Intelligent Control Methods

Lecture 10: Fuzzy Control (1)

Slovak University of TechnologyFaculty of Material Science and Technology in Trnava

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Introduction Classical control theory:

mathematical description of processes(differential equations)

Fuzzy control: normally used by people, based on experiences and

expert-knowledge, (which are described by linguistic tools, not by equations, mathematical tools are replaced by fuzzy logic).

Examples: reverted pendulum (described by 4 non-linear differential

equations!) car parking: turn the driving wheel just a bit to the left and

turn back (not: turn the wheel 16o33“ and drive back 2,675 m)

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h(t) e(t) u(t) y(t)

-

R(s) S(s)

Klassical controller states (calculates) the control action u(t) according to e(t): For example PI-controller:

dtteItPetu )()()( )()()( teKteKtu pi

Fuzzy controller states the actuating signal according to control strategy based on rules IF – THEN. F.E:

IF difference is big and difference of difference is small THEN difference of control action is big.

Usual by people decision and performance:(If a car rides faster than we want (e) and it reduced

gently (Δe), we brake stronger (Δu)).

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Control strategy:

Rules IF – THEN (in a form similar to normal speech).

Derived according to some type of classical controller.

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P-controller: u(t) = KP e(t) u(t) – control action

e(t) – control difference

Fuzzy P-controller:

IF e is Ae THEN u is Bu

Ae, Bu – linquistic expressions giving the value of control difference

and control action.

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PD-controller:

u(t) = KP e(t) + KD Δe(t)

Fuzzy PD-controller:

IF e is Ae AND Δe is AΔe THEN u is Bu

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PI-controller:

Δu(t) = KI e(t) + KP Δe(t)

Fuzzy PI-controller:

IF e is Ae AND Δe is AΔe THEN Δu is BΔu

Often case, Δu is more native for people (valve or gas pedal opening or closing) than the absolute value u (valve open 62 %, pedal pressed 16o).

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PID-controller:

Δu(t) = KI e(t) + KP Δe(t) + KD Δ2e (t)

Fuzzy PID-controller:

IF e is Ae AND Δe is AΔe AND Δe2 is AΔ2e

THEN Δu is BΔu

Assigned for non-linear and unstabil processes. Problem with great number of antecedents combinations.

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Matematical background of fuzzy control (1):

Clasical (crisp) sets:

A1 = {ball, cylinder, cube} set of figures given by elements listing

A2 = {x Z / 6 < x < 10} set of numbers given by property

T 1 for x A charakteristic function of ax A A(x) = set A

F 0 for x A (gives membership of elements to the

set A)

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Matematical background of fuzzy control (2):

U

A

negation

U

A B

intersection

U

A B

union

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Matematical background of fuzzy control (3):

Pojem relácie (v prípade ostrých množín):

Let X and Y are definition scopes and let their cartesian product is U = X x Y. Then a binary relation R is each subset R U.

(the same definition is valid for n-dimensional relations)

Example:

X = {Jana, Iveta, Eva} and Y = {Peter, Ján, Milan, Igor} are definition scopes (universes)

R = {(Jana, Igor), (Iveta, Peter), (Eva, Ján)} is relation „married couples“ defined on X x Y.

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Matematical background of fuzzy control (4):

ii

n

iF uu /)(

1

Fuzzy set definition:

Fuzzy set is the set of elements, which can belong into the set partially.

The membership of element into the set is given by membership function (what is generalised characteristic function of the set).

F: U <0,1>

F = {(u,F(u)/uU} = F(u1)/u1 + F(u2)/u2 + ... +F(un)/un

uuUu

F /)(

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Matematical background of fuzzy control (5):

Fuzzy set example:

Let the temperature in a room is <0,30> (oC), i.e.

U = <0,30>

Membership functions into sets Cald, Good, Hot are:

1

0 15 30

c(25) = 0,0 g(25) = 0,3 H(25) = 0,7

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Matematical background of fuzzy control (6):

Typical membership functions (linear, therefore simple):

(u)1 0 for u

(u,,) = (u-)/(-) for u 1 for u

u

(u)1 1 for u

L(u,,) = (-u)/(-) for u 0 for u

u

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Matematical background of fuzzy control (7):

Typical membership functions:(u)1 0 for u

(u,,,) = (u-)/(-) for u

(-u)/(-) for u u 1 for u(u) 0 for u1 (u-)/(-) for u

(u,,,,) = 1 for u (-u)/(-) for u

u 0 for u

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Matematical background of fuzzy control (8):Often (general) case of description of definition scope by fuzzy sets without considering the physical parameters:

1 NB NM NS Z PS PM PB

-6 -4 -2 0 2 4 6 u

NB (Negative Big): L(u,-6,-4) NM (Negative Medium): (u,-6,-4,-2)NS (Negative Small): (u,-4,-2,0) Z (Zero): (u,-2,0,2) PS (Positive Small): (u,0,2,4) PM (Positive Medium): (u,2,0,4) PB (Positive Big): (u,4,6)

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Operations with fuzzy sets: A B

x

A’

x

Complement (negation):

A’(x) = 1 - A(x)

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Operations with fuzzy sets: A B

x

A B

x

Intersection:

AB(x) = min (A(x), B(x))

Union:

AB(x) = max (A(x), B(x))

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Fuzzy relation:

),/(),( vuvuRUxV

R

Let U and V are definition scopes and let it is given the function R: UxV 0,1. Binary fuzzy relation R is fuzzy set of ordered couples

If the definition scopes are continuous, then:

),/(),( vuvuRUxV

R

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Fuzzy relation (example):

X = {Jana, Iveta, Eva} and Y = {Peter, Ján, Milan, Igor} are definition scopes.

Relation „Friends“ defined on X x Y:

  Peter Ján Milan Igor

Jana 0,8 0,9 0,1 0,3

Iveta 0,5 0,6 0,3 0,7

Eva 0,2 0,1 0,8 0,4

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Operations with fuzzy relations: (intersection and union)

Let R and S are binary relations defined on X x Y. Then membership functions for intersection and union of relations R and S are defined for all x,y as follow:

Intersection: RS(x,y) = min (R(x,y), S(x,y))

Union: RS(x,y) = max (R(x,y), S(x,y))

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Operations with fuzzy relations (example for intersection and union):

  Peter Ján Milan Igor

Jana 0 0 0 1

Iveta 1 0 0 0

Eva 0 1 0 0

X = {Jana, Iveta, Eva} and Y = {Peter, Ján, Milan, Igor} are definition scopes.

Relations „Married couples“ and „Friends“ defined on X x Y:

Married couples (M): Friends (F):  Peter Ján Milan Igor

Jana 0,8 0,9 0,1 0,3

Iveta 0,5 0,6 0,3 0,7

Eva 0,2 0,1 0,8 0,4

  Peter Ján Milan Igor

Jana 0 0 0 0,3

Iveta 0,5 0 0 0

Eva 0 0,1 0 0

  Peter Ján Milan Igor

Jana 0,8 0,9 0,1 1

Iveta 1 0,6 0,3 0,7

Eva 0,2 1 0,8 0,4

Married c. and friends (MF(x,y)) M.c. or friends (MF(x,y))

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Operations with fuzzy relations (2): ProjectionLet R is binary relation defined on X x Y. Then projection R into Y is fuzzy set

yyxprojRnaY RxY

/),(max

I.e.: Projection R into Y means the finding of maximal value R in each column y1, y2, ... yn in the table and assignment of this value to element yj.

  Peter Ján Milan Igor

Jana 0,8 0,9 0,1 0,3

Iveta 0,5 0,6 0,3 0,7

Eva 0,2 0,1 0,8 0,4

Proj R in Y = 0,8/Pe + 0,9/Já + 0,8/Mi + 0,7/Ig

Proj R in X = 0,9/Ja + 0,7/Iv + 0,8/Ev

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Operations with fuzzy relations (3): ExtensionOpposit operation for projection:

Let F is a fuzzy set defined on Y. Then cylindric extension F to X x Y is the set of all couples (x,y) X x Y with membership function CE(F)

(x,y), i.e.:

),/()()( yxyFceXxY

F

I.e.: Cylindric extension means the building of a table from the function.

  Peter Ján Milan Igor

Jana 0,8 0,7 0,3 0,6

Iveta 0,8 0,7 0,3 0,6

Eva 0,8 0,7 0,3 0,6

F = 0,8/Pe + 0,7/Já + 0,3/Mi + 0,6/Ig