lecture 3 fuzzy reasoning 1. inference engine core of every fuzzy controller the computational...

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