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FUZZY CONTROL

Main bibliography

� J.M.C. Sousa and U. Kaymak. Fuzzy Decision Making

in Modeling and Control. World Scientific Series in

Robotics and Intelligent Systems, vol. 27, Dec. 2002.

� Fakhreddine O. Karray and Clarence De Silva. Soft

Computing and Intelligent Systems Design. Addison

Wesley, 2004.

� Michael Negnevitsky. Artificial Intelligence: A Guide to

Intelligent Systems. Addison-Wesley, Pearson

Education, 2002.

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Fuzzy control

� Controller designed by using If-Then rules instead of

mathematical formulas (knowledge-based control).

� Early motivation: mimic experienced operators.

� Fuzzy reasoning: interpolation between discrete

outputs.

� Currently: also controllers designed on the basis of a

fuzzy model (model-based fuzzy control).

� A fuzzy controller represents a nonlinear mapping

(but completely deterministic!).

392

Fuzzy control: history

1965 First publication on fuzzy sets (Zadeh)

1974 Fuzzy control applied to a laboratory system

(Mamdani)

1982 First industrial application of fuzzy control (to a

cement kiln)

1985 Sendai subway train control, consumer products

(Japan)

200? Large number of (micro)controllers: fuel injection,

cameras, washing machines, etc.

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Fuzzy control schemes

� PID fuzzy control (nonlinear)

� Fuzzy supervisory control

� Fuzzy model-based control

394

Low-level fuzzy control

395

Fuzzy LogicController Process

yr

d

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Fuzzy control: basic elements

396

1 1 2 2: is is is

is

k k k k

n nk

R x A x A x A

u B

If and and and

then

…

Fuzzy PD controller rule table

397

∆e

e(k) NB NS ZE PS PB

NB NB NB NS NS ZE

NS NB NS NS ZE PS

ZE NS NS ZE PS PS

PS NS ZE PS PS PB

PB ZE PS PS PB PB

8: is NS is ZE is NSR e e u∆If and then

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Mapping of FLC

� Fuzzy rules associate fuzzy regions in the antecedent

space with fuzzy regions in the consequent space.

398

Fuzzy inference mechanism

1. Establish fuzzy relation

2. Inference: sup-t composition

3. Defuzzification

399

1( , ) ( ) ( ), 1, ,k

nk k

j j uj

u x u k Kµ µ µ=

= ∧ =∧x …

R

1( , ) ( , )

RR

== ∨

Kk

ku uµ x x

[ ]( ) sup ( ) ( , )R′ ′

∈

= ⊗B AX

y uµ µ µx

x x

( )

( )

Bcog u U

Bu U

u u duu

u du

µ

µ

′∈

′∈

=∫∫

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Membership functions

400

Max-min inference

401

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Max-product inference

402

Comparison of inferences

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Design of a fuzzy controller

1. Determine input(s) and output(s).

2. Determine membership functions.

3. Define the rule base, based on e.g. expert knowledge.

4. Test the controller for typical test signals.

5. Fine-tune the controller (the designer can go back to

step 1 if necessary).

404

Types of PID fuzzy controllers

� PD fuzzy controller

� PI fuzzy controller

� PID fuzzy controller

405

: is is is ∆∆i i i

e uR e e A u AIf and then

: is is is ∆ ∆∆ ∆i i i

e uR e e A u AIf and then

2

2: is is is is ∆ ∆∆∆ ∆ ∆i i i i

e ueR e e A e A u AIf and and then

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Example: demo sltank

1. If level is OK then no_change valve

2. If level is low then open fast valve

3. If level is high then close fast valve

4. If level is OK and rate is positive then close slow valve

5. If level is OK and rate is negative then open slow valve

406

Rh

PID control

407

0 50 100 150 200 250 3000.4

0.6

0.8

1

1.2

1.4

1.6

time

level

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PD fuzzy control

408

0 50 100 150 200 250 3000.4

0.6

0.8

1

1.2

1.4

1.6

1.8

time

level

Example: proportional control

� Controller's input-output mapping

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Proportional control: rules

1. If error is Negative Big then control input is Negative Big.

2. If error is Zero then control input is Zero.

3. If error is Positive Big then control input is Positive Big.

410

Example: friction compensation

� DC motor with static friction.

� Fuzzy rules to represent “normal” proportional

control.

� Additional rules to prevent undesirable states.

Model of the DC motor

411

1

out_1J.s+b

1

Load

1

s

Dead Zone

L.s+R

K(s)

Armature

K

1

in_1

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Proportional controller

412

MuxMotor

Control u

Angle

P

Proportional controller output

413

0 5 10 15 20 25 30

-0.1

-0.05

0

0.05

0.1

0.15

time [s]

shaft

an

gle

[ra

d]

0 5 10 15 20 25 30-1.5

-1

-0.5

0

0.5

1

1.5

time [s]

co

ntr

ol

inp

ut

[V]

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Fuzzy control rule base

� Proportional Rules:

1. If error is Negative Big then control input is Negative Big.

2. If error is Zero then control input is Zero.

3. If error is Positive Big then control input is Positive Big.

� Additional rules:

4. If error is Negative Small then control input is not

Negative Small.

5. If error is Positive Small then control input is not Positive

Small.

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Input-Output mapping

415

-0.5

0.5

1

-0.15 -0.1 -0.05 0 0.05 0.1 0.15 e

u

local nonlinearity

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Fuzzy control results

416

0 5 10 15 20 25 30

-0.1

-0.05

0

0.05

0.1

0.15

time [s]

shaft

an

gle

[ra

d]

0 5 10 15 20 25 30-1.5

-1

-0.5

0

0.5

1

1.5

time [s]

co

ntr

ol

inp

ut

[V]

Supervisory fuzzy control

� Example: If y is low then reduce Kp and increase Kd.

417

PIDController Process

yr

FuzzySupervisor