A First Course in FUZZY and NEURAL CONTROL

136 CHAPTER 4. FUZZY CONTROL

The look-up table

θ\θ^0 NB NM NS PS PM PB

NB NB NB NB NM NS PS

NM NB NB NM NS PS PS

NS NB NM NS PS PS PM

PS NM NS NS PS PM PB

PM NS NS PS PM PB PB

PB NS PS PM PB PB PB

summarizes 36 possible rules in a compact form.
The input(θ,θ^0 )to each ruleRj will result in a fuzzy subset ofU.This
fuzzy subset is often taken to be the minimum:

φj(u)=min{Aj(θ),Bj(θ^0 ),Cj(u)}

The fusion of rules, via the maximum, produces a fuzzy subset ofUrepresenting
a control action:

Ψ(u)=max{φj(u):j=1, 2 ,...,k}

If, for example, the measurements areθ=− 8 ◦andθ^0 =2◦/sthen the fuzzy
sets depicted above give the values

A 2 (−8) = 0. 17

A 3 (−8) = 0. 88

B 4 (2) = 0. 6

B 5 (2) = 0. 82

and all otherAiandBiare zero at this point. Thus the only rules pertinent for
this input are the four
θ\θ^0 NM NS
PS NS NS
PM NS PS

as all others give the value zero for this input. These four rules are said to ìfireî
for this input.
Combining these rules,we have the fuzzy set

Ψ(u)=[A 2 (−8)∧B 4 (2)∧C 3 (u)]∨[A 2 (−8)∧B 5 (2)∧C 3 (u)]

∨[A 3 (−8)∧B 4 (2)∧C 3 (u)]∨[A 3 (−8)∧B 5 (2)∧C 4 (u)]

=[0. 17 ∧ 0. 6 ∧C 3 (u)]∨[0. 17 ∧ 0. 82 ∧C 3 (u)]

∨[0. 88 ∧ 0. 6 ∧C 3 (u)]∨[0. 88 ∧ 0. 82 ∧C 4 (u)]

=[0. 6 ∧C 3 (u)]∨[0. 82 ∧C 4 (u)]

and the aggregated output is the following fuzzy subset ofU: