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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 11, NO. 1, FEBRUARY 2003 135

Self-Organizing Neuro-Fuzzy Systemfor Control of Unknown Plants

Chunshien Li and Chun-Yi Lee

AbstractA cluster-based self-organizing neuro-fuzzysystem (SO-NFS) is proposed for control of unknown plants.The neuro-fuzzy system can learn its knowledge base frominputoutput training data. A plant model is not required fortraining, that is, the plant is unknown to the SO-NFS. Using newdata types, the vectors and matrices, a construction theory isdeveloped for the organization process and the inference activitiesof the cluster-based SO-NFS. With the construction theory, acompact equation for describing the relation between the inputbase variables and inference results is established. This equationnot only gives the inference relation between inputs and outputsbut also specifies the linguistic meanings in the process. Newpseudo-error learning control is proposed for closed-loop controlapplications. Using a cluster-based algorithm, the neuro-fuzzy

system in its genesis can be generated by the stimulation ofinput/output training data to have its initial control policy(IFTHEN rules) for application. With the well-known randomoptimization method, the generated neuro-fuzzy system can learnits data base for specific applications. The proposed approachcan be applied on control of unknown plants, and can levitate thecurse of dimensionality in traditional fuzzy systems. Two examplesare demonstrated.

Index TermsClustering, neuro-fuzzy control, self-learning,self-organization.

I. INTRODUCTION

SINCE fuzzy set theory was proposed by Zadeh [32], fuzzy

logic has been widely applied on various areas, such as pat-tern recognition [3], image processing [20], signal processing

[21], speech recognition [10], decision analysis [33] and con-

trol. Fuzzy logic controllers (FLCs) have been successfully ap-

plied to various control problems, for example, parking control

[24], wastewater treatment [28], servomotor position control

[15], and many others. In most fuzzy control systems, IFTHEN

rules were derived from human experts. Obviously, it is dif-

ficult for a human expert to create appropriate IFTHEN rules

for a complex system by only observing input/output data. To

cope with this difficulty, a new approach for self-organization

of system structure and self-adjustment of system parameters

is proposed in this paper. In addition, input space partition is

an important factor of deciding the number of fuzzy rules in afuzzy system. The most convenient way is to partition the input

space into grids. The major problem of such kind of partition

is that the number of fuzzy rules increases exponentially if the

number of input variables or that of partition increases. This is

Manuscript received April 11, 2000; revised April 19, 2001 and April 26,2002. This work was supported by the National Science Council, Taiwan, Re-public of China, under Grant NSC89-2218-E-002.

The authors are with the Department of Electrical Engineering, Chang GungUniversity, Tao-Yuan 333, Taiwan, R.O.C.

Digital Object Identifier 10.1109/TFUZZ.2002.805898

Fig. 1. Closed-loop learning control.

the so-called problem of curse of dimensionality. To cope with

the curse problem of dimensionality, a clustering method is em-

ployed in this article. The cluster-based algorithm provides a

more flexible way for space partition to avoid drastic increaseof fuzzy rules and thus generates the corresponding rule base

with appropriate number of rules.

The backpropagation (BP) learning algorithm is commonly

used for feedforward neural networks [31]. However, the BP

method, due to its derivative nature, can become trapped at a

local minimum. The convergence of BP is slow, and there are

some limitations in the BP algorithm, for example, reception

and activation functions of neurons in a neural network must

be differentiable. These drawbacks make the BP algorithm in-

efficient to use, especially when the plant is unknown. Instead,

the well-known random optimization (RO) learning algorithm

[12], [13] is used for parameter identification. The RO method

ensures convergence to the global minimum of a cost functionwith probability 1 in a compact set [2], [14], [17], [23]. This

method does not require the derivative of a cost function. It

can use both nondifferentiable and differentiable functions in

a neural network. Moreover, this method is quite useful when

the dimensions of parameters become large [1]. The concept of

pseudo-error is proposed to construct and train the neuro-fuzzy

system to serve as a controller.

The paper is organized as follows. In Section II, we describe

the learning control scheme. The cluster-based self-organization

process is presented in Section III. The architecture of SO-NFS

is described in Section IV. In Section V, the SO-NFS is applied

on two control problems. Discussions are given in Section VI.

Finally, conclusions are summarized.

II. LEARNING CONTROL SCHEME

A feedback control diagram is shown in Fig. 1, in which the

controller is a neuro-fuzzy inference system called SO-NFS

in this article. There are no rules in the SO-NFS at beginning,

and the controlled plant is unknown to the SO-NFS. The

inputoutput training data as the priori knowledge about the

unknown plant is collected and used to construct the SO-NFS.

The pseudo-error learning is proposed for control applications.

1063-6706/03$17.00 2003 IEEE
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136 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 11, NO. 1, FEBRUARY 2003

Fig. 2. Generation of pseudo-errors.

A. Pseudo-Error Learning Scheme

In the learning scheme, the input to the SO-NFS is the error

between theplant outputand thedesired. To obtain training data,

the so-called pseudo-errors are generated and collected. Pseudo-

errors are potential errors that could be occurred in application.

The information of dynamic behavior of a controlled plant canbe unveiled with pseudo-errors. Using pseudo-error distribu-

tion, the application domain of interest in the input space can be

known in advance. This is a very attractive merit to the learning

control approach. The pseudo-errors are generated by the way

that pseudo-targets served temporarily as the desired outputs are

used to compare with plant outputs. The pseudo-error data are

designed using appropriate pseudo-targets to sufficiently cover

the application domain of interest. Pseudo-target is important in

the use of pseudo-error learning approach. Pseudo-targets are

potential targets that could be used in application. To cover the

potential application domain as possible as the SO-NFS can,

the pseudo-targets should be designed so that the pseudo-er-

rors can be scattered on the domain of potential application. Thepseudo-errors collected are then used to construct the SO-NFS

in self-organization phase. The pseudo-error generation process

is shown in Fig. 2. After obtaining the structure of the SO-NFS,

the well-known RO algorithm [12], [13], [17] is used for param-

eter learning to fine-tune the SO-NFS. The objective of param-

eter learning is to find a set of parameters to minimize a cost

function . At each time step, the errors between the de-

sired output and the plant output are calculated and

contributed to the cost function. By repeating the process, we

can define the cost function by accumulating the squared errors

for steps, given as

(1)

In (1), there are training patterns involved in the

training process and time steps to calculate the cost function

[8], [13], [18], in which and are weighting

factors and indicates when to change weighting factor

from to .

B. RO for Pseudo-Error Learning

The well-known RO method was proposed by Matyas [17]

and modified by Solis and Wets [23] who proposed a formula

to adjust the mean vector at each search step to increase its

efficiency. The RO method does not require the derivative of

a cost function. The RO method searches randomly around a

point for a better solution that can minimize the cost function forthe application purpose. In general, the RO randomly generates

two points around the current point in the parameter space and

moves to the better point with a smaller cost value, then around

the new point, the search for better solution continues. The RO

algorithm is further improved by designing interpolation-point

checking and escape-away mechanism to enhance its searching

efficiency. For the th iteration in the learning process, in addi-

tion to the current parameter point and the two randomly

generated points, , the improved RO may use two

more interpolation points in the parameter space

to search for a better solution, when needed. However, when the

RO method menders around a point too long, it can jump ran-

domly to a new point and restarts searching process. The ROmethod ensures convergence to the global minimum of a cost

function with probability 1 in a compact set. The proof

of convergence can be found in literature [2], [14], [17], [23].

The improved RO algorithm used in the paper is given as fol-

lows.

Step 1: (Initial settings)

select an initial set of parameters,

,

set variances for Gaussian random vari-

ables,

set mean vector of Gaussian random

vector to zero,

failure_count 0,

local_search_number some integer

value,

t_iteration total number of iteration,

set , the acceptable error.

Step 2: (Generation of Gaussian random

numbers)

Generate Gaussian random vector with

variance, .

Step 3: (update of system parameters and

mean vector), , and

If is minimum, then

, ,

failure_count 0, and goto Step 6.

elseif is minimum, then

, ,

failure_count 0, and goto Step 6.

else goto Step 4.

Step 4: (Interpolation-point checking,

update of system parameters and mean

vector)
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LI AND LEE: SELF-ORGANIZING NEURO-FUZZY SYSTEM FOR CONTROL OF UNKNOWN PLANTS 141

is the normalized firing strength of the th fuzzy control rule.

Thus, the normalized firing strength vector is described by

......

(40)

From (36), (37), and (40), the normalized firing strength vector

is expressed by

...(41)

where the normalized firing strength of the th rule, , is

expressed by

(42)

for . In the consequents of the th rule, the

coefficients in the combination are given in vector form, i.e.,

(43)

for and . The consequents

are then written as

(44)

where is the th rule action from the th rule for

and . The th fuzzy inferenceaction at time is obtained as follows:

(45)

for . Let the rule action vector for the th rule

be

(46)

and let

(47)

which is called therule action matrix for the fuzzy outputs from

the rules. Using (38), (40) and (47), the fuzzy inference ac-

tion vector is given as

...(48)

Fig. 4. Neuro-fuzzy architecture of SO-NFS.

The final fuzzy inference result in matrix form is given by

(49)

Equation (49) is called the dynamic fuzzy inference system

equation, which describesthe inputoutput relation between the

input crisp vector and the inferenced results . It ex-

presses how the fuzzy inference proceeds, and it shows the par-

ticipation of the crisp value vector , the fuzzy basis set

, the rule base linguistic value set and the conse-

quents , in computing the fuzzy inference

action vector . Let

(50)

The parameters , and in (9), (10) and (50) are called the

fuzzy system parameters. Let the fuzzy system parameters be

collected together to form the fuzzy system parameter set ,

i.e.,

(51)

The rule base is established by (28) for antecedents and by

(47) for consequents. The data base and the rule base are incor-

porated in (49) to form the knowledge base through which the

fuzzy system works.

IV. NEURO-FUZZY ARCHITECTURE OF SO-NFS

The SO-NFS is a neuro-fuzzy inference system imple-

mented in the framework of neural network and fuzzy logic

that combines the learning ability of neural network and the

human-like fuzzy reasoning ability. A six-layer structure is

used to implement the proposed neuro-fuzzy system. The

architecture of SO-NFS is shown in Fig. 4. Explanation for the

six layers in detail is specified as follows.

Layer 0: This layer is the input layer. Each node in this layer

corresponds to an input crisp variable

for

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Fig. 6. Pseudo-errors and generated clusters for pseudo-error learning.

TABLE IMEMBERSHIP FUNCTIONS AND CONSEQUENT PARAMETERS BEFORE ANDAFTER TRAINING FOR PSEUDO-ERROR LEARNING SYSTEM OF EXAMPLE 1

to obtain sufficient error information. The sampling period

was set to 10 s. With and , the generated

clusters and the collected pseudo-errors are shown in Fig. 6.

There are three fuzzy rules (clusters) in therule base of SO-NFS.

To design a cost function in (1), the parameters were set

to , , , , and

. Five training target temperatures 35, 45, 55,65, and 75 C were used in parameter identification process.

The values of the parameters for the antecedent membership

functionsand theconsequentsin thethree fuzzy control rules are

listed in Table I for both before and after 3000 learning cycles.

The learning curve is shown in Fig. 7, and the result of control

is shown in Fig. 8.

For inverse learning control, the input was randomly

generated from 0 V to 5 V at time step , and was the

response of plant to the random input. There were 58 training

patterns collected from 25 C to 80 C to sufficiently cover the

output space. The , and +1) for were

collected as training pattern pairs, . U sing

Fig. 7. Learning curve of pseudo-error learning.

Fig. 8. Response of temperature control (pseudo-error learning).

Fig. 9. Input training patterns and generated clusters for inverse learning.

and in (8), the 58 collected training pat-

terns and thegenerated clusters are shown in Fig. 9. In parameter

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Fig. 10. SO-NFS output and desired output.

TABLE IIMEMBERSHIP FUNCTIONS AND CONSEQUENT PARAMETERS BEFORE AND

AFTER TRAINING FOR INVERSE CONTROL OF EXAMPLE 1

learning phase of inverse learning, and were in-

putted to the SO-NFS, and was the corresponding target.

With the BP algorithm, the error between the SO-NFS output

and the target was accumulated, squared, and then used to form

the cost function, and the learning process was proceeded. After

BP learning, the result is shown in Fig. 10, and the parameters

before and after learning are listed in Table II. The result for in-

verse temperature control of the water-bath system is shown in

Fig. 11.

For the purpose of comparing the pseudo-error learning to

the inverse learning, the ramp-tracing ability, and the rejecting

Fig. 11. Response of temperature control (inverse learning).

Fig. 12. Tracking and disturbance rejection (inverse learning).

ability of disturbance were performed. The reference curve is

given as follows:

Initial condition: C

Reference curve:

Temperature

C min

min

C min

Artificial disturbances:

C at min and min

In Fig. 12, the inverse controller does not trace the reference

curve very well, while the excellent performance is observed in

Fig. 13 with thepseudo-error learning. To compare the proposed

approach to the inverse learning approach, the steady state error,

settling time, and mean squareerrorare summarized in Table III.

Excellent performance with the proposed approach is observed.

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Fig. 13. Tracking and disturbance rejection (pseudo-error learning).

TABLE III

STEADY-STATE ERROR, SETTLING TIME, AND MEAN SQUARE ERROR

Example 2Elevator System Control: An elevator with

loaded and unloaded conditions is used to illustrate the per-

formance of the proposed SO-NFS for motion control. The

force limitation is limited fin between 0 to 50 500 Newtons.

The SO-NFS controls the elevator to trace a reference curve

generated automatically with respect to distance. The reference

curve is designed to have five regions showed in Fig. 14. Theengineering specification for the elevator motion control is

given in Table IV. The total distance for the elevator to move

is known in advance and can be written as

- (52)

To describe the relationship between speed, acceleration, and

distance, the Newtons motion law is used. With the Newtons

motion law for the motion of elevator in Regions I and III, we

have

(for Region I) (53)

(for Region II) (54)

TABLE IVSPECIFICATIONS AND REFERENCE CURVE PARAMETERS FOR MOTION

CONTROL OF ELEVATOR

where and are accelerations in Regions I and III, re-

spectively. Usually, acceleration and deceleration in each di-

rection are given to the same value in specification. For sim-

plicity, we assume that they are the same and are denoted as

. By equating (53) to (54), we have

(55)

With (52) and (55), the can be obtained by

-

(56)

With (56), is calculated using (53). According to (53) and

(56), if - is equal to the minimum of - given

in specifications,thenthe and willbe maximized, i.e.,

and . This can be written

by

(57)

If maximum speed given in specifications

then

maximum speed in the specifications

(58)

else (59)

After the and are determined, the lengths of and

- can be obtained as follows:

-

(60)

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Fig. 14. Distancespeed reference curve.

Therefore, the distancespeed reference curve can be designed

and calculated using (57)(60) and Table IV. The motion equa-

tion of an elevator system is given as

direction sign

when

for upward motion

or when

for downward motion.

otherwise.

(61)

(62)

(63)

direction sign

upward motion

downward motion,

mass of elevator

kg when elevator is loaded

kg when elevator is unloaded

where is the input force to elevator at time ,

the inferred force from SO-NFS, the sampling

time 0.025 s, and the time constant of delay 0.07 s. In

the elevator control, the friction is considered as a constant,

which is denoted as and is set to 50 Newtons. When the

elevator motion is upward, direction_sign is set to 1, otherwise

it is set to 1. When the input force to the elevator

can overcome the gravity and friction forupward motion or when the is greater than for

downward motion to prevent the elevator from falling too fast,

the SO-NFS controls the elevator to move either upward or

downward. When the desired condition for either upward or

downward motion is not satisfied, the brake is always applied

on the elevator to keep it motionless. Whenever the desired

condition is satisfied, the brake is released and the elevator

is controlled by the SO-NFS. The limitations for upward and

downward motion can avoid the elevator from moving in

undesired direction and can restrain the downward motion in

a reasonable acceleration value. The braking force for upward

and downward motion is expressed as

for upward motion

for downward motion

when

otherwise

and

braking constant (64)

For motion control in up and down directions, mass of

elevator for loaded and unloaded conditions, time constant of

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Fig. 15. Pseudo-error data and generated clusters of example 2.

delay, and braking constant are listed in Table IV. Error and its

derivative are selected as the input linguistic variables of the

SO-NFS to control the elevator system. The base variables for

the error and its derivative are denoted by , , where

the is defined by the difference between the reference

speed and elevator speed , i.e., .

The reference curve is used to provide pseudo-target patterns

for pseudo-error generation with a variety of accelerations

and decelerations. After the pseudo-errors are generated and

collected, the clustering algorithm performs the partition of

input space for the initial knowledge base of the SO-NFS.

The spreads for each newly generated cluster are set to [0.1,

0.01] for dimensions of error and its derivative, respectively.The threshold for generating new cluster is set to .

Six rules were generated after the self-organization learning.

The pseudo-error data and the clustering result are shown

in Fig. 15. The fed back error is accumulated to form the

cost function given in (1), and the set of parameters of the

SO-NFS is modified by the RO algorithm to minimize the cost

function. For the cost function in (1), the parameters

were set to , , and ,

. After 5000 learning cycles, the learned results

for the knowledge base of the SO-NFS are listed in Table V.

After learning, the SO-NFS is used to the motion control of

the elevator system. The control scheme is shown in Fig. 16.

The distance of the elevator at time is used toobtain the reference speed from the reference curve.

The is compared to the elevator speed

to produce the error which is fed into the SO-NFS with its

derivative. The inferred result of SO-NFS is multiplied with a

coefficient which is set to 0.12. A saturation function is used

to comply with force limitation. The upward motion control of

the elevator with for the distances of 3.5 meters and 35 meters

and the downward motion for the distances of 3.5 meters and

21 meters are conducted under both loaded and unloaded con-

ditions. The elevator weights 5 10 kg for loaded condition

and 3.5 10 kg for unloaded condition. The control results

of both upward motion for 35 meters and downward motion

TABLE VSO-NFS PARAMETERS FOR MOTION CONTROL OF ELEVATOR SYSTEM

for 3.5 meters are shown in Figs. 17 and 18, respectively. The

final position and positioning accuracy for all above motion

control are summarized in Table VI. Excellent performance of

the SO-NFS is observed in the motion control of the elevator

system. Positioning accuracy is less than 2 10 meters for

various loading conditions in varying distances.

VI. DISCUSSIONS

In the proposed method, the input space of SO-NFS is

formed by pseudo-errors. Pseudo-errors are collected in the

region of interest that errors may be potentially occurred during

the control process. By the pseudo-errors and training patterns,

the SO-NFS can learn to control the dynamic behavior of aplant. The SO-NFS is trained to fine-tune the system param-

eters and control the plant, by minimizing the cost function

formed by the errors between the plant outputs and the desired

outputs. In inverse learning scheme, the SO-NFS learns the

inverse dynamic behavior from observation of the relationship

between inputs and outputs of a plant. The major premise for

the inverse learning is the existence of inverse dynamic of a

plant, which is not generally valid. Using this learning scheme,

the SO-NFS needs to learn the entire inverse dynamic behavior

of the plant in such a way that the plant input can correspond

to the desired output. If the SO-NFS cannot learn the entire

dynamic behavior, the ability of SO-NFS to control the plant

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Fig. 16. Closed-loop elevator control with SO-NFS.

(a) (b)

(c) (d)

Fig. 17. Upward motion control of the loaded elevator system ( m = 5 2 1 0 kg) for 35 m. (a) Timespeed curve. (b) Distancespeed curve. (c) Timeforcecurve. (d) Distanceforce curve.

will be limited or even cannot control the plant. In example 1,

the comparison for the approaches of both the proposed method

and the inverse learning shows that the former is better than the

latter, as shown in Table III and Figs. 8, and 1113.

The SO-NFS is capable of learning the control policy

to a given unknown plant. Training of the SO-NFS and its

eventual control activities are based on fuzzy logic description.

The number of fuzzy control rules can be determined by

the cluster-based learning algorithm and can be limited to a

reasonable range by giving suitable parameters in self-organ-

ization learning. A construction theory for the cluster-based

neuro-fuzzy inference system is presented to provide a the-

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(a)

(b)

Fig. 18. Downward motion control of the unloaded elevator system (m =5 2 1 0 kg) for 3.5 m. (a) Distancespeed curve. (b) Distanceforce curve.

TABLE VIFINAL POSITION AND POSITIONING ACCURACY

oretical foundation for describing how the proposed system

can be established and how the inference activities proceed.

In the construction theory, both the linguistic information

and the numerical calculation are preserved so that the whole

fuzzy activities can be understood for its linguistic meanings,

although the linguistic meanings in the fuzzy inference isusually ignored in most fuzzy systems in the literature. We

think that this is important to originate to fuzzy concept and to

understand what meanings the fuzzy activities are.

The improved RO algorithm is simple to use, which is based

on the Gaussian distribution. It is not complex at all and its con-

vergence has been proved in the literature [2], [14], [17], [23].

The parameters in the RO are variance for Gaussian random

numbers, threshold to determine if the learning is trapped at

a local minimum, acceptable error, and total iterations for

learning. Only two points in the parameter space are checked

in each iteration in the improved RO and two interpolation

points may be checked if needed. The RO method for learning

is better than the BP algorithm in calculation simplicity and

practical model-free approach, especially when the system is

complex and the system parameter set is large. Besides, the BP

is not suitable for model-free learning approach because signals

cannot back-propagate through the controlled plant [18].

VII. CONCLUSION

The proposed SO-NFS combines fuzzy logic and neural

network and takes the advantages of fuzzy data representation,

fuzzy inference, parallel processing, and learning ability. A

fuzzy system construction theory is presented for both the

cluster-based self-organization process and the fuzzy inference

process. The SO-NFS is capable of learning the control policy

to a given unknown plant. Using the observation of input and

output behavior for a plant, a model of the plant is not required

during training process. The random optimization used in the

learning process allows imposing control action constraints.

The training patterns served as the training targets or initial

conditions during control process provide the information for

the SO-NFS to learn the similarities among these trainingpatterns. Once these pattern characteristics have been learned,

the SO-NFS can accommodate various initial conditions or

targets that are never learned before. The two examples show

that the proposed pseudo-error learning method is an excellent

model-free approach, and that the proposed SO-NFS is a

practical learning approach for control.

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Chunshien Li received the B.S. degree in naval ar-chitecture from the National Taiwan Ocean Univer-sity, Taiwan,R.O.C.,in 1984, twoM.S.degreesin en-gineering mechanics and electrical engineering andcomputer science, and the Ph.D. degree in electricalengineering and computer science from the Univer-sity of Illinois, Chicago, in 1990, 1993, and 1996, re-spectively.

Since February 1998, he has been with the Depart-ment of Electrical Engineering, Chang Gung Univer-sity, Tao-Yuan, Taiwan, R.O.C. His current research

interests are fuzzy systems, neural networks, intelligent systems and control,fuzzysystem on a chip,chip/processor-based real-time system applications, andlearning systems.

Chun-Yi Lee was born in Taiwan, R.O.C., in1972. He received the B.S. degree in biomedicalengineering from Chung-Yuan Christian University,Chungli, Taiwan, R.O.C., and the M.S. degree inelectrical engineering from Chang Gung University,Tao-Yuan, Taiwan, R.O.C., in 1996 and 2000,respectively.

He is currently with Programmable Microelec-tronics (Taiwan) Corporation, Hsinchu, Taiwan,R.O.C. His research interests include fuzzy logic,neural networks, fuzzy control, intelligent control,

learning algorithm, and neuro-fuzzy inference systems.

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