1. School of Industrial and Systems Engineering, Georgia Institute of Technology 1 Defuzzification Filters and Applications to Power System Stabilization Problems Augustine O. Esogbue Qiang Song Warren E. Hearnes II Intelligent Systems and Control Lab School of Industrial and Systems Engineering Georgia Institute of Technology

2. School of Industrial and Systems Engineering, Georgia Institute of Technology 2 Outline Brief Review of Defuzzification Optimality Principle of Defuzzification Defuzzification Filters Application to Power System Stabilization Conclusions

3. School of Industrial and Systems Engineering, Georgia Institute of Technology 3 Fuzzy Control System Architecture Plant Fuzzy Controller Feedback Calculation input output + - u Defuzzification

4. School of Industrial and Systems Engineering, Georgia Institute of Technology 4 Defuzzification Defuzzification: a process or procedure to convert a fuzzy quantity to a crisp number Characteristics –Solely a process of converting a fuzzy quantity to a real number –No system performance in mind –No objective criterion –Could become very arbitrary

5. School of Industrial and Systems Engineering, Georgia Institute of Technology 5 Common Defuzzification Methods Center of Gravity Method Mean of Maxima Method

6. School of Industrial and Systems Engineering, Georgia Institute of Technology 6 Yager and Filev’s Work (Yager & Filev, 1993) Assuming the true or ideal defuzzification values are available, the parameters in the transformation are determined in such a way that the difference between the actual defuzzification values and the true values are minimized. Advantages –With a goal in mind. –No longer arbitrary. –Optimization oriented. Disadvantages –How to obtain the true defuzzification value?

7. School of Industrial and Systems Engineering, Georgia Institute of Technology 7 Mabuchi’s Work (Mabuchi, 1993) A predetermined criterion is chosen for selecting the crisp defuzzified value. The criterion is also a function between the difference of the true defuzzification value and the actual one. A sensitivity function (criterion) is used to compare and select the defuzzified value by comparison with the true defuzzified value. Advantages Disadvantages

8. School of Industrial and Systems Engineering, Georgia Institute of Technology 8 Difficulty in Yager and Filev’s and Mabuchi’s Work Very hard or almost impossible to obtain the true defuzzification value without any system or environment information. Defuzzification for the sake of defuzzification is meaningless.

9. School of Industrial and Systems Engineering, Georgia Institute of Technology 9 A Solution It is common that in a system environment, the performance is evaluated based on the system output. The purpose of system design is to optimize a predefined system performance index. If system output can be converted into input, then the true defuzzification value could be obtained.

10. School of Industrial and Systems Engineering, Georgia Institute of Technology 10 Optimality Principle of Defuzzification (Song and Leland, 1996) In any applications of fuzzy systems, there should be at least one system performance index; the purpose of defuzzification is therefore to select a crisp value based on the fuzzy quantity to eventually optimize the system performance index. Features –A principle –Environment is considered in defuzzification –An optimization process

11. School of Industrial and Systems Engineering, Georgia Institute of Technology 11 Defuzzification Filter A defuzzification filter may be defined as any dynamic system which is used for the purpose of compensating the effect of noise in the defuzzified control signal. Any defuzzification method whose output is the input to a defuzzification filter is called a preliminary defuzzification method, and the corresponding defuzzified value is called preliminary defuzzified values.

12. School of Industrial and Systems Engineering, Georgia Institute of Technology 12 Characteristics Dynamic System –linear or nonlinear –continuous or discrete Input from fuzzy controller, output to plant Used to enhance system performance May have different types, or designs for the same application

13. School of Industrial and Systems Engineering, Georgia Institute of Technology 13 Design of Defuzzification Filters Notations –  (k), a parameter vector at time k –u p (k), a preliminary defuzzified value at time k –u a (k), the actual defuzzified value at time k, also the output of the defuzzification filter –u o (k), the desirable defuzzified value at time k Steps in designing a defuzzification filter –Determine a proper model –Determine the parameters in the model according to the desired defuzzified values and the actual defuzzified values –Derive the inverse of the plant to obtain the desired defuzzified values

14. School of Industrial and Systems Engineering, Georgia Institute of Technology 14 Model of Defuzzification Filters

15. School of Industrial and Systems Engineering, Georgia Institute of Technology 15 Model in Time Domain u a (k) = W T (k)  (k) where W(k) = (-u a (k-1),-u a (k-2), …, -u a (k-m), u p (k-1), u p (k-2), …, u p (k-n)) T

16. School of Industrial and Systems Engineering, Georgia Institute of Technology 16 A Recursive Formulae for  (k)

17. School of Industrial and Systems Engineering, Georgia Institute of Technology 17 The Desired Defuzzification Values u o (k) When the model of the plant is known, the desired defuzzification values can be derived. This is done through the inverse of the plant model. The inverse is obtained by means of Lie-derivatives.

18. School of Industrial and Systems Engineering, Georgia Institute of Technology 18 The Plant Model

19. School of Industrial and Systems Engineering, Georgia Institute of Technology 19 The Inverse where u the ideal control, y d is the desired output, c i are coefficients of a Hurwitz polynomial, and

20. School of Industrial and Systems Engineering, Georgia Institute of Technology 20 Application to Power Stabilization Architecture of the system Fuzzy Neural Self-Learning Controller Plant Defuzzification Filter Preliminary Defuzzifier

21. School of Industrial and Systems Engineering, Georgia Institute of Technology 21 The Plant A synchronized machine with an exciter and a stabilizer connected to an infinite bus. Synchronized Machine Stabilizer Voltage Regulator and Exciter V ref u  + +

22. School of Industrial and Systems Engineering, Georgia Institute of Technology 22 The Plant(2) Linearized model of 6th order 6 state variables –  :angular velocity deviation –  : torque angle deviation –e q : q-axis component voltage behind transient reactance change –V t : terminal voltage change –V o : infinite bus voltage change –e fd : equivalent excitation voltage change

23. School of Industrial and Systems Engineering, Georgia Institute of Technology 23 The New System Synchronized Machine Voltage Regulator and Exciter V ref u  + + Fuzzy Neural Self-Learning Controller Defuzzification Filter

24. School of Industrial and Systems Engineering, Georgia Institute of Technology 24 Simulation Results Case 1: The fuzzy neural self-learning controller can stabilize the system. –With the defuzzification filter, the settling time is improved.

25. School of Industrial and Systems Engineering, Georgia Institute of Technology 25 Simulation Results(2) Case 2: The fuzzy neural self-learning controller cannot stabilize the system. –With the defuzzification filter, the system is stabilized.

26. School of Industrial and Systems Engineering, Georgia Institute of Technology 26 Simulation Results(3) Case 3: The fuzzy neural self-learning controller can stabilize the system, but the settling time is long. –With the defuzzification filter, the settling time is much shorter.

27. School of Industrial and Systems Engineering, Georgia Institute of Technology 27 Conclusions By using the system performance index, we can avoid otherwise the unknown true defuzzification value. The true defuzzification value can be generated by using the inverse of the plant model(this may be hard to obtain in some cases). Defuzzification filter is based on the idea of Optimality Principle of Defuzzification. If applied in the system environment, Yager and Filev’s defuzzifier, and Mabuchi’s can be both regarded as defuzzification filters. –But are different filter designs

28. School of Industrial and Systems Engineering, Georgia Institute of Technology 28 Conclusions(2) Defuzzification filter improves system performances otherwise controlled by a fuzzy neural self-learning controller. Further research should find more applications for defuzzification filter, design nonlinear filters, and find better design methods.