142 CHAPTER 4. FUZZY CONTROL
each having value 1 on one of the pieces of the mapping and 0 on the other.
-3 -2 -1^01 x 2 3A 1 (x)solid line,A 2 (x)dashed line
and rules
R 1 :IfxisA 1 thenf 1 (x)=1+x
R 2 :IfxisA 2 thenf 2 (x)=2+x
Now
A 1 (x)f 1 (x)+A 2 (x)f 2 (x)=
2+2x if x≤− 1
1
2 +3
2 x+x(^2) if − 1 ≤x≤ 1
−1+4x if 1 ≤x
and
A 1 (x)+A 2 (x)=1
giving the following plot:
-20
-10
10
-4 -2 2 x 4
y=
P 2
j=1Aj(x)fj(xj).P
2
j=1Aj(x)These models can also be used as a nonlinear interpolator between linear
systems. Sugeno proposes rules of the form
Ri:Ifz 1 isCi 1 and ... andzpisCipthenx ̇i(t)=Ai(x(t)) +Biu((t)) (4.5)fori =1, 2 ,...,r. In this rule, Sugeno uses a canonical realization of the
system known in classical control theory as the ìcontroller canonical form.î
Here, x(t)=(x 1 (t),...,xn(t))is then-dimensional state vector,u(t)=
(u 1 (t),...,um(t))is them-dimensional input vector,Ai,Bi,i =1, 2 ,...,r,