Now, let's consider a particular scenario, with 0 = -1.5, and de = 2. STEP 1: Fuzzification STEP 3: Aggregation negative zero positive 21012 Hpos(0)=0 Uzero(0) = 0.25 Uneg(0) = 0.75 positive negative e -5 0 5 Hpos(de) = 0.4 STEP 2: Rule Evaluation We evaluate every rule in the FAM: Uzero(de) = 0.6 Uneg(de) = 0 IF 0-pos (0) IF e-pos (0) IF 0 pos (0) AND de pos (.4) AND de-zero (.6) AND de neg (0) force BP (0) force = P (0) -force-Z (0) 1F0-zero (.25) IF 0-zero 1.25 IF 0-zero (.25) AND de pos (.4) AND de-zero (16) AND de neg (0) -force-P (.25) -force-Z (.25) force = N (0) IF 9-neg (.75) IF 0-neg (.75) IFA-neg (.75) AND de pos (14) AND de-zero (-6) AND de neg (0) force = Z (-4) force = N 1.6) force = BN (0) big neg 20 pos big pos. -16-12-8-40 48 12 16 force -16 +16 This is a fuzzy result (obviously!) STEP 4: Defuzzification Find the centroid of the above figure. Or, at what point along the x-axis would such an object "balance"? This can be calculated exactly using integration by parts (not easy!). Or it can be estimated satisfactorily using a Riemann Sum, such as: (-16*0)+(-12*.5)+(8*. 6) + (4*. 6) + (0* .4) + (4* . 4) + (8*.25)+(12*.25)+(16*0) Note that rule evaluation is done using the fuzzy definition of AND: X AND Y = MIN ((X). H(Y)) The circled rules have a non-zero output membership - they "fire". But they are contradictory: force P (.25) force force (.25) ??? (.4) ??? How do we resolve this? force N (.6) + .4 + .25 +,25 + 0 Which equals -2.2 in this case. This is a crisp result. The controller thus sends a small negative force of -2.2 to the cart. This is reasonable, since the inputs indicated that the pole was leaning heavily to the left, but was also moving to the right at a modest speed. The correction that is already underway probably isn't quite enough to straighten the pole completely, so a slight additional correction by moving the cart a bit to the left makes sense.