1

SOFTWARE FOR FUZZY LOGIC CONTROL

D.H.F. Liu (1994), J. Abonyi (2005)

In 1965, Professor L.A. Zadeh of University California, Berkley, presented his paper

outlying fuzzy theory. In about 1970, fuzzy theory began to produce results in Japan,

Europe, and China. Fuzzy control has found applications in cement kinks,

papermaking machines, and polymerization reactors. Nowadays, fuzzy controllers are

also used to control consumer products, such as dishwashers, washing machines,

video cameras, and rice cookers, even though this fact is not always advertised.

Among the main characteristics of these processes are their time-varying, nonlinear

behavior and the low frequency of their measurements.

Fuzzy logic is a very human concept, potentially applicable to a wide range of

processes and tasks that require human intuition and experience. Fuzzy logic control

can be applied by means of software, dedicated controllers, or fuzzy microprocessors

emdebbed in digital products. Application flexibility combined with inherent

simplicity and wide ranges of capabilities give fuzzy logic technology great potential

for growth.

2

INTRODUCTION

Today’s manufacturing processes present many challenging control problems; among

these are nonlinear dynamic behavior, uncertain and time varying parameters, and

unmeasured disturbances. In the past decade, the control of these systems has received

considerable attention in both academia and industry. While advanced control

strategies — coupled with cheaper and powerful computer systems — have made

nonlinear process control much more practical, there is still a considerable gap

between control theory and industrial practice. It is frustrating for the control theory

community that elegant and comprehensive frameworks for system analysis and

design are seldom applied in the process industry, which still applies the well-known

PID controller (90% of the control-loops), and relies on manual control in complex

process situations.

Conventional control theory may fall short if the model of the process is difficult to

obtain, (partly) unknown, or highly nonlinear. The design of controllers for seemingly

easy everyday tasks such as driving a car or grasping a fragile object continues to be a

challenge for robotics, while human beings easily perform these tasks. Similarly,

some of the industrial processes are controlled by one of the following ways.

• The process is controlled manually by experienced operators.

• The process is controlled by an automatic control system that needs manual

on-line “trimming”.

Many industrial processes operated by humans cannot be automated using

conventional control techniques, since the performance of these controllers is often

inferior to that of the operators. One of the reasons is that linear controllers, which are

commonly used in conventional control, are not appropriate for nonlinear plants.

Another reason is that humans aggregate various kinds of information and combine

3

control strategies, which cannot be integrated into a single analytic control law. The

underlying principle of knowledge-based (expert) control is to capture and implement

experience and knowledge available from experts (e.g., process operators).

A specific type of knowledge-based control is the fuzzy rule-based control, where the

control actions corresponding to particular conditions of the system are described in

terms of fuzzy if-then rules.

To investigate the current application issues of advanced control techniques, a survey

was made in Japan in 1995 [1]. This query asked the number and the evaluation of

control applications, key factors for successful and failed implementations, etc. The

investigated control techniques were classified as follows:

• Advanced PID: I-PID and two degrees of freedom PID, decoupling PID, dead-

time compensation, gain scheduling, and auto tuning control

• Modern Control Theory: LQG regulator, observer, Kalman filter, model

predictive control (MPC), adaptive control, H-infinity control, repetitive

control, exact linearization control, optimization control

• FAN (Fuzzy, Artificial intelligence, Neural network): Fuzzy control, rule-based

control, neural network control.

The survey showed that advanced PID type control is widely applied and about 30%

of the 110 respondents have already used this type of control in their factories. MPC

and fuzzy control are the most widely used of the modern control and FAN techniques.

Slightly less than 40% of the factories have applied these solutions. Compared to a

survey in 1989, the applications of decoupling PID, dead-time compensation, Kalman

filter, model predictive control, rule-based control, fuzzy control and optimization are

increasing more and more. Although modern control theory is rarely applied, except

model-predictive control, 60-70% of the respondents are satisfied with the results and

4

the satisfaction with FAN is going up. These tendencies indicate that there is a huge

demand in the industry for new fuzzy control solutions.

The objective of this section is to present the principle of fuzzy modeling and control,

identify and explain the various design choices, and overview the implementation

tools for engineers.

PRINCIPLE OF FUZZY SYSTEMS

Fuzzy control is a control method based on fuzzy logic. Fuzzy logic is not really

„fuzzy”. Just as fuzzy logic can be described simply as ’’computing with words rather

than numbers’’, fuzzy control can be described simply as ’’control with sentences

rather than equations’’.

For many real world applications a great deal of information is provided by human

experts, who do not reason in terms of mathematics but instead describe the system

verbally through vague or imprecise statements like,

IF The Temperature is Big THEN The Pressure is High.

Because so much human knowledge and expertise come in terms of such verbal rules,

one of the sound engineering approaches is to try to integrate such linguistic

information into the modeling process. A convenient and common approach of doing

this is to use fuzzy logic concepts to cast the verbal knowledge into a conventional

mathematical representation.

Fuzzy logic facilitates the representation in digital computers of this kind of

knowledge, which subsequently can be fine-tuned using process experiments (e.g.

based on input-output process data). From this basis, fuzzy system is a computation

framework based on the concepts of fuzzy sets, fuzzy if-then rules, and fuzzy

reasoning. This section will introduce the reader to the structure of fuzzy models. It

5

will not attempt to provide a broad survey of the field. For such a survey the reader is

referred to “An Introduction to Fuzzy Control” by Driankov, Hellendoorn, and

Reinfrank [2] or “Fuzzy Control” by K.M. Passino and S. Yurkovic [3], or “A course

in Fuzzy Systems and Control“ by L.X. Wang [4].

Fuzzy Sets

Conventional set theory is based on the premise that an element either belongs to or

does not belong to a given set. Fuzzy set theory takes a less rigid view and allows

elements to have degrees of membership of a particular set such that elements are not

restricted to either being in or out of a set but are allowed to be “somewhat” in. In

many cases this is a more natural approach. For example, consider the case of a

person describing the atmospheric temperature as being “hot”. If one was to express

this concept in conventional set theory one would be forced to designate a distinct

range of temperatures, such as 25°C and over, as belonging to the set hot. That is

hot = [25;∞)°C. This seems contrived because any temperature which falls just

slightly outside this range would not be a member of the set, even though a human

being may not be able to distinguish between it and one which is just inside the set.

In fuzzy set theory, a precise representation of imprecise knowledge is not enforced

since strict limits of a set are not required to be defined, instead a membership

function is defined. A membership function describes the relationship between a

variable and the degree of membership of the fuzzy set that correspond to particular

values of that variable. This degree of membership is usually defined in terms of a

number between 0 and 1, inclusive, where 0 implies total absence of membership, 1

implies complete membership, and any value in between implies partial membership

of the fuzzy set. This may be written as follows: A(x) = [0; 1] for x ∈ U, where A(x)

6

is the membership function and U is the universe of discourse which defines the total

range of interest over which the variable x should be defined.

For example, to define membership of the fuzzy set, hot, a function which rises from

0 to 1 over the range 15°C to 25°C may be used, i.e.,

( )

°>°≤≤

°<−=

Cxx

CxxxA

252515

15

110

150

. 2.31(1)

This implies that 15°C is not hot; 20°C is a bit hot; 23°C is quite hot; and 30°C is

truly hot. Specific measurable values, such as 15 and 20 are often referred to as crisp

values or fuzzy singletons , to distinguish them from fuzzy values, such as hot, which

are defined by a fuzzy set. Fuzzy values are sometimes also called linguistic values.

As Figure 2.31a illustrates, this definition is more reflective of human or linguistic

interpretations of temperatures and hence better approximates such concepts.

10 15 20 25 300

0.5

1

Temperature [deg °C]

Mem

bers

hip

/ Tru

th

Fuzzy

Crisp

10 15 20 25 300

0.5

1

Temperature [deg °C]

Mem

bers

hip

/ Tru

th

Fuzzy

Crisp

FIG. 2.31a

Fuzzy set representation of high temperature.

While seeming imprecise to a human being, fuzzy sets are mathematically precise in

that they can be fully represented by exact numbers. They can therefore be seen as a

method of tieing together human and machine knowledge representations. Given that

such a natural method of representing information in a computer exists, information

processing methods can be applied to it with the use of fuzzy systems.

7

Fuzzy Systems

A fuzzy controller has a set of rules that it uses to calculate the final control action.

Each rule is a linguistic expression about the control action to have taken in response

to a given set of process conditions. The rules are in the familiar if-then format, and

formally the if-side is called the condition and the then-side is called the conclusion

(more often, perhaps, the pair is called antecedent - consequent or premise -

conclusion)

IF (CONDITION) THEN ACTION

Take for instance a typical fuzzy controller with two rules:

1. IF error is Neg and change in error is Neg then output is NB

2. IF error is Neg and change in error is Zero then output is NM

The input value ’’Neg’’ is a linguistic term short for the word Negative the output

value ’’NB’’ stands for Negative Big and ’’NM’’ for Negative Medium. The computer

is able to execute the rules and compute a control signal depending on the measured

inputs error and change in error.

The collection of such rules is called a rule base. The basic idea of fuzzy logic control

is to formulate sets of rules for automatic operation based on practical experience and

knowledge about manual control of the process. In fuzzy logic, the condition of a rule

is fulfilled to a certain degree, and each rule will influence the result of the set of rules

in accordance with its grade of fulfillment.

To illustrate this procedure the basic configuration of a fuzzy system is shown in

Figure 2.31b.

8

FIG. 2.31b

Fuzzy controller in a closed-loop configuration (top panel) consists of dynamic filters

and a static map (middle panel). The static map is formed by the knowledge base,

inference mechanism and fuzzification and defuzzification interfaces.

As it is depicted in this figure, a fuzzy system involves the following components [5]:

Data preprocessing processes the controller’s inputs in order to obtain the inputs of

the static fuzzy system. It will typically perform some of the following operations on

the input signals:

• Signal Scaling. The physical values of the input of the fuzzy system may

differ significantly in magnitude. By mapping these to proper normalized (but

interpretable) domains via scaling, one can instead work with signals roughly

of the same magnitude. This is accomplished by normalization gains which

scale the input into the normalized domain [-1,1]. Values that fall outside the

normalized domain are mapped onto the appropriate endpoint.

9

• Dynamic Filtering. In a fuzzy PID controller, for instance, linear filters are

used to obtain the derivative and the integral of the control error, e. Nonlinear

filters are found in nonlinear observers, and in adaptive fuzzy control where

they are used to obtain the fuzzy system parameter estimates.

• Feature Extraction. Through the extraction of different features numeric

transformations of the controller inputs are performed. These transformations

may be Fourier or wavelet transforms, coordinate transformations or other

basic operations performed on the fuzzy controller inputs.

Fuzzification maps the crisp values of the preprocessed input of the model into

suitable fuzzy sets represented by membership functions.

The Rule base is the cornerstone of the fuzzy model. The expert knowledge, which is

assumed to be given as a number of if-then rules, is stored in a fuzzy rule base.

According to the consequent proposition, there are three distinct classes of fuzzy

models:

– Fuzzy linguistic models (Mamdani models) where both the antecedent and

consequent are fuzzy propositions. Hence, a general rule of a linguistic or

Mamdani fuzzy model is given by

jjnnjj BisyAisxAisxR thenandandIf ,,11: K , where Rj denotes

the j-th rule, rNj ,...,1= , and rN is the number of the rules. The antecedent

variables represent the input of the fuzzy system, ix . jiA , and jB are fuzzy

sets described by membership functions.

– Fuzzy relational models are based on fuzzy relations and relational equations

[6]. These models can be considered as a generalization of the linguistic

10

model, allowing one particular antecedent proposition to be associated with

several different consequent propositions via a fuzzy relation.

– Takagi — Sugeno (TS) fuzzy models where the consequent is a crisp function

of the input variables, ( )xjf , rather than a fuzzy proposition [7].

( )xthenandandIf jjnnjj fyAisxAisxR =,,11: K 2.31(2)

Mamdani (linguistic) fuzzy models with either fuzzy or singleton consequents are usually

used as a direct closed-loop controller, while Takagi–Sugeno (TS) fuzzy systems are

typically used as a supervisory controllers of fuzzy models of dynamical processes.

Inference engine. The inference mechanism or inference engine is the computational

method which calculates the degree to which each rule fires for a given fuzzified input

pattern by considering the rule and label sets. A rule is said to fire when the conditions

upon which it depends occur. Since these conditions are defined by fuzzy sets which

have degrees of membership, a rule will have a degree of firing or firing strength. The

firing strength is determined by the mechanism which is used to implement the and in

the expression. There are different methods for implementing each of the logical

operators and the reader is referred to [2] for the details on these.

Please, place Figure 2.31c here.

As can be seen on Figure 2.31c, in the presented example the logical product (AND

connection) of a group of grades is simply the minimum value of all of the grades.

Thus, in rule 5, the logical product of 1 for the control error and 0.5 for the change of

the control error is 0.5. This procedure is repeated for each of the rules, as shown in

Figure 2.31c. The logical sum combines the results of the rules. This is shown in the

bottom of the corner of Figure 2.31c.

Defuzzification. A defuzzifier compiles the information provided by each of the rules

and makes a decision from this basis. In linguistic fuzzy models the defuzzification

11

converts the resulted fuzzy sets defined by the inference engine to the output of the

model to a standard crisp signal. The centroid method of defuzzification takes a

weighted sum of the designated consequences of the rules according to the firing

strengths of the rules. In the example shown in Figure 2.31c, which is the special case

of Takagi-Sugeno fuzzy models, this type of defuzzification results in the following

simple form:

( ) ( )∑=

=rN

jjj fy

1

xxβ 2.31(3)

where ( )xjβ is the normalized degree of fulfillment (firing strength) of the rules:

∑∏

∏

= =

==rN

i

n

jiji

n

jiji

j

xA

xA

1 1,

1,

1

)(

)(

)(xβ . 2.31(4)

Postprocessing. The preprocessing step gives the output of the fuzzy system based on

the crisp signal obtained after defuzzification. This often means the scaling of the

output. Operations this post-filter may perform include:

• Signal Scaling. A denormalization gain can be used which scales the output of

the fuzzy system to the physical domain of the actuator signal.

• Dynamic Filtering. In some cases, the output of the fuzzy system is the

increment of the control action. The actual control signal is then obtained by

integrating the control increments. Of course, other forms of smoothing

devices and even nonlinear filters may be considered.

12

Example: Fuzzy PI controller

The previously presented decomposition of a controller to a static map and dynamic

filters can be done for most classical control structures. To see this, consider a

Proportional-Integral (PI) controller, biasdtteT

teKtu +

+= ∫ )(1)()( which can be

expressed in discrete form by a simple difference equation: )()()( 21 keqkeqku ∆+=∆ ,

where )1()()( −−=∆ kekeke and 21,qq are the parameters of the controller that define a

hyperplane shown in Figure 2.31d.

AND

Aggregation Accumulation

Defuzzification

Activation

β 4

β 5

e ∆e∆u

AND

Aggregation Accumulation

Defuzzification

Activation

β 4

β 5

e ∆e∆u

FIG. 2.31c

Example for a fuzzy inference.

This linear form can be generalized to a nonlinear function: ( )xfku =∆ )( , where the

inputs of this static mapping are )(1 kex = and )(2 kex ∆= . In a fuzzy controller this

nonlinear function f is represented by a fuzzy mapping.

Since the rule base represents a static mapping between the antecedent and the

consequent variables, external dynamic filters must be used to obtain the desired

dynamic behavior of the controller (Figure 2.31b). Furthermore, as the PI controller

is defined in incremental form, signal processing is required both before and after the

fuzzy mapping to obtain the control signal: )()1()( kukuku ∆+−= .

The set of rules of the PI fuzzy controller can be expressed with the following form:

13

)()()()(: ,2,1,2,1 keqkequTHENAiskeandAiskeIFR jjjjjj ∆+=∆∆ 2.31(5)

where ju∆ denotes the output of the j-th implication, and )2,1(, =pq jp are consequent

parameters of the sub-PI controllers.

When the rule consequents are constants instead of local PI controllers, the resulted

Mamdani fuzzy system can be viewed as defining a piecewise constant function with

extensive interpolation.

-1000

100

-1000

100-200

0

200

e

∆u

∆e-1000

100

-1000

100-200

0

200

e

∆u

∆e

FIG. 2.31d

Control surface of a linear PI controller.

TABLE 2.31e

Rule base of a simple fuzzy PI controller.

e ∆e N ZE P

N -200 -100 0

ZE -100 0 100

P 0 100 200

14

An example of such a rule base is given in Table 2.31e. Three linguistic terms are

used for each variable, (N – Negative, ZE – Zero, P – Positive). Each entry of the

table defines one rule, e.g. R11: “If e is N And ∆e is N then ∆u= - 200”. When the

fuzzy inference shown in Figure 2.31b is used to calculate the output of the fuzzy

model, the resulting control surface obtained by plotting the inferred control action for

discretized values of e and ∆e is shown in Figure 2.31f.

-1000

100

-100

0

100-200

-100

0

100

200

∆ ee

∆u

-1000

100

-100

0

100-200

-100

0

100

200

∆ ee

∆u

FIG. 2.31f

Nonlinear control surface of a fuzzy PI controller.

DIRECT AND SUPERVISORY FUZZY CONTROL

Since the first applications of PID-type fuzzy controllers the application range of

fuzzy systems has widened substantially. Although in most cases a fuzzy controller is

used for direct feedback control, it can also be used on the supervisory level, e.g., as a

self-tuning device in a conventional PID controller. Also, fuzzy control is no longer

only used to directly express a priori process knowledge. For example, a fuzzy

controller can be derived from a fuzzy model obtained through system identification.

Therefore, a very general definition of fuzzy control can be given: A fuzzy controller

15

is a controller that contains a (nonlinear) mapping that has been defined by using

fuzzy if-then rules. There are four main sources for finding such fuzzy rules:

• Expert experience and control engineering knowledge: One classical example

is the operator’s handbook for a cement kiln [8]. The most common approach

to establishing such a collection of rules of thumb, is to question experts or

operators using a carefully organized questionnaire.

• Based on the operator’s control actions: Fuzzy if-then rules can be deduced

from observations of an operator’s control actions or a log book. In this case

the rules express input-output relationships.

• Based on a fuzzy model of the process: A linguistic rule base may be viewed as

an inverse model of the controlled process. Thus the fuzzy control rules might

be obtained by inverting a fuzzy model of the process. This method is

restricted to relatively low order systems, but it provides an explicit solution

assuming that fuzzy models of the open and closed loop systems are available

[9]. Another approach is fuzzy identification or fuzzy model-based control (see

the following subsection).

• Based on learning (Adaptive fuzzy control): Adaptive or learning control

system is used to cut down the amount of required human intervention, to

increase the flexibility of the algorithm, to improve the control performance,

and to reduce the initial design time and cost. Learning ability is one of the

essential attributes of intelligent controllers. Being intelligent means that the

controller can improve its future performance based on information what it has

gained in the past. Because of the recent surge of these techniques, one of the

recent research directions is to design fuzzy systems that have learning

capabilities. The first adaptive fuzzy controller was proposed by Procyk and

16

Mamdani (1979) [10]. This self-organizing algorithm was the basis of the

algorithms of Árva (1991) [11], Renders (1997) [12], and others. Nowadays, a

promising approach is the combination of neural networks and fuzzy logic

systems. Artificial neural networks (ANN) are easy to train and have known

convergence properties [13]. Because they are opaque, the resulted model is

not interpretable and it is difficult to incorporate linguistic and/or first-

principle knowledge into the identification procedure. Conversely, fuzzy

models provide transparency through they linguistic interpretability. The

combination of these methods resulted in neuro-fuzzy systems [14, 15]. The

rule base of these neuro-fuzzy models is initialized by expert knowledge, and

the model parameters are determined via similar methods that are used for the

training of neural networks. As the neuro-fuzzy model can initialize and learn

linguistic rules, the modeling framework can be considered as a direct transfer

of knowledge. This is the powerful advantage of adaptive fuzzy systems in

comparison with other nonlinear model structures (e.g. neural networks).

Classical fuzzy control algorithms

The early work in fuzzy control was motivated by a desire to mimic the control

actions of an experienced human operator (knowledge-based part) and obtain smooth

interpolation between discrete outputs that would normally be obtained (fuzzy logic

part). Hence, these fuzzy controllers performs just as well as the operators whose

knowledge was used for control design. Note, this approach is not only for control

applications. If the target system to be emulated is a human physician or a credit

analyst, then the resulting fuzzy inference systems become fuzzy expert systems for

diagnosis and credit analysis, respectively.

17

ProcessPID

Fuzzy

Scheme a

ProcessProcessPIDPID

FuzzyFuzzy

Scheme a

ProcessPID

Fuzzy

Scheme b

ProcessPID

Fuzzy

ProcessProcessPIDPID

FuzzyFuzzy

Scheme b

ProcessPID

Fuzzy

Scheme c

ProcessProcessPIDPID

FuzzyFuzzy

Scheme c

FIG. 2.31g

Fuzzy controller configurations. Fuzzy replaces PID (a), fuzzy adjusts PID

parameters (b), and fuzzy adds to PID output (c).

As it is illustrated in Figure 2.31g, beside this direct control scheme fuzzy controllers

are being used in various control schemes:

Scheme a, In this configuration, the operator may select between a fuzzy control

strategy and conventional control loops. Often the conventional loops represent an

existing control scheme, which has been controlling the process before installation of

a high level strategy. The operator has to decide which of the two alternatives is the

most likely to produce the best control performance.

Scheme b. In this configuration, the high level strategy is used for adjustments of the

parameters of the conventional control loops. A common problem with linear PID

controllers used for control of highly nonlinear processes is that the set of controller

18

parameters produces satisfactory performance only when the process is within a small

operational region. Outside this region, other controller settings are necessary, and

these adjustments may be done automatically by a high level strategy. This is called

gain scheduling since such algorithms were originally used to change process gains. A

gain scheduling controller contains a linear controller whose parameters are changed

as a function of the operating point in a preprogrammed way. It requires thorough

knowledge of the plant, but it is often a good way to compensate for nonlinearities

and parameter variations. Additional sensor measurements can be used as scheduling

variables that govern the change of the controller parameters, often by means of a

look-up table.

Scheme c. can be viewed as a collaboration of linear and nonlinear control actions; the

controller may be a linear PID controller, while the fuzzy controller is a

supplementary nonlinear controller. Normally, conventional control systems which

are based on PID controllers are capable of controlling the process when the operation

is steady and close to normal conditions. However, if sudden changes occur or if the

process enters abnormal situations, then this configuration may be useful to bring the

process back to normal operation as fast as possible. This scheme can also be

interpreted as a feedforward-feedback control, where a measurable disturbance is

being compensated. Generally, this solution requires good model, but if a

mathematical model is difficult or expensive to obtain, a fuzzy model may be useful.

These approaches resulted in many successful applications as summarized in Table

2.31h.

19

TABLE 2.31h

Application of conventional fuzzy controllers.

Start-up of a Catalytic Reactor by Fuzzy Controller Yamashita, 1988 [16]

Fuzzy Control of a Polymerization Reactor Roffel, 1991[17]

Fuzzy Control of an Iron Ore Sintering Plant Arbeithuber, 1995 [18]

Self-Tuning of a PI Controller using Fuzzy Logic, for a Construction Unit Testing Apparatus Berger, 1996 [19]

Adaptive Fuzzy Control of the Molten Steel Level in a Strip-Casting Process Lee, 1996 [20]

Temperature Control of Highly Exothermic Batch Polymerization Reactors Ni, 1997 [21]

pH Fuzzy Control of Automated Industrial Electroplating of Gold Jinxiang, 1997 [22]

Fuzzy Self-Organizing pH Control of an Ammonia Scrubber Ylen, 1997 [23]

Fuzzy Logic Control of an Unstable Biological Reactor Inamdar, 1997 [24]

A Study of Gain Scheduling Control Applied to an Exothermic CSTR Lagerberg, 1997 [25]

Modeling and Control of Continuous Stirred Tank Reactor for Thermal Copolymerization Hwan, 1997 [26]

Fuzzy Control of Batch Polymerization Reactor Abonyi, 1997 [27]

Application example: Fuzzy Control of a Polymerization reactor

In this example the fuzzy control of the batch production of Expanded Polystyrene

(EPS) is investigated. The typical unit of batch processes in polymer industries is an

autoclave with a heating-cooling system, where the reactor temperature is controlled

by adjusting the temperature of the water recirculating thought the jacket of the

reactor. Compared to continuous processes the control of batch processes is more

difficult due to complex sequential control tasks and the time-varying operating

conditions. In this case study a further difficulty arises from the fact that vinyl

polymerization is an exothermic auto-catalytic reaction.

20

Several researches have addressed control problems in reactor systems of this type.

Juba and Hamer [28], and Berber [29] provide an excellent overview on the

challenges in batch reactor control and suggest several control strategies. Davidson

proposes a control algorithm which combines knowledge of the process with the logic

used by the skilled operator and compares the result with standard PI controllers [30].

Examples for the applications of fuzzy control to chemical reactors can be found in

the literature [9].

In the following the benefits of the application of the previously presented fuzzy

control schemes will be presented.

The aim of the development of fuzzy controller was to obtain better control

performance than the performance of the PID controller. The parameters of the

designed controllers were determined based on simulator experiments. The simulator

of the process has been built using MATLAB and C based on the first-principle

mathematical model of the process [31,32,33]. During the tuning of the controller the

following performance index was minimized by Sequential Quadratic Programming

∑∑==

∆+=N

k

N

k

kukeQ1

2

1

2 )()( λ where λ is a weighting factor to maintain the dynamics of

control signal (0.2).

As we have shown in the previous example, conventional control solutions (PID

algorithms, gain-scheduling) can be realized by choosing the proper fuzzy structure

and can be improved by applying Takagi-Sugeno fuzzy models. Among the several

possibilities Takagi-Sugeno fuzzy controllers were designed, The rules of the fuzzy

PI controller were formulated by Equation 2.31(1). As Table 2.31i demonstrates, the

extension of the classical control schemes with nonlinearities defined by fuzzy

systems resulted 5-30% improvement of the performance index characterizing the

operation of the controllers.

21

TABLE 2.31i

Comparison of the performances of various controllers.

Control algorithm Performance Index

Optimal PI 1.883 Optimal gain-scheduled PI 1.633 Optimal PI type fuzzy 1.579 Optimal fuzzy supervised PI controller

1.509

40

60

80

100

120

0.0

0.5

0.9

1.4

1.9

2.4

2.8

3.3

3.8

4.3

4.7

5.2

5.7

6.1

6.6

7.1

7.6

8.0

8.5

9.0

9.4

9.9

time (hour)

T (°

C)

0

10

20

30

40

50

60

70

80

90

100

konv

( - )

Tr

Tin

Tout

Tw

conversion

FIG 2.31j

Performance of the optimal fuzzy supervised PI –controller.

This example showed that fuzzy control is not only useful to handle heuristic control

knowledge, but it can be a proper tool where the already existing linear control

algorithm should replaced by a better, more effective nonlinear control algorithm.

22

MODEL BASED FUZZY CONTROL

In spite of the practical successes, conventional fuzzy control has some drawbacks.

Firstly, fuzzy control rules are experience oriented and/or obtained after time-

consuming trial-and-error experiments. Secondly, when a significant change in the

system occurs that is outside of the operator experience, re-tuning of the fuzzy

controller is necessary. Moreover, it is much easier to obtain information on how a

process responds to particular inputs than to record how and why an operator responds

to particular situations. These difficulties explain the recent surge of interest in model

based design of fuzzy controllers, where fuzzy or inverse fuzzy models are used to

design the controller [34, 35, 36].

Fuzzy modeling can be effectively used in the identification of process dynamics in

order to cope with nonlinear and complex systems [9, 36,37,38]. The applicability of

fuzzy models and model-based control techniques are illustrated by several laboratory

scale or industrial examples overviewed in Table 2.31k.

23

TABLE 2.31k

Application of fuzzy techniques in model based control.

Fuzzy Modelling and Control of Multilayer Incinerator Sugeno, 1986 [39]

A Model Based Controller Postlethwaite, 1994 [40]

Fuzzy Model Based Control: Stability, Robustness, and Performance Issues Johansen, 1994 [41]

Predictive Control of Batch Fermentation Process Foss, 1995 [42]

Application of Fuzzy Relational Modeling to Industrial Product Quality Control Qian, 1995 [43]

Fuzzy-Neural-Net-Based Inferential Control for a High-Purity Distillation Column Luo, 1995 [44]

Fuzzy Models in Process Optimal Control Keil, 1995 [45]

Predictive Fuzzy Control Applied to the Sinter Strand Process Hu, 1996 [46]

Building a Model Based Fuzzy Controller Postlethwaite, 1996 [47]

Fuzzy Relational Model Based Control Applying Stochastic and Iterative Methods for Model Identification

Sing, 1996 [48]

Long-Range Predictive Control Using Fuzzy Process Models Linkens, 1996 [49]

A New Indentation Algorithm for Fuzzy Relational Models and its Applications in Model Based Control

Postlethwaite, 1997 [50]

Model Based Design of Fuzzy Control Systems Abonyi,2003 [9]

Inverse Fuzzy Model Based control

The simplest way to control a process using a fuzzy model is to use inverse process

model as a controller. Several methods can be applied to obtain the inverse fuzzy

model of a given process, but the following two are the most utilized:

• Identification of the inverse model from input-output data.

• Inversion of the original model.

If an ideal model of the process is available, the control is perfect and input-output

stable [51]. This situation of perfect control is impossible to achieve because an exact

inversion of the process can only be found in special situations and the model is never

identical to the process, resulting in model-plant mismatch.

24

The Internal Model Controller (IMC) structure consists of a controller, a process

model, a filter, and the process itself as shown in Figure 2.31l.

Fuzzy Controller

Process

Fuzzy Model

Fuzzy model inversion

Filter

w y

-

-

+

+

em

ef

u

FIG 2.31l

Fuzzy model in IMC architecture.

In this control arrangement, the process model is placed in parallel with a real system.

The difference between the system and its model represents the modeling error and

unmodeled process disturbances. This filtered difference is used by the controller to

compensate the disturbance and the effects of the modeling error. The IMC structure

provides a direct method for the design of feedback controllers. According to the

above mentioned properties of the control scheme, if a good model of the plant is

available, the closed-loop system will achieve exact setpoint following despite

unmeasured disturbances acting the plant.

In [52] an adaptive fuzzy controller has been designed based on fuzzy relational

model and applied to a laboratory-scale liquid level rig. Posthlethwaite has also

applied fuzzy relational models for the identification and control of nonlinear, mainly

first-order processes [50]. Contrary to these solutions, special types of Takagi-Sugeno

fuzzy models can be analytically inverted. Such controllers were applied for the

control of a radial industrial-cooling blast designed for air conditioning of buildings

[53] and pressure control of a fermenter [36].

When the process and control variables are subjected to level and rate constraints,

and/or the system has significant time-delay or nonminimum-phase behavior, the

inversion must be done for p-step-ahead [54, 55].

25

Fuzzy model based predictive control

The p-step-ahead (p-inverse) controller can be seen as a special case of model

predictive control (MPC). The nonlinear fuzzy models can be incorporated into this

MPC scheme as it is depicted in Figure 2.31m.

u(.) y(.)w(.)Optimizer Process

Fuzzymodel

MPC Algorithm

FIG. 2.31m

Fuzzy model in model predictive control architecture.

Among the several possible solutions of MPC, mainly the linearization-based

approach has been utilized in the literature, since when a linear model is extracted

from a fuzzy model, the controller can be based on classical linear model-based

control algorithms [56, 57]. Such local design techniques derived from model

predictive control has been applied in [9, 58, 59, 60, 61, 62].

26

Operating regime based modeling

The modeling framework that is based on combining a number of local models, where

each local model has a predefined operating region in which the model is valid is

called operating regime based model. The basic idea of operating regime based

modeling is the following.

Every model has a limited range of validity. This may be restricted by the modeling

assumptions for a mechanistic model, or by the experimental conditions model which

the identification data was logged for an empirical model. The model that has a range

of validity less than the operating regime of the process is called local model, as

opposed to a global model that is valid in the full range of operation. Global modeling

is a complicated task because of the need to describe the interactions between a large

number of phenomena that appear globally. Local modeling, on the other hand, may

be considerably simpler, because locally there may be a smaller number of

phenomena that are relevant, and their interactions are simpler. The local models can

be combined into a global model using an interpolation technique.

The main advantage of this framework is its transparency. Both the concept of

operating regimes, and the model structure, are easy to understand. This is important,

since the model structure can be interpreted in terms of operating regimes, but also

quantitatively in terms of individual local models. Furthermore, the operating regimes

can be represented as a fuzzy set [36, 38]. This representation is appealing, since

many systems change behavior smoothly as a function of the operating point, and the

soft transition between the regimes introduced by the fuzzy set representation captures

this feature in an elegant fashion. An example of such fuzzy models is the Takagi-

Sugeno fuzzy model.

Takagi-Sugeno fuzzy models of dynamical systems

It is assumed, that the MIMO dynamic process can be represented by the following

nonlinear vector function: ))1(),...,(),1(),...,(()1( +−−−+−=+ bdda nnknknkkfk uuyyy ,

where TNyyy ],...[ 1=y is an yN dimensional output vector, T

Nuuu ],...[ 1=u is an uN

dimensional input vector, an and bn are maximum lags considered for the outputs and

inputs, respectively, dn is the minimum discrete dead time, and f represent the

nonlinear model.

27

This MIMO Nonlinear Autoregressive with eXogenous Input (NARX) model can be

represented by a Takagi-Sugeno fuzzy model,

jd

n

i

ji

n

i

ji

j

jnnjj

nikikk

AiszAiszRba

cuByAy

thenandandIf

++−−++−=+ ∑∑==

)1()1()1(

:

11

,,11 K

2.31(6)

Where ji

ji BA , matrices and the jc vector represent the j-th linear multivariable

model, where the operating regime of this model in defined by the antecedent part of

the j-th fuzzy rule. In these rule antecedents )(, iji zA represents the j-th antecedent

fuzzy sets for the i-th input, where ],...[ 1 nzz=z is a “scheduling” vector, which is

usually a subset of the previous process inputs and outputs, )}1(),...,1(),...,(),1(),...,1(),...,({ 1111 +−−+−−−+−+−= bdNubddaNya nnknnknknknkk uuuyyyz

The proposed fuzzy model can be seen as a multivariable linear parameter varying

(LPV) system model, where at the z operating point, the fuzzy model represents the

following linear time-invariant (LTI) model

)()1()()1()()1(11

zcuzByzAy ++−−++−=+ ∑∑==

d

n

ii

n

ii nikikk

ba

2.31(7)

with

)()(1∑=

=rN

j

jiji AzzA β

,

)()(1∑=

=rN

j

jiji BzzB β

,

)()(1∑=

=rN

j

jj czzc β

, where

1)(0 ≤≤ zjβ is the normalized truth value of the j-th rule are calculated by Equation

2.31(4).

28

Example: Fuzzy Model based Control of a Distillation Column

The examined process is a first-principle model of a binary distillation column. The

column has 39 trays, a reboiler and a condenser. The simulation model was developed

by Skogestad [63]. The simulated system covers the most important effects for the

dynamic of a real distillation column. The studied column operates in LV

configuration with two manipulated variables (reflux and boilup rate, u1=L,V, u2=V)

and two controlled variables (top and bottom impurities, y1=1-xD, y2= xB).

The goal of this example was improve the performance of the process by replacing the

utilized PID controller to an advanced control strategy. Firstly a linear model

predictive controller (MPC) has been designed, but the control performance of the

control loop has not been improved significantly. In order to reach much better

performance, the linear model of the process has been replaced by a nonlinear

operating-regime based model of the distillation column. The identified Takagi-

Sugeno fuzzy model was used to represent the nonlinear dynamic system that is based

on the interpolation between local linear MIMO ARX models. As the process gain

varies in direct proportion to the concentrations, the fuzzy sets - the operating regions

of the local models - are defined on the domain of the product impurities. This result

in the following MIMO TS fuzzy model structure given by Equation 2.31(6).

The order of the local models is chosen to be na=nb=2. The process was identified

without time delay, nd=0. This rule base defines a grid-type partition of the operating

regime. As Figure 2.31n shows this fuzzy model consists of 16 multivariable local

models.

29

FIG. 2.31n

Example for operating regions of local linear models defined by fuzzy sets.

Because of the utilized nonlinear model, in the resulted MPC a nonlinear optimization

problem had to be solved. To avoid this, a linear model is extracted from the nonlinear

fuzzy model at each sampling time based on the linear-parameter-varying model

interpretation of the Takagi-Sugeno fuzzy system. The multivariable generalized

predictive controller (GPC) [64] is based on a linear ARX model extracted from the

fuzzy model at each sampling time [9].

Hence, based on the y(k) measured outputs, the previous values of the u control

signal, the future set-points w, and the )(),(),( zczBzA jj parameters of the actual

linear approximation of the nonlinear system (see Figure 2.31o), the multivariable

general predictive controller (GPC) calculates the future control sequence vector

{ })( jk +u at time k that minimizes a quadratic cost function.

30

PC

D, xD L

MD

LC

p

F, zF

MB

B, xB

LCV

V

L

1-xD

xB

TS fuzzy model

A(z), B(z), c(z)

w1 w2

Linear MPC

FIG. 2.31o

Fuzzy model based predictive control of a distillation column controlled with LV-

configuration.

The achieved results are promising even from the industrial application perspective

(Figure 2.31p).

31

FIG. 2.31p

Performance of the constrained discrete PID (thick -), the constrained linear MPC

(thin -) and the constrained nonlinear MPC (thick -- -- ) for the distillation column.

200 400 600 800 1000 1200 1400Time [min]

0 200 400 600 800 1000 1200 14000

0.005

0.01

y 1

0 200 400 600 800 1000 1200 14000

0.005

0.01

y 2

0 200 400 600 800 1000 1200 14002

3

4

u 1

0 3

4

5

u

32

SOFTWARE AND HARDWARE TOOLS

The development of fuzzy controllers relies on intensive interaction with the designer.

Therefore, special developer tools have been introduced by various software (SW) and

hardware (HW) suppliers such as Omron, Siemens, Aptronix, Inform, National

Semiconductors, etc. Fuzzy control is also gradually becoming a standard option in

plant-wide control systems, such as the systems from Honeywell. An extensive list of

these HW and SW tools is given in www.fmt.vein.hu/softcomp/fuzzy. Although the

user interfaces of these tools have been developed for different tasks, most of them

consist of the following blocks.

33

Project Editor

The heart of the user interface is a graphical project editor that allows the user to

build a fuzzy control system from basic blocks. An example for such editor is given in

Figure 2.31q.

FIG. 2.31q

Example for a project editor. Input and output variables can be defined and connected to the fuzzy inference unit

either directly or via pre-processing or post-processing elements such as dynamic

filters, integrators, differentiators, etc. The functions of these blocks are defined by

the user, using the C language or its modification. Several fuzzy inference units can

be combined to create more complicated (e.g., hierarchical or distributed) fuzzy

control schemes.

34

Rule base and membership functions

The fuzzy logic rules determine how the system reacts to input conditions. A rule

based controller is easy to understand and easy to maintain for a non-specialist end-

user. The rule base and the related fuzzy sets (membership functions) are defined

using the rule base and membership function editors. This is where the designer can

implement the control and optimization strategies for his/her solution. To best serve

the needs of different applications, most of the tools provide several different rule

editors. The Spreadsheet Rule editor follows the familiar style of a table. The user can

define complete rule bases just by point&click. Many experienced users prefer the

Matrix Rule editor for complex rule blocks. Entering rules in FTL (Fuzzy Technology

Language) format enables the rule definition in the most flexible way. Either way the

rules are defined, the designer can always use any other editor to process or analyze

them further.

The membership functions editor is a graphical environment for defining the shape

and position of the membership functions. Figure 2.31q gives an example of the

various interface screens.

Analysis and simulational tools

After the rules and membership functions have been designed, the function of the

fuzzy controller can be tested using tools for static analysis and dynamic simulation.

Input values can be entered from the keyboard or read from a file in order to check

whether the controller generates expected outputs. The degree of fulfillment of each

rule, the adjusted output fuzzy sets, the results of rule aggregation and defuzzification

can be displayed on line or logged in a file. For a selected pair of inputs and a selected

output the control surface can be examined in two or three dimensions. Some

35

packages also provide function for automatic checking of completeness and

redundancy of the rules in the rule base. E.g. fuzzyTECH's 2D and 3D Analyzers let

the designer to analyze the transfer characteristics of a fuzzy logic system or its

components in any possible way. Any modification of the system is immediately

reflected in all analyzers. This facilitates "interactive" development of a solution.

Dynamic behavior of the closed loop system can be analyzed in simulation. While

these tools comprises all simulation features for the fuzzy logic system under design,

in some softwares it is also possible to use standard simulation software packages

such as Matlab/Simulink™, MatrixX™.

Code generation and communication links

Once the fuzzy controller is tested using the software analysis tools, it can be used for

controlling the plant either directly from the environment (via computer ports or

analog inputs/outputs), or through generating a run-time code. Most of the programs

generate a standard C-code and also a machine code for a specific hardware, such as

microcontrollers or programmable logic controllers (PLCs). In this way, existing

hardware can be also used for fuzzy control. Besides that, specialized fuzzy hardware

is marketed, such as fuzzy control chips or fuzzy coprocessors for PLCs.

An example of a fuzzy processor is Motorola 68HC12 MCU. This 16-bit

microcontroller family includes four fuzzy logic instructions in addition to the

memory and on-chip peripheral functions. A fuzzy inference kernel on the HC12

takes 1/5 as much code space and executes more than 10 times faster compared to an

HC11 general purpose MCU.

36

It should be noted, that after the design of the fuzzy system an equivalent controller

could be implemented using conventional techniques, e.g. in fact, any fuzzy controller

could be represented by a classical look-up table.

37

SUMMARY AND CONCLUDING REMARKS

From the control engineering perspective, a fuzzy controller is a nonlinear controller.

The linguistic nature of fuzzy control makes it possible to handle expert knowledge

concerning how the process should be controlled or how the process behaves. The

interpolation aspect of fuzzy control has led to the viewpoint where fuzzy systems are

seen as smooth function approximation schemes. According to Michael Athans, fuzzy

control methods are “parasitic”; they simply implement interpolations of control

strategies obtained by other means [65]. This is partly true. Most of the fuzzy control

algorithms can be interpreted as gain-scheduled controllers. Therefore, some people

attack the fuzzy control community by stating that the final control and/or modeling

algorithm just boils down to a nonlinear gain schedule which could actually be

obtained by other interpolation methods, too.

Fuzzy control is a new technique that should be seen as an extension to existing

control methods and not their replacement. It provides an extra set of tools which the

control engineer has to learn how to use where it makes sense. Nonlinear and partially

known systems that pose problems to conventional control techniques can be tackled

using fuzzy control. Fuzzy techniques provide a man–machine interface, which

facilitates the acceptance, validation and transparency of the process model or

controller very much [66]. In this way, the control engineering is a step closer to

achieving a higher level of automation in places where it has not been possible before.

38

REFERENCES

1. Takatsu H. and Itoh T, “Future Need for Control Theory in Industry

– Report of the Control Technology Survey in Japanese Industry”,

IEEE Transactions on Control Systems Technology, Vol. 7, 1989, pp.

298-305

2. Driankov D. and Helerndoorn H. and Reinfrank M., An Introduction

to Fuzzy Control, Springer-Verlag Berlin, 1993

3. Passino K.M. and Yurkovic S., Fuzzy Control, Addison-Wesley, New York, SA,

1998

4. Wang L.X., A course in Fuzzy Systems and Control, Prentice Hall, New York,

USA, 1997

5. Yager R. R. and Filev D. P. Essentials of Fuzzy Modelling and

Control, John Wiley, New York, 1994

6. Yi S.Y. and Chung M.J., “Identification of Fuzzy Relational Model

and its applications to Control”, Fuzzy Sets and Systems, 59, 1, 25-

33, 1993

7. Takagi T. and Sugeno M., “Fuzzy Identification of Systems and Its

Applications to Modelling and Control”, IEEE Transactions on

Systems, Man, and Cybernetics, 15, 1, 116-132, 1985

8. Holmblad, L. P. and Østergaard, J.-J. (1982). Control of a cement

kiln by fuzzy logic, L. Gupta, and Sanchez (eds), Fuzzy Information

and Decision Processes, North-Holland, Amsterdam

9. Abonyi, J., Fuzzy Model Identification, Birkhauser Boston, MA,

USA, 2003

10. Mamdani E.H., “Advances in the linguistic synthesis of fuzzy

controllers”, International Journal of Man-Machine Studies, Vol. 8,

1976, pp. 669-678.

11. Arva P. and Nemeth S., “Learning Algorithm in Fuzzy Control”,

Annales Univ. Sci. Budapest., Sect. Comp. Vol. 12, 1991, pp. 25-32.

39

12. Renders J.M. and Saerens M. and Hugues, “Fuzzy Adaptive control

of a Certain Class of SISO Discrete-time Processes”, Fuzzy Sets and

Systems, Vol. 85, 1997, pp. 49-61.

13. Narendra K.S. and Parthasarathy K., “Identification and Control of

Dynamical Systems”, IEEE Transactions on Neural Networks, Vol.

1, 1990, pp. 4-27.

14. Jang J.- S. R and Sun C.-T., “Neuro-Fuzzy Modelling and Control”,

Proceedings of the IEEE, Vol. 83, 1995, pp. 378-406.

15. Jang J.- S. R., “ANFIS: Adaptive-Network-based Fuzzy Inference

Systems”, IEEE Transactions on Systems, Man and Cybernetics Vol.

23(3), 1993, pp. 665-685.

16. Yamashita Y. and Matshumoto S. and Suzuki M., “Start-up of a

Catalytic Reactor by Fuzzy Controller”, Journal of Chemical

Engineering of Japan, Vol. 21, 1988, pp. 277-281.

17. Roffel B. and Chin P.A., “Fuzzy Control of a Polymerization

Reactor”, Hydrocarbon Processing, June 1991, pp. 47-49.

18. Arbeithuber C. and Jorgl H.P. and Raml H., “Fuzzy Control of an

Iron Ore Sintering Plant”, Control Engineering Practice, Vol. 3(12),

1995, pp. 1669-1674.

19. Berger M., “Self-Tuning of a PI Controller using Fuzzy Logic, for a

Construction Unit Testing Apparatus”, Control Engineering

Practice, Vol. 4(6), 1996, pp. 785-790.

20. Lee D. and Kang T., “Adaptive Fuzzy Control of the Molten Steel

Level in a Strip-Casting Process”, Control Engineering Practice,

Vol. 4(11), 1996, pp. 1511-1520.

21. Ni H. and Debelak K. and Hunkeler D., “Temperature Control of

Highly Exothermic Batch Polymerisation Reactors”, International

Journal of Applied Polymerisation Science, Vol. 63, 1997, pp. 761-

772.

22. Jinxiang Z. and Zaiman Z. and Yanbin Q. and Linru Y. and Suqin W.

and Lianyu L., “pH Fuzzy Control of Automated Industrial

40

Electroplating of Gold”, Chemical Engineering Technology, Vol. 20,

1997, pp. 576-580.

23. Ylen J.P. and Jutila P., “Fuzzy Self-Organising pH Control of an

Ammonia Scrubber”, Control Engineering Practice, Vol. 5(9), 1997,

pp. 1233-1244.

24. Inamdar S.R and Chiu M.S.,”Fuzzy Logic Control of an Unstable Biological

Reactor”, Chemical Engineering Technology, Vol. 20, 1997, pp. 414-418.

25. Lagerberg A. and Breitholtz C., “A Study of Gain Scheduling

Control Applied to an Exothermic CSTR”, Chemical Engineering

Technology, Vol. 20, 1997, pp. 435-444.

26. Hwang W.H. and Chey J.I. and J.I. Rhee J.I., “Modelling and Control

of Continuous Stirred Tank Reactor for Thermal Copolymerisation”,

International Journal of Applied Polymerisation Science, Vol. 67,

1998, pp. 921-931.

27. Abonyi J. and Nagy L. and Szeifert F., “Fuzzy Control of Batch

Polymerization Reactors”, IEEE International Conference on

Intelligent Systems, INES'97, 1997, pp. 251-255.

28. Juba M.R and J.W Hamer, “Progress and challenges in batch process

control,” In Proceedings of the CPC III,CHACE, 1986, Amsterdam,

Netherlands.

29. Berber R. “Control of Batch Reactors: Review,” Trans IChemE,

vol.74 Part A, pp. 3-20, 1996.

30. Davdison, R.S. “Using Process Information to Control a

Multipurpose Batch Chemical Reactor,” Computers Chem. Engng,

vol.13, pp. 83-85, 1989.

31. G.D. Cawthon and K.S. Knaebel, “Optimization of Semi-batch

Polymerization Reactors,” Computers chem. Engng, vol.13, pp. 63-

72, 1989.

32. S. Nemeth and F.C. Thyrion, “Study of the Runaway Characteristics

of Suspension Polymerization of Styrene,” Chem. Eng. Technol, vol.

18, pp. 315-323, 1985.

41

33. K.Y. Choi and G.D. Lei, “Modeling of Free-Radical Polymerization

of Styrene by Bifunctional Iniciators,” AIChE Journal., vol.33, pp.

2067-2076, 1987.

34. Babuška R., H.A.B. te Braake, Krijgsman A.J. and Verbruggen H.B.,

„Comparison of intelligent control schemes for real-time pressure control”,

Control Engineering Practice, Vol. 4(11)., 1996, pp. 1585-1592.

35. Palm R. and Driankov D. and Hellendoorn H., Model Based Fuzzy

Control, Springer Verlag, 1996

36. Babuška R., Fuzzy Modelling for Control, Kluwer Academic

Publishers, Boston, 1998

37. Babuška R. and Verbruggen H.B., „An overwiew of fuzzy modeling for

control”, Control Engineering Practice, Vol. 4(11), 1996, pp. 1593-1606.

38. Hellendoorn H. and Driankov D., editors. Fuzzy Model Identification:Selected

Approaches, Springer, Berlin, Germany, 1997

39. Sugeno M. and Kang G.T., “Fuzzy Modelling and Control of

Multilayer Incinerator”, Fuzzy Sets and Systems, Vol. 18, 1986, pp.

329-346.

40. Posthlethwaite B., “A Model-based Fuzzy Controller”, Trans

IChemE, Vol. 72(A1), 1994 , pp. 38-46.

41. Johansen T.A., “Operating Regime Based Process Modelling and

Identification”, Ph.D Thesis, University of Trondheim, The

Norvegian Institute of Technology, 1994

42. Foss B.A. and Johansen T.A. and Sorensen A.V., “Nonlinear

Predictive Control Using Local Models - Applied to a Batch

Fermentation Processes”, Control Engineering Practice, Vol. 3, 1995

, pp. 389-396.

43. Qian Yu and Tessier J.C., “Application of Fuzzy Relational

Modelling to Industrial Product Quality Control”, Chemical

Engineering Technology, Vol. 18, 1995, pp. 330-336.

42

44. Luo R.F. and Zhang Z.J., “Fuzzy-Neural-Net-Based Inferential

Control for a High-Purity Distillation Column”, Control Engineering

Practice, Vol. 3, 1995, pp. 31-40.

45. Keil F. and Stoyanov S. and Shunova E., “Fuzzy Models in Process

Optimal Control”, Hungarian Journal of Industrial Chemistry, Vol.

23, 1995, pp. 201-206.

46. Hu J.Q. and Rose E., “Predictive Fuzzy Control Applied to the Sinter

Strand Process”, Control Engineering Practice, Vol. 5, 1997, pp.

247-252.

47. Posthlethwaite B., “Buliding a Model-Based Fuzzy Controller”,

Fuzzy Sets and Systems, Vol. 79, 1997, pp. 3-13.

48. Sing C. H. and Postlethwaite B., “Fuzzy Relational Model-Based

Control Applying Stochastic and Iterative Methods for Model

Identification”, Trans IChemE, Vol. 74(A1), 1996, pp. 70-76.

49. Linkens D.A. and Kandiah S., “Long-Range Predictive Control Using

Fuzzy Process Models”, Chemical Engineering Research and Design,

Vol. 74, 1996, Part A, pp. 77-88.

50. Posthlethwaite B. and Brown M. and Sing C. H., “A New

Identification Algorithm for Fuzzy Relational Models and its

Applications in Model-based Control”, Trans IChemE, Vol. 75(A1),

1997, pp. 453-458.

51. Garcia C. E and Morari M., “Internal Model Control. 1 A Unifying

Review and Some New Results” Ind. Eng. Chem. Process Des. Dev.,

Vol. 21, 1982, pp. 308-323.

52. Graham P. and Newel R. B., “Fuzzy Adaptive Control of a First

Order Process”, Fuzzy Sets and Systems, Vol.31, 1989, pp. 47-65.

53. Fischer M. and Nelles M., “Predictive Control Based on Local Linear

Fuzzy Models”, International Journal of Systems Science, Vol. 29,

1998, pp. 679-697.

54. Hernandez E. and Arkun Y.,”Study of Control-Relevant Properties of

Backpropagation Neural Network Models of Nonlinear Dynamical

43

Systems”, Computers and Chemical Engineering, Vol.16(4), 1992,

pp. 227-240.

55. Sistu P.B. and Bequette B.W., “Nonlinear predictive control of

uncertain processes: Application to a CSTR”, AICHE Journal, Vol.

37(11)., 1991, pp. 1711-1722.

56. Driankov D. and Palm R., editors. Advances in Fuzzy Control,

Springer, Heidelberg, Germany, 1998

57. Palm R., and Rehfuess U., “Fuzzy Controllers as Gain Scheduling

Approximations”, Fuzzy Sets and Systems, Vol. 85, 1997, pp. 233-

246.

58. Abonyi J. Nagy L. and Szeifert F., “Fuzzy model based predictive control by

instantaneous linearization”, Fuzzy Sets and Systems, Vol. 120(1), 2001, pp.

109-201.

59. Biassizo K. K. and Skrjanc I. and Matko D., “Fuzzy Predictive

Control of Highly Nonlinear pH process”, Comp. Chem. Eng., Vol.

21/S, 1997, pp. 613-618.

60. Espinosa J.J. and Vandewalle J., “Predictive Control Using Fuzzy

Models”, Advances in Soft Computing - Engineering Design and

Manufacturing (Eds.), R. Roy, Springer – London, ISBN 1-852-

33062-7, 1998

61. Fisher M. and Isermann R., “Inverse Fuzzy Process Models for

Robust Hybrid Control”, In Advances in Fuzzy Control, Eds.

(Driankov D. and Palm R.), 103-128, Physica-Verlag, Heilderberg,

New York, 1998

62. Linkens D.A. and Kandiah S., “Long-Range Predictive Control Using

Fuzzy Process Models”, Chemical Engineering Research and Design,

Vol. 74, Part A, 1996, pp. 77-88.

63. Skogestad S., “Dynamics and control of distillation columns”, Chem. Eng. Res.

Des. (Trans IChemE), Vol. 75, 1997, pp. 539-562.

64. Clarke D.W. and Mohtadi C., “Properties of generalized predictive control”,

Automatica, Vol. 25(6), 1989, pp. 859-875.

44

65. Athans M, Debate with Prof. Lofti A. Zadeh, 7th European

Conference on Intelligent Techniques and Soft Computing (EUFIT

'99), Aachen, Germany, Sep 1999

66. Verbruggen H.B. and Brujin P.M., “Fuzzy Control and Conventional

Control: What is (and can be) the Real Contribution of Fuzzy

Systems ?”, Fuzzy Sets and Systems, Vol. 90, 1997, pp. 151-160.