional-order DEsDenoteOriginal equation changed to
  • 13. Friday, 2010-7- 2,16:05:56Slide 13 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010From G-L definitionAndThe closed-form solution can be obtained
  • 14. Friday, 2010-7- 2,16:05:56Slide 14 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010MATLAB Code and SyntaxCodeSyntax
  • 15. Friday, 2010-7- 2,16:05:56Slide 15 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010ExampleFractional-order differential equationwith step input u(t)MATLAB solutions
  • 16. Friday, 2010-7- 2,16:05:56Slide 16 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/20101.4 Analytical Solutions to LinearFractional-order Differential EquationsLaplace transform propertySpecial cases: Impulse input:Step inputs:
  • 17. Friday, 2010-7- 2,16:05:56Slide 17 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Partial fraction expansion ofCommensurate-order SystemsDefinitionTransfer functionAfter partial fraction expansion, step responses
  • 18. Friday, 2010-7- 2,16:05:56Slide 18 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Example:Partial fractional expansionStep response, theoretical
  • 19. Friday, 2010-7- 2,16:05:56Slide 19 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Also works for the cases with multiple polesFor more complicated systemsAnalytical solutions are too complicated
  • 20. Friday, 2010-7- 2,16:05:56Slide 20 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/20102 Fractional-Order Transfer Functions--- MATLAB Object ModellingMotivated by the Control Systems ToolboxSpecify a system in one variable G,use of * and +, and step(G), bode(G), convenientOutlines in the sectionDesign of a FOTF ObjectModeling Using FOTFsStability Assessment of FOTFsNumerical Time Domain AnalysisFrequency Domain Analysis
  • 21. Friday, 2010-7- 2,16:05:56Slide 21 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Fractional-Order Transfer FunctionsFive parameters:Possible to design a MATLAB objectCreate a @fotf folderEstablish two essential functionsfotf.m (for creation), display.m (for display object)
  • 22. Friday, 2010-7- 2,16:05:56Slide 22 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Object creationSyntax
  • 23. Friday, 2010-7- 2,16:05:56Slide 23 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Display function
  • 24. Friday, 2010-7- 2,16:05:56Slide 24 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Model Entering ExamplesExample1Example 2Example 3:
  • 25. Friday, 2010-7- 2,16:05:56Slide 25 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/20102.2 Modelling of FOTF SystemsSeries connection: G1*G2Overload functions are needed for mtimes.mSimilarly other functions can be writtenplus.m, feedback.m, uminus.m, mrdivide.msimple.m, mpower.m, inv.m, minus.m
  • 26. Friday, 2010-7- 2,16:05:56Slide 26 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Theoretical ResultsSeries connectionParallel connectionFeedback Connection
  • 27. Friday, 2010-7- 2,16:05:56Slide 27 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Modelling ExamplesPlantControllerUnity negative feedback connectionClosed-loop system
  • 28. Friday, 2010-7- 2,16:05:56Slide 28 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/20102.3 Analysis of Fractional-Order SystemsStability regions for commensurate-order TFsMATLAB functionExample: the previousclosed-loop system
  • 29. Friday, 2010-7- 2,16:05:56Slide 29 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/20102.4 Numerical Time Domain AnalysisBased on fode_sol function discussed earlier,overload functions step and lsim are writtenStep responseTime response to arbitrary inputsNo restrictions. Reliable numerical solutionsValidate the results
  • 30. Friday, 2010-7- 2,16:05:56Slide 30 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010ExamplesClosed-loop modelModel with input
  • 31. Friday, 2010-7- 2,16:05:56Slide 31 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/20102.5 Frequency Domain AnalysisOverload functionsBode.mNyquist.mNichols.mVia ExamplesSlopes. Not integer times of 20dB/sec
  • 32. Friday, 2010-7- 2,16:05:56Slide 32 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/20102.6 Norm Measures of FOTFsNorms2-normInfinity normOverload functionsExamples
  • 33. Friday, 2010-7- 2,16:05:56Slide 33 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/20103 Simulation Studies of Fractional-order Nonlinear SystemsProblems of Existing methodsGrunwald-Letnikov definitions and others onlyapplies to the cases where input to a fractional-order D/I is knownStep and lsim functions only works for FOTFobjects, not nonlinear systemsFor nonlinear control systems, a block diagrambased approach is needed.A Simulink block is needed for FO-D
  • 34. Friday, 2010-7- 2,16:05:56Slide 34 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Filters for Approximating FO-DsContinued fraction approximationOustaloup’s filterModified Oustaloup’s filterMasking a Simulink block with the Oustaloup’sfilter and othersSimulation of nonlinear frcational-ordersystems with examplesValidation of simulation results
  • 35. Friday, 2010-7- 2,16:05:56Slide 35 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/20103.1 Continued FractionsMath formFor s^0.5
  • 36. Friday, 2010-7- 2,16:05:56Slide 36 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/20103.2 Oustaloup’s FilterIdea of Oustaloup’s FilterMethod
  • 37. Friday, 2010-7- 2,16:05:56Slide 37 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010MATLAB ImplementationMATLAB codeSyntaxExample
  • 38. Friday, 2010-7- 2,16:05:56Slide 38 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/20103.3 Modified Oustaloup’s FilterMethodCodeSyntax
  • 39. Friday, 2010-7- 2,16:05:56Slide 39 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/20103.4 Simulink ModellingMask a Simulink block --- the key elementPossibly with a low-pass filter
  • 40. Friday, 2010-7- 2,16:05:56Slide 40 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Example 1: Linear modelDenoteSimulinkmodellingc10mfode1.mdl
  • 41. Friday, 2010-7- 2,16:05:56Slide 41 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Example 2: Nonlinear systemRewrite the equationSimulink modelc10mfod2.mdl
  • 42. Friday, 2010-7- 2,16:05:56Slide 42 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Example 3: fractional-order delay systemRewriteSimulink modelcxfdde1.mdlControl loops can beestablishedWith Simulink,complicated systemscan be studied.
  • 43. Friday, 2010-7- 2,16:05:56Slide 43 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/20103.6 Validations of Simulation ResultsNo analytical solution. Indirect methods:Change parameters in equation solver, such asRelTol, and see whether consistent results canbe obtainedChange simulation algorithmsChange Oustaloup’s filter parametersThe frequency rangeThe order NThe filter, Oustaloup, modified, and others
  • 44. Friday, 2010-7- 2,16:05:56Slide 44 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/20104 Optimal Controller DesignWhat Criterion is Suitable for AddressingOptimality of Servo Control Systems:Criterion SelectionsMATLAB/Simulink based Optimal ControllerDesign ProceduresOptimum Fractional-Order PID Controllers:Parameter Setting via Optimization ThroughAn Example
  • 45. Friday, 2010-7- 2,16:05:56Slide 45 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/20104.1 Optimal Criterion SelectionsWhat kind of control can be regarded asoptimal? Time domain optimization is goingto 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 otherFinite-time ITAE is to be used
  • 46. Friday, 2010-7- 2,16:05:56Slide 46 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Why Finite-Time ITAETwo criteria:Which oneis better?ITAE type ofcriteria aremeaningful
  • 47. Friday, 2010-7- 2,16:05:56Slide 47 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Selection of finite-timeTested in an example
  • 48. Friday, 2010-7- 2,16:05:56Slide 48 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/20104.2 Design Examples withMATLAB/SimulinkPlant model, time-varyingSimulink
  • 49. Friday, 2010-7- 2,16:05:56Slide 49 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Optimum DesignEstablish a MATLAB objective functionDesign via optimizationAllow nonlinear elements and complicatedsystems, constrained optimizations possible
  • 50. Friday, 2010-7- 2,16:05:56Slide 50 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/20104.3 Optimal FO PID DesignController with 5 parametersDesign Example, Plant
  • 51. Friday, 2010-7- 2,16:05:56Slide 51 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010MATLAB objective functionOptimal controller designOptimal Controller found
  • 52. Friday, 2010-7- 2,16:05:56Slide 52 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/20105 Implementation of FO ControllersContinuous ImplementationOustaloup’s filterModified Oustaloup’s filterOther implementationsDiscrete ImplementationVia Step/Impulse Response InvariantsFrequency Domain FittingSub-Optimal Integer-Order Model Reduction
  • 53. Friday, 2010-7- 2,16:05:56Slide 53 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Continuous ImplementationsAs Discussed EarlierApproximation to Fractional-order operators(differentiators/integrator) only. Suitable forFO-PID type of controllersFunctions to use
  • 54. Friday, 2010-7- 2,16:05:56Slide 54 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Discrete-Time ImplementationsFIR Filter, ’s workAgain for fraction-order operatorsAlso possible, Tustin’s approximation
  • 55. Friday, 2010-7- 2,16:05:56Slide 55 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Step/Impulse Response InvariantsApproximation ModelsThe following functions can be used,Dr Yangquan Chen’s workExample
  • 56. Friday, 2010-7- 2,16:05:56Slide 56 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Discrete-Time Approximation toMATLAB solutions, due to Dr Chen’s codeExampleRewrite asMATLAB solutions
  • 57. Friday, 2010-7- 2,16:05:56Slide 57 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/20105.3 Frequency Response Fitting ofFractional-Order ControllersCriterionMATLAB FunctionExample
  • 58. Friday, 2010-7- 2,16:05:56Slide 58 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010A complicated controllerController, with QFT methodMATLAB Implementation
  • 59. Friday, 2010-7- 2,16:05:56Slide 59 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Integer-order fitting modelComparisons
  • 60. Friday, 2010-7- 2,16:05:56Slide 60 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/20105.5 Rational Approximation toFractional-Order Transfer FunctionsOriginal modelFitting integer-order modelFitting criterionwhere
  • 61. Friday, 2010-7- 2,16:05:56Slide 61 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Model Fitting Algorithm1. Select an initial reduced model2. Evaluate an error3. Use an optimization (i.e., Powells algorithm)to iterate one step for a better estimatedmodel4. Set , go to Step (2) until anoptimal reduced model is obtained5. Extract the delay from , if any
  • 62. Friday, 2010-7- 2,16:05:56Slide 62 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010MATLAB Function ImplementationFunction callExampleFinding full-order approximationReduction
  • 63. Friday, 2010-7- 2,16:05:56Slide 63 of 63 Computational Aspects of Fractional-Order Control ProblemsDingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010Concluding RemarksMATLAB code are prepared for fractional-order systems, especially useful for beginnersHandy facilities can also be used byexperienced users, for immediate acquisitionof plots and research resultsCode available fromhttp://mechatronics.ece.usu.edu/foc/wcica2010tw/