Fuzzy Rules Membership11

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16 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 14, NO. 1, FEBRUARY 2006

A Simplified Version of Mamdanis Fuzzy Controller:The Natural Logic Controller

Alejandro Aceves-Lpez, Member, IEEE, and Joseph Aguilar-Martin

AbstractThis paper proposes the natural logic controller(NLC) that it comes through a very important simplification ofthe Mamdanis fuzzy controller (MFC) allowing easy-design forsingle-inputsingle-output (SISO) regulation problems. Usually,fuzzy controllers are built with two classical signals of process:The error and its rate of change. They use a moderate numberof fuzzy subsets and fuzzy rules. The main features of the NLCapproach are that use the minimal fuzzy partition (only two fuzzysubsets per variable) and it use the minimal fuzzy rule base (onlytwo rules). The nonlinear resulting fuzzy controller is the simplestone with an analytically well-defined, input-output mapping andaccepting a linear approximation at origin. It allows easy extension

to more than two signals of process. Some properties of nonlinearmapping of NLC are analyzed and some results are also presentson testing stability when NLC is used on a linear process. A specialattention is addressed to the two inputs NLC case, where stabilitycan be tested using the circle criterion. Finally, two applicationexamples are discussed in details.

Index TermsFuzzy control, mixed fuzzy-logic connective, nat-ural logic controller (NLC), stability analysis.

I. INTRODUCTION

WHEN AN expert understands qualitatively the processdynamics, it is possible to specify a qualitative controlstrategy using linguistic rules, named fuzzy rules. A classicalexample of those fuzzy control configurations is shown in Fig. 1,

where and are the signals used by the expert to decide the

intensity of the control action applied to the process. This

PD-fuzzy controller yields a standard methodology to construct

nonlinear feedback functions.

Roughly speaking, fuzzy controllers of this type are dynamic

nonlinear functions of error signal, while fuzzy rule-based func-

tions are static, memory-less, and nonlinear. The type of non-

linearity built with a fuzzy rule-based function depends on the

fuzzy subset shape, the fuzzy rule-base and the fuzzy operators

(like fuzzy-logic association operator, inference operator, and

defuzzyfication method).The theoretical property of those fuzzy rule-based functions,

called the universal approximator property (UAP), was proved

in 1992 by [1] and [2], it tells that any continuous nonlinear real

function can be approximated by means of a fuzzy rule-based

function. Notice that UAP applied to the control structure of

Manuscript received December 24, 2001; revised July 8, 2004 and June 8,2005.

A. Aceves-Lpez is with the ITESM-CEM, CP.52926 Mexico City, Mexico(e-mail: [email protected]).

J. Aguilar-Martin is with the LAAS-CNRS, 31077 Toulouse, France (e-mail:[email protected]).

Digital Object Identifier 10.1109/TFUZZ.2005.861603

Fig. 1. Classical fuzzy-control configuration.

Fig. 1 assumes the existence of a nonlinear function able to

control, or at least stabilize, the process in closed loop. If this

nonlinear function exists then the fuzzy rule-based mapping has

an important number of rules and therefore a high complexity.

The high combinatorial complexity of such fuzzy rule-base ap-

proach explains why only a few multiple-inputmultiple-output

(MIMO) fuzzy control applications are found.

To overcome the above mentioned complexity problem,

many alternatives have been proposed in the literature, such as

fuzzy-decoupling/decentralized controllers [3], self-learning

fuzzy controllers [4], fuzzy-model identification and control[5], reduced complexity fuzzy-controllers [6][11], only to

mention a few.

This paper explores the opposite of UAP idea of increasing

complexity to fulfill performances; it conversely proposes a way

to build the simplest fuzzy rule-based function.

The main idea is to limit wisely the degree of freedom of

fuzzy rule-based function by limiting the number of subsets

and the number of fuzzy rules [6][11] in order to obtain an

easier design. Two main structural simplifications to the orig-

inal Mamdanis fuzzy controller (MFC) are considered. Thefirst

one consists on defining only two fuzzy subsets for each signal

from the process, and the second one consists on defining only

two extreme fuzzy-rules. The consequence of those reductions

is an analytically well-defined family of nonlinear functions ac-

cepting a local linear approximation at the origin. By loosing the

UAP property, an easier design for single-inputsingle-output

(SISO) fuzzy control applications is proposed and it gives rise

to a suitable approach for MIMO applications.

This paper is divided as follows: Section II defines this

new controller, Section III shows some useful mathematical

properties, Section IV shows some results about stability of

this nonlinear controller by using mainly the circle criterion,

in Section V the proposed NLC is applied to two different

1063-6706/$20.00 2006 IEEE


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ALEJANDRO AND JOSEPH: SIMPLIFIED VERSION OF MAMDANIS FUZZY CONTROLLER 17

Fig. 2. Control configuration.

processes and its performances are discussed, and finally,

conclusions are discussed in Section VI.

II. DEFINITION OF NLC

In order to simplify the presentation, a process with one input

signal and one output signal is first considered (Fig. 2). The

difference between the observation and the reference signal

is called signal. Dependency with time will be omitted

for simplicity proposes.

Let us assume that an expert define the -block in order to

compute all experts meaningful signals

from the . The outputs of the -block are gathered in the

column vector .

Frequently, the -block splits into three independent blocks:

Proportional, integral, and derivative but this is not restrictive

and there are many other choices, for instance, it can reconstruct

the process state vector by means of an observer. In the NLC

approach we will accept any proposed -block by the expert.

A. Normalizing Gains

Without loosing generality, all signals of fuzzy rule-based

function will be normalized

for

The choice of the normalizing gains is usually made by

heuristic methods, whereas in the NLC approach they will be

related to natural constraints of signals. Obviously, any real

system has physical constraints, whenever they are compulsoryor simply imposed and it is essential to take them into account.

Two types of constraints will be distinguished here: the satu-

ration of the control action and the acceptable domains of the

sensor signals.

On one hand, control saturation implies that the effective con-

trol action can only take its values in a compact (often sym-

metric) interval , in the fuzzy logic frame-

work this interval is called universe of discourse. On the other

hand, the sensors acceptable domains are natural to the process

because it has either physical limits or security operation regions

and they may be different from one process to another. Consider

as examples the limited beams length in a ball-beam prototypeor the extreme safe temperature in a chemical reactor. They de-

Fig. 3. Fuzzy partition.

fine the respective universes of discourse and they should be

respected. It must be remarked that the universe of discourse of

is a closed domain, whereas the universe of discourse of each

signal is not closed because they might take values outside

acceptable domain, although it is not physically saturated.

Let us consider for example, and suppose that

some expert-knowledge can define an acceptable domain for

this signal. If is in the specified domain the process is con-

sidered in a normal situation, but if is outside that domain,

the maximal energy must be applied in order to return the

-signal back to that domain. This rule can be represented

as

should belong to

otherwise must be maximal

As an extension of this idea, it can be supposed that an expert

is able to define all natural constraints for each component of

signal vector . For instance, if is the rate of change of error

then correspond to the maximal admissible rate of change

of error

should belong to

otherwise must be maximal

Those considerations give an easy way to tune the normal-

izing gains by using the natural constrains

B. Fuzzy SubsetsIn order to reduce the complexity of fuzzy rule-based func-

tion, the simplest fuzzy partition is imposed, i.e., only two fuzzy

subsets as shown in Fig. 3. They represent the nega-

tive values and the positive values of normalized process sig-

nals. The corresponding membership functions are defined, for

, as

(1)

(2)

where

(3)


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In the case of the control action, the two membership func-

tions will be defined as

C. Fuzzy Control Rules

Last section defined fuzzy bi-partitions in the signal space,

therefore a specific system situation, called here as , can be

characterized as a collection of fuzzy elementary propositions

of type: is where . Such a complete rule-

base has , where the first and the last rules are

First rule If is is then is

Last rule If is is then is

The second fundamental simplification consists in using onlythe first and the last rules instead of the complete rule base.

This may seem an abusive simplification, its interest will find

a justification whenever

1) the fuzzy rules-base is symmetry with respect to their

input variables (most rule bases exhibit that property as

in MacVicarWhelan rules bases [12] or BuckleyYing

linear rules bases [13]);

2) the two extreme rules correspond exactly to the well

known and often proposed on-off controller.

Obviously, assuming a reduced fuzzy rules-base may produce

violent control actions, as is often the case in the on-off con-

trol case. In order to obtain a smoother control action, a com-pound fuzzy-logic association operator will be proposed here.

Two attitudes can be taken for the interpretation of an-

tecedents association of each rule: The conjunction (AND)

of all elementary fuzzy propositions or the disjunction (OR)

of them. The first case corresponds to the strictest attitude,

which means, all elementary antecedents must be true to infer

an extreme control action. The second one corresponds to the

weakest attitude; that means that only one elementary fuzzy

proposition needs to be true to infer an extreme control action.

Each of these two attitudes can be used and their choices depend

on the degree of exigency. In order to soften the inferences

proposed by the NLC, a linear convex interpolation betweenthese two extreme choices will be considered by using a mixed

linearly compensated connective (MLCC), [20] such as

In the definition of MLCC, is an -dimension column

vector where each element is noted as , let an iterated trian-

gular norm, is its dual norm and . Calling the unit

interval, that is , -norms are associative mappings

from to , and their iterated form is defined straightfor-

ward, notwithstanding the MLCC are not associative.

In order to give another degree of freedom for the choice ofthe MLCC in NLC, the Franks -norms family [14] will be

considered. The Franks family of continuous -norms depends

on parameter and it can be written as follows:

The dual -conorm is obtained straightforward by

DeMorgans law

It must be noticed that the three more frequently used -norms

are included in this family [21] with specific choices of param-

eter .

1) Zadehs -norm:

2) Product (probabilistic -norm):

3) Lukasiewiczs -norm:

By the association property of any fuzzy-logic operator, the

Franks -norm can also be extended to a -length vector and

define an dimensional -norm: .

The following two properties of Franks -norm will be used

later.

1) For any pair -norm, -conorm in the Franks family

holds for any and valid choice of and .

2) For any valid choice of , it holds that

D. Fuzzy-Logic Inference Method

The NLC approach remains in the Mamdanis fuzzy-logic

controller class, therefore, the truth values of each extreme-rule

is expressed as follows:

and the truth values of antecedents are calculated with the

MLCC

(4)

(5)

where the vectors are


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The antecedent strength of each rule is a combination gov-

erned by parameter that becomes a tuning parameter influ-

encing the truth values of the antecedents as follows.

When , the association of elementary antecedents

is made by an AND-association.

When , the association of elementary antecedents

is made by an OR-association. Otherwise, the corresponding association is a linear

convex interpolation between those extreme choices.

E. Defuzzyfication Method

In the Mamdanis fuzzy-logic controller class, the classical

center-of-average (COA) defuzzyfication is applied; it would

give here the following normalized control action:

Nevertheless, in order to preserve the association type im-

posed by the MLCC, adjusted by the parameter , the following

defuzzyfication method is proposed:

(6)

This nonstandard choice is called complements-center-of-

average (CCA). The following lemma justifies that choice.

Lemma 1: Let us call extreme case where at least one

. The CCA defuzzyfication method preserves the type of asso-

ciation imposed by , i.e., the strength of the control action in

the extreme case is an increasing function of the parameter .

Proof: Without loss of generality let us fix andtake (AND-association). If the classic COA defuzzyfica-

tion method were used, the resulting control action will had the

expression

Suppose , then . By using the neutral-ele-

ment property of -norms [15], it will be concluded that be-

comes equal to one (the maximum positive value) for any value

of . This result is opposite to expected exigency of the AND-

association. In fact, this result was expected for the OR-associa-tion. Consider now the CCA defuzzyfication method. The con-

trol action is

Which is equal to one (the maximum positive value) if and only

if and simultaneously. This result is now coherent

with the expected exigency of AND-association. The situation

with , OR-association, is handled similarly.

The NLC controller is formally defined by the following

theorem.

Theorem 1: NLC: Let for all from 1 to ,and . Let us define the vectors

and using

(1)(6). The NLC is a nonlinear, static, memory-less function

, structurally depending on and , defined

by

Proof: Straightforward from (1)(6).

The NLC can be considered as a two-rules Mamdanis fuzzy

controller, but the Natural Logic Controller label has been

chosen to express that, in this controller, all plant observations

are combined by means of a MLCC parameterized by , based

on the notion of exigency to adjust the strength of the control

actions. Its design depends uniquely on the natural constraints,

imposed by realistic control actions and acceptable intervals

for all observable-signals of process.

The NLC approach avoids the number of rules combinatorial

explosion found in most fuzzy approaches, therefore it allowsshorter computing times and easier design. Nevertheless, spe-

cial attention must be taken when a closed-loop configuration is

designed with NLC, because the performances cannot be guar-

anteed. Fortunately, by taking advantage of some mathematical

properties, stability can be analyzed in advance.

III. MATHEMATICAL PROPERTIES

In this section, we present two kinds of properties. The first

five properties are general to the NLC for any dimension of

, the last four are specific to bidimensional NLC case

.

A. General Properties

Property 1: Symmetry Condition: The NLC is an odd func-

tion respect to all its inputs

Proof: Using the membership functions definition by

(1)(3) it is easy to prove

Using those two identities and the DeMorgans law, it is possible

to show that

Start now from the definition of and use the last

two identities.

Property 2: Origin Condition: If all are zero

then the control action produced by the NLC is alsozero.


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Fig. 4. Control surfaces produced by the bidimensional NLC.

Proof: All implies .

From the Franks -norms definition, it follows easily

that

. Applying this result to the NLC definition gives:.

Property 3: Local linear approximation around the origin

for . If all are in a neighborhood of the origin,

there exists a compact region where the control

action produced by the NLC is as close to a linear functionas desired.


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Proof: The -norm associated to is the product

and by imposing all , it is possible to write the

continuous nonlinear function from NLC definition

with . Applying Taylors series first-approxima-

tion around origin

where

This linear equivalence will be useful in three situations

1) for comparison studies with respect to linear techniques,

in order to identify advantages (or disadvantages) of spe-

cific nonlinearity of NLC;

2) in iterative optimization methods of the normalizing gains

for which the starting point of algorithm could be the

equivalent linear gains;

3) to analyze local stability when a linear process is in

closed-loop with a NLC;

Property 4: Monotony Condition: The module of the control

action produced by the NLC varies monotonically with .

Proof: Let start from . Knowing that inequality

is always true for any vector ,

then it can be written

By multiplying the last inequality by ,

we have

Definition 1: In a decision-making framework, two variables

are on strict contradiction if they have opposite qualities. It

means that there exist at least one variable larger than its max-imum value and another variable lower than its minimum value.

Property 5: Zero-Control Action: The control action pro-

duced by the NLC is zero if at least two input variable of NLC

are on strict contradiction.

Proof: Let us consider two variables with opposite qual-

ities: , and for .

These assumptions lead us to

The resulting control action of the NLC becomes equal tozero.

Fig. 5. Lyapunov ellipsoidal region.

This last property is a direct consequence of reducing the

number of rules and it suggests that by reducing the set of non-

linear functions attainable by the NLC, some processes families

may not be closed-loop stable with this controller.

B. Particular Properties of Two-Input NLC

Fig. 4 shows seven different nonlinear control surfaces

when . Three different values for were considered

and three different values for were taken

. Notice the resemblance of these control surfaces

with the typical fuzzy control surfaces produced by a MFC

with MacVicarWhelan rule-base or BuckleyYing linear

rule-base. It shall be remarked that the function is the same

for and any Frank s -norm.

The next four Properties of NLC are capital for the stability

analysis.

Definition 2: Sector Property: A static memory-less function

is said to belong to the sector if

there exist two finite numbers such that

or equivalent

for any and . If the compact region

is a subspace of then it is said that

locally. If the compact region is equal to then it is said

that globally.

Property 6: Global Sector for : Let

and with . TheNLC function globally.

Proof: The proof is divided in three steps.

i) Starting from ,

and using the Franks -norm

property , the following

inequality can be obtained:

ii) It can easily be proved that

where .


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Fig. 6. Ball and beam device connected to a PC.

iii) Combining last two inequalities, we get

Using (1)(3) and knowing that

then

where .

Property 7: Global Sector for : Let

and with . The NLC function

globally.

Proof: This proof is equivalent to Property 6 and will be

omitted.

Property 8: Local Sector for : Let

and with .

The NLC function locally in the subspace

.

Proof: The proof is divided in three steps.

i) Starting from and using the Frank s

-norm property , the following

inequality can be obtained:

ii) If then .

iii) Combining last two inequalities

Property 9: Local Sector for : Letand with .

The NLC function locally in the subspace

.

Proof: This proof is equivalent toproperty 8 and will be

omitted.

IV. STABILITY ANALYSIS OF NLC

Stability is the most important characteristic of any control

system and it determines the viability of a control configura-

tion. If the controller has a reduced structure, the stability is still

more important. When an NLC is used, it is capital to specify

under which conditions the closed-loop system remains stable.

For this reason, an accurate stability study of the NLC has been

developed. Two types of results have been obtained. The first

one comes from the linear approximation of the NLC presented

in Property 3. The second is obtained using the same idea de-

veloped in [22] where the circle criterion is used to analyze the

stability of a linear system in closed-loop with a generic fuzzy

controller, considered as an unknown nonlinear memory-less

feedback function verifying some sector condition. The speci-

ficity of the present stability analysis is the proposition of appro-

priated gains and for the NLC based on Properties 6

and 7 of previous section.

Consider the control configuration shown on Fig. 2, where

the process is modeled by

Dependency with time was omitted for simplicity. Let uspropose the following model for the -block that includes

observers, filters and other usual dynamic part of feedback

compensators:

The nonlinear static feedback part of NLC is given by


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Fig. 7. Ball and beam control configuration with a NLC.

Let us force a zero reference and nonzero initial conditions.

These considerations lead us to a Lures canonical form

(7)

where is expressed by Theorem 1, , ,

, , , and

being a minimal realization of the serial connection of linear

model of process and the linear model of

-block . The next results present stability

conditions of closed-loop System 7.

Theorem 2: SISO Linear Local Stability: Let stand for

a -length column of ones, stand for a -length column of

zeros and . For any fixed , the origin of System

7 is local asymptotically stable in a small compact set

if is Hurwitz, where

Proof: It comes straightforward from Property 3 of

NLC.Theorem 3: Global Stability of Bidimensional NLC: Let us

consider the following conditions: 1) There exist a specific

so that verifies a sector condition globally, 2) matrix

of system 7 is Hurwitz, and 3) there exist a symmetric positive-

defined matrix solution of

where and is a small enough

positive constant. If the three previous conditions hold then the

origin of System 7, with , is global asymptot-

ically stable.

Proof: It comes directly from Properties 6 and 7 and thecircle criterion.

Theorem 4: Local Stability of Bi-Dimensional NLC: Under

the following conditions: 1) There exist a specific so

that verifies a sector condition locally, 2) matrix

is Hurwitz, and 3) there exist a symmetric

positive-defined matrix solution of

where and is a small enough

positive constant. If the three previous conditions hold, then the

origin of System 7, with , is local asymptoti-

cally stable.

Proof: It comes directly from Properties 8 and 9 and the

direct application of circle criterion.

Theorem 3 guaranties stability in the whole state-space but its

conditions are restrictive: The NLC must have only two input

variables, the linear system must be stable in open loop and a

matrix solution of the Riccati-like equation must be found,

and this is not always easy.

Theorem 4 guaranties stability only in a region of the

statespace with less restrictive conditions: the linear system

must be stable with a linear output feedback. From this result,

an estimation of the stability region can be done as follows.

Corollary 1: Stability Region Estimation: Under conditions

of Theorem 4, any initial condition verifying

produces an asymptotic path to the origin, where

Proof: If System 7 is local asymptotically stable in thesense of Theorem 4, then there exists an ellipsoidal Lyapunov


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domain contained in the subspace where the sector condition is

verified. The biggest ellipsoidal Lyapunov domain in that sub-

space is obtained solving the following optimization problem:

under and

As shown in Fig. 5, the solution of that problem is equivalent to

the solution of

under

under

The solution to that problem is given by

where and are the Lagrange variable. Finally, from the-

orem 4, any initial condition verifying pro-

duces an asymptotic path to origin.

Finally, theorem 2 can be extended to the following multi-

variable linear configuration. Consider an MIMO linear system

with outputs and independent NLC

... (8)

where , , , ,

, and

.

Theorem 5: MIMO Linear Local Stability: Let stand

for a full matrix of ones, stand for a -length column of zeros

and . For any fixed , the origin of system 8 is

local asymptotically stable in a small compact set ifis Hurwitz, where

for

Proof: It comes straightforward from Property 3 of NLC

and stability analysis of linear systems.

V. ILLUSTRATIVE EXAMPLES

Two examples will be discussed. The first one is the classic

nonlinear ball and beam device, found in many undergraduate

control laboratories. The main objective is to show how the NLCworks, illustrating the performances that can be obtained and

Fig. 8. Real-time performance when = 1

.

showing how to analyze its stability. The second example is the

classic linear model of a distillation column given by [16]. The

main objective of that application is to showhowthe NLC can be

used in an MIMO process, in which several time delays appear,and what are the expected performances.


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Fig. 9. Real-time performance when = 0 .

A. Ball and Beam Example

The experiment is depicted in Fig. 6. The ball is free to roll

along the beam, with one degree of freedom, and the beam canrotate only in a vertical-plane by applying a torque with an elec-

Fig. 10. Schematic diagram of the column.

Fig. 11. Decentralized PID control configuration.

Fig. 12. Decentralized PID control configuration.

trical device. The goal is to achieve a desired balls position on

the beam.

Previous knowledge about this process suggests us to use the

output signals and its rate of change as input signals to the con-

troller. Then, the -block proposed is proportional-derivative.

The control configuration with a NLC is depicted in Fig. 7.

The natural constraints of this process are

a) control action is saturated: volts;

b) balls sensor position has physical limits:

;

c) balls speed can not be bigger than . (obtainedexperimentally);


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Fig. 13. Temporary responses to a step on distillate.

d) beams sensor angle has physical limits:

;

e) beams s peed c an n ot b e bigger t hat . ( obtained

experimentally).These natural constraints of ball and beam device were used

to guide the choice of normalization gains. The following values

have been chosen.

a) , directly taken from the saturation limits of

.

b) , because a difference between the balls desired

position and the balls real position larger than

will not be tolerated and then the maximum available en-

ergy will be applied.

c) , because the module of the ball speed should not

exceed 0.1 m/s.

d) , directly taken from the physical constraint ofthe beams angle.

e) , because the module of beam speed should not

exceed 0.8 rad/s.

A linear model was obtained by a frequency-domain identifi-

cation method where matrix , and are:

The stability was analyzed as a first step using the Theorem 2.It was concluded that the proposed NLC feedback configuration


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Fig. 14. Temporary responses to a step on bottom product.

is locally stable, because the following matrix is Hurwitz for any

valid value of :

In a second step, some real-time experiments were per-

formed, one with and another with . The desired

balls position was selected close to the beams border in order

to force the system to exhibit its nonlinear effects.

Figs. 8 and 9 show t he b ehaviors w hen and . B oth

cases achieve the desired balls position, nevertheless rise-time,

overshoot, setting-time and control action, are very different.

Notice that better performances were obtained with more energy

consumption. Fig. 8 shows a strong resemblance between on-offcontrol and the NLC when . In Fig. 9, when ,

the strict contradiction effect is more evident because there are

some intervals of time where control action is zero.

An enormous advantage was archived in terms of complexity

reduction. Consider a traditional fuzzy controller with four sig-

nals as inputs having each one five fuzzy subsets, the complete

base will have 625 rules. With the proposed NLC only two rules

were needed. A tremendous reduction of fuzzy rules, from 625to 2, was achieved whereas the performances are similar.

B. Distillation Column Model Application

In this section, a comparison between the NLC approach and

a PID decentralized control configuration is presented, both ap-

plied to a same MIMO (2 2) control problem. The process is

the binary, eight plates, distillation column reported by [16] that

is shown in Fig. 10. The linear model is


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Fig. 15. Temporary responses to a step of perturbation.

where input signals are the reflux flow rate and the steam

flow rate , the output signals are the top product composi-

tion in mole fraction and the bottom product composi-

tion also in mole fraction. The influence of the feed flow

rate (in mole fraction) was taken from [17]. This linear

model is valid around the set point: , ,

, , and .Two control configurations will be compared in this section,

the PID decentralized control configuration proposed by [18]

(see also [19]) and the corresponding NLC configuration pro-

posed here. See figures Figs. 11 and 12.

The natural interval of two NLC were defined as

It must be pointed out that the NLC could easily be com-

pleted with other variables to take advantage of multivariable

aspect but we restricted the NLC variables to the error and its

rate of change, for sake of comparison with the PID configura-

tion method.

The temporary responses are shown in Figs. 1315. They

were made under the three following regulation situations:

1) a variation of on the top composition;

2) a variation of 0.01 on the bottom composition;

3) a perturbation of 0.051 b/min on the feed flow.

As expected, all temporary behaviors were strongly affected

by interactions between both decentralized control loops. Nei-

ther PI nor NLC configuration can reduce interaction to zero,

but NLC with keeps it much smaller than PI, or NLC

with . In the case of , the NLC shows always less

oscillations because it applies smoother control actions. Those

extremes, and , are both interesting for different

operating situations of the column. Therefore, a specific choiceof could be left to a supervisor.


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VI. CONCLUSION

This paper presents a new fuzzy controller, named NLC,

which results from important structural simplifications of the

MFC. The consequence of those simplifications is a nonlinear

function, analytically well-defined mapping, with a linear

approximation around the state space origin. In other words,

by reducing the degree of freedom of the MFC, we obtained a

suitable controller with a few of parameter to tune, allowing

easier design for SISO regulation problems.

The main mathematical properties of this new nonlinear func-

tion are: the origin condition, the symmetry condition, the ex-

istence of a linear approximation and the strict contradiction.

It was proposed the utilization of circle criterion for stability

analysis of linear systems in feedback with a NLC. It is impor-

tant to notice that the usage of circle criterion for fuzzy control

was early presented by [22] for the general case but any spe-

cific proposition of appropriated , values can be donewithout an exact knowledge of the fuzzy function. The origi-

nality of our works is that the NLC has a very specific struc-

ture then appropriated sector conditions were proposed to prove

stability.

Two examples have been presented in this paper. They show

that NLC can be used in regulation problems reducing the neces-

sary time to build up a fuzzy-like controller. The first example

was the real-time ball and beam apparatus. The obtained be-

havior showed an effective regulation by considering only the

natural constraints of process. The second example was a sim-

ulation of the distillation column of [16]. A comparison with

the conventional PID technique was done. The simulation re-

sults validate the usefulness of the NLC approach because it

has shown better performances. An interesting alternative was

opened for supervisory control thanks to the parameter. As

future work we propose to continue this research by developing

self-tuning methods for the NLC.

REFERENCES

[1] B. Kosko, Fuzzy systems as universal approximators, in Proc. IEEEInt. Conf. Fuzzy Systems, San Diego, CA, Mar. 1992, pp. 11431162.

[2] L.X. Wang, Fuzzy systems are universalapproximators, in Proc.IEEEInt. Conf. Fuzzy Systems, San Diego, CA, Mar. 1992, pp. 11631170.

[3] K. S. Ray and Dutta-Majuider, Fuzzy logic control of a nonlinear mul-tivariable steam generating unit using decoupling theory, IEEE Trans.Syst., Man, Cybern., vol. SMC15, no. 4, pp. 539558, Jul. 1985.

[4] Linkens and Nie, Constructing rule bases for multivariable fuzzycontrol by self-learning part I&II, Int. J. Syst. Sci., vol. 24, no. 1, pp.111127, 1993.

[5] T. Takagi and M. Sugeno, Fuzzy identification of systems and its appli-cations to modeling and control, IEEE Trans. Syst., Man, Cybern., vol.SMC-15, no. 1, pp. 116132, Jan. 1985.

[6] H. Ying, W. Siler, and J. J. Buckley, Fuzzy control theory: A nonlinearcase, Automatica, vol. 26, no. 3, pp. 513520, 1990.

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[8] B. Vidolov and C. Melin, An approach to the obtaining of knowledgebases for MIMO fuzzy control: Application to a non linear thermal

system control, in Proc. 6th IEEE Conf. Fuzzy Systems, Jul. 15, 1997,pp. 99103.

[9] J. Aguilar-Martn and J. C. Hernandez, Conectivos mixtos de la lgicaborrosa para la coordinacin en control fuzzy, in I Jornadas SobreTransferencia de Tecnologa Fuzzy: Universidad de Murcia, 1995.

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design of regulators based on fuzzy logic compensated connectives,BISC Seminar Lecture, Berkeley, CA, Rapport LAAS 96 447, 1996.

[11] J. C. Hernndez, J. Aguilar-Martin, and J. Q. Casin, Natural logic ap-proach to fuzzy PID regulation, in Journes HispanoFranaises surles Systmes Intelligents et le Contrle Avanc, Barcelona, Spain, Nov.

1213, 1996.[12] P. J. MacVicar-Whelan, Fuzzy sets for man machine interaction, Int.

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[14] E. P. Klement, Characterization of fuzzy measures constructed bymeans of triangular norms, J. Math. Anal. Appl., vol. 86, pp. 345358,1982.

[15] L.-X. Wang, A Course in Fuzzy Systems and Control. Upper Saddle

River, NJ: Prentice-Hall, 1998.

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Engineers. New York: Mc Graw-Hill, 1990.

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Alejandro Aceves-Lpez (M04) was born inMxico City, Mexico, in 1970. He received theengineers degree in electronics and the M.Sc.degree in manufacturing systems from the InstitutoTecnolgico y de Estudios Superiores de MonterreyCampus Estado de Mxico (ITESM-CEM), in 1991and 1994, respectively, where he workedas a TeacherAssistant for two years. He joined the LaboratoiredAnalyze et Architecture des Systmes du CentreNational de la Recherche Scientifique, France, aPh.D. student in 1996. He received the Ph.D. degree

in automatics from the Universit Paul Sabatier, Toulouse, France, in 2000.Since 2001, he has been with the ITESM-CEM as a Full Professor in the

Mechatronic Department teaching control systems and fuzzy control. Since2001, he has been participating in the four-legged futbol robotic project. Hestarted the humanoide research project in 2004. He has been the Director of

the Mechatronic Master program since 2003. He focuses his research on fuzzycontrol of dynamic system, stability analysis of nonlinear systems, and controlof robots.


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Joseph Aguilar-Martin was born in Barcelona,Catalonia, Spain, in 1939. He received his engineer sdegree in 1962 from the Ecole Nationale SuprieuredElectrotechnique dElectronique et dHydraulique,Toulouse, France, the M.Sc. degree in 1967 fromthe Imperial College of the University of London,U.K., and the Doctorats Sciences Physiques degreein 1974 from the Universit Paul Sabatier, Toulouse,

France. He also received a degree in Linguisticsfrom the Universit de Toulouse-le-Mirail.In 1968, he joined the Laboratoire dAutomatique

et dAnalyze de Systmes of the Centre National de la Recherche Scientifique,Toulouse, France, where he has remained with short interruptions since that

time. He was one of the introductors of Kalman Bucy filtering in France, andlater developed many industrial applicationsof nonlinear filtering.He was anin-vited Senior Scientist at the Electronics Research Laboratory at the Universityof California, Berkeley, in 1978. He has also been invited for shorter periods as aVisiting Scientist, in 1980, by the Polytechnic University of Catalonia, Spain, in1980, by theEcalePolytechnique de Montreal PQ,Canada, and1982 by theUni-versidade Estadual de Campinas, UNICAMP, Sao Paulo, Brazil. More recently,he participated in the setting up of the Centre dEstudis Avanats de Blanes,Catalonia, Spain. His research interest includes applications of stochastic andfuzzy systems theory to industrial processes control and to environmental sys-tems simulation. He presently focuses his research on expert supervision of dy-namic processes, including qualitative and nonstandard logic for simulation and

recognition under imprecision and uncertainty. He has published more than 40papers in his fields of interest, mostly on-stochastic and fuzzy systems for Con-trol Systems.

Dr. Joseph is a member of the Socit Francophone de Classification and ofthe Socit de Mathmatiques Appliques et Industrielles.

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