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Fuzzy Control
• Fuzzy Sets
• Design of a Fuzzy Controller– Fuzzification of inputs: get_inputs()– Fuzzy Inference– Processing the Rules: find_rules()– Centroid Defuzzification– Output Defuzzification: find_output()– A Fuzzy Control Example --
• Floating Ping-Pong Ball
" So far as the laws of mathematics refer to reality, they are not certain,
And so far as they are certain, they do not refer to reality."
Albert EinsteinGeometrie und Erfahrung
Fuzzy Logic
Fuzzy Sets
The sentence on the other sideof the line is false
The sentence on the other sideof the line is false
Is this sentence true or false?
Probabiltiy vs. Fuzziness
Probability describes the uncertainty of an event occurrence.
Fuzziness describes event ambiguity.
Whether an event occurs is RANDOM.
To what degree it occurs is FUZZY.
Probability:There is a 50% chance of an applebeing in the refrigerator.
Fuzzy:There is a half an apple in therefrigerator.
Fuzzy Control
• Fuzzy Sets
• Design of a Fuzzy Controller– Fuzzification of inputs: get_inputs()– Fuzzy Inference– Processing the Rules: find_rules()– Centroid Defuzzification– Output Defuzzification: find_output()– A Fuzzy Control Example --
• Floating Ping-Pong Ball
INPUTS
OUTPUT
Map to Fuzzy Sets
FUZZY RULES
If A AND B then L
• • •
Defuzzification
get_inputs();
fire_rules();
find_output();
A Fuzzy Controller
Fuzzy Control
• Fuzzy Sets
• Design of a Fuzzy Controller– Fuzzification of inputs: get_inputs()– Fuzzy Inference– Processing the Rules: find_rules()– Centroid Defuzzification– Output Defuzzification: find_output()– A Fuzzy Control Example --
• Floating Ping-Pong Ball
Fuzzy Control
• Fuzzy Sets
• Design of a Fuzzy Controller– Fuzzification of inputs: get_inputs()– Fuzzy Inference– Processing the Rules: find_rules()– Centroid Defuzzification– Output Defuzzification: find_output()– A Fuzzy Control Example --
• Floating Ping-Pong Ball
Fuzzy Inference
if x1 is A1 and x2 is B1 then y is L1 rule 1
if x1 is A2 and x2 is B2 then y is L2 rule 2
Given the fact that
x1 is A' and x2 is B' fact
the problem is to find the conclusion
y is L' conclusion
Fuzzy Control
• Fuzzy Sets
• Design of a Fuzzy Controller– Fuzzification of inputs: get_inputs()– Fuzzy Inference– Processing the Rules: find_rules()– Centroid Defuzzification– Output Defuzzification: find_output()– A Fuzzy Control Example --
• Floating Ping-Pong Ball
Fuzzy Control
• Fuzzy Sets
• Design of a Fuzzy Controller– Fuzzification of inputs: get_inputs()– Fuzzy Inference– Processing the Rules: find_rules()– Centroid Defuzzification– Output Defuzzification: find_output()– A Fuzzy Control Example --
• Floating Ping-Pong Ball
14.2.4 Centroid Defuzzification
The last step in the fuzzy controller shown in Figure 14.7 is defuzzification. Thisinvolves finding the centroid of the net output fuzzy set L' shown in Figures 14.15 and14.16. Although we have used the MIN-MAX rule in the previous section we will beginby deriving the centroid equation for the sum rule shown in Figure 14.16. This willilluminate the assumptions made in deriving the defuzzification equation that we willactually use in the fuzzy controller.
Let Li(y) be the original output membership function associated with rule i where yis the output universe of discourse (see Figure 14.15.). After applying rule i thismembership function will be reduced to the value
mi(y) = wiLi(y) (14.1)
where wi is the minimum weight found by applying rule i. The sum of these reducedoutput membership functions over all rules is then given by
M(y) = i=1
Nmi(y) (14.2)
where N is the number of rules.
The crisp output value y0 is then given by the centroid of M(y) from the equation
y0 = yM(y)dy
M(y)dy (14.3)
Note that the centroid of membership function Li(y) is given by
ci = yLi(y)dy
Li(y)dy (14.4)
But
Ii = Li(y)dy (14.5)
is just the area of membership function Li(y). Substituting (14.5) into (14.4) we can write
yLi(y)dy = ciIi (14.6)
Using Eqs. (14.1) and (14.2) we can write the numerator of (14.3) as
yM(y)dy = y
i=1
NwiLi(y) dy
= i=1
N
ywiLi(y) dy
= i=1
N wiciIi (14.7)
where (14.6) was used in the last step.
Similarly, using (14.1) and (14.2) the denominator of (14.3) can be written as
M(y)dy =
i=1
NwiLi(y) dy
= i=1
N
wiLi(y) dy
= i=1
N wiIi (14.8)
where (14.5) was used in the last step. Substituting (14.7) and (14.8) into (14.3) we canwrite the crisp output of the fuzzy controller as
y0 =
i=1
N wiciIi
i=1
N wiIi
(14.9)
Eq. (14.9) says that we can compute the output centroid from the centroids, ci, of theindividual output membership functions.
Note in Eq. (14.9) the summation is over all N rules. But the number of outputmembership functions, Q, will, in general, be less than the number of rules, N. This meansthat in the sums in Eq. (14.9) there will be many terms that will have the same values of ciand Ii. For example, suppose that rules 2, 3, and 4 in the sum all have the outputmembership function Lk as the consequent. This means that in the sum
w2c2I2 + w3c3I3 + w4c4I4
the values ci and Ii are the same values ck and Ik because they are just the centroid andarea of the kth output membership function. These three terms would then contribute thevalue
(w2 + w3 + w4)ckIk = WkckIk
to the sum, where
Wk = (w2 + w3 + w4)
is the sum of all weights from rules whose consequent is output membership function Lk.
This means that the equation for the output value, y0, given by (14.9) can be rewritten as
y0 =
k=1
Q
WkckIk
k=1
Q
WkIk
(14.10)
If the area of all output membership functions, Ik are equal, then Eq. (14.10) reduces to
y0 =
k=1
Q
Wkck
k=1
Q
Wk
(14.11)
Eqs. (14.10) and (14.11) show that the output crisp value of a fuzzy controller can becomputed by summing over only the number of output membership functions rather thanover all fuzzy rules. Also, if we use Eq. (14.11) to compute the output crisp value, thenwe need to specify only the centroids, ck, of the output fuzzy membership functions. Thisis equivalent to assuming singleton fuzzy sets for the output.
We will always use singleton fuzzy sets for the output represented by thecentroids, ck. We will also use the MIN-MAX inference rule described in the previoussection. It should be clear from Figure 14.16 that in this case the centroid y0 will still begiven by Eq. (14.11) where Wk is now the output array, Out(k), shown in Figure 14.18and computed by the word firerules given in Figure 14.19.
Fuzzy Control
• Fuzzy Sets
• Design of a Fuzzy Controller– Fuzzification of inputs: get_inputs()– Fuzzy Inference– Processing the Rules: find_rules()– Centroid Defuzzification– Output Defuzzification: find_output()– A Fuzzy Control Example --
• Floating Ping-Pong Ball
Fuzzy Control
• Fuzzy Sets
• Design of a Fuzzy Controller– Fuzzification of inputs: get_inputs()– Fuzzy Inference– Processing the Rules: find_rules()– Centroid Defuzzification– Output Defuzzification: find_output()– A Fuzzy Control Example --
• Floating Ping-Pong Ball