11 Inverted Pendulum Emily Hamilton ECE Department, University of Minnesota Duluth December 21, 2009 ECE Fall 2009

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

333 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

44 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}

55 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}

66 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}}

77 Fuzzy Operations Union – represented by AUB –AUB = max(A(x), B(x))

88 Fuzzy Operations Intersection – represented by A ∩ B –A∩B = min(A(x), B(x))

99 Fuzzy Operations Complement – represented by A’(x) –A’(x) = 1 – A(x)

10 Common Membership Functions Triangular Trapezoidal Gaussian

11 Common Membership Functions Bell Sigmoid

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

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

20 Inverted Pendulum

21 Inverted Pendulum

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

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.

24 Inverted Pendulum Rule Table

25 Inverted Pendulum Fuzzy Controller

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: –A possmall (e(t)) = 1 –A zero (d/dt( e(t)) ) = A possmall (d/dt( e(t)) ) = 0.5

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

28 Angle Membership Functions

29 Velocity Membership Functions

30 Output Membership Functions

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.

32 Inference Engine 1.Determines relevance of each rule in the given situation using the premises 2.Draws information using the rule base and the inputs 32

33 Defuzzification 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.

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

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

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

37 Questions?