Fuzzy Control

– Configuration – Design choices– Takagi-Sugeno controller

Direct Control

Deviations Actions OutputsRef

Controller

End-user

Inferenceengine

Rulebase

Plant

Building Blocks

Fuzzy controller

Inferenceengine

Rulebase Defuzzi

-ficationPostpro-cessing

Fuzzi-fication

Prepro-cessing

Nonlinear Input Scaling

-5 0 5

-100

-50

0

50

100

measured input

sca

led

inp

ut

If-Then Rule Base

1. If error is Neg and change in error is Neg then output is NB

2. If error is Neg and change in error is Zero then output is NM

3. If error is Neg and change in error is Pos then output is Zero

4. If error is Zero and change in error is Neg then output is NM

5. If error is Zero and change in error is Zero then output is Zero

6. If error is Zero and change in error is Pos then output is PM

7. If error is Pos and change in error is Neg then output is Zero

8. If error is Pos and change in error is Zero then output is PM

9. If error is Pos and change in error is Pos then output is PB

Relational Rule Format

Error Change in error Control

Pos Pos PB

Pos Zero PM

Pos Neg Zero

Zero Pos PM

Zero Zero Zero

Zero Neg NM

Neg Pos Zero

Neg Zero NM

Neg Neg NB

Tabular Rule Format

Change in error

Neg Zero Pos

Neg NB NM Zero

Error Zero NM Zero PM

Pos Zero PM PB

Connectives

)(),(max

)(),(min

xxBA

xxBA

BA

BA

)()()()(

)()(

xxxxBA

xxBA

BABA

BA

minimum

maximum

algebraic product

probabilistic sum

FLS I/O Families

-1 -0.5 0 0.5 10

0.5

1

Input

Mem

bers

hip

-1 -0.5 0 0.5 10

0.5

1

Output

Mem

bers

hip

NegZero

Pos

Examples Of Primary Sets

-100 0 1000

0.5

1

(a)-100 0 1000

0.5

1

(d)-100 0 1000

0.5

1

(g)-100 0 1000

0.5

1

(j)-100 0 1000

0.5

1

(m)

-100 0 1000

0.5

1

(b)-100 0 1000

0.5

1

(e)-100 0 1000

0.5

1

(h)-100 0 1000

0.5

1

(k)-100 0 1000

0.5

1

(n)

-100 0 1000

0.5

1

(c)-100 0 1000

0.5

1

(f)-100 0 1000

0.5

1

(i)-100 0 1000

0.5

1

(l)-100 0 1000

0.5

1

(o)

Inference And Terminology

AND

Aggregation

Accumulation

Defuzzification

Activation

4

5

Defuzzification

0 50 100

0

0.5

1

RM

BOACO

G

MOM

LM

Rule Based Controllers

1. If error is Neg then control is Neg

2. If error is Zero then control is Zero

3. If error is Pos then control is Pos

Mamdani Inference

-100 0 1000

0.5

1error

-100 0 1000

0.5

1control

-100 0 1000

0.5

1

-100 0 1000

0.5

1

-100 0 1000

0.5

1

-100 0 1000

0.5

1

-100 0 1000

0.5

1

u = -25.7

FLS Inference

-100 0 1000

0.5

1error

-100 0 1000

0.5

1control

-100 0 1000

0.5

1

-100 0 1000

0.5

1

-100 0 1000

0.5

1

-100 0 1000

0.5

1

-100 0 1000

0.5

1

u = -29.7

Sugeno Inference

-100 0 1000

0.5

1error

-100 0 1000

0.5

1control

-100 0 1000

0.5

1

-100 0 1000

0.5

1

-100 0 1000

0.5

1

-100 0 1000

0.5

1

-100 0 1000

0.5

1

u = -36.3

Singleton Output

1. If error is Pos then control is 10

2. If error is Zero then control is 0

3. If error is Neg then control is -10

First Order Output

1. If error is Pos then control is a2*error + b2

2. If error is Neg then control is a1*error + b1

Interpolation (Takagi-Sugeno)

0 50 1000

50

100

150

(a)

outp

ut

1

2

0 50 1000

0.5

1

(b)

mem

bers

hip

Rule Base To Table

Look-Up Table

Change in error

-100 -50 0 50 100

Error

100 0 40 100 100 200

50 -40 0 61 121 160

0 -100 -61 0 61 100

-50 -100 -121 -61 0 40

-100 -200 -160 -100 -40 0

Control Surface

-100

0

100

-100

0

100-200

0

200

ECE

u

-100 -50 0 50 1000

0.2

0.4

0.6

0.8

1

input family

me

mb

ers

hip

Linear Controller

-100

0

100

-100

0

100-200

0

200

ECE

u

-100 -50 0 50 1000

0.2

0.4

0.6

0.8

1

input family

me

mb

ers

hip

Linear Rule Base

Conditions For Linearity

• Triangular sets, crossing at = 0.5• Rules: complete -combination• Define as *• Use conclusion singletons, positioned at sum of input

peak positions• Use sum-accumulation and COGS defuzzification

Simplification of 4 rules

1. If error is Neg and change in error is Neg then control is NB3. If error is Neg and change in error is Pos then control is Zero7. If error is Pos and change in error is Neg then control is Zero9. If error is Pos and change in error is Pos then control is PB

PBPosPos CEEu )1(

is

Simplification of 9 rules1. If error is Neg and change in error is Neg then output is NB2. If error is Neg and change in error is Zero then output is NM3. If error is Neg and change in error is Pos then output is Zero4. If error is Zero and change in error is Neg then output is NM5. If error is Zero and change in error is Zero then output is Zero6. If error is Zero and change in error is Pos then output is PM7. If error is Pos and change in error is Neg then output is Zero8. If error is Pos and change in error is Zero then output is PM9. If error is Pos and change in error is Pos then output is PB

is

PBNegPosNegPos CECEEEu 2

1

Summary Of Choices

• Rule-base related choices: # of inputs and outputs, rules, universes, continuous or discrete, # of membership functions, their overlap and width, singleton conclusions

• Inference engine choices: Connectives, modifiers, activation operation, aggregation operation, accumulation operation

• Defuzzification method: COG, COGS, BOA, MOM, LM, RM

• Pre- and postprocessing: Scaling, quantization, sampling time