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372 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 14, NO. 3, JUNE 2006

Genetic Algorithm-Based Optimal Fuzzy ControllerDesign in the Linguistic Space

Chih-Hsun Chou

AbstractIn this paper, a genetic algorithm (GA) based optimalfuzzy controller design is proposed. The design procedure is ac-complished by establishing an index function as the consequentpart of the fuzzy control rule. The inputs of the controller, afterscaling, are utilized by the index function for computing the outputlinguistic value. This linguistic value can then be used to map thesuitable fuzzy control actions. This proposed novel fuzzy controlrule has crisp input and fuzzified output characteristics. The indexfunction plays a role in mapping thedesired fuzzy sets for defuzzifi-cation resulting in a controlled hypersurface in the linguistic spaceformed by the input fuzzy variables. Two types of index functions,both linear and nonlinear, are introduced for controlling systems

with different degrees of nonlinearity. The parameters of the indexfunction are obtained by applying a simple GA with a suitablefitness function. Various controlled systems result in various pa-rameter sets depending on their dynamics. Under the acquired op-timal parameter set the optimal index function can be used to gen-erate the desired control actions. Several simulation examples aregiven to verify the performance of the proposed GA-based fuzzycontroller.

Index TermsControl hypersurface, fuzzy controller, geneticalgorithms (GAs), index function, linguistic space.

I. INTRODUCTION

MANY complex control problems can often be trans-formed into numerical type functional optimization

problems. If the solution space of the optimization problem

is simple, conventional nonlinear optimization techniques are

capable of obtaining a satisfactory solution set. In many cir-

cumstances, however, the solution space is highly dimensional,

discontinuous or noisy. In such a case, conventional optimiza-

tion techniques easily arrive at a locally optimal solution and

usually lack robustness. One technique that is both robust

and global over a broad spectrum of problems is based on

genetic algorithms (GAs). GAs are search procedures based on

the mechanics of natural genetics that combine a Darwinian

survival-of-the-fittest strategy to eliminate unfit characteristics

with random information exchange exploiting knowledgecontained in old solutions.

Modern fuzzy controllers exhibit superior applicability

[1][3] and display considerable robustness [4][6] in com-

parison to the traditional ones. The former is based on control

Manuscript received November 19, 2002; revised June 17, 2004 andSeptember 12, 2005. This work was supported in part by the National ScienceCouncil of Taiwan, Republic of China, under Grant NSC90-2213-E-216-011.

The author is with the Department of Computer Science and Informa-tion Engineering, Chung-Hua University, Taiwan 300, R.O.C. (e-mail:[email protected]).

Digital Object Identifier 10.1109/TFUZZ.2006.876329

TABLE IMAMDANI TYPE FUZZY CONTROL RULES

rules consisting of conditional linguistic statements based

on the relationship between input and output variables. This

gives it the enticing advantages of emulating the behavior of a

human operator as well as dealing with model uncertainty. In

the design of a fuzzy controller the definition of membership

functions and the establishment of control rules are usually

subjective. Adding the difficulty of having to tune the scaling

factors (SFs), to an unsystematic design procedure usually

makes it impossible to achieve adequate performance because

the controlled process is so complex. To overcome this flaw,

a lot of research uses GAs to be able to optimally set the

parameters of the fuzzy controller.GAs were first introduced by J. Holland as search algorithms

and have since been analyzed and extended by De Jong, Gold-

berg, et al. [7], [8]. One of these extensions is its use in optimal

fuzzy controller design [9][11]. The main purpose of using

GAs in the design of a fuzzy controller is not only to use the

robust and global benefits of GAs, but also to develop a system-

atic design approach for the fuzzy controller.

The main steps in the design of a fuzzy controller in-

clude constructing the control rules, stating the membership

functions, and tuning the SFs. To accomplish these steps,

various approaches such as trial-and-error, heuristic, model

following, and neural network based methods have been pro-posed [12][15]. Regarding GA-based design approaches,

fuzzy controllers can be classified as Mamdani type or TSK

type depending on the fuzzy control rules. An example of the

Mamdani type fuzzy control rules with two inputs and a single

output is shown in Table I. For this type of control rules, both

condition and consequent parts are described by fuzzy sets such

as NB (negative big), NM (negative medium), NS (negative

small), ZE (zero), PS (positive small), PM (positive medium),

and PB (positive big). If the consequent parts of the fuzzy rules

in Table I are replaced by a linear equation of the inputs such

as , where denotes

the th input with coefficient , then the rules are the TSK

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CHOU: GENETIC ALGORITHM-BASED OPTIMAL FUZZY CONTROLLER DESIGN 373

Fig. 1. PD type fuzzy controller.

type. Hence, a fuzzy-PID (PD, PI) controller with the following

control rules:

if is and is and is

then output is (1-1)

where , and denote the error, change in error and sum of

error between the desired output and the actual output, respec-

tively, can be regarded as a special TSK type fuzzy controller.

In this paper, a new type of GA-based fuzzy controller which

has the characteristics of crisp input and fuzzified output is pre-

sented. The rest of this paper covers the following: Section IIgives a literature survey and the problem formulation. In Sec-

tion III, the structure of the developed fuzzy controller is de-

scribed. The proposed GA-based design procedure, both linear

and nonlinear, is introduced in Sections IV and V, giving several

simulation examples. Section VI gives the conclusion.

II. LITERATURE SURVEY AND PROBLEM FORMULATION

In this section, the two types of GA-based fuzzy controller de-

sign mentioned in Section I are briefly surveyed and the problem

that this paper deals with is formulated.

A. A Brief Survey of the GA-Based Fuzzy Controller DesignIn the Introduction, the GA-based fuzzy controller design was

categorized into the Mamdani type and the TSK type, in the

following the related studies are examined.

1) GA-Based Mamdani Type Fuzzy Controller Design: The

three most commonly used membership functions are the trian-

gular, trapezoidal and Gaussian shape membership functions.

The first two are defined by the parameters of center point,

edge point and width [16], [17], while the third is defined by its

mean and standard deviation. In the case of the Mamdani type

fuzzy rules with two inputs and a single output, for example,

if a Gaussian shape membership function is applied, each rule

can then be specified by the three pairs of mean and standarddeviation. Putting these six parameters together forms a mem-

bership function chromosome [9], [18][31], making it prac-

tical for determining the optimal membership functions by GAs.

In the search of an optimal fuzzy rule table the names of the

output fuzzy sets from NB to PB used in the consequent part of

the fuzzy rules are indexed by integers from 1 to 7. By coding

each of these seven integers with three bits, a rule chromosome

with 147 bits can be used to represent the 49 fuzzy rules in the

rule table such as the one shown in Table I. In this way, an op-

timal rule table can be found by applying GAs [23], [25], [30],

[32][40]. Determining the SFs by GA is straightforward. The

SFs are usually coded directly into a string for evolution [ 41].

Determining only some of the parameters as above usuallyonly leads to a local optimal solution. A chromosome that can

be used to simultaneously search these parameters is much more

practical. Hence, combinations of the previous chromosomes

are presented. In [25], [42][47], the membership function chro-

mosome and the rule chromosome are combined. Combinations

of the scaling factor chromosome with the rule chromosome can

be found in [48] and [49], while integrating all of them is pro-

posed in [50] and [51]. In addition to the previous studies, sim-

ilar ideas were also applied to determine the condition parts of

the fuzzy rules [52], [53] or to select the necessary fuzzy rules

[54], [55].

2) GA-Based TSK Type Fuzzy Controller Design: A TSK

type fuzzy rule has the following form:

if is and is and and is

then output is

(2-1)

where are f uzzy sets. D esign o f this t ype of c on-

troller can be accomplished by coding the membership func-

tions in the condition part and the coefficients in the consequent

part into a chromosome. Coding of the membership functions

is the same as that of the Mamdani type, the coding of the co-

efficients is analogous to that of the SFs. In this way, the best

standard deviation of the applied Gaussian membership func-tions can be found [56], the optimal coefficients can be searched

[57][59],and both[60][66]. As stated before, if the fuzzy rule

is in the form of (1-1), then the TSK type fuzzy controller be-

comes a fuzzy-PID (PD, PI) controller. The design procedure

of such a controller is the same as that of the TSK type. Hence,

the previous idea is also applied in [67] to find the membership

functions and coefficients of a fuzzy-PID controller.

Both input and output variables are fuzzified for the Mamdani

type; fuzzy input variables and crisp output variable are utilized

for the TSK type. Nevertheless, if the fuzzy sets of the output

variable of the Mamdani type fuzzy controller are defined as

fuzzy singletons, and the coefficients , in the

consequent part of the TSK type fuzzy controller are all set as0, then there is no difference between these two controllers.

B. Problem Formulation

Consider a Mamdani PD-type discrete-time fuzzy controller

whose inputs and (Fig. 1) are defined as

(2-2a)

(2-2b)

where and denote the applied set point and the plant output

of the discrete-time system, respectively. The fuzzy variables for

the input are and and output variable is . The controlrules established by using these variables are shown in Table I.

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Fig. 2. Applied membership functions.

The seven fuzzy sets used for all three fuzzy variables are de-

fined by the membership functions shown in Fig. 2.

The control rules in Table I were based on the characteris-

tics of the step response, stating the state-action relationship

in a linear way. The membership functions in Fig. 2 are con-

structed intuitively. Furthermore, the tuning of the scaling fac-

tors in Fig. 1 is usually done without using a formula. In order to

systematically design an optimal fuzzy controller, the GAs were

integrated in the design procedure as mentioned in the previous

sections. The GA-based design procedure proposed in this study

requires a novel fuzzy rule structure, which will be described inthe next section.

III. PARAMETRIC TYPE FUZZY CONTROL RULES

What follows is a Mamdani PD-type fuzzy control rule

example

if is and is then is (3-1)

This type of fuzzy rule, though friendly to human beings, is not

practical for tuning. So, this linguistic type fuzzy control rule

was modified into a parametric form


where denotes undefined to be explained later and is a

linear or a nonlinear function of . Then, can be utilized as

an index function for mapping the desired control action. For

example, , then the

fuzzy rule is a Mamdani type; if , then

it is a TSK type.

The main operations of a fuzzy controller include fuzzifica-

tion, inference and defuzzification. During the control process,

the controller inputs are scaled for fuzzification before the rule

inference, so the subscript of . was defined as a function of

the input scaling factors and as well as the controller

inputs as follows:


can then be used to map the appropriate output

fuzzy sets. For e xample, i f equals 4 , according

to the output membership functions defined in Fig. 2, it corre-

sponds to the rule:


If equals 3.2, the two fuzzy rules


and


are mapped with membership grades of about 0.4 and 0.8 in-

dividually. The defuzzification procedure, for example of the

center average defuzzifier, is then computed by

(3-7)

in which 2 and 4 are the center values of the output fuzzy sets

appearing in the consequent parts of (3-5) and (3-6). This de-

fuzzified result is then scaled by the output , say , for

generating the desired control signal .From the previous description it can be seen that only the

output variable was fuzzified. The input values, after being

scaled by the input s, are utilized by the index function for

computing the output linguistic value. This linguistic value can

then be used to map the suitable fuzzy control actions. Hence,

the proposed novel fuzzy control rule has crisp input and

fuzzified output characteristics which differ from the Mamdani

and TSK types. The fuzzification procedure for the inputs is

not necessary, therefore the fuzzy set in the condition part of

the fuzzy rule is replaced by , which means it is undefined,

as shown in (3-2). At the same time, the consequent part of

the proposed method is a real-valued function of the scalingresults of and , named and , rather than a fuzzy

set (Mamdani type) or a function of and (TSK type). For

this method two adequate fuzzy sets, as well as the individual

membership grades, were mapped by the real value index

function. These two fuzzy sets, along with the corresponding

membership grades facilitate defuzzification. Fuzzy rules that

map inputs to two output fuzzy sets can also be called rules

with double consequence as proposed in [68] and [69].

IV. GA-BASED PARAMETRIC FUZZY CONTROLLER

DESIGNTHE LINEAR CASE

As stated in Section III, the subscript in is

a function of , , and . For simplicity, in this paper,

we define it as the following linear function:

(4-1)

where and are the scaled re-

sults of the input variables and , respectively. Because the

universal discourses of the input fuzzy variables and are

both defined as [ ], and are also limited by [ ].

Function can be viewed as a linear function defined in the lin-guistic space formed by and .

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TABLE IIRANGES OF THE PARAMETERS USED IN THE LINEAR INDEX FUNCTION

A. The Linearly Parametric Fuzzy Controller

In the linear case, the index function was

defined as

(4-2)

This index function plays a role to map the output fuzzy sets

defined by the membership functions shown in Fig. 2, it is also

limited by the following limiter:

if

if

if .

(4-3)

In this case, designing the fuzzy controller becomes

an issue of searching the optimal parameter set

. In the following, GAs

are applied to determine the parameter set.

B. GA-Based Linearly Parametric Fuzzy Controller Design

Focusing on the performance of the proposed novel structure,

the simple GA without any improvement was applied. Since the

simple GA works with binary coding chromosomes, the six pa-

rameters , , , , and were encoded into a

binary string. The corresponding searching ranges of these pa-rameters had to be defined. For , because itmappedthe input

variable to the linguistic variable error , a larger

would map the same to a larger , resulting in a larger con-

trol action. Hence, reflected the degree oferror tolerance

of the fuzzy controller. Similarly, mapped the input vari-

able to the linguistic variable change in error , if the

desired output was defined as a fixed set point, the change in

error was equivalent to the output variation. This variation mea-

sured the speed with which the plant output approached the

set point. A larger mapped the same to a larger ,

hence, reflected the degree ofoutput variation tolerance

of the fuzzy controller. The SF was used to determine themagnitude of the control signal. The setting of this parameter

depended on the controlled plant and the set point. Finally, the

coefficients , and were the weights of the three terms

, , and bias in the index function. The searching

ranges of these parameters (Table II) were set according to the

above. Setting suitable search ranges had the advantage of im-

proving the search efficiency.

Once the search ranges were restricted, setting the chro-

mosome length depended on the precision required. Short

length saves computation time while long length provides

higher precision. In this study, six bits were used to code each

parameter. On examining the performance of the proposed

fuzzy controller, the simple GA without any enhancement wasapplied to determine the parameters. New chromosomes were

TABLE IIIPARAMETERS OF THE APPLIED SIMPLE GA

generated by crossover, mutation and random replacement with

every generation. The selection policy for crossover was ac-

complished by selecting the top-ranking of the chromosomes

and the arbitrary of the remaining chromosomes. This pair

chromosomes were used to generate pairs of new chromo-

somes by uniform crossover. Three basic crossover operstors

including single-point, two-point and uniform crossovers are

widely used. The uniform crossover applied in this study was

shown to be better than the other two [70]. chromosomes

were randomly picked from these new chromosomes (one

chromosome from each pair) to replace the arbitrary selected

chromosomes. Each chromosome, except the top-ranking

and the worst ones had a probability for mutation. A randomly

selected bit of each parameter was mutated. For random re-

placement, all the bits of the worst chromosome were replaced

by random values. In binary GA, the crossover operator plays

the major role in producing new chromosome in the search

space. The mutation process introduces randomness into the

reproduction process providing an opportunity for escaping

from local optimal. Hence, the crossover probability of a binary

GA is usually much higher than the mutation probability [ 71].The determining of the population size, ranging from 30 to

110, was done according to the suggestion of [72] and [73].

The parameters of the applied simple GA are summarized in

Table III. In Section V, some comparison simulations are given

to demonstrate the sensitivities of these parameters.

C. About the Fitness Function

In this study, the applied fitness function was defined as:

fitness

(4-4)

Function (4-4), when , becames

error

error (4-5)

This special case of , error , is analo-

gous to the square-of-error fitness function. To see the difference

between error and , consider the fol-

lowing corresponding fitness equations:

errorerror (4-6)

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TABLE IVOPTIMAL PARAMETERS FOR THE OSCILLATORY SYSTEM

(4-7)

Because , , and are always nonnegative, their ranges

for error are all . For , the range

of is also , while the ranges of , and are all

instead. In other words, the ranges of , , and

can be determined by varying . Under the same fitness value

, , when and , often ex-

hibits an output response with smaller , and and larger

than that of error . In this way, the goal of lower degrees

of undershoot , overshoot and steady state error

compared to the rise time can be stated by setting the pa-rameters , , , and in (4-4).

D. Simulation Example

For the linear case, a second-order oscillatory system, with

oscillating property near the steady state, is applied. The con-

troller had to handle high degrees of overshoot, undershoot and

steady state error. The transfer function of this plant was [74]

(4-8)

The following fitness function was utilized:

(4-9)

The obtained parameter set and the figure of the corre-

sponding index function are shown in Table IV and Fig. 3,

respectively. Because the index function used in this section

was the linear type, the acquired index function resulted in

linear plane in the linguistic space formed by and .

The output response of this system is shown in Fig. 4. Asshown in the figure, almost no oscillation phenomenon occurs

in the output trajectory. This smooth trajectory is achieved

by the control signal trajectory shown in Fig. 5, which keeps

at a specific value at steady state. To check the utility of the

modified fitness function (4-9), the simulation results by using

both fitness functions (4-9) and (4-5) were compared as shown

in Fig. 6. In this comparison, both approaches were run for

ten times. The ten output trajectories controlled by using the

modified fitness function (4-9) are shown in Fig. 6(a), while

Fig. 6(b) shows the results by using the fitness function (4-5).

It can be found that the trajectories shown in Fig. 6(b) exhibit

more degrees of overshoot, undershoot and steady state error

than those in Fig. 6(a). This comparison gives a verification ofthe benefit of using the modified fitness function.

Fig. 3. Graph of 9 corresponding to the parameters inTable IV.

Fig. 4. Output response of the oscillatory system.

Fig. 5. Control signal trajectory for the oscillatory process.

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Fig. 6. Output responses of the oscillatory system controlled by using(a) fitness function (4-9) and (b) fitness function (4-5).

V. GA-BASED PARAMETRIC FUZZY CONTROLLER

DESIGNTHE NONLINEAR CASE

The index function (4-2) was defined as a linear function ofand . Note that, although this function was linear with

respect to and , the overall fuzzy controllerwasnonlinear

to the inputs. For the nonlinear case, the index function was

defined asa nonlinear functionof and . Beforepresentingit, the dynamics of the linear index function were analyzed.

Property 1: The dynamics of (4-2) is a linear combination of

the dynamics of and , in which the coefficients of the linearcombination are proportional to the parameters , , ,

and .

Proof:

1) For , , .

2) For

if and

if andif and

otherwise

The dynamics of the index value in (4-2) were linearly related

to the dynamics of the inputs. A controlled plant with high non-

linearity usually exhibits dynamics that are highly nonlinear re-

lated to the dynamics of its inputs. So, it was necessary to con-

struct a nonlinear type index function.

A. GA-Based Nonlinearly Parametric Fuzzy Controller Design

The linear index function resulted in a linear hyperplane in

the linguistic space formed by and . To have nonlineardynamics, this hyperplane had to be nonlinear. To generate such

Fig. 7. Plots of the basis functions withs

equal to0

0.8, 0 and 5.

Fig. 8. Various nonlinear index functions.

an index function, the th basis function in the linguistic space

was defined as

if

if(5-1)

in which

(5-2)

Theparameter was theupperlimitof theindexfunction, which

equaled the upper limit of , was a parameter belonging to

[ ] and . is given in (4-3). Examples of basis functions

corresponding to various were plotted as in Fig. 7. In Fig. 7,

the basis function transformsthelinearindex function

into the nonlinear type. The dynamics of depend on the

value of as well as . For equals 0, , being the

linearindex function. For larger (smaller) , had a larger

(smaller) slope when nearing the origin and a smaller (larger)

slope when nearing the boundaries. This slope-varying property

gave the possibility of designingan index function withnonlinear

dynamics.The basisfunctions, however, always had nonnegative

slopes. Hence, two or more appropriate basis functions wereused to construct the desired nonlinear index function. In this

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TABLE VSEARCHING RANGES OF THE PARAMETERS IN THE NONLINEAR INDEX FUNCTION

TABLE VI

OPTIMAL PARAMETERS OF THE NONLINEAR INDEX FUNCTION FOR THE DAMPING SYSTEM

study, two basis functions were used to construct the index

function shown in (5-3) at the bottom of the page, in which deg

is a parameter. The linear index function in (4-2) is a special

case of this index function ( , ).

Equation (5-3) could generate much more desirable nonlinear

dynamics under suitable settings of , and deg (Fig. 8). Re-

ferring to Fig. 8, the nonlinear index function was

highly nonlinear related to , which means that the dynamicsof were highly nonlinear with respect to the dynamics

of and . More basis functions could be used to construct a

more fittingindex function.Using morebasis functions,however,

resulted in more parameters to search. The three extra parame-

ters in (5-3) were also searched by GAs and the total number of

searching parameters became 9. The searching ranges of these

nine parameters are listed in Table V. The parameters of the

applied simple GA were the same as those used in Section IV.

B. Simulation Results

Compared with the linear index function, the nonlinear type

consumed three more parameters for the nonlinear dynamics.

These three parameters, introducing an additional 18 bits in thechromosome, made the searching space times that of the

linear type. Which, however, did not consume much searching

time because the nonlinear index function provided a higher

degree of freedom to acquire the desired dynamics as observed

in the subsequent simulations. The simulation examples ap-

plied included a second-order damping system and an inverted

pendulum system. This subsection focuses on examining the

control results of the proposed nonlinear type method. In the

next subsection, we give the comparisons between the proposed

GA-based fuzzy controller and other approaches.

Simulation Example 1: The Second-Order Damping

SystemThe damping system applied here has transfer func-

tion of the form

(5-4)

whose damping property complicates the controller

design [74]. The fitness function (4-9) is also used

Fig. 9. Nonlinear index function for controlling the second-order dampingsystem.

Fig. 10. Output response of the damping system controlled by using thenonlinear index function.

here. First, the obtained parameters of the nonlinear

index function are shown in Table VI. The graph of this

index function is displayed in Fig. 9, exhibiting a high

degree of nonlinear dynamics. The output response

(5-3)

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Fig. 11. Variation of the minimum fitnessthe damping process. (a) Linear case. (b) Nonlinear case.

TABLE VIIOPTIMAL PARAMETERS OF THE NONLINEAR INDEX FUNCTION FOR THE INVERTED PENDULUM SYSTEM

when using this index function is shown in Fig. 10.

To check the searching time, the required evolution

generations for obtaining the optimal parameters were

observed. The variations of the minimum fitness when

using both the linear and the nonlinear index functions

are shown in Fig. 11. Note that the ranges of -axes

in both cases are [0 200] and [0 300]. Fig. 11 displaysthe less required computation time of the nonlinear

type, it is because the three extra parameters gave the

nonlinear index function a greater degree of freedom

for constructing a suitable control hypersurface.

Simulation Example 2: The Inverted Pendulum SystemThe nonlinear index function was introduced for

dealing with the nonlinear dynamics of the controlled

plant. So, an inverted pendulum system of the form

(5-5)

with highly nonlinear dynamics was applied for sim-

ulation. In (5-5), denotes the angle of the inverted

pendulum, is the acceleration of gravity, (mass

of cart) equals 1.0 kg, (mass of pole) equals 0.1 kg,

(half length of pole) is 0.5 m, and is the applied

force in newtons. The fitness function (4-9) is still ap-plied. For checking the ability of treating nonlinear

dynamics, the initial angle of the inverted pendulum

was set as a degree of 50. The parameters of the non-

linear index function obtained by the applied simpleGA are listed in Table VII. The graph of this index

Fig. 12. 3-D graph of the nonlinear index function for controlling the inverted

pendulum system.

function is shown in Fig. 12. The smooth hypersur-face shown in Fig. 12 reflects the smooth nonlineardynamics (of the trigonometric functions) of the in-

verted pendulum system. In Fig. 12, the inferred index

value at every control step is marked. For example, the

mark5(4.9394, 1.1695, 1.869) indicates that at control

step 5, the values of and , 4.9394 and 1.1695,

were mapped to an index value of 1.869 according to

the index function. These marks form a control signal

trajectory on the control hypersurface in the linguistic

space, as shown in the figure. These index values, after

the defuzzification procedure and the scaling by ,resulted in a control signal trajectory in the time space,

as plotted in Fig. 13. Finally, the output response of theinverted pendulum system is shown in Fig. 14.

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Fig. 13. Control signal trajectory in the time space.

Fig. 14. Output response of the inverted pendulum system.

C. Comparison Simulations

In Section II, the Mamdani and TSK types GA-based fuzzy

controller design was briefly discussed. In this subsection, some

comparisons are made among the proposed method and these

two approaches. The inverted pendulum system was used as the

controlled plant because of its nonlinear property. For objec-tivity, every method was run 50 times, and each run lasted 200

generations. In addition, both the population size and the se-

lection policy were taken into consideration. The comparisons

were made by examining the following four items:

the minimum fitness of the chromosomes in the popu-

lation at the end of each run;

the average fitness of the chromosomes in the popula-

tion at the end of each run;

the average of the minimum fitness values of all

runs;

the average of the average fitness values of all

runs.

Case 1: The Proposed Linear-Type Method With DifferentPopulation Sizes and Selection Policies: When applying the

TABLE VIIIVALUES OF

f

AND f

FOR VARIOUS POPULATION SIZES( l = 1 )

TABLE IXVALUES OF

f

AND f

FOR VARIOUS POPULATION SIZES( l = 5 )

Fig. 15. Curves off

andf

with various population sizes( l = 1 )

.1

:Linear-type; : TSK type; : Mamdani type.

GAs, the population size and the selection policy are two impor-tant factors influencing the searching efficiency. The population

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Fig. 16. Curves of f and f with various population sizes ( l = 5 ) . 1 :Linear-type; : TSK type; : Mamdani type.

size, according to the suggestion of [72] and [73], ranged from

about 30 to 110. So, population sizes of 20, 40, 80, and 120

were applied in the simulations for comparison. For selection

policy, the number of top-ranking chromosomes , as mentioned

in Section IV-B, was set as 1 and 5 for the sake of comparison.

Table VIII shows and with respect to various popula-

tion sizes. It can be found that, as the population size increased,

decreased slighly while increased. The reason is that

the crossover of 1 pair with every generation is not enough for

a population size of 80 or 120.

The crossover of one pair only does not seem enough for a

larger population size, so was also set as 5 in the simulations.

The values of for various population sizes are shown in

Table IX. Comparing Table VIII and Table IX, it was found that

the values of corresponding to different population sizes

under the settings of and , are similar. On theother hand, the value of , under , are much smaller

TABLE XVALUES OF

f

AND f

UNDER VARIOUS POPULATION SIZES

for the larger population sizes. This shows that is more

compatible with population size of 80 or 120 than . It

concludes that a larger population size requres a larger value

of .

Case 2: Comparisons of the Mamdani Type, TSK Type and

Proposed Linear-Type Methods: The performances of the pro-

posed linear-type method with the Mamdani and TSK types

were compared first. The comparison of the nonlinear-type is

given next. Figs. 15 and 16, corresponding to the parameter

and , show the curves of and for the

three methods under various population sizes. The curves of thelinear-type method exhibit smaller variations than that of the

other two methods, which means that the linear-type method is

more stable than the other two. The values of and , cor-

responding to and , are shown in Table X. It displays

that the linear-type method is superior to the Mamdani type, and

the Mamdani type is better than the TSK type. The TSK type,

however, exhibits better stability than the Mamdani type.

Usually, a larger searching space requires more evolution

generations, which is time-consuming. In our simulations the

number of evolution generations is set as 200. The searching

space of the proposed linear-type method is much smaller than

the other two because of its shorter chromosome length. So,

more evolution generations seem practical for the other two

methods. To check that, the simulation results of the linear-type

and Mamdani type methods were compared with population

size and generation number 400. In this case, the

variations of and for both methods are smaller than

before, as shown in Fig. 17. Table XI, as well as Fig. 17,

displays the better performance of the proposed linear-type

method. This simulation gives a verification that the linear-type

method can not only achieve a better performance but also

reduce the size of search space.

Case 3: Comparisons of the Mamdani Type, TSK Type

and Proposed Nonlinear-Type Methods: Case 2 verifies the

improvement of the linear-type method, the improvement,however, is not so obvious especially the average fitness. On

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Fig. 17. Curves off

andf

withm = 1 2 0

and generation number 400.1

: Linear-type; : Mamdani type.

TABLE XI

VALUES OF f AND f WITH m = 1 2 0 AND GENERATION NUMBER 400

the other hand, the nonlinear-type index function needs three

extra parameters to get more flexible dynamics of the control

hypersurface. To examine the improvement, the control per-

formances of the nonlinear-type and the Mamdani type and

TSK type methods were also compared. Under the same GA

parameters as before, the curves of and , Fig. 18 andFig. 19, are shown to be corresponding to and . In

this case, the two curves of the nonlinear-type method display

much smaller variations than the other two methods. Mean-

while, the values and of the nonlinear-type method,

as summaried in Table XII, are also much smaller than that

of the other two. These expertment results reveal the worth of

developing a nonlinear index function for the fuzzy controller.

Comparing with the linear-type case, only three extra parame-

ters are required for the nonlinear-type case, which benefits the

search procedure of the GA.

VI. DISCUSSION AND CONCLUSION

Many studies have been done on the GA-based fuzzy con-

troller design. Most of them can be categorized as either a

Mamdani type or a TSK type design approach. Regardless of

the type, applying the genetic algorithms to the design pro-

cedure aims at setting membership functions, tuning scaling

factors and establishing the fuzzy rules of the controller. To

accomplish these jobs, it is necessary to code the parameters

of the three components into the chromosome for evolution. In

this study the coding methods for the Mamdani and TSK type

fuzzy controllers were explored, and a new design approach is

now presented. The proposed GA-based fuzzy controller uses

a novel structure for setting the consequent part of the fuzzycontrol rule by an index function. This index function, with

Fig. 18. Curves of f and f with various population sizes ( l = 1 ) . 1 :The nonlinear-type; : TSK type; : Mamdani type.

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Fig. 19. Curves of f and f with various population sizes ( l = 5 ) . 1 :The nonlinear-type; : TSK type; : Mamdani type.

parameters optimally set by GAs, results in a control hypersur-

face in the linguistic space formed by the input fuzzy variables.

Two types of index functions, both linear and nonlinear, were

developed. The proposed novel approach has the following

characteristics.

The input variables are not fuzzified, making the fuzzifi-

cation process unnecessary.

The index function is constructed in the linguistic space

rather than the real number space.

The index function can be designed as a linear or nonlinear

type depending on the required dynamics of the control

hypersurface.

The dynamics of the index function can be viewed by ex-

amining the control hypersurface constructed in the lin-

guistic space.

Only 6 to 9 parameters are needed to code into the chro-

mosome for the search. The chromosome is much shorter

than that of the Mamdani or TSK type, reducing searchingspace and saving computation time.

TABLE XIIVALUES OF

f

AND f

UNDER VARIOUS POPULATION SIZES

The search ranges of the parameters are discussed.

This novel approach is easily integrated with the improved

GA.

To examine the applicability of the proposed new structure,

the simple GA without any improvement was applied to find

the optimal parameters. Three systems including an oscillatory

system, a damping system, an oscillatory system and an inverted

pendulum system, were used as the control plants for simu-

lation. In addition, many comparison simulations of the pro-

posed structure, the Mamdani type and the TSK type methods

are given. In this study, the optimal index functions were ob-tained by applying the simple GA in an offline manner. Also, the

controlled plants in the simulations were all SISO cases. In fur-

ther studies, multiple index functions can be utilized for dealing

with the MIMO system, and the fuzzy neural network structure

can be used for acquiring the parameters of the index function

online.

ACKNOWLEDGMENT

The author would like to thank the reviewers, the Asso-

ciate Editor, and the Editor for their invaluable comments and

suggestions.

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Chih-Hsun Chou received the B.S. degree from

the Department of Electronic Engineering, TamkangUniversity, Taipei, Taiwan, R.O.C., in 1985, and thePh.D. degree from the Department of Electrical En-gineering, Ta-Tung Institute of Technology, Taipei,Taiwan, R.O.C., in 1994.

He is currentlyan AssociateProfessorwith theDe-partment of Computer Science and Information En-gineering, Chung-Hua University, Hsinchu, Taiwan,R.O.C. His current research interests include artifi-cial intelligence, intelligent control, audio signal pro-

cessing, and mobile systems.

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