NEURO-FUZZY CONTROL OF NONLINEAR SYSTEMS

Application in a Ball and Beam System

Marconi Cˆamara Rodrigues, F´abio Meneghetti U. Ara´ujo and Andr´e Laurindo Maitelli

Departamento de Engenharia de Computac¸˜ao e Automac¸˜ao

Universidade Federal do Rio Grande do Norte, Natal, RN, Brazil

Keywords:

Control, Hybrid Systems, Fuzzy Systems, Artiﬁcial Neural Networks, NEFCON, ANFIS.

Abstract:

This study shows both the development and characteristics of some of the main techniques used to control

nonlinear systems. Starting from a fuzzy controller, it was possible to apply similar learning techniques to

those used in Artiﬁcial Neural Networks (ANNs), and evolve to ANFIS and NEFCON neurofuzzy models.

These neurofuzzy models were applied to a real ball and beam plant and both their adaptations and their results

were discussed. For each controller developed the input variables, the parameters used to adapt the variables

and the algorithms applied in each one are speciﬁed. The tests were performed in a ball and beam plant and

the results are directed toward obtaining a comparison between the initial and ﬁnal evolution phase of the

neuro-fuzzy controllers, as well as the applicability of each one according to their intrinsic characteristics.

1 INTRODUCTION

Controllers such as PIDs and those based on the re-

alignment of states, are simple to implement and in-

expensive to operate. However, adjusting their pa-

rameters can take considerable time and their per-

formance is usually limited. A number of automatic

adjustment techniques for PID controller parameters

were developed (Oliveira, 1994; Coelho and Coelho,

1999). These techniques, in addition to raising opera-

tional cost, do not enable PID controllers to resolve

problems, such as controlling nonlinear systems or

maintaining their performance in the presence of un-

certainties or parametric variations.

The need for more encompassing techniques in

the control area led to the emergence of intelligent

techniques such as artiﬁcial neural networks, fuzzy

systems, genetic algorithms and other techniques

based on reinforcement learning.

The fuzzy systems and Artiﬁcial Neural Networks

(ANNs) are very useful tools for nonlinear systems

control, with or without mathematical models. These

two techniques will be the main focus of this paper,

since it was from their structures that the hybrids pre-

sented here were developed.

ANNs attempt to reproduce the capacity of learn-

ing and generalizing the knowledge of the cerebral

structure of living beings. Based on simple known

structures such as artiﬁcial neurons, the data prop-

agate from neuron to neuron via synapses, whose

weights can be adjusted over the course of the learn-

ing process (Haykin, 2001).

Around thirty years after the introduction of the

fuzzy set theory by Lotﬁ Zadeh (Zadeh, 1965), the

researcher Jyn-Shing Roger Jang published an article

in which fuzzy parameters were calculated using the

backpropagation technique, widely used to adjust the

synaptic weights of ANNs (Jang and S., 1995). This

technique of associating fuzzy with artiﬁcial neu-

ral networks became known as neuro-fuzzy and the

model implemented by Jang was called the Adaptive-

Network-Based Fuzzy Inference System, or ANFIS.

Other neuro-fuzzy models were also developed and,

among these, the NEFCON model (neuro-fuzzy con-

troller) stands out for having easy-to-implement char-

acteristics in real time.

This study was developed because of the potential

applicability of ANFIS and NEFCON hybrid intelli-

gent controllers, and showed characteristics of imple-

mentation and of use, the intrinsic characteristics of

each model, as well as a number of advantages and

disadvantages.

This study is divided as follows: section 2, dis-

cusses the neuro-fuzzy implementation approaches

proposed by the ANFIS (Qiang et al., 2008) and NE-

FCON (Shujaec et al., 2002) models. Section 3,

presents the control structure used to obtain the re-

sults of the application, in real time, of the hybrid

201

Câmara Rodrigues M., Meneghetti U. Araújo F. and Laurindo Maitelli A. (2009).

NEURO-FUZZY CONTROL OF NONLINEAR SYSTEMS - Application in a Ball and Beam System.

In Proceedings of the 6th International Conference on Informatics in Control, Automation and Robotics - Intelligent Control Systems and Optimization,

pages 201-206

DOI: 10.5220/0002202502010206

Copyright

c

SciTePress

techniques in a beam and ball plant. The last sec-

tion, 4, contains the conclusions about the techniques

presented and possible future studies.

2 NEURO-FUZZY

CONTROLLERS

Neuro-Fuzzy controllers can aggregate different char-

acteristics of fuzzy controllers and artiﬁcial neural

networks in a single structure. Thus, the controller

based on neuro-fuzzy models will have easy-to-

interpret control actions, promoted by the fuzzy con-

trollers, and a learning stage, which is the main char-

acteristic of neural networks. This section presents

two neuro-fuzzy controllers, ANFIS and NEFCON.

2.1 ANFIS Model

The ANFIS model uses a fuzzy controller as its basic

structure, which can be interpreted as a 6-layer neural

network, in which learning techniques such as back-

propagation can be applied (Jang et al., 1997).

To simplify, consider a fuzzy system with two in-

puts (x and y), two membership functions (MFs) for

each input variable and one output z. Figure 1 illus-

trates the structure of the ANFIS model considered.

Layers:0

1

2

1 1

1

z

N

N

N

W

1

W

2

W

3

W

1

W

2

W

3

1

1

3 4

5

X

X

Y

N

W

4

W

4

Y

Figure 1: ANFIS with two inputs (x and y) and one output

(z).

Data ﬂow can be analyzed layer by layer, as shown

below:

• Layer 0: Represents the inputs of the model.

• Layer 1: The neurons of this layer represent the

MFs of the input; that is, the fuzziﬁcation phase.

• Layer 2: Represents the number of rules. The

outputs of the previous layer are operated accord-

ing to the inference phase of the fuzzy system con-

sidered. Multiplication is an interesting option,

since it is easy to derive and simple to implement.

• Layer 3: The output of this layer will be the

output of normalized neurons from the previous

layer; that is, the output of each neuron of the pre-

vious layer divided by the sum of the output of all

the neurons in this same layer.

• Layer 4: The function associated to the neurons

of this layer will be the function f(x, y), used by

the Sugeno model, in which x and y are the system

inputs and p, q and r are the adjustable parameters

of the Sugeno function.

• Layer 5: In this layer the sum of the neuron out-

puts of the previous layer occurs, obtaining thus

the control signal for the system.

From the neural network presented, it can be

clearly observed that the neurons that need learning

are present in layers 1 and 4, since layer 1 contains the

input MFs and layer 4 the Sugeno polynomial, which

deﬁne the implications of the rules.

The adjustment of controller parameters in the

ANFIS model can be obtained using adaptive tech-

niques such as the backpropagation algorithm. The

derivatives of equations are found in (Jang et al.,

1997). To improve the convergence speed of the al-

gorithm, the η-adaptive technique was used (Rezende

and Maitelli, 1999), in addition to backpropagation.

2.2 NEFCON Model

NEFCON is a neuro-fuzzy controller model based on

the generic architecture of an ANN, but speciﬁcally

of a 3-layer perceptron network (ﬁgure 2). Making a

parallel between the 3-layer perceptron networks and

the fuzzy systems, we have (Nrnberger et al., 1999):

Layer0

x

y

Layer1

Layer2

f(a,b)=T (a,b)

f(.)=(a,b,c,d)

z

T

Antecedent

Consequent

A1 A2

B1 B2

C1C2C3

Figure 2: NEFCON with 2 inputs (x and y) and 1 output (z).

• Layer 0: This layer represents the inputs. The

MFs are found in the synapses that link this layer

to the following layer and it is in these synapses

that fuzziﬁcation occurs.

• Layer 1: This layer (intermediary layer) abstracts

the fuzzy system rules and it is where the actu-

ation level of each output membership function

(MF) is found. In the synapses between this layer

ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics

202

and the following layer, the actuation level found

acts on the consequent MFs.

• Layer 2: Layer 2 is responsible for defuzziﬁca-

tion. The algorithm proposed by (Nrnberger et al.,

1999), suggests the use of the mean of the maxi-

mums method for the defuzziﬁcation stage, which

reduces the consequent MFs to simple impulses

located in their maximums.

The learning algorithm of the NEFCON model is

based on the idea of reinforcement learning (Sutton

and Barto, 1998). This algorithm can be divided into

two main phases: creating rules and optimizing the

MFs (Rodrigues et al., 2004; Rodrigues et al., 2006).

To create rules the algorithm may or not receive

a set of initial rules. If a set of initial rules is re-

ceived, then the rule creation phase will optimize it.

It is necessary to specify both the actuation intervals

for each input and the actuation interval for the out-

put, since these intervals enable the interval to com-

pare different-sized inputs and create a rule to reduce

the input of greatest error.

To illustrate this idea, consider the actuation inter-

vals of two inputs (E1 and E2) are [5, 5] and [-0.5,

0.5], respectively. The input E1 error is 1, and the E2

error is 0.3. To calculate the error of an input in terms

of its actuation interval, the value of the input error is

divided by the size of the actuation interval. Thus, the

error of E1 will be 0.1, whereas the error of E2 will be

0.3. Observe that even though it has a smaller abso-

lute value, the value of input E2 is greater than that of

input E1, a situation that leads the algorithm to create

a rule with a tendency to minimize the error of input

E2. Thus, the algorithm will discover which rule from

the set of rules is activated with greatest strength and

greatest µ. This rule will activate an output MF that

will be obtained by comparing the input in its actua-

tion interval and the output interval divided into out-

put MFs.

Different from that suggested by the algorithm

that inspired this work (Nrnberger et al., 1999), a pre-

viously created rule can be modiﬁed at any moment if

a different rule is found for the situation. There is also

no need to modify the structure of the MFs, given that

they will be treated in a later phase.

To optimize the MFs, the algorithm uses a strategy

similar to that used by reinforcement learning. When

an input MF is activated, it contributes to reducing or

increasing the error. If the action produced provokes

an increase in system error, this MF has its actuation

ﬁeld reduced. Otherwise, the MF has its actuation in-

terval increased. A similar situation occurs with the

output MFs. However, with these, the gain or loss

occurs in their intensity; that is, the MF that collab-

orates with the increased error will have its intensity

reduced; otherwise, the intensity will be increased.

Mathematically, we have that the plant E error

is found according to the insertion of inputs into the

plant. The contribution, tr, of each rule for the output

is estimated and error Er is calculated for each rule

unit, according to the following equation,

Er = µ · E · sgn(tr) (1)

in which:

sgn(tr) = tr signal

With these data, the consequent modiﬁcations can

be represented by:

∆b

i

= η· µ · E (2)

And the antecedent modiﬁcations by:

∆a

(i)

j

= −η· Er · (b

(i)

j

− a

(i)

j

) (3)

∆c

(i)

j

= η· Er· (c

(i)

j

− b

(i)

j

) (4)

in which:

η = Learning coefﬁcient

a, b, c = Vertices of the membership functions

It can be easily observed that the algorithm does

not alter the position of the MF of the antecedents. Its

base is only increased or decreased proportionally on

both sides; that is, if the MF has a positive contribu-

tion, there will be greater likelihood of its occurring

again.

A number of restrictions were inserted to avoid

the overlapping of more than two MFs and to avoid

the emergence of gaps between them. Therefore, an

overlap between zero and 50% must be guaranteed;

that is, the vertices of a triangle must be contained in

the interval corresponding to the middle of the base of

neighboring triangles.

3 APPLICATION IN THE BALL

AND BEAM SYSTEM

The ball and beam system, in which the controllers

were tested, is composed basically of a beam-ball sys-

tem, a servo motor with a reducer gearbox and a ruler

with a ball of reference. There are three sensors, one

to measure the position of the reference ball, another

for the position of the ball to be controlled and one to

measure the angular position of the servo motor (ﬁg-

ure 3).

The aim of the controller is to makethe ball placed

on the beam to follow the pathway speciﬁed by the

reference ball. Thus, a control system is designed to

NEURO-FUZZY CONTROL OF NONLINEAR SYSTEMS - Application in a Ball and Beam System

203

Referenceball

sensor

Controlledball

sensor

Servomotor

sensor

Figure 3: Sensor location in the ball and beam system.

Ball-Beam

Intelligent

Controller

PID

Ball

Reference

Ref Ang

Angle

Motor Tension

Figure 4: Flowchart illustrating the ball and beam control

loops.

send a voltage to the servo motor, which, upon mov-

ing, raises or lowers one of the ends of the beam, caus-

ing the ball to move.

The control system designed for the ball and beam

is divided into two loops (ﬁgure 4): one external,

where an intelligent controller is responsible for re-

ceiving the position of the ball to be controlled and

the position of the reference ball and from this, pro-

vide the reference angle for the second loop, where a

PID controller generates a control signal, which is re-

quired for the servo motor to position itself according

to the reference angle supplied by the ﬁrst controller.

The use of the intelligent controller to substitute

the two control loops of the plant is a considerable

challenge. Thus, it was decided to use hybrid tech-

niques to substitute only the external loop, given that

the internal loop functions sufﬁciently well with the

PID designed and the external loop is a sufﬁcient chal-

lenge for the proposal.

3.1 Results with ANFIS

The ANFIS model will be optimized by backpropaga-

tion. To achieve this, training pairs (points) must be

obtained. A PID (Proportional Integrative Derivative)

controller was used to obtain the training pairs, substi-

tuting the intelligent controller (ﬁgure 4). The choice

of three inputs for the ANFIS model was based on the

characteristics of the controller to be copied, namely

the PID. The three inputs considered were the errors:

current and previous of the ball to be controlled with

respect to the reference ball, and the previous refer-

ence angle.

The training point capture stage was followed by

the training stage of the ANFIS model. The training

algorithm used the input-output pairs obtained with

the PID controller to adapt the adjustable parameters

of the system.

Seen as a fuzzy system, it has 5 bell-shaped MFs

for each input variable. Thus, the intersections be-

tween the MFs form 125 possible rules that reference

ﬁrst-order polynomials (px+ qy + rz+ s), as a func-

tion of input variables x = current error, y = previous

error and z = previous reference angle. The variables

p, q, r and s are the adjustable parameters of the poly-

nomials and all were initiated with a value of 0 (zero).

The learning coefﬁcient for this system was ini-

tiated with η = 10

−6

, the initial values for the MFs

were uniformly distributed within the intervals and

the sample period considered was 0.05 s.

The initial value for variable η and the intervals in

which the MFs were distributed were two of the great-

est difﬁculties in implementing the model. When the

η value was elevated (around 10

−4

for this model) or

when the intervals were very different from the real-

ity of the system to be controlled, the programbecame

unstable.

An association with the fuzzy system shows that

the equivalent neural network will have the following

number of neurons in each layer: 3 - layer 0; 15 -

layer 1; 125 -layers 2, 3 and 4; 1 - layer 5.

For training, a total of 800 points were collected

from the PID of the external loop of the control sys-

tem that acts on the ball and beam. After training, the

input MFs have optimized shapes, as shown in ﬁg. 5.

−10 −5 0 5 10

0

0.5

1

Fuzzy error

−10 −5 0 5 10

0

0.5

1

Previous fuzzy error

−6 −4 −2 0 2 4 6

0

0.5

1

Previous reference angle

Figure 5: Membership functions after ANFIS adjust.

For an effective learning validation, the controller

trained in the ball and beam plant was used for a se-

quence of preestablished reference points, which dis-

pensed with the use of the reference beam. The ef-

fect of the controller on the plant can be observed

in ﬁgure 6. First, the reference received a pseudo

ramdom signal (ﬁgure 6). Notice that in the signal

changes, the reference is not followed with much per-

fection by the neuro-fuzzy controller. Analysis of the

control signal from ANFIS showed that the beam re-

ceives the angle which, in theory, would take the ball

to the position of reference. However, the inﬂuence of

dry attrition causes the ball to be controlled to remain

ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics

204

0 10 20 30 40 50 60 70

0

10

20

30

40

Time (s)

Position on the beam (cm)

Figure 6: Neuro-fuzzy controller of the ANFIS model

(dashed), reference (solid) and PID (dashdot).

immobile. The PID controller has a control signal

with many more oscillations than does ANFIS and,

thus, manages to avoid interference from dry attrition.

When the reference received a sine signal, the neuro-

fuzzy controller follows the reference as well as the

PID controller and sometimes more accurately. How-

ever, at low velocities (extremes above and below the

graphs), accuracy is reduced by dry attrition.

3.2 Results with NEFCON

For this model two inputs with ﬁve triangular func-

tions each were used, and one output, the angle of

reference, with seven singleton-type MFs. The input

MFs are initiated with 50% overlap and symmetri-

cally divided within their actuation interval, which is

supplied by the designer. The maximum number of

active rules for the system is 25. The learning con-

stant for the controller was considered to be 10

−3

and

the sampling period for this model was 0.02 seconds.

This value is due to the fact that the NEFCON model

has a lower computational cost than that of the ANFIS

model.

The position error of the ball and its variation were

deﬁned as system inputs. The value desired for both

inputs was deﬁned as zero and the execution range

used for these variables was between -20 and 20 cm

for the error and between -0.6 and 0.6 cm/s for the er-

ror variation. With respect to the output variable, the

reference angle can vary from -45 to 45 degrees. The

initial rule base was considered empty, except for the

central rule, which received the output value equiva-

lent to the zero reference angle.

As previously mentioned, NEFCON has its learn-

ing phase in real time. To show the development of

this learning while the algorithm attempts to control

the plant, a sine wave in the reference beam was used

as a learning sequence. The rule creation phase was

the ﬁrst to enter into operation and its result is pre-

sented in table 1, where NH = Negative high, NL =

Negative low, ZE = Zero, PL = Positive low, PH =

Positive high, NE = Negative and PO = Positive.

Table 1: Rules found after 35 seconds of rule creation.

Error variation

NH NL ZE PL PH

E NH - - - - -

R NL NH - NE PO PH

R ZE NH NL ZE PL -

O PL NH NL PL PH PH

R PH - - - - -

−20 −10 0 10 20

0

0.5

1

1.5

Fuzzy error

−0.4 −0.2 0 0.2 0.4 0.6

0

0.5

1

1.5

Fuzzy error variation

−50 0 50

0

0.5

1

1.5

Fuzzy outputs

Figure 7: Membership functions after NEFCON adjust.

The second part of the algorithm functions as a

ﬁne adjustment of the fuzzy parameters by only mov-

ing the MFs while searching for an optimal adjust-

ment. The MFs have the shape illustrated in ﬁg. 7.

Observe that a situation of non symmetry occurred

in the central MF (corresponding to membership ZE)

of the second input (Error variation) of the neuro-

fuzzy controller. Even though the algorithm tends to

adjust the two sides symmetrically, the restrictions to

gap formation between the MFs enable this asymmet-

ric condition. We can observe also the compensation

supplied by the algorithm for the output MF to cali-

brate the zero angle of the plant.

The result obtained with the reference sequence is

shown in ﬁgure 8.

A numerical comparison between the results ob-

tained for this second reference sequence is presented

in table 2.

This table, despite showing the superiority of the

0 10 20 30 40 50 60 70

0

10

20

30

40

Time (s)

Position on the beam (cm)

Figure 8: Neuro-fuzzy controller of the NEFCON model

(dashed) and reference (solid).

NEURO-FUZZY CONTROL OF NONLINEAR SYSTEMS - Application in a Ball and Beam System

205

Table 2: Numerical comparison between the controllers.

METHOD NEFCON ANFIS PID

IAE 2.13 5.55 5.29

ISE 10.82 49.12 41.94

controller that used the NEFCON model, does not

serve as a comparison parameter,since ANFIS copied

the PID and NEFCON is optimal by its own structure.

4 CONCLUSIONS

This study showed how to implement and apply two

neuro-fuzzy techniques to generate controllers (AN-

FIS and NEFCON). A number of applications were

developed and discussed in such a way that both tech-

niques could be applied successfully.

The controller based on the ANFIS model was de-

veloped with learning based on the backpropagation

algorithm, using training pairs extracted from a PID

controller and, in addition to controlling the ball and

beam system, eliminated the high control signal vari-

ations in the learning stage. This fact makes ANFIS

more susceptible to the effects of dry attrition present

in the system. The controller based on the NEFCON

model used a learning technique based on reinforce-

ment learning with a number of alterations, and also

achieved satisfactory results, since it successfully im-

plemented the task of controlling the ball and beam

system.

When compared to ANFIS, NEFCON is less com-

plex to use, requiring it only to inform the algorithm

on the admissible intervals for the input and output

variables of the controller. This greater facility en-

ables learning that is more directed toward user needs.

Optimization techniques such as genetic algo-

rithms and ant colonies, among others, are some other

techniques that may promote an even greater im-

provement in the hybrid models and might include

future characteristics such as the mutation or evolu-

tion of the various forms associated to the structures.

Another factor that could be analyzed is the compati-

bility of the models developed, in order to apply both

in a same plant, in such a way that these models would

collaborate or agree with each other.

ACKNOWLEDGEMENTS

The authors would like to thank CAPES-Brasil and

CENPES-Petrobras for the ﬁnancial support.

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