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Engineering Applications of Artificial Intelligence 18 (2005) 875–880

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Design of Mamdani fuzzy logic controllers with rule baseminimisation using genetic algorithm

K. Belarbia,�, F. Titela, W. Bourebiaa, K. Benmahammedb

aLaboratoire d’Automatique et de Robotique, Faculty of engineering, University of Constantine, Constantine AlgeriabDepartment of electronics, Faculty of engineering, University of Setif, Setif, Algeria

Received 21 June 2004; accepted 11 March 2005

Available online 5 May 2005

Abstract

This paper presents a design procedure for Mamdani fuzzy logic controller including rule base minimisation. The rules are

modelled with binary weights on which constraints are imposed in order to ensure consistency. A genetic algorithm is used for

finding stabilising controllers that minimise the number of rules. The cost function includes a stability/performance coefficient which

insures that stable, performance satisfying controllers are given the highest possible fitness. The number of fuzzy sets for the input

and the control variables are set by the user and the design procedure is concerned only with the rule base and the distribution of the

fuzzy sets in the universes of discourses. Two examples were studied: the control of the pole and cart system and the control of the

concentration in CSTR. In both cases, the fuzzy sets were isosceles triangles evenly distributed, in the universe of discourses.

r 2005 Elsevier Ltd. All rights reserved.

Keywords: Mamdani fuzzy logic controllers; Genetic algorithm; Rule base minimisation

1. Introduction

The degrees of freedom of a fuzzy logic controller,FLC, are related to its structure and the associatedparameters. The structure is defined by the fuzzy inputand output variables and the number and shape of thefuzzy sets for each variable whereas the parameters arerelated to the distribution of the membership functionson the universe of discourse and the rule base (Iserman,1998; Lee, 1990; Passino and Yurkowich, 1998). Usually,in solving the FLC design problem, the structure is fixedbefore hand and one is only concerned with finding theparameters that give optimality and/or closed loopstability. A variety of methods have been proposed,ranging from genetic algorithms (Bastian, 2000; Homai-far and McCormick, 1995; Kang et al., 2000; Karr, 1991;Kim and Zeigler, 1996; Linkens and Nyongesa, 1995;Wang and Yen, 1999; Wu and Yu, 2000) to more

e front matter r 2005 Elsevier Ltd. All rights reserved.

gappai.2005.03.003

ing author. Tel.: +213 318 19040; fax: +213318 19010.

ess: kbelarbi@yahoo.com (K. Belarbi).

classical techniques such as gradient optimisation, LMI-based methods or sliding mode (Athalye et al., 1993; Lamet al., 2000; Palm, 1994; Tanaka and Sugeno, 1992; Wanget al., 1996; Wu and Lin, 2000).

Genetic algorithms have become a standard fordesigning fuzzy logic controllers. The basic approachis to design the controller through closed loop simula-tion with the objective of minimising the output errorfor a sufficiently long time horizon.

The optimisation concerns the parameters of themembership functions of the premise and/or theconsequences and the rule base. All the induced rulesare implemented and rule base reduction is not includedin the genetic design.

In this work, a procedure for designing a Mamdanifuzzy logic controller including rule base reduction isintroduced. The rules are modelled with binary weightson which constraints are imposed in order to ensureconsistency. A simple genetic algorithm is used forfinding stabilising controllers that minimise the numberof rules.

ARTICLE IN PRESSK. Belarbi et al. / Engineering Applications of Artificial Intelligence 18 (2005) 875–880876

This paper is organised as follows: the second sectionintroduces a model of Mamdani FLC suitable for rulebase minimisation, the third section develops the geneticsolution procedure and the fourth section presents twoexamples.

2. Model of the Mamdani fuzzy controller

In order to design a Mamdani fuzzy logic with acompact rule base, a rule Ri is written as:

If X 1 is A1j1 and X 2 is A2j2 . . . then U is wikBk

Aljiand Bk, l ¼ 1 . . . n, jl ¼ 1 . . . nl , k ¼ 1 . . .m, are,

respectively, fuzzy sets associated with the n fuzzy inputvariables, each partitioned into nj fuzzy sets, and thefuzzy output variable, partitioned into m fuzzy sets.

wik, is a binary variable that determines the conse-quence of the rule, where subscript i refers to the ruleand subscript k refers to the output fuzzy set, i ¼ 1 . . . r,k ¼ 1 . . .m, with r ¼ n1 � n2 � � � � � nn; the total num-ber of candidate rules.

Thus, if wik ¼ 08k, k ¼ 1 . . .m, rule i has noconsequence and will not be included in the controllerrule base, in this case

Xm

k¼1

wik ¼ 0 8i ¼ 1 . . . r (1)

and if wik ¼ 1 for some k then the consequence of i is Bk,in this case

Xm

k¼1

wik ¼ 1 8i ¼ 1 . . . r. (2)

Given that a rule has at most one consequence,combination of (1) and (2) yields the followingconstraints on the wik:Xm

k¼1

wikp1 8i ¼ 1 . . . r. (3)

Since every time wik ¼ 1, a rule is included in the rulebase, the total number of rules included in the controllerrule base is given by

J ¼Xr

i¼1

Xm

k¼1

wik: (4)

The crisp output u of the fuzzy controller is computedwith the centre of gravity defuzzification formula

u ¼

Pmk¼1mkmBk

ðuÞPmk¼1mBk

ðuÞ(5)

with mk the centre of fuzzy set Bk.Using the product maximum inference mechanism

mBkðuÞ ¼ maxðmR1

w1k;mR2w2k;mR3

w3k; . . . ;mRrwnkÞ (6)

and

mRi¼ mA1j1

ðx1ÞmA2j2ðx2Þ . . . mAnjn

ðxnÞ. (7)

After multiplication by wik, the brackets of the left handside of Eq. (6) will contain only the rules withconsequence Bk.

In the case of triangular membership function,Aða; b; cÞ, the grade of membership can be written as

mAljðxÞ ¼ max 0;min

x � a

b � a;c � x

c � b

� �� �. (8)

When the fuzzy controller is introduced in a closed loop,the stability of the closed loop system depends on thetriplets (a, b, c) associated with the input variables fuzzysets, the centres of the output variable fuzzy sets dk, andthe binary variables wi,k defining the rule base. Thecontroller design problem can be expressed as finding allthese parameters in order to achieve stability, someperformance design criteria and compactness of the rulebase. This can be formulated as

minallða;b;c;dk ;wikÞ

Stabili sin gcontrollers

J ¼Xr

i¼1

Xm

k¼1

wik

!(9)

The stability of a given controller is determined throughclosed-loop simulation of the system for a sufficientlylong time.

In this work, the granularity of the input and outputfuzzy variables is left as an alternative to the designer.Indeed, since the alternatives are few, the designer maychoose these iteratively until a satisfactory solution isobtained. For instance, a 333 controller will have threefuzzy sets for all variables, a 553 controller will have fivefuzzy sets for the input variables and three for theoutput variables and so on.

3. The genetic solution

When implementing a simple genetic algorithm anumber of issues must be addressed (Michalewicz,1994), these include the coding procedure, the selectionprocedure, crossover, mutation. In the problem con-sidered here the handling of constraints (3) on the wik

must also be addressed.

3.1. Chromosome and coding

The chromosome is composed of two sub-chromo-somes: the first contains the parameters of the ante-cedents, here the triplets (a, b, c) associated with theinput variables fuzzy sets, and the centres of the outputvariable fuzzy sets dk, while the second contains thebinary weights wik. The chromosome is thus structuredas in Table 1. The chain wi1;wi2; . . .wim; in Table 1,which will be called rule chain in the sequel, defines theconsequence of rule i and at most one of its binary

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Table 1

Chromosome representation

First sub-chromosome Second sub-chromosome

a1; b1; c1; a2; b2; c2; . . . ; an; bn; cn; d1; d2; . . . ; dm, w11;w12; . . . ;w1m=w21;w22; . . . ;w2m= . . . =wn1;wn2; . . . ;wnm

m m m

K. Belarbi et al. / Engineering Applications of Artificial Intelligence 18 (2005) 875–880 877

components can be set to one. In Table 1 rule chains areseparated by a slash /.

A two-point crossover is used: the first point fallswithin the first sub-chromosome and the second pointwithin the second. In order to handle constraints (3),crossover and mutation in the second sub-chromosomeare carried out as follows:

�

Crossover: when crossover occurs in the second sub-chromosome, it is enforced at the beginning of a rulechain wi1;wi2; . . .wim; indicated by the up arrow inTable 1. There is hence an exchange of rule chainswithout altering their contents. This ensures thatconstraints (3) are satisfied provided they are satisfiedbefore crossover is carried out.

�

Mutation: If the mutation points falls within thesecond sub-chromosome, when a one is mutated to azero, constraint (3) is not violated, however when azero is mutated to a one, it may be violated. In orderto keep it satisfied, if there is already a one in the samerule chain, it is set to zero. The following example,with m ¼ 3, clarifies the procedure:

Consider the following chromosome before mutation:First sub-chromosome |010/100/yy./100.If mutation occurs within the second sub chromosome,then if mutation point is at k ¼ 2 of the first rule chain,we obtain after mutation:First sub-chromosome |000/100/yy./100.(3) is satisfied.However if mutation occurs at k ¼ 1 of the same rulechain, after mutation:First sub-chromosome |110/100/yy./100.(3) is violated, in order to maintain it satisfied, the bit atk ¼ 2, is set to zero, and after this correction:First sub-chromosome |100/100/yy./100.

3.2. Cost function calculation

Taken as it is, cost function (9) is meaningless, sincethe search procedure may always produce a controllerwith zero rules. In order to ensure that the search takesinto account closed-loop stability, it is changed to

minStabili sin gcontrollers

J ¼ CXm

i¼1

Xm

k¼1

wik

!, (10)

where C is a stability/performance coefficient.

The cost function is computed in two steps:

(i)

For every chromosome in the current population,the sum part of (9) is computed by summing all thew of the consequence part of the chromosome, thisgives the number of rules in the rule base. If the sumis zero, then the cost is given a high prohibitivevalue, M0.

(ii)

The coefficient C is determined through closed-loopsimulation and may take three values:

�

for a controller with an unstable closed-loop systemC ¼ M1,

�

for a controller with a stable closed-loop system thatdoes not satisfy some minimum performance re-quirements C ¼ M2,

�

for a controller with a stable closed-loop systemsatisfying the minimum performance requirementsC ¼ M3.

Clearly M3oM2oM1, and choosing M1 very highrelatively to M2 and M3, an unstable controller willhave a very high cost function. On the other hand,among controllers producing the same closed-loopperformances, the one with the least number of ruleswill always be privileged.

4. Simulation results

4.1. Example 1: Control of the pole and cart system

The pole and cart system is a classical benchmarkproblem in the fuzzy controller design literature(Bastian, 2000; Karr, 1991). The control objective is tobalance the pole by applying a force on the basis of thecart. Although simple in nature, it has interestingfeatures for controller benchmarking: it is highly non-linear when far from the vertical equilibrium and issensitive to parameters variations such as initial condi-tions, pole length and mass.

The controller has two inputs, the pole angle yðtÞ andits variation DyðtÞ ¼ yðtÞ � yðt � 1Þ and the output is theforce F to be applied to the cart.

The input and output variables fuzzy sets are equalisosceles triangles evenly distributed in the universe ofdiscourse, and centred at the origin as in Fig. 1, thesewill be referred to by ITED. This partitioning iscommon in fuzzy control and ensures both distinguish-ability and completeness of the fuzzy inference system.

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40°

K. Belarbi et al. / Engineering Applications of Artificial Intelligence 18 (2005) 875–880878

Clearly, it suffices to have the basis of one triangle foreach fuzzy variable to derive the distribution of all fuzzysets on the universe of discourse. For the pole angle yðtÞ,the basis of the triangular fuzzy sets is Ly, for itsvariation, LDy, and for the force LF. Three fuzzy sets:NEGATIVE, ZERO, POSITIVE were used for all fuzzyvariables. The first sub-chromosome will contain Ly,LDy, and LF, the parameters defining the fuzzy sets, thesecond sub-chromosome will contain the wik. Each rulechain is composed of three binary variables, wi1wi2wi3,corresponding to the three output fuzzy sets. Given arule I, the following situations may arise:

If wi1 ¼ 1, wi2 ¼ 0, wi3 ¼ 0, then the fuzzy consequenceof the rule is NEGATIVE.If wi1 ¼ 0, wi2 ¼ 1, wi3 ¼ 0, then the fuzzy consequenceof the rule is ZERO.If wi1 ¼ 0, wi2 ¼ 0, wi3 ¼ 1, then the fuzzy consequenceof the rule is POSITIVE.If wi1 ¼ 0, wi2 ¼ 0, wi3 ¼ 0, then the rule is not includedin the rule base.

For the genetic algorithm, the minimisation problem istransformed into a maximisation:

Max f ¼1000

1þ J(11)

with M0 ¼ M1 ¼ 103, M2 ¼ 10, M3 ¼ 1.The unstable controllers will have a smaller fitness

compared with the stable ones, and a lower probabilityof reproduction.

The performance constraints are on the maximumovershoot, set to 5%, and the maximum rise time set to2 s. The cost function is computed through closed loopsimulation as indicated in Section 3, with a nominalmodel having a pole with mass m ¼ 0:1Kg and length1 ¼ 1m and a cart with mass mc ¼ 1Kg. The initialconditions are: yð0Þ ¼ 30 and _yð0Þ ¼ 0=s:

The genetic algorithm was run for 100 generationswith roulette–wheel selection. The two best individualsof the current population are automatically inserted in

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0Li

Fig. 1. Distribution of the fuzzy sets for the fuzzy controller.

the next. The initial population was generated such thatconstraints (3) are satisfied for all individuals. Theparameters were specified as:

Population size, Psize ¼ 50,Crossover probability, pc ¼ 0:85,Mutation probability pm ¼ 0:01.

The last generation contained a set of stable andperformances satisfying solutions with controllers hav-ing five to nine rules. A five rules controller wasextracted for analysis. The parameters of the controllersare: Ly ¼ 37:91; LDy ¼ 6:015; LF ¼ 300:0.

The five rules are:

If yðtÞ is NEGATIVE and its variation is ZERO thenForce is NEGATIVE.If yðtÞ is ZERO and its variation is NEGATIVE thenForce is NEGATIVE.If yðtÞ is ZERO and its variation is ZERO then Force isZERO.If yðtÞ is POSITIVE and its variation is ZERO thenForce is POSITIVE.If yðtÞ is POSITIVE its variation is POSITIVE thenForce is POSITIVE.

The controller was tested for conditions other than thenominal. The results are summarised in Fig. 2. Thecontroller was particularly sensitive to the pole lengthwith success for 0:4oLo1:5m and the initial conditionswith success for �181oyð0Þo351.

4.2. Example 2: Concentration control in a CSTR

The exothermic continuously stirred tank reactor,CSTR, considered here is taken from (Shacham et al.,1994) and described by the following two nondimen-sional mass and heat balances:

_x1 ¼ �x1 þ Dað1� x1Þex2=1þx2=g (12)

30°

20°

10°

0°

-10°

-20°0 2.5 5 7.5 10

time in seconds

pole

ang

le in

deg

rees

12.5 15 17.5 20

Fig. 2. The pole angle for various conditions.

ARTICLE IN PRESS

0.7

0.65

0.6

0.55

0.5

0.45

0.4

0.35

conc

entr

atio

n

K. Belarbi et al. / Engineering Applications of Artificial Intelligence 18 (2005) 875–880 879

_x2 ¼ �x2 þ BDað1� x1Þex2=1þx2=g � bðx2 � qcÞ (13)

assuming a first-order exothermic reaction A ! B;where x1 is the concentration of reactant A; x2 istemperature inside the reactor and qc is the coolant flowrate. Table 2 gives the nomenclature list. The parametersare given as B ¼ 21:5, g ¼ 28:5, D ¼ 0:036 and b ¼ 25:2.All variables are dimensionless and normalised (Shac-ham et al., 1994).

The objective is to control the concentration x1 bymanipulating the coolant flow rate qc. Integral action isadded to the fuzzy controller in order to eliminate offset.The inputs to the controller are the error eðtÞ betweenthe measured concentration and the reference and itsvariation DeðtÞ ¼ eðtÞ � eðt � 1Þ: The output is thecontrol increment DqcðtÞ. As in the previous example,three ITED fuzzy sets, NEGATIVE, ZERO, POSI-TIVE, are used for all variables, with Le, LDe, LDqc

;respectively, the basis of the triangular fuzzy sets for theerror, its variation and the control increment.

The parameter C in the cost function was set to thesame values as in the previous example. The maximumovershoot was set to 5% and the maximum rise time setto 2min.

The set point was varied in three steps: x1 ref ¼f0:6; 0:7; 0:6g; each reference step lasted 10min. Thegenetic algorithm was run with the same setting andparameters as in the first example. The following fiverules controller was extracted from the last generation:

Le ¼ 0:216; LDe ¼ 0:100000; LDqc¼ 0:9152;

with rule base:

If error is NEGATIVE and its variation is ZERO thenthe control increment is NEGATIVE.

Table 2

Nomenclature list for the CSTR

C Reactant concentration

Cf Feed concentration of reactant

x1 Dimensionless concentration x1 ¼ ðCf � CÞ=Cf

T Reaction temperature

T c Coolant temperature

T f Feed temperature

Tj0 Nominal feed temperature

g Dimensionless activation energy g ¼ Ea=RTj0

x2 Dimensionless temperature x2 ¼ ðT � Tj0Þg=Tj0

qc Dimensionless coolant temperature qc ¼ ðT c � Tj0Þg=Tj0

B Dimensionless heat of reaction B ¼ �DHCfg=CpTj0, DH heat of

reaction

Da Damkohler number Da ¼ Vkaeg=Qf

V Reactor volume

b Dimensionless cooling rate b ¼ UA=QfCp

A Heat transfer surface area

Cp Heat capacity

U Overall heat transfer capacity

Qf mass feed flow rate

If error is ZERO and its variation is NEGATIVE thenthe control increment is NEGATIVE.If error is ZERO and its variation is ZERO then thecontrol increment is ZERO.If error is POSITIVE and its variation is ZERO then thecontrol increment is POSITIVE.If error is POSITIVE its variation is POSITIVE then thecontrol increment is POSITIVE.

The extracted controller was tested in closed loop with areference x1 ref ¼ f0:4; 0:5; 0:6; 0:7; 0:6; 0:5g; and withparameter variation: b ¼ 28 at time equal 40min. Figs. 3and 4 show respectively the concentration and thecoolant flow rate. There is no offset and the rise timeand overshoot are practically the same for all set points,which indicate good robustness property. An under-shoot is apparent in the concentration after theparameters variation indicating a change of the closedloop dynamics.

0 10 20 30time in minutes

40 50 60

Fig. 3. The concentration in the CSTR.

5

4.5

4

3.5

3

2.5

20 10 20 30

time in minutes

cool

ant f

low

rat

e

40 50 60

Fig. 4. The coolant flow rate.

ARTICLE IN PRESSK. Belarbi et al. / Engineering Applications of Artificial Intelligence 18 (2005) 875–880880

5. Conclusion

A design procedure for Mamdani fuzzy logic con-troller including rule base minimisation is proposed. Therules are modelled with binary weights on whichconstraints are imposed in order to ensure consistency.A genetic algorithm is used for finding stabilisingcontrollers that minimise the number of rules. The costfunction includes a stability/performance coefficientwhich insures that stable performance satisfying con-trollers are given the highest possible fitness. Thenumber of fuzzy sets for the input and the controlvariables are set by the user and the design procedure isconcerned only with the rule base and the distribution ofthe fuzzy sets in the universes of discourses. Twoexamples were studied: the control of the pole and cartsystem and the control of the concentration in CSTR. Inboth cases, the fuzzy sets were isosceles triangles evenlydistributed, in the universe of discourses.

The optimisation procedure produced compact fuzzycontrollers with five rules having relatively goodrobustness properties.

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