lecture 3 fuzzy reasoning 1. inference engine core of every fuzzy controller the computational...
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Lecture 3
Fuzzy Reasoning
1
inference enginecore of every fuzzy controller
the computational mechanism with which decisions can be inferred even though the knowledge may be incomplete.
fuzzy inference engines perform an exhaustive search of the rules in the knowledge base to determine the degree of fit for each rule for a given set of causes.
only one unique rule contributes to the final decision
fuzzy propositional implication defines the relationship between the linguistic
variables of a fuzzy controller.
Cartesian product
Using the conjunctive operator (min)
algebraic product the Cartesian product
fuzzy propositional implication example
X Y
321
4321
yyyY
xxxxX
),(),(),(
),(),(),(
),(),(),(
),(),(),(
342414
332313
322212
312111
yxyxyx
yxyxyx
yxyxyx
yxyxyx
YX
Two fuzzy sets ，
Cartesian product ：
fuzzy propositional implication example
fuzzy propositional implication example
relational matrix
fuzzy propositional implication example
relational matrix
The Cartesian product based on the conjunctive operator min is much simpler and more efficient to implement computationally and is therefore generally preferred in fuzzy controller inference engines. Most commercially available fuzzy controllers in fact use this method.
3.1 The Fuzzy Algorithm Assume that
where ψ is some implication operator and
3.1 The Fuzzy Algorithm the membership function for N rules in a fuzzy algorithm is given by
3.2 Fuzzy Reasoning two fuzzy implication inference rules
1. Generalized Modus Ponens (or GMP) 广义取式（肯定前提）假言推理法
简称为广义前向推理法
For the special case
Α’=Α and Β’=Β then GMP reduces to Modus Ponens.
肯定前提的假言推理
use in all fuzzy controllers.
3.2 Fuzzy Reasoning two fuzzy implication inference rules
2. Generalized Modus Tollens (or GMT)
For the special case
广义拒式（否定结论）假言推理法
广义后向推理法
application in expert systems
then GMT reduces to Modus Tollens
否定结论的假言推理
3.2 Fuzzy Reasoning
Boolean implication
Lukasiewicz implication
Zadeh implication
Mamdani implication
Larsen implication
3.2 Fuzzy Reasoning Boolean implication
For the case of Ν rules,
3.2 Fuzzy Reasoning
Lukasiewicz implication
For the case of Ν rules,
Bounded sum
3.2 Fuzzy Reasoning
Zadeh implication
difficult to apply in practice
3.2 Fuzzy Reasoning
Mamdani implication
For a fuzzy algorithm comprising N rules
3.2 Fuzzy Reasoning Larsen implication
For a fuzzy algorithm comprising N rules
3.3 The Compositional Rules of Inference
Given, for instance
The composition of these two rules into one can be expressed as:
rule composition
3.2 The Compositional Rules of Inference the membership function of the resultant compositional rule of inference
Mamdani implication
Larsen implication
3.3 The Compositional Rules of Inference
the procedure for determining the consequent (or effect), given the antecedent (or cause). Given
and the compositional rule of inference:
if the antecedent is
consequent ？？
3.3 The Compositional Rules of Inference the max-min operators
max-product operators:
3.3 The Compositional Rules of Inference
example
Slow
Fast
determine the outcome if A = ‘slightly Slow’ for which there no rule exists
3.3 The Compositional Rules of Inference The first step is to compute the Cartesian product and using the min operator
3.3 The Compositional Rules of Inference
The second step using the fuzzy compositional inference rule:
the Mamdani compositional rule
3.3 The Compositional Rules of Inference
The final operation
3.3 The Compositional Rules of Inference using the max-product rule of compositional inference:
The first step
3.3 The Compositional Rules of Inference using the max-product rule of compositional inference:
The second step
3.3 The Compositional Rules of Inference using the max-product rule of compositional inference:
The maximum elements of each column are therefore:
the Mamdani compositional rule
3.3 The Compositional Rules of Inference
43212
43211
16.02.00~
04.00.17.0~
aaaaA
aaaaA
3212
3211
14.01.0~
06.00.1~
bbbB
bbbB
43213
2.00.16.03.0~
aaaaA
3
~B
Given If inputs：
then outputs：
if input
Then output ?
example
3.3 The Compositional Rules of Inference
)~~
(~
111 BAR 0.06.00.1
0.0
4.0
0.1
7.0
0.00.00.0
0.04.04.0
0.06.00.1
0.06.07.0
)~~
(~
222 BAR 0.14.01.0
0.1
6.0
2.0
0.0
0.14.01.0
6.04.01.0
2.02.01.0
0.00.00.0
0.14.01.0
6.04.04.0
2.06.00.1
0.06.07.0
0.14.01.0
6.04.01.0
2.02.01.0
0.00.00.0
0.00.00.0
0.04.04.0
0.06.00.1
0.06.07.0
~~~21 RRR
3.3 The Compositional Rules of Inference
0.14.01.0
6.04.04.0
2.06.00.1
0.06.07.0
2.00.16.03.0~~~
33 RAB
321
6.06.06.06.06.06.0
2.06.02.00.02.04.06.03.01.04.06.03.0
])0.12.0()6.00.1()2.06.0()0.03.0(
),4.02.0()4.00.1()6.06.0()6.03.0(
),1.02.0()4.00.1()0.16.0()7.03.0(
bbb
The unit set
