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Inverted Pendulum

Inverted Pendulum. Emily Hamilton ECE Department, University of Minnesota Duluth December 21, 2009. ECE 5831 - Fall 2009. 1. Overview. Fuzzy sets Fuzzy operations Conventional controller Performance objectives and evaluations Fuzzy controller Defining a rule base Fuzzification

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Inverted Pendulum

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  1. Inverted Pendulum Emily Hamilton ECE Department, University of Minnesota Duluth December 21, 2009 ECE 5831 - Fall 2009 1

  2. Overview • Fuzzy sets • Fuzzy operations • Conventional controller • Performance objectives and evaluations • Fuzzy controller • Defining a rule base • Fuzzification • Inference engine • Defuzzification • Conclusion 2 2

  3. Fuzzy Controller • Parts of fuzzy controller • Rule base: set of If-Then rules • Inference mechanism: combines rules to obtain the best control • Fuzzification interface: transforms linguistic inputs into fuzzy inputs • Defuzzification interface: transforms fuzzy outputs into linguistic terms 3 3

  4. Fuzzy Sets • Fuzzy set A defined by: A = {(x, A(x)) | x is in X} • Where X is the set of elements in the set: X = {0, 1,… , n} 4

  5. Fuzzy Sets • X is also known as the universe of a fuzzy set • A(x) is the membership function of x • Grade of membership of the set • Values in the range {0,1} 5

  6. Fuzzy Operations Some Common Operations • Union • Intersection • Complement Take the fuzzy sets A and B for these examples. A = {{1,0.3}, {2,0.7}, {3,0.6}} B = {{1,0.4}, {2,0.1}, {3,0.9}} 6

  7. Fuzzy Operations • Union • represented by AUB • AUB = max(A(x), B(x)) 7

  8. Fuzzy Operations • Intersection • represented by A ∩ B • A∩B = min(A(x), B(x)) 8

  9. Fuzzy Operations • Complement • represented by A’(x) • A’(x) = 1 – A(x) 9

  10. Common Membership Functions • Triangular • Trapezoidal • Gaussian 10

  11. Common Membership Functions • Bell • Sigmoid 11

  12. Conventional Control System 12

  13. Performance Objectives • Disturbance rejection properties • Insensitivity to plant parameter variations • Stability • Rise-time • Overshoot • Settling time • Steady-state error 13

  14. Technical Constraints • Cost • Computational complexity: • Manufacturability • Reliability • Maintainability • Adaptability • Understandability • Politics 14

  15. Performance Evaluation • Mathematical Evaluation • To prove that all performance objectives have been met • Relies on accuracy of mathematical model • Complex nonlinear mathematical models do not exist yet • Can be used to enhance confidence that control system will work properly 15

  16. Performance Evaluation • Simulation-Based Analysis • Simulation of actual system is built and tested with the control system • Can be more accurate than the mathematical model because system constraints and changes can be applied easily • Not perfectly accurate 16

  17. Performance Evaluation • Experimental Evaluation • Implementing the control system in the actual process • Can be helpful to find problems that would not have been found elsewhere • Can be risky 17

  18. Fuzzy Control System 18 18

  19. Defining a Rule Base • Choose inputs and outputs • Put knowledge into rules: • Use linguistic descriptions from experts for inputs and outputs • Relate the inputs and outputs with the experts’ knowledge • Create a table representing the rule base 19 19

  20. Inverted Pendulum 20 20

  21. Inverted Pendulum 21 21

  22. Inverted Pendulum Rules • If the angle q is positive and the velocity q ' is positive, then decrease a lot. • If the angle q is positive and the velocity q ' is zero, then decrease . • If the angle q is positive and the velocity q ' is negative, then do not apply . • If the angle q is zero and the velocity q ' is positive, then decrease . • If the angle q is zero and the velocity q ' is zero, then do not apply 22 22

  23. Inverted Pendulum Rules • If the angle q is zero and the velocity q ' is negative, then increase • If the angle q is negative and the velocity q ' is positive, then do not apply . • If the angle q is negative and the velocity q ' is zero, then increase . • If the angle q is negative and the velocity q ' is negative, then increase a lot. 23

  24. Inverted Pendulum Rule Table 24 24

  25. Inverted Pendulum Fuzzy Controller 25 25

  26. Membership Functions • Evaluate the certainty of the linguistic values • We use the certainties of the linguistic value to create out membership functions. • Input values: • e(t) = π/4 • d/dt(e(t)) = π/16 • Membership function values: • Apossmall(e(t)) = 1 • Azero(d/dt(e(t))) = Apossmall(d/dt(e(t))) = 0.5 26 26

  27. Fuzzification • Example: Certainty of NegAngle • Angle, θ, is 45˚. F(45 ˚) = 0 • θ = -45 ˚. F(-45 ˚) = 0.4 • θ = -95 ˚. F(-95 ˚) = 1 27 27

  28. Angle Membership Functions 28 28

  29. Velocity Membership Functions 29 29

  30. Output Membership Functions 30 30

  31. Inference Engine • Premise: the certainty of a rule in a situation. Ex: P(θ, d/dt(θ)) = 0.5 • A rule is “on” if its certainty is greater than zero. • The inference engine combines the recommendations of all rules that are “on” to find the control output. 31 31

  32. Inference Engine • Determines relevance of each rule in the given situation using the premises • Draws information using the rule base and the inputs 32 32

  33. Defuzzification transfers the fuzzy output into a crisp value. This equation is used to defuzzifiy the outputs of the inference engine in the Center of Gravity (Area) inference engine. Each inference engine has its own equation for defuzzification. Defuzzification 33 33

  34. The crisp output value we receive using the Center of Area inference engine is -21.3063 units of force. Inverted Pendulum 34 34

  35. Conclusion • Fuzzy sets • Control system should meet performance objectives and pass evaluations • Use expert knowledge to create inputs, outputs, and rule base. • Fuzzify crisp inputs • Inference engine uses rule base to decide control output • Defuzzify output to crisp value 35 35

  36. References • [1] K. Passino, S. Yurkovich. Fuzzy Control. 1998. Addison Wesley Longman, Inc. • J. Jang, C. Sun. “Neuro-Fuzzy Modeling and Control.” 36 36

  37. Questions? 37 37

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