Fuzzy Control

Jan Jantzenjj@inference.dk

www.inference.dk2013

2

Summary

• Configuration of controller• Design choices• The Takagi-Sugeno controller

3

Direct Control Configuration

Deviations Actions OutputsRef

Controller

End-user

Inferenceengine

Rulebase

Plant

Could be a multi-input-multi-output controller

4

Example: Tank Level ControlV1

Control valve

Inlet stream

Outlet stream

Tank

5

Control Rules

1. If the level is low, then open V1

2. If the level is high, then close V1

Actually, a single rule might be sufficient. Which?

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High Level

This requires a level sensor, which is able to measure how full the tank is. Ultrasound, for instance.

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High And Low Levels

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Choice of Rule Base

1. 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

Two inputs with three values each (Neg, Zero, Pos) results in nine rules.Two inputs with two values each (Neg, Pos) results in four rules.

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Choice of Connectives

)(),(max

)(),(minxxBAxxBA

BA

BA

)()()()()()(

xxxxBAxxBA

BABA

BA

minimum

maximum

product

probabilistic sum

Choose product and probabilistic sum, if you wish to achieve linearity.

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Choice Of Primary Sets

Classical set

Singleton

Universal setTrapezoidalTriangular

An input family

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Inference in a Fuzzy PD Controller

Input error enters here

Input change in error enters here

Membership of input error of this set

Membership of input change in error of this set

The AND of the two memberships is carried forward as a weight on the control signal singleton

All rules contribute, and the final control signal is the weighted average of the contributions from the rules.

Sugeno type of rule base with singleton outputs

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Singleton Output

1. If error is Pos then control is 102. If error is Zero then control is 03. If error is Neg then control is -10

Equivalent to a singleton placed in the position -10 in the universe. It is simpler than a full membership function, and more intuitive too.

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Sugeno Inference

1. If error is Pos then control is 102. If error is Zero then control is 03. If error is Neg then control is -10

Rule 3

Rule 2

Rule 1

The weighted average

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First Order Output

1. If error is Large then control is a1*error + b1

2. If error is Small then control is a2*error + b2

The equation for a line L2, which depends on the coefficients a2 and b2.

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Interpolation (Takagi-Sugeno)

Line 1

Line 2

Interpolant

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Rule Base to Table

Choose discrete values, for instance one for every 10 %

Same

Calculate the result for every combination of the two inputs

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Look-Up TableChange 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

Five discrete points were chosen in each input universe, resulting in 25 pre-calculated values of the control signal

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Control Surface

1. If E is Neg and CE is Neg then u = -2002. If E is Neg and CE is Pos then u = 03. If E is Pos and CE is Neg then u = 04. If E is Pos and CE is Pos then u = 200

Neg PosA mesh plot of the control table

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Linear Rule Base

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Linear Control Surface

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Conditions For Linearity• Triangular sets, crossing at = 0.5• Rules: complete -combination• Define as multiplication (×)• Use conclusion singletons, positioned at sum of input

peak positions• Calculate the control signal as the weighted average

These five conditions settle many design choices. Very easy!

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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)• Pre- and postprocessing (scaling, quantization, sampling

time)

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ADVANCED SECTION*

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Building Blocks

Fuzzy controller

Inferenceengine

Rulebase Defuzzi

-ficationPostpro-cessing

Fuzzi-fication

Prepro-cessing

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Nonlinear Input Scaling

-5 0 5

-100

-50

0

50

100

measured input

scal

ed in

put

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Relational Rule FormatError Change in error ControlPos Pos PBPos Zero PMPos Neg ZeroZero Pos PM Zero Zero ZeroZero Neg NMNeg Pos ZeroNeg Zero NMNeg Neg NB

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Tabular Rule Format

Change in error

Neg Zero Pos

Neg NB NM Zero

Error Zero NM Zero PM

Pos Zero PM PB

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Defuzzification

0 50 1000

0.5

1

RM

BOA

COGMOM

LM

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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

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Inference And TerminologyAND

Aggregation

Accumulation

Defuzzification

Activation

a 4

a 5

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Rule Base To Table

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Rule Based Controllers

1. If error is Neg then control is Neg2. If error is Zero then control is Zero3. If error is Pos then control is Pos

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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

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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

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Simplification of 4 rules1. 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

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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 21