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

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

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  1. 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

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

  23. Display function Slide 23 of 63 Computational Aspects of Fractional-Order Control Problems Dingyü Xue for WCICA’ 2010, Jinan, P R China, 07/2010

  24. 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

  25. 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

  26. 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

  27. 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

  28. 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

  29. 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

  30. 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

  31. 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

  32. 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

  33. 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

  34. 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

  35. 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

  36. 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

  37. 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

  38. 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

  39. 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

  40. 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

  41. 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

  42. 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

  43. 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

  44. 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

  45. 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

  46. 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

  47. 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

  48. 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

  49. 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

  50. 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

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