A First Course in FUZZY and NEURAL CONTROL

4.2. MAIN APPROACHES TO FUZZY CONTROL 143

are state and input matrices, respectively, andz(t)=(z 1 (t),...,zp(t))is the
p-dimensional input to the fuzzy system. The output is

x ̇(t)=

Pr

i=1[AixP(t)+Biu(t)]τi(z(t))

r

i=1τi(z(t))

=

Pr

Pi=1rAiτi(z(t))

i=1τi(z(t))

x(t)+

Pr

Pi=1r Biτi(z(t))

i=1τi(z(t))

u(t)

where
τi(z(t)) = 4 pk=1Cik(zk(t))

for some appropriate t-norm 4 .Inthespecialcaser=1, the antecedent is a
standard linear system:
x ̇(t)=Ax(t)+Bu(t)

In the general case, such a fuzzy system can be thought of as a nonlinear inter-
polator betweenrlinear systems.

Example 4.2Suppose thatz(t)=x(t),p=n=m=1,andr=2.Takethe
two fuzzy sets

C 1 (z)=







1 if z≤− 1

1 −z

2 if −^1 ≤z≤^1

0 if 1 ≤z

and

C 2 (z)=







0 if z≤ 0

1+z

2 if −^1 ≤z≤^1

1 if 1 ≤z

-3 -2 -1^01 x 2 3

C 1 (z)solid line,C 2 (z)dashed line
and rules

R 1 :IfzisC 1 thenx ̇ 1 =−x 1 +2u 1

R 2 :IfzisC 2 thenx ̇ 2 =− 2 x 2 +u 2

soA 1 =− 1 ,B 1 =2,A 2 =− 2 ,andB 2 =1.Sinceτ 1 (z)+τ 1 (z)=C 1 (z)+
C 2 (z)=1, the output of the system is

x ̇=(−C 1 (z(t))−C 2 (z(t)))x+(2C 1 (z(t)) +C 2 (z(t)))u

Thus, we have

x ̇=







−x+2u if z≤− 1

−x+

° 3 −z

2

¢

u if − 1 ≤z≤ 1

− 2 x+u if 1 ≤z