Complexity Reduction in Fuzzy Systems Using Functional Principal Component Analysis - Fuzzy Modeling and Control: Theory and Applications - page 60

Tu(k−2) Tu(k−1)

1 1

Neg
Zero
Pos

Neg
Zero
Pos

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0

−10 −5 0 5 10 −10 −5 0 5 10

N·m N·m

Angle(k−2) Angle(k−1)

1 1

Neg
Zero
Pos

Neg
Zero
Pos

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0

−100 −50 0 50 100 −100 −50 0 50 100

Position(º) Position(º)

Fig. 3.8 Membership functions for the mechanical system FIS

With 81 rules, after FPCA, we obtain just one rule with

⎛ ⎞

− 0 . 00006

⎝ ⎠

0 . 00130

g ˜ ( x ) = − 0 . 01961 · ξ( x ) (3.47)

0 . 04195

− 0 . 00037

In the Fig. 3.9 not seen hardly any difference between the fuzzy model and fuzzy
simplified. Measurement error between the simplified system and the original, in this
time interval (3332 samples), is

99 . 92

2 dt

10 − 11 e = e ( t ) = 3 · (3.48)

0

Giving random values to all inputs of the two fuzzy sets, we get the result shown
in Fig. 3.10 . In this case, the error according to equation ( 3.48 ) is 34.835.