Homework 5
Problem 1 Write the formula for the COA defuzzification strategy when we have a finite universe of
discourse X = {x1, x2, x3,…xn} where x1<…<xn.
𝜇(𝑥)𝑖𝑠 𝑡ℎ𝑒 𝑚𝑒𝑚𝑏𝑒𝑟 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑏𝑦 𝑋
𝑋𝐶𝑂𝐴 =
∑𝑛𝑖=1 𝜇(𝑥𝑖 )𝑥𝑖 𝑑𝑥
∑𝑛𝑖=1 𝜇(𝑥𝑖 ) 𝑑𝑥
Problem 2 Use COA, BOA and MOM defuzzification strategies to find the representative values for the
overall implied fuzzy set A defined by µA(x)=trapezoid(x, 10, 30, 50, 90)
1
0.5
0
𝑋𝐶𝑂𝐴
0
10
20
30
40
50
60
70
80
90
30 𝑥 − 10
50
90 90 − 𝑥
∫10 (30 − 10) 𝑥 𝑑𝑥 + ∫30 (1)𝑥 𝑑𝑥 + ∫50 (90 − 50) 𝑥 𝑑𝑥
=
=
= 𝟒𝟔
30 𝑥 − 10
50
90 90 − 𝑥
∫𝑋 𝜇𝐴 (𝑥) 𝑑𝑥
(1)
(
)
𝑑𝑥
+
𝑑𝑥
+
(
)
𝑑𝑥
∫10 30 − 10
∫30
∫50 90 − 50
∫𝑋 𝜇𝐴 (𝑥)𝑥 𝑑𝑥
𝑋𝐵𝑂𝐴
𝑋𝐵𝑂𝐴 => ∫
max(𝑥)
𝜇𝐴 (𝑥)𝑑𝑥 = ∫
min(𝑥)
30
∫
10
𝜇𝐴 (𝑥)𝑑𝑥
𝑋𝐵𝑂𝐴
𝑋𝐵𝑂𝐴
50
90
𝑥 − 10
90 − 𝑥
(1)𝑑𝑥 = ∫ (1) 𝑑𝑥 + ∫ (
(
) 𝑑𝑥 + ∫
) 𝑑𝑥
30 − 10
90
− 50
30
𝑋𝐵𝑂𝐴
50
10 + 𝑋𝐵𝑂𝐴 − 30 = 50 − 𝑋𝐵𝑂𝐴 + 20
50 + 20 + 30 − 10
𝑋𝐵𝑂𝐴 =
= 𝟒𝟓
2
𝑋𝑀𝑂𝑀 =
∫𝑋′ (1)𝑥 𝑑𝑥
∫𝑋′ (1) 𝑑𝑥
50
=
∫30 (1)𝑥 𝑑𝑥
50
∫30 (1) 𝑑𝑥
=
800
= 𝟒𝟎
20
100
Problem 3
%
%
%
%
%
%
Simplified Product Fuzzy Inference Engine with Fuzzy Singleton Inputs
-4 Rule, 2 Input
-Product Inference Engine
-Gaussian and Sigmoidal membership functions
-MFs are normal (there is a max that equals 1)
-Center Average Deffuzificaition
% X and Y universes
X_universe = -5:.1:5;
Y_universe = -5:.1:5;
% Initialize output space
z_out = zeros(length(X_universe), length(Y_universe));
% Generic equations for the problem
sigmoid=@(x,a,c) (1/(1+exp(-a*(x-c))));
gaussian=@(x,s,c) exp(-1/2*((x-c)/s)^2);
% Input Membership functions
ux_small=@(x) sigmoid(x, -3, 0);
ux_large=@(x) sigmoid(x, 3, 0);
uy_small=@(y) sigmoid(y, -0.9, 0);
uy_large=@(y) sigmoid(y, 0.9, 0);
% Rule output Centers
c_negLarge=-3.5;
c_negSmall=-1.7;
c_posSmall=1.7;
c_posLarge=3.5;
% Intermediate Arrays to hold values during calculations
w=zeros(1,4); % Rule Weights
rc=zeros(1,4); % Rule Contribution
% Calculate the Rule surface by traversing through X and Y
for xIndex=1:length(X_universe),
for yIndex=1:length(Y_universe),
x=X_universe(xIndex);
y=Y_universe(yIndex);
% Rule 1 if x is small and y is small them z is negative large
w(1) = ux_small(x) * uy_small(y);
rc(1) = w(1)* c_negLarge;
% Rule 2 if x is small and y is large them z is negative small
w(2) = ux_small(x) * uy_large(y);
rc(2) = w(2)* c_negSmall;
% Rule 3 if x is large and y is small them z is positive small
w(3) = ux_large(x) * uy_small(y);
rc(3) = w(3)* c_posSmall;
% Rule 4 if x is large and y is large them z is positive large
w(4) = ux_large(x) * uy_large(y);
rc(4) = w(4)* c_posLarge;
% Deffuzify
z_out(yIndex, xIndex) = sum(rc) / sum(w);
end
end
figure(1);
mesh(X_universe, Y_universe, z_out);
axis([-5 5 -5 5 -4 4]);
box off;
title({'Product Inference Engine with Center Average Deffuzification', '4
Rules, 2 Inputs'});
xlabel('X');
ylabel('Y');
zlabel('Fuzzy System Rule Surface');