Constantinos SiettosSchool of Applied Mathematics & PhysicsNational Technical University of Athens, Greece

OUTLINE

• What is Fuzzy Reasoning

• Basic elements of Fuzzy Set Theory

• Basic structure of a fuzzy inference system

• Fuzzy control systems

•Fuzzy Decision Making: Classification and Clustering

FUZZY REASONING

The Butterfly Problem

Χ2

Χ1

: Classify the points in 2D into 4 sets

…. But what happens for points in between?

FUZZY REASONING

The theory of fuzzy sets was introduced by him in 1965.

“as the complexity of a system increases, our ability to make precise and yet significant statements about its behavior diminishes until a threshold is reached beyond which precision and significance (or relevance) become almost mutually exclusive characteristics”.

L.A.Zadeh, 1973

Fuzzy Reasoning, based on the theory of fuzzy sets, is a new approach to complex systems theory and decision processes.

It encompasses Artificial Intelligence, information processing and theories from logic to pure and applied mathematics.

Its applications range from production, finance, marketing and other decision-making problems

to micro-controller based systems in home appliances and large-scale process control systems

BENCHMARKS IN THE HISTORY OF FUZZY REASONING

First paper on fuzzy systems (Zadeh, 1965)Linguistic approach (Zadeh, 1972)Fuzzy Logic controller (Assilian & Mamdani, 1974)Table-Based Controller (Mamdani,1977)Heat Exchanger based on fuzy logic (Οstergaard, 1977)Self-organizing fuzzy controller (Mamdani, 1977, Procyk & Mamdani, 1979)Fuzzy logic control for cement production (Holmblad & Ostergaard, 1982)Fuzzy controllers on Tokyo subway shuttles (Hitachi, 1984)Fuzzy Chip (Togai & Watanabe, 1985)Ηardware implementation of fuzzy system (Yamakawa & Miki, 1986)Hybrid Neural-Fuzzy systems (Kosko, 1990)

FUZZY LOGIC: BASIC CONCEPTS

A classical set A is defined as a collection of elements or objects. Any element or object x either belongs or does not belong to A. The membership μΑ(x) of x in A is a mapping: μA : X{0, 1}, that is, it may take the value 1 or 0, which represent the truth value of x in A. It follows that, if Ā is the complement set of A and ∩ represents intersection of sets, then:

AA .

Fuzzy logic is a logic based on fuzzy sets, i.e. sets of elements or objects characterized by truth-values in the [0,1] interval rather than crisp 0 and 1, as in the conventional set theory. The function that assigns a number in [0,1] to each element of the universe of discourse of a fuzzy set is called membership function.

THE NOTION OF MEMBERSHIP

Let X denote the universe of discourse of a fuzzy set A. A is completely characterized by its membership function μA :

μA : X[0, 1]

and is defined as a set of pairs:

A = {(x, μA (x))}.

The most commonly used membership functions are the following (Dubois and Prade, 1980, Zimmermann, 1996; Pappis & Siettos, 2004):

Triangular membership functionTrapezoid membership functionLinear membership functionSigmoidal membership functionΠ – type membership functionGaussian membership function

α0

0.5

1

0

0.5

1

α β γ β γ

THE NOTION OF MEMBERSHIP:MOST COMMON TYPES

0.

0.5

1

0

0.5

1

α β α β γ δ

γ> x 0

γ< x < β α-γ

γ-x2

β< x < α α-γ

α-x2-1

α< x 1

=) γβ, α, S(x; 2

2

γ> x 1

γ< x < β α-γ

γ-x2-1

β < x < α α-γ

α-x2

α< x 0

= ) γβ, α, S(x; 2

2

γ> x 0

γ< x< β β-γ

γ-x-

β< x<α α-β

a-xα< x 0

=) γβ, α, Tri(x;

δ > x 0

δ< x< γγ-δ

δ-x-

γ< x< β 1

β< x<α α-β

α-xα< x 0

=δ) γ,β, α, Tra(x;

THE SET OF “TALL” PERSONS

a=1.50cm, β=1.75, γ=1.90

THE SET OF “SHORT” ONES

“COMFORTABLE ROOM TEMPERATURE

a=10, β=18, γ=23, δ=30 a=10, β=30

An example: The Set of “Fast” Cars

150 x0,

150x100 1,

100x7525

75-x75x0 0,

μ(x),

Lets use the Power (in PS) as a measure. Then we Could assign the following membership to the set of“Fast Cars”

A question: What about cars with Horse-Power more than150 PS? Why they have a zero membership in the FuzzySet “Fast Cars”?

….These Cars are not longer “Fast” but …..“Very Fast”

An example: Room Temperature

The problem: For an Air-Conditioning System Design it isasked the Description of the variable“In-Room Temperature”

Task #1: What is the “Universe of Discourse” for thevariable “In-Room Temperature”

Lets say it is 0<T<40

Task # 2: Define the number and the character of theFuzzy sets that will be used to define the variable

Let us choose 5 fuzzy sets with triangular MFsΠ Χ Χ Α Ζ Π Ζ

0 12 18 26 40 Τ ( C )ο

T-F F N C T-C

FUZZY SET OPERATIONS

...................(y)(x)μμ

2

(y)μ+(x)μ

xμ,xμmin

μ

BA

BA

ΒΑ

ΒΑ

Two fuzzy sets A and B on the universe of discourse X are equal if their membership functions are equal for each x X x X : μΑ(x)= μΒ(x). A fuzzy set A is a subset of B (AB) if x X : μΑ(x) μΒ(x). For the operation of intersection of two fuzzy sets A and B, there is a plethora of definitions in the bibliography. The choice is application depended.

FUZZY SET OPERATIONS

The union of two fuzzy sets A and B is also defined in several ways:

xX:

..........................................

(y)μ (x)μ-(y)μ+(x)μ

6

(y)μ (x),μ4max+(y)μ(x),μ2min

(y)μ(x),μmax

μ

ΒΑΒΑ

ΒΑΒΑ

ΒΑ

ΒΑ

The complement A of a fuzzy set A is defined as: x X : )(x μ-1)(x μ ΑA .

Examples on Fuzzy Operations

Intersection

Let two Fuzzy Sets

Α={0/1+ 0.2/2 + 0.8/3 + 1/ 4 + 1/ 5}, Β={ 0.1/1 + 0.4/2 + 0.5/3 + 0.7/4 + 0.3/ 5)

Α Β= {0/1 + 0.08/2 + 0.4/3 + 0.7/4 + 0.3/5} (Product operation)

Α Β ={0/1 + 0.2/ 2 + 0.5/ 3 + 0.7/4 + 0.3/5 ) (Min operation)

Union A Β={0.1/1 + 0.4/2 + 0.8/3 +1/4 + 1/5} (Max operator)

Complement A’ = {1/1 + 0.8/2 + 0.2/ 3 + 0/4 + 0/5}

Β’ ={0.9/1 +0.6/2 +0.5/3 + 0.3/4+ 0.7/5)

(A Β)’ = ? = A’ Β’= {1/1 + 0.8/2 + 0.5/3 + 0.3/4 + 0.7/5) (min operation)

TRANSFORMATION OPERATORS

The transformation operator acts on a membership function to modify the concept of the linquistic term that describes the fuzzy set.

For example, in the clause “number very close to 10”, the transformation

operator “very” acts on the linguistic term “close to 10” which corresponds to a fuzzy set.

Very nΑΑ

(x)μ=(x)μ ~ , n>1

more/ less nΑΑ(x)μ=(x)μ ~ , 0<n<1

more than (lt) Μlt(A)(x) = 0 for x?x0, x0: μA(x0) = max μΑ(x) = 1-μΑ(x) for x<x0

more/ less (mt) μmt(A)(x) = 0 for x?x0, x0: μA(x0) = max μΑ(x) = 1-μΑ(x) for x>x0

Cartesian Inner Product of Fuzzy Sets

If Α1, Α2,..., Αν are fuzzy sets defined in U1, U2,..., Uν, their Cartesian inner product is a fuzzy set F=Α1xΑ2x...xΑν in U1x U2x,...x Uν with membership function: μF(u1,u2,...,uν)= i=1,ν μΑι (ui)

e.g. μF(u1,u2,...,uν)=min{μΑ1(u1), μΑ2(u2),....., μΑν(uν)} or μF(u1,u2,...,uν)= μΑ1(u1) μΑ2(u2)........... μΑν(uν).

Example on Cartesian Inner Product of Fuzzy Sets

P1 P2 P3

T1 0.1 0.1 0.1

T2 0.5 0.6 0.1

T3 0.5 0.9 0.1

T4 0.2 0.2 0.1

A X B =

Let U1 and U2 be two universes of discourse and the membership

function μR: U1xU2 ->[0,1].

Then a fuzzy relation R on U1xU2 is defined as (Zimmerman,

1995; Pappis and Siettos, 2005):

Fuzzy Relations

)u,(u)/ u,(uμ 2121

BU

R

)u ,/(u)u,(uμ 21UxV

21R

if U1 ,U2 are continuous

if U1, U2 are discrete.

R=

R=

An example Lets the Universe of Discourse U= {2, 3, 4}

The Fuzzy Relation: “Almost Equal numbers”

R= 1 (2,2) + 1(3,3) +1 (4,4) + 0.6 (2, 3) + 0.6 (3,2) + 0.6 (3,4 ) + 0.6 (4, 3) + 0.2 (2, 4) + 0.2 (4, 2)

2 3 4

2 1 0.6 0.2

3 0.6 1 0.6

4 0.2 0.6 1

or: R =

Fuzzy Composition

Let R1 and R2 be two fuzzy relations on U1xU2 and U2xU3 respectively,

Then the composition C of R1 and R2 is a fuzzy relation defined as follows:

33221121R21R3121 Uu Uu ,Uu ))},u ,(uμ)u ,(u(μ ),u ,{(uRRC21

,

kjikpkpjipjij RRRRRRRR 2121.....2121 1221i1

An example R =

4090

6020

..

.. S =

105080

30401

...

..

Τ= R S =

4090

6020

..

..

105080

30401

...

..=

304090

205060

...

...

Τ(1,1) = max{min( 0.2, 1), min(0.6, 0.8)}

FUZZY IMPLICATION

Let Α and Β be two fuzzy sets in U1 ,U2 respectively. The implication I:Α=>Β U1 xU2 is defined as (Ross, 1994, Zimmerman, 1995):

I=AxB = )u , /(u)(uμ)(uμ 212Β1

xUU

Α

21

The rule: “If the error is negative big then control output is positive big” is an implication : error x implies control action y.

Let the two discrete fuzzy sets A= {(ui, μA (ui)), i=1,..,,n} defined on U and

B= {(vj, μB (vj)), j=1,…,m} defined on V.

Then the implication Α=>Β is a fuzzy relation R R= {((ui,vj), μR (ui,vj)), i=1,..,n, j=1,…,m }

defined on UxV, whose membership function μR (ui,vj) is given by:

)(vμ)(uμ...........)(vμ)(uμ)(vμ)(uμ

...................................................

)(vμ)(uμ..........)(vμ)(uμ)(vμ)(uμ

)(vμ)(uμ..........)(vμ)(uμ)(vμ)(uμ

)(vμ.......)(vμ)(vμ

)(uμ

.........

)(uμ

)(uμ

mΒnΑ2ΒnΑ1ΒnΑ

mΒ2Α2Β2Α1Β2Α

mΒ1Α2Β1Α1Β1Α

mΒ2Β1Β

nΑ

2Α

1Α

Example on FUZZY IMPLICATION

)(vμ)(uμ...........)(vμ)(uμ)(vμ)(uμ

...................................................

)(vμ)(uμ..........)(vμ)(uμ)(vμ)(uμ

)(vμ)(uμ..........)(vμ)(uμ)(vμ)(uμ

)(vμ.......)(vμ)(vμ

)(uμ

.........

)(uμ

)(uμ

mΒnΑ2ΒnΑ1ΒnΑ

mΒ2Α2Β2Α1Β2Α

mΒ1Α2Β1Α1Β1Α

mΒ2Β1Β

nΑ

2Α

1Α

If e(t) is Positive Big Then u(t) is Positive Big

Let the universe of discourses of e(t) and u(t)

U= [-5, -1, 0, 1, 5] (mvolts) and

Ω =[-1,0,1] (mvolts)

The FS: “Positive Big” is defined for e(t) as

PS(e) = [0, 0, 0, 0.4, 1]

The FS: “Positive Big” is defined for u(t) as

PS(e) = [0, 0, 1].

Then the implication R=AxB=

0 0 1

0 0 0 0

0 0 0 0

0 0 0 0

0.4 0 0 0.4

1 0 0 1

So if e(t0)= 5 mvolts

u(t0) = {-1/0, 0/0, 1/1}

If e(t0)= 1 mvolts

u(t0) = {-1/0, 0/0, 1/0.4}

INFERENCE RULES

Let R be a fuzzy relation on U1xU2 and Α be a fuzzy set in U1. The composition

ΑοR=B

is a fuzzy set in U2, which represents the conclusion made from the fuzzy set Α (fact) based on the implication R (rule).

Let a multiple input - single output (MISO) rule base with Ν rules. The i-th rule is given by If Αi1 and Αi2 and ...and Αin then Βi, where

n =the number of input variables xi Αij =fuzzy set of input variable xj in i-th rule Βi =fuzzy set of output variable yj in i-th rule. The i-th rule is the implication Ii=Ai=>Bi, Αi= Ai1 Ai2 …Ain =

ijni A1 .

Then the implication Itot of N rules is given by:

Itot=R1 R2 … RN= iiN

1iiN

1i BAR

An example on INFERENCE RULESLet a multiple input - single output (MISO) rule base with Ν rules. The i-th rule is given by If Αi1 and Αi2 and ...and Αin then Βi, where

n =the number of input variables xi Αij =fuzzy set of input variable xj in i-th rule Βi =fuzzy set of output variable yj in i-th rule. The i-th rule is the implication Ii=Ai=>Bi, Αi= Ai1 Ai2 …Ain =

ijni A1 .

Then the implication Itot of N rules is given by:

Itot=R1 R2 … RN= iiN

1iiN

1i BAR

R1: IF e(t) is Positive THEN u(t) is Positive

R2: IF e(t) is Zero THEN u(t) is Zero

R3: IF e(t) is Negative THEN u(t) is Negative

e(t): U= [-5, -1, 0, 1, 5] (mvolts)

u(t): Ω =[-5, -1, 0, 1, 5] (mvolts)

P(e)=[-5/0, -1/0, 0/0, 1/0.4, 5/1]

Z(e)=[-5/0, -1/0.5, 0/1, 1/0.5, 5/0]

N(e)=[-5/1, -1/0.4, 0/0, 1/0, 5/0].

P(u)=[-5/0, -1/0, 0/0, 1/0.4, 5/1]

Z(u)=[-5/0, -1/0.5, 0/1, 1/0.5, 5/0]

N(u)= [-5/1, -1/0.4, 0/0, 1/0 , 5/0].

Fuzzy Sets for e(t) and u(t)

Let e(t) = -1 Then the Fuzzy Set N(e) takes the value of 0.4

Hence: u(t0) = [-5/0.4, -1/0.5, 0/0.5, 1/0.5, 5/0]

An example on INFERENCE RULESLet a multiple input - single output (MISO) rule base with Ν rules. The i-th rule is given by If Αi1 and Αi2 and ...and Αin then Βi, where

n =the number of input variables xi Αij =fuzzy set of input variable xj in i-th rule Βi =fuzzy set of output variable yj in i-th rule. The i-th rule is the implication Ii=Ai=>Bi, Αi= Ai1 Ai2 …Ain =

ijni A1 .

Then the implication Itot of N rules is given by:

Itot=R1 R2 … RN= iiN

1iiN

1i BAR

R1: IF e(t) is Positive THEN u(t) is Positive

R2: IF e(t) is Zero THEN u(t) is Zero

R3: IF e(t) is Negative THEN u(t) is Negative

e(t): U= [-5, -1, 0, 1, 5] (mvolts)

u(t): Ω =[-5, -1, 0, 1, 5] (mvolts)

P(e)=[-5/0, -1/0, 0/0, 1/0.4, 5/1] Z(e)=[-5/0, -1/0.5, 0/1, 1/0.5, 5/0] N(e)=[-5/1, -1/0.4, 0/0, 1/0, 5/0].

P(u)=[-5/0, -1/0, 0/0, 1/0.4, 5/1] Z(u)=[-5/0, -1/0.5, 0/1, 1/0.5, 5/0] N(u)= [-5/1, -1/0.4, 0/0, 1/0 , 5/0].

Fuzzy Sets for e(t) and u(t)

Let e(t) = -1 and the Implication R3: [-5/0.4, -1/0.4, 0/0, 1/0, 5/0]

For R1: [-5/0, -1/0, 0/0, 1/0, 5/0]

For R2: [-5/0, -1/0.5, 0/0.5, 1/0.5, 5/0]

BASIC STRUCTURE OF A FUZZY INFERENCE SYSTEM

A data base defining the number, labels and types of the membership functions the fuzzy sets used as values for each system variable

A rule base, which essentially maps fuzzy values of the inputs to fuzzy values of the outputs. This actually reflects the decision-making policy

Rule i: IF x is Ai and y is Bi THEN z is Ci

The fuzzy reasoning unit performs various fuzzy logic operations to infer the output (decision) from the given fuzzy inputs.

AN EXAMPLE OF FUZZY INFERENCE SYSTEMAssume that there are two input variables, e (error) and ce (change of error), one output variable, cu (change of output) and two rules:

Rule1: If e is Α1 AND ce is Β1 THEN the cu is C1Rule2: If e is Α2 AND ce is Β2 ΤHEN cu is C2.

In the Max-Min inference method, the fuzzy operator AND (intersection)

means that the minimum value of the antecedents is taken:μΑ AND μΒ = min { μΑ, μΒ},

while for the Max-product one the product of the antecedents is taken:

μΑ AND μΒ = μΑ* μΒ

AN EXAMPLE OF FUZZY INFERENCE SYSTEM

MAX-MIN

MAX-DOT

1 1B

2A 2B 2C

1C

21 CC

e ce

e ce

u

u

u

0e 0ce

0e 0ce

1 1B

2A 2B

1C

2C

21 CC

e ce

e ce

u

u

u

0e 0ce

0e 0ce

Defuzzification Unit

Deffuzification typically involves weighting and combining a number of fuzzy sets resulting from the fuzzy inference process in a calculation which gives a single a single risp value for each output.

The most prevalent and physically appealing among the defuzzification methods are those of mean of maximum, centroid, and center of sum of areas. (Lee, 1990, Ross, 1995, Lee, 1990; Drianov et al., 1993)

Mean of maximum defuzzification technique

where n is the number of rules in a MISO system, Ηi be the maximum value of the membership function of the output fuzzy set which corresponds to rule I, αi is the degree that the rule i is fired.

ii

n

1=iiii

Hα

xHα=x

Centroid deffuzification technique

This is the most prevalent and physically appealing among the defuzzification methods (Lee, 1990, Ross, 1995).

This method takes the center of gravity of the final fuzzy space in order to produce a result sensitive to all rules; it is described by the following equation (Ross, 1995):

n

=1iii

n

=1iii

Aα

Mα

=u

where Μi is the value of the membership function of the output fuzzy set of rule i, A is corresponding surface, αi is the degree that the rule i is fired. Note that the overlapping areas are merged (figure 7a).

In the case of continuous space the output value is given by (Ross, 1994; Taprantzis et al., 1997)

u=

U U

U U

du (u)μ

du (u)μu

μ(U)

U

Center of sums deffuzification technique

U

μ(U)

A similar to the centroid technique but computationally more efficient in terms of speed technique is that if the center of sums. The difference is that the overlapping-betewwn the output fuzy sets-areas are not merged (figure 7 b). The discrete value of the output is given by (Lee, 1990; Drianov et al., 1993):

l

1i

n

1kik

n

1=kik

l

1ii

uμ

uμuu

FUZZY CONTROL RULE BASETwo are the main approaches in the design of rule bases (Yan et al., 1994):

The Heuristic approach

The systematic approach

Heuristic approaches (Yan et al., 1994, King and Mamdani, 1977) provide a convenient way to build fuzzy control rules in order to achieve the desired output response, requiring only qualitative knowledge for the behaviour of the system under stydy. For a two-input (e and ce) one-output variable (cu) system these rules are of the form:

IF e is P (Positive) AND ce is N (Negative) THEN cu is P (Positive)

IF e is N (Negative) AND ce is P (Positive) THEN cu is N (Negative)

The reasoning for the construction of the fuzzy control rules can be summarized as follows:

If the system output has the desired value and the change of the error (ce) is zero then keep the control action constant.

If the system output diverges from the desired value then the control action changes with respect to the sign and the magnitude of the error e and the change of error ce.

Dimensionless ConcentrationDimensionless Concentration

Dimensionless TemperatureDimensionless Temperature

χ1

χ2

Damkohler NumberDamkohler Number

Reaction HeatReaction Heat

Heat Transfer coefficientHeat Transfer coefficient

Dα

Β

b

))exp(1( 2111 xxDaxx

22122 )exp( )1( bxxxDaBxx

An example: Fuzzy control of a CSTR

The control objective is to maintain the control variable, which is the composition of the reacting mixture, within the desired operational settings eliminating mostly input concentration disturbances.

The manipulated variable is taken to be the coolant temperature.

Fuzzy control of a CSTR: The controller Design

Variables:

e(t) = r(t) - y(t), ce(t)=e(t)-e(t-1), cu(t)= [u(t)-u(t-1)],

where: r(t) is the set point at time t, y(t) is the process output at time t,

e(t), ce(t) are the error and the change of error at time t.

FUZZIFICATION:

cenb nm ns ze ps pm pb

pb ze ze nm nb nb nb nbpm ps ze ns nm nm nb nb

e ps pm ps ze ns ns nm nb

ze pb pm ps ze ns nm nbns pb pm ps ps ze ns nmnm pb pb pm pm ps ze zenb pb pb pb pb pm ze ze

THE FUZZY RULES

Given the fact that a reduction in the coolant temperature decreases the output concentration, and inversely, The output is below

the set point

.. And the error is getting smaller

Fuzzy control systems

inG outG

PI-like Fuzzy controller

PD-like Fuzzy controller

1-cz - 1

1K + 1K =

e(z)

u(z) = )(zC

PI-like Fuzzy controller

The PI controller in the z-Domain has the following form (Stephanopoulos, 1984):

In the time domain the above can be rewritten as cu = Kc ce + (KcK)e

1-cz - 1

1K + 1K =

e(z)

u(z) = )(zC

where ce the change in error, and u the control output signal.

In order to generate an equivalent fuzzy controller, the same inputs e, ce and the same output, cu, will be used in its design.

Based on the above, a two-input-single-output FLC is derived with the following variables:input variables: e(t) = r(t) - y(t)

ce(t) = e(t) - e(t-1)output variable: cu(t) = u(t)-u(t-1)

In a general form the control action cu can be represented as a nonlinear function of the input variables e(t), ce(t):

cu = f(e', ce', t) = f (GE e, GCE ce, t)

For small perturbations δe, δce around an equilibrium, the above equation is approximated by

Finally one obtains the simplified discretized equation

cu(t) = GE e(t) + GCE ce(t)

δceece

f + δe

cee

fcu

PD-like Fuzzy controller

u(t) = GE e(t) + GCE ce(t)

A case study: Fuzzy control of a plug-flow tubular reactor

C o ola n t, T C

F,T ,C

1

A 1

F, T , C , C

2

A 2 B 2

C o olin g ja ck et

C z

Cu

t

Ck

A

ACR

DH-TC

Tt

hAt

TρuA

pc

t

TρA

pc k

RT

EexpOkk

The control objective: to maintain the control variable, which is the composition of the reacting mixture at the output of the reactor, within the desired operational settings and particularly to keep the A reactant concentration at the output, below its nominal steady state value, eliminating mostly input concentration disturbances

The manipulated variable is taken to be the coolant temperature. The incremental fuzzy controller, a two-input-single-output FLC is derived with the following variables:

e(t)=r(t)-y(t), ce(t)=e(t)-e(t-1), cu(t)=u(t)-u(t-1),

where: r(t) is the set point at time t (set point moisture), y(t) is the process output at time t (output moisture), e(t), ce(t) are the error and the change of error at time t.

Fuzzification

For the fuzzification of the input -output variables, seven fuzzy sets are defined for each variable,

e(t), ce(t) and cu(t) with fixed triangular shaped membership functions normalized in the same universe of discourse as it is shown in the figure

RULE -BASE

Given the fact that a reduction in the coolant temperature decreases the output concentration, and inversely, the reasoning for the construction of the fuzzy control rules is as follows:

Keep the output of the FLC constant if the output has the desired value and the change of error is zero.

Change the control action of the FLC according to the values and signs of the error, e and the change of error, ce:

.If the error is negative (the process output is above the set point) and the change of error is negative (at the previous step the controller was driving the system output upwards), then the controller should turn its output downwards. Hence, considering negative feedback, the change in control action should be positive, i.e. cu>0, since u(t)=u(t-1) +cu.

.If the error is positive (the process output is below the set point) and the change of error is positive (at the previous step the controller was driving the system output downwards), then the controller should turn its output upwards. Hence, considering negative feedback, the change in control action should be negative, i.e. cu<0, since u(t)=u(t-1) +cu.

.If the error is positive (the process output is below the set point) and the change of error is negative, implying that at the previous step the controller was driving the system output upwards, trying to correct the control deviation, then the controller need not to take any further action.

.If the error is negative (the process output is above the set point) and the change of error is positive, implying that at the previous step the controller was driving the system output downwards, then the controller need not to take any further action.

cenb nm ns ze ps pm pb

pb ze ze nm nb nb nb nbpm ps ze ns nm nm nb nb

e ps pm ps ze ns ns nm nb

ze pb pm ps ze ns nm nbns pb pm ps ps ze ns nmnm pb pb pm pm ps ze zenb pb pb pb pb pm ze ze

0

0.1

0.2

0.3

0.4

0.5

5 10 20

% step changes

IAE

(ca

)

FLC

PI 1

PI 2

0

70

140

210

280

5 10 20

% step changes

IAE

(T

c)

0

0.1

0.2

0.3

0.4

5 10 20

% step changes

ISE

(ca

)

0

20000

40000

60000

80000

5 10 20

% step changesIS

E (

Tc)

0

0.7

1.4

2.1

2.8

5 10 20

% step changes

ITA

E (

ca)

0

250

500

750

1000

5 10 20

% step changes

ITA

E (

Tc)

Performance Comparison of the Fuzzy and the PI controller tuned by the process reaction method (PI 1), by minimising the IAE criterion (PI 2)