Constantinos Siettos School of Applied Mathematics & Physics National Technical University of Athens, Greece

Constantinos Siettos

School of Applied Mathematics & Physics

National 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

Χ

₂

Χ

₁

: 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

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 function Trapezoid membership function Linear membership function Sigmoidal membership function Π – type membership function Gaussian membership function

0 α

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 - 2 x

β

< x

<

α α

- γ

α - 2 x - 1

α

<

x 1

= ) γ β, α,

S(x; ₂

2

















 







 





γ

>

x 1

γ

< x

<

β α - γ

γ - 2 x - 1

β

< x

<

α α

- γ

α - 2 x

α

<

x 0

= ) γ β, α,

S(x; ₂

2















γ

>

x 0

γ

<

x

<

β β - γ

γ - -x

β

<

x

<

α α - β

a - x

α

<

x 0

= ) γ β, α, Tri(x;















δ

>

x 0

δ

<

x

<

γ γ - δ

δ - -x

γ

<

x

<

β 1

β

<

x

<

α α - β

α - x

α

<

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 x

0,

150 x

100 1,

100 x

25 75 75 -

x 0, 0 x 75

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 than 150 PS? Why they have a zero membership in the Fuzzy Set “Fast Cars”?

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

An example: Room Temperature

The problem: For an Air-Conditioning System Design it is asked the Description of the variable

“In-Room Temperature”

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

Lets say it is 0<T<40

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

Let us choose 5 fuzzy sets with triangular MFs

Π Χ Χ Α Ζ Π Ζ

0 1 2 1 8 2 6 4 0 Τ ( C )^ο

T-F F N C T-C

FUZZY SET OPERATIONS

   

 





























 

...

...

(y) (x)μ μ

2

(y) μ + (x) μ

x μ , x μ min

μ

B A

B A

Β Α

Β Α

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 (A



B) 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

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 μ_Α^~(x) =



μ_Α(x)



ⁿ, n>1 more/ less ^μ_Α^{~(x) =}



^μ_Α^(x)



ⁿ , 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. μ_F(u₁,u₂,...,u_ν)=min{μ_Α1(u₁), μ_Α2(u₂),..., μ_Αν(u_ν)}

or μ_F(u₁,u₂,...,u_ν)= μ_Α1(u₁) μ_Α2(u₂)... μ_Αν(u_ν).

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 μ_R: U₁xU₂ ->[0,1].

Then a fuzzy relation R on U₁xU₂ is defined as (Zimmerman, 1995; Pappis and Siettos, 2005):

Fuzzy Relations

) u , (u )/

u , (u

μ

_{1 2 1 2}

B U



R



) u , /(u ) u , (u

μ _{1 2}

UxV

2 1



R

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:

3 3

2 2

1 1

2 1 R

2 1 R

3 1 2

1

R {(u , u ), (μ (u , u ) μ (u , u ))}, u U , u U u U

R

C    

1



2

  , 

 ^{R 1}

_i1

^{ R 2}

_1j

  ^{ R 1}

_i2

^{ R 2}

_{2 j}

 ^{ ... }  ^{R 1}

_ip

^{ R 2}

_pj

 ^{ }

_k^p_1

^{R 1}

_ik

^{ R 2}

_kj

An example R =





 





4 0 9 0

6 0 2 0

. .

.

.

S =





 





1 0 5 0 8 0

3 0 4 0 1

. .

.

. .

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 = μ (u₁) μ_Β(u₂) /(u₁ ,u₂)

xU U

Α

2 1



 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:

 

























































) (v μ ) (u μ . ...

) (v μ ) (u μ ) (v μ ) (u μ

..

...

...

....

...

...

...

) (v μ ) (u μ ...

) (v μ ) (u μ ) (v μ ) (u μ

) (v μ ) (u μ ...

) (v μ ) (u μ ) (v μ ) (u μ

) (v μ ...

) (v μ ) (v μ ) (u μ

...

) (u μ

) (u μ

m Β n Α 2

Β n Α 1 Β n Α

m Β 2 Α 2

Β 2 Α 1 Β 2 Α

m Β 1 Α 2

Β 1 Α 1 Β 1 Α

m Β 2

Β 1 Β n

Α 2 Α

1 Α

Example on FUZZY IMPLICATION

 

























































) (v μ ) (u μ . ...

) (v μ ) (u μ ) (v μ ) (u μ

..

...

...

....

...

...

...

) (v μ ) (u μ ...

) (v μ ) (u μ ) (v μ ) (u μ

) (v μ ) (u μ ...

) (v μ ) (u μ ) (v μ ) (u μ

) (v μ ...

) (v μ ) (v μ ) (u μ

...

) (u μ

) (u μ

m Β n Α 2

Β n Α 1 Β n Α

m Β 2 Α 2

Β 2 Α 1 Β 2 Α

m Β 1 Α 2

Β 1 Α 1 Β 1 Α

m Β 2

Β 1 Β n

Α 2 Α

1 Α

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 x_i

Α_ij =fuzzy set of input variable x_j in i-th rule Βi =fuzzy set of output variable yj in i-th rule.

The i-th rule is the implication I_i=A_i=>B_i, Α_i= A_i1 Ai2 …  Ain =

ij n i_1 A

 .

Then the implication Itot of N rules is given by:

Itot=R1R2…RN=_i^N_1 R_i _i^N_1A_i  B_i

An example on INFERENCE RULES

Let a multiple input - single output (MISO) rule base with Ν rules. The i-th rule is given by

If Α_i1 and Α_i2 and ...and Α_in then Β_i,where n =the number of input variables x_i

Α_ij =fuzzy set of input variable x_j in i-th rule Β_i =fuzzy set of output variable y_j in i-th rule.

The i-th rule is the implication I_i=A_i=>B_i, Α_i= A_i1 A_i2 …  A_in =

ij n i_1 A

 .

Then the implication I_tot of N rules is given by:

I_tot=R₁ R₂ …  R_N=_i^N_1 R_i _i^N_1A_i B_i

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 RULES

Let a multiple input - single output (MISO) rule base with Ν rules. The i-th rule is given by

If Α_i1 and Α_i2 and ...and Α_in then Β_i,where n =the number of input variables x_i

Α_ij =fuzzy set of input variable x_j in i-th rule Β_i =fuzzy set of output variable y_j in i-th rule.

The i-th rule is the implication I_i=A_i=>B_i, Α_i= A_i1 A_i2 …  A_in =

ij n i_1 A

 .

Then the implication I_tot of N rules is given by:

I_tot=R₁ R₂ …  R_N=_i^N_1 R_i _i^N_1A_i B_i

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 SYSTEM

Assume 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 C1 Rule2: 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 B¹

A2 B₂ C₂

C1

2

1 C

C 

e ce

e ce

u

u

u

e0 ce₀

e0 ce₀

1 B₁

A2 B₂

C1

C2

2

1 C

C 

e ce

e ce

u

u

u

e0 ce₀

e0 ce₀

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, Η_i be the maximum value of the membership function of the output fuzzy set which corresponds to rule I, α_i is the degree that the rule i is fired.

i i n

1

=

i i i i

H α

x H α

= 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

=1 i

i i n

=1 i

i i

A α

M α

= u

where Μ_i is the value of the membership function of the output fuzzy set of rule i, A is corresponding surface, α_i 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) μ

u

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

1 i

n 1 k

i k n

1

= k

i k l

1 i

i

u μ

u μ u

u

FUZZY CONTROL RULE BASE

Two 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 Concentration Dimensionless Concentration Dimensionless Temperature Dimensionless Temperature

χ₁ χ₂

Damkohler Number Damkohler Number

Reaction Heat Reaction Heat

Heat Transfer coefficient Heat Transfer coefficient Dα

Β b

) )exp(

1

(

_{1 2}

1

1

x Da x x

x     

2 2

1 2

2

x B Da ( 1 x ) exp( x ) bx

x      

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:

ce

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e ps pm ps ze ns ns nm nb ze pb pm ps ze ns nm nb ns pb pm ps ps ze ns nm nm pb pb pm pm ps ze ze nb 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

Gin G_out

PI-like Fuzzy controller

PD-like Fuzzy controller

 

 





1 c -

z - 1 K 1 + 1 K e(z) =

= u(z) )

(z

C

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 = K

_c

ce + (K

_c

K)e



 





1 c -

z - 1 K 1 + 1 K e(z) =

= u(z) ) (z C

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)

δce ce e

+ f δe e ce cu f









 







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 o l a n t , T ^C F , T ,

C

1 A 1

F , T , C , C

2

A 2 B 2

C o o l i n g j a c k e t

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.

ce

nb nm ns ze ps pm pb

pb ze ze nm nb nb nb nb

pm ps ze ns nm nm nb nb

e

ps pm ps ze ns ns nm nb

ze pb pm ps ze ns nm nb

ns pb pm ps ps ze ns nm

nm pb pb pm pm ps ze ze

nb 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 (Tc)

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 changes

ISE (Tc)

0 0.7 1.4 2.1 2.8

5 10 20

% step changes

ITAE (ca)

0 250 500 750 1000

5 10 20

% step changes

ITAE (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)