A First Course in FUZZY and NEURAL CONTROL

140 CHAPTER 4. FUZZY CONTROL

Cjx,j=1,...,N. In the Mamdani and Larsen methods, this is done by taking
the maximum. In the Mamdani method, for eachx∈X=X 1 ◊∑∑∑◊Xn,this
gives the fuzzy subset

Cx(u)=C(u|x)= max

j=1,...,N

μ

min

i=1,...,n

{Aji(xi),Bj(u)}

∂

(4.3)

ofU. Equation 4.3 is calledMamdani synthesis.
In the Larsen method, for eachx∈X, this gives the fuzzy subset

Cx(u)=C(u|x)= max

j=1,...,N

μμ

min

i=1,...,n

{Aji(xi)}∑Bj(u)

∂∂

(4.4)

ofU. Equation 4.4 is calledLarsen synthesis.
The overall output is, for eachx, a fuzzy subsetCxofU.Whatisthe
meaning ofCx? When the inputxis given, each control actionuis compatible
with degreeCx(u). We need a single numerical outputu∗=u∗(x)for the
control law, and we need to get this from the fuzzy subsetCx.Inotherwords,we
need todefuzzifythe fuzzy subsetCx. In the Mamdani and Larsen approaches,
we defuzzify Cxby using the center of area (center of gravity or centroid)
procedure, namely

u∗(x)=

R

UuC

x(u)du

R

UC

x(u)du
Of course, there are many different ways to defuzzifyCx. See Section 3.9
for a discussion of defuzzification methods. The complexity, or speed, of the
computation will be affected by the method of defuzzification. The question of
how to choose a defuzzification method should be settled by the performance of
the induced controller.

4.2.2 Model-basedfuzzycontrol..................

Another useful approach to fuzzy control is a functional form of fuzzy system,
due to Takagi and Sugeno, in which defuzzification is not needed. This es-
sentially model-based fuzzy control method can be used when it is possible to
describe the dynamics of a plant locally in approximate terms.
Therulebaseinthiscaseisoftheform
Rj:Ifx 1 isAj 1 and ... andxnisAjnthenuj=fj(x 1 ,x 2 ,...,xn)

forj=1, 2 ,...,r,wherethexiare observed values of the input variables, thefj
are functions, and theAijform a fuzzy partition of the input space. Taking the
product of theAjiís we can express the rules in the simpler form

Rj:ìIfxisAjthenuj=fj(x 1 ,x 2 ,...,xn)î

Thefiring degreeof each ruleRj isAj(x), and the overall output control
value is taken to be

u(x)=

Xr

j=1

Aj(x)fj(x)

, r

X

j=1

Aj(x)