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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

Inverted Pendulum

Emily Hamilton

ECE Department, University of Minnesota Duluth

December 21, 2009

ECE 5831 - Fall 2009

1

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

2

2

fuzzy controller
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

fuzzy sets
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

fuzzy sets5
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

fuzzy operations
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

fuzzy operations7
Fuzzy Operations
  • Union
    • represented by AUB
    • AUB = max(A(x), B(x))

7

fuzzy operations8
Fuzzy Operations
  • Intersection
    • represented by A ∩ B
    • A∩B = min(A(x), B(x))

8

fuzzy operations9
Fuzzy Operations
  • Complement
    • represented by A’(x)
    • A’(x) = 1 – A(x)

9

common membership functions
Common Membership Functions
  • Triangular
  • Trapezoidal
  • Gaussian

10

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

13

technical constraints
Technical Constraints
  • Cost
  • Computational complexity:
  • Manufacturability
  • Reliability
  • Maintainability
  • Adaptability
  • Understandability
  • Politics

14

performance evaluation
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

performance evaluation16
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

performance evaluation17
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

defining a rule base
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

inverted pendulum rules
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

inverted pendulum rules23
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

membership functions
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

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26

fuzzification
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

inference engine
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

inference engine32
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

defuzzification
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

conclusion
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

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

36

36