Fuzzy Control - ctrl.cinvestav.mxyuw/file/sa14.pdf · De–nition: fuzzy set De–nition (fuzzy...
Transcript of Fuzzy Control - ctrl.cinvestav.mxyuw/file/sa14.pdf · De–nition: fuzzy set De–nition (fuzzy...
Fuzzy Control
Wen Yu
Departamento de Control AutomáticoCINVESTAV-IPN
A.P. 14-740, Av.IPN 2508, México D.F., 07360, México
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History
Lofti A. Zadeh. Fuzzy sets, Inf. Control 8, 338-353, 1965
US and certain parts of Europe ignored it. Mathematic?
fuzzy logic was excepted with open arms in Japan, China and mostOriental countries.
The world�s largest number of fuzzy researchers are in China withover 10,000 scientists.
The popularity of fuzzy logic in the Orient re�ects the fact thatOriental thinking more easily accepts the concept of "fuzziness".
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De�nition: fuzzy set
De�nition(fuzzy set) Let X be a nonempty set. A fuzzy set A in X is characterizedby its membership function µA : X ! [0, 1] and µA is interpreted as thedegree of membership of element x in fuzzy set A for each x 2 X .
Crisp set assigns value {0,1} to members in X
Fuzzy set assigns value [0,1] to members in X
Frequently we will write A(x) instead of µA(x).
A fuzzy set is totally characterized by a membership function (MF).
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Examples
ExampleThe membership function of the fuzzy set of real numbers �close to 1�, iscan be de�ned
A(t) = e� (t�1)2
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Examples
ExampleSomebody is "tall"
MFs
Heights5’10’’
.5
.8
.1
“tall”in Asia
“tall”in the US
“tall”in NBA
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Triangular membership function
µA = A (x) =
8<:1� a�x
α α� a � x � a1� x�c
β a � x � a+ β
0 otherwise
supp (A) = [a� α, a+ β] , [A]γ = [a� (1� γ) α, a+ (1� γ) β],γ 2 [0, 1].
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Trapezoidal membership function
µA = A (x) =
8>><>>:1� a�x
α α� a � x � a1 a � x � b
1� x�cβ b � x � b+ β
0 otherwise
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Gaussian membership function
4 2 0 2 4
0.5
1.0
x
y
µA (x) = e� (x�1)2
4 = e�(x�a)2
σ2
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Bell membership function
µA (x) =1
1+�� x�cb
��2b
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Advantages of Fuzzy Systems
Conceptually straightforward to understand.
The mathematical concepts behind fuzzy reasoning are simple.
Flexible:You can modify and add on fuzzy rules without starting fromscratch.
Tolerant of imprecise data. Everything is imprecise if you look closelyenough, but more than that, many things are imprecise even at �rstglance
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Advantages of Fuzzy Systems
Can model nonlinear functions of arbitrary complexity.
Can create a fuzzy system to match any set of input/output data.
Can be built on top of the experience of experts.
close to natural language.
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Disadvantages of Fuzzy Systems
Creating the fuzzy rules base can be troublesome
It is di¢ cult to create the fuzzy rules base from input/output data ifno fuzzy rule extraction technique is used
Accuracy of the inference depends directly on the number of fuzzyrules used in complex problem
Increasing input variables and fuzzy membership fns used will increasethe number of fuzzy rules exponentially.
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Fuzzy IF-THEN rules: product rule
IF <fuzzy proposition>, THEN <fuzzy proposition>
Atomic:x is A
Compound (AND, OR, NOT)
x is A AND x is not B
Compound fuzzy proposition is fuzzy relations
AND ! intersections
OR ! unions
NOT ! complents
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Fuzzy IF-THEN rules
Example
F = (x is A and x is not B) or (x is C )
its membership function is
µF = s ft [µA, c (µB )] , µC g
for exampleµF = µA � (1� µB ) + µC
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Clsssical IF-THEN operation
For classical proposition
If pressure is high THEN volume is small
Classical method (production rule)
(IF p THEN q)) (p ! q)
By logic table (p ! q) = p _ q or ((p ^ q) _ p)
p q p ! q1 1 10 1 10 0 11 0 0
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Fuzzy IF-THEN operation
IF p THEN q
how to operate µp and µq1 Larsen
µQM = µpµq2 Mamdani
µQM = min�
µp , µq
�or µQM = µpµq
3 ×ukasiewiczµQM = min
�1� µp , µq
�4 Kleene-Dienes
µQM = max�1� µp , µq
�5 Godel
µQM =
�1 µp � µq
µq otherwise
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Fuzzy rule base
De�nitiona fuzzy rule base consists of a set of fuzzy IF-THEN rules
Rl : IF x1 is Al1 and � � � and xn is Aln, THEN y is B l
where Ali and Bl are fuzzy sets in Ui and V , respectively, and xi and y are
linguistic variable, l = 1, � � � ,M. It is also called canonical fuzzy IF-THENrules
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Fuzzy rule base: Lemma
LemmaThe canonical fuzzy IF-THEN rules include
1 Partial rules: m < n
Rl : IF x1 is Al1 and � � � and xm is Alm , THEN y is B l
2 Or rulesRl : IF x1 is Al1 or x2 is A
l2, THEN y is B
l
3 Single fuzzy statement : Rl : y is B l
4 Gradual rules
Rl : The samller the x, the bigger the y
5 No fuzzy rules: conventional product rules
Rl : IF x1 is Al1 and � � � and xn is Aln, THEN y is B l(CINVESTAV-IPN) Fuzzy Control March 9, 2020 18 / 43
Fuzzy rule base: proof
1 it is equivalent to
Rl : IF � � � xm+1 is I and � � � and xn is I , THEN y is B l
where I si fuzzy set with µI = 12 it is equivalent to two rules
Rl :IF x1 is Al1 THEN y is B
l
IF x2 is Al2, THEN y is Bl
3 it is equivalent to
Rl : IF x1 is I and � � � and xn is I , THEN y is B l
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Fuzzy rule base: proof
4 it is equivalent to
Rl : IF x is S THEN y is B
where µS =1
1+e5�(x+2), µB =
11+e�5(y�2)
52.502.55
1
0.75
0.5
0.25
0
x
y
x
y
52.502.55
1
0.75
0.5
0.25
0
x
y
x
y
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Fuzzy rule base: proof
5 if Ali and Bl can only take values 1 or 0, they become non-fuzzy rules
Rl : IF x1 is Al1 and � � � and xn is Aln, THEN y is B l
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Property: complete
De�nitionA set of fuzzy IF-THEN rules is complete if for any x 2 U, there exists atleast one rule such that µAli
6= 0
Example2-input 1-output fuzzy system, 3 membership function for x1, 2membership function for x2, in order for a fuzzy rule base to be complete,it must contain 3� 2 = 6 rules
R1 : IF x1 is S1 and x2 is S2, THEN y is B1
R2 : IF x1 is S1 and x2 is L2, THEN y is B2
R3 : IF x1 is M1 and x2 is S2, THEN y is B3
R4 : IF x1 is M1 and x2 is L2, THEN y is B4
R5 : IF x1 is L1 and x2 is S2, THEN y is B5
R6 : IF x1 is L1 and x2 is L2, THEN y is B6
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Property: complete
x� = (1, 0) when R2 is missed
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Property: consistent, continuous
De�nitionA set of fuzzy IF-THEN rules is consistent if there are no rules with sameIF parts, but di¤erent THEN parts
De�nitionA set of fuzzy IF-THEN rules is continuous if there do not exist suchneighboring rules whose THEN part fuzzy sets have empty intersection(smooth)
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Compositional rule of inference
If we know y = f (x) , x = a, then b = f (a)If we know "IF x is A THEN y is B"
µQ = µAµB
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Compositional rule of inference
! If x is A0, what is B 0
µA\Q = t�µA, µQ
�the projection of A\Q on V is µB 0
µB 0 = supx2U
t�µA, µQ
�
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Fuzzy inference engine
Combine the fuzzy IF-THEN rules into a mapping from fuzzy set A in Uto a fuzzy set B in V
Composition based inference
Individual-rule based inference
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Fuzzy inference engine: Composition based inference
All rules are combined into a single fuzzy IF-THEN rule
Union: the rules are independentIntersection: the rules are strongly coupled (strange)
Rl : IF x1 is Al1 and � � � and xn is Aln, THEN y is B l
1 For rule l , the "and" has fuzzy relation
µAl1�����Aln = µAl1� . . . � µAln
where "�" represents t-norm operator2 implication "!", Mandani µAl1�����AlnµB l3 Union
QM = [Ml=1R l , µQM = µR 1+ . . . +µR l
where "+" represents s-norm operator4 For any input C , the fuzzy inference engine give output D as(Mandani)
µD = supx2U
thµC , µQM
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Fuzzy inference engine: Individual-rule based inference
Each rule determines an output, step 1 and setp 2 are the same
3 Generalized modus ponens
µD l = supx2U
t [µC , µR l ]
where "+" represents s-norm operator
4 For any input C , the fuzzy inference engine give output D as(Mandani)
µD = µD 1+ . . . +µDM
Product inference engine
µD = maxl
(supx2U
"(µC )
n
∏i=1
µAli
!(µB l )
#)
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Product inference engine
LemmaIf the fuzzy set C is a fuzzy singleton,
µC =
�1 x = x�
0 otherwise
then
µD = maxl
( n
∏i=1
µAli
!(µB l )
)
The most di¢ cult supx2U dispears
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Fuzzi�ers
From real valued point x� 2 U to a fuzzy set A in U
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Fuzzi�ers
Singleton
µA =
�1 x = x�
0 otherwise
Gaussian
µA = e(x�x�)2
σ2
Triangular
µA =
(1� jx�x �j
α jx � x�j � a0 otherwise
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Fuzzi�ers: summary
The singleton fuzzi�er greatly simpli�es the computation
If the membership function are Gaussian or traiangular, Gaussian ortraiangular fuzzi�er alos simplify the computation
Gaussian or traiangular fuzzi�er can suppress noise in the input
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Defuzzi�ers
From a fuzzy set B in V to crisp point y � 2 V (best represents the fuzzyset B), similar to the mean value of a random variable
Center of gravity
y � =
RyµBdyRµBdy
Center average
y � =∑Ml=1 y
lwl∑Ml=1 wl
Maximum, y � is the point at which µB archieves its maximum value
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Example
R1 : IF x1 is A1 and x2 is A2, THEN y is A1R2 : IF x1 is A2 and x2 is A1, THEN y is A2
where
µA1 =
�1� jx j jx j � 10 otherwise
µA2 =
�1� jx � 1j 0 � x � 2
0 otherwise
If (x�1 , x�2 ) = (0.3, 0.6) , y?
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Example
We use singleton
µB 0 = maxl
( n
∏i=1
µAli
!(µB l )
)= max
nµA1 (0.3) µA2 (0.6) µA1 (y) , µA2 (0.3) µA1 (0.6) µA2 (y)
o= max
n0.42µA1 (y) , 0.12µA2 (y)
oCenter average
y � =0 � 0.42+ 1 � 0.120.42+ 0.12
= 0.22
Center of gravity: 0.187Maximum: 0.5
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Example
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Fuzzy system as nonlinear mappings
LemmaFuzzy set has the form of
Rl : IF x1 is Al1 and � � � and xn is Aln, THEN y is B l
the center of B l is y l , then with product inference engine, singletonfuzzi�er, and center average defuzzi�er, the fuzzy system has the followingform
y =
∑Ml=1 y
l
n
∏i=1
µAli
!
∑Ml=1
n
∏i=1
µAli
!
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Fuzzy system as nonlinear mappings: poof
Product inference engine with singleton fuzzi�er
µB 0 =Mmaxl
( n
∏i=1
µAli
!(µB l )
)
The center of B l is y l , for x�i the height of k�th fuzzy set isn
∏i=1
µAli(x�) µB l
�y l�=
n
∏i=1
µAli(x�), the center average defuzzi�er
y =∑Ml=1 y
lw l
∑Ml=1 w l
It islingustic rules ! nonlinear mapping
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Fuzzy system: example
Fuzzy rules
R j : IF x1 is Aj1 AND � � � xn is Ajn THEN y is B j
(1) Fuzzi�er (singleton)
µ =
�1 if x = x�
0 otherwise
so �xi is A
ji
�! µAji
(x�i )
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Fuzzy system: example
(2) Fuzzy operation: (x1 is Aj1 AND� � � xn is A
jn)
n
∏i
µAji(x�i )
Mamdani implications (product inference engine)
n
∏i
µAji(x�i ) µB j (y)
l�rules p is "OR"
µp = maxl
"n
∏i
µAji(x�i ) µBj (y)
#
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Fuzzy system: example
(3)Defuzzi�er, suppose y j is center of Bj , the height of the lthe fuzzy set
isn
∏i
µAji(x�i ) µB j
�y j�, and µB j
�y j�= 1 (normal fuzzy set), so center
average is
y � =
m
∑j=1y j
n
∏i
µAji(x�i )
!m
∑j=1
n
∏i
µAji(x�i )
!
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Fuzzy system as universal approximators
TheoremThe fuzzy system with product inference engine, singleton fuzzi�er, andcenter average defuzzi�er, and Gaussian membership functions areuniversal approximators
supx2U
jf (x)� g (x)j < ε
for any real countinuous function g (x) , any ε > 0
f (x) =
∑Ml=1 y
l
n
∏i=1ali exp
��� xi�x li
σli
�2�!
∑Ml=1
n
∏i=1ali exp
��� xi�x li
σli
�2�!
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