Fuzzy modeling and control based on cell structures

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STELLINGEN

behorende bij het proefschrift van F. van der Rhee

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1. Het is verantwoord de database van het vage model en van de vage regelaar een knowledge structure te noemen.

2. Een belangrijke voortzetting van de ontwikkeling van de vage regelaar zoals beschreven in het proefschrift is het ontwikkelen van knowledge structures.

3. Bij het onderzoek op het gebied van Kunstmatige Intelligentie is het beter te denken in termen van communicerende cellen, structuren en leerprocessen, dan te denken in termen van "slimme" programma's.

4. Vaagheid speelt een belangrijke rol in de relatie tussen werkelijkheid en menselijke waardering.

5. Uiterste tolerantie impliceert intolerantie ten aanzien van het Christelijk geloof.

6. Voor de produktie van basislevensbehoeften dient naar de beginselen van de ecologie en de kleinschaligheid te worden gehandeld.

7. Op de scholen dient het scheppings- en zondvloed model als minstens gelijkwaardig aan het model van de evolutie theorie te worden behandeld.

8. De symboliek van het Heilig Avondmaal wordt meer recht gedaan bij het gebruik van volkorenbrood, dan bij het gebruik van wittebrood. Dit sluit waarschijnlijk ook beter aan bij de historische instelling ervan.

9. De verontreiniging van het milieu van de aarde is een indicatie dat de beschaving daalt, i

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Fuzzy modeling and controL.

based on cell structures

An expert system for modeling and control

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus, prof. drs. P.A. Schenck, in het openbaar te verdedigen ten overstaan van een commissie

door het College van Dekanen daartoe aangewezen,

op donderdag 3 november 1988 te 16.00 uur door

Floor van der Rhee,

geboren te Alblasserdam, elektrotechnisch ingenieur.

TR diss

1679

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Dit proefschrift is goedgekeurd door de promotoren: Prof. ir. H.R. van Nauta Lemke

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Summary 1

SUMMARY

A new knowledge-based method has been developed for modeling and control, based on a fuzzy reasoning method. The method has been based on some ideas about human brain function.

A knowledge structure is built up with cells. This knowledge structure is used for both fuzzy modeling and fuzzy control. Before the model and the controller can be used, a learning phase is required. In this phase a number of time responses of a process are stored in the cells of the knowledge structure. After the learning phase, the system enters an application phase, in which the obtained knowledge is used.

The knowledge structure has been organized according to two fundamentally different types of relations: relations in time and relations in value. These relations (connections between the cells) are built up in the learning phase. The values of the input and the output of a process, stored in the cells, and the relations in time preserve the process behavior. The relations in value reflect information about similar responses.- They are used to search for the responses which are applied in the fuzzy model and in the fuzzy controller.

A two-dimensional knowledge structure has been developed, which is based on a (u,y) plane. The (u,y) plane is divided (with a non-fuzzy division) in classes with about the same u and the same y. A class is used as the beginning of a class list, which is built up with the relations in value. All the cells in these class lists together are just all the cells of the knowledge structure. It is not necessary to search for the responses in the whole (u,y) plane. It is sufficient only to search in a part of it. This part, which is called the two-dimensional search region, is basically described by a set of inequalities. It appeared possible to describe the search region explicit, using a number of equations.

An n-dimensional knowledge structure has been developed, which offers an improved organization. A straightforward extension of the two-dimensional (u,y) plane to an n-dimensional plane is not practical, because of the large memory space which would be required. Instead of an n-dimensional plane a new type of relations are introduced: search relations. It appeared to be possible to develop an n-dimensional knowledge structure which uses the memory space very efficiently. The search region could be easily extended to an n-dimensional search region.

The fuzzy model and the fuzzy controller are based on a comparison of the responses in the knowledge structure with the . actual response. For this comparison a fuzzy comparison set is used, which is enlarged if necessary,

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11 Fuzzy modeling and control based on cell structures

dependent on the amount of information which is found in the knowledge structure. In the comparison of the responses a factor of linearity is used, so that the responses with the same shape are compared, regardless of their amplitudes. The factor of linearity is bounded by a fuzzy linearity set. This fuzzy linearity set reflects to some extent the nonlinearity of the process.

A response in the knowledge structure can be divided into two parts: the actual cell and the cells backwards in time, which are called the past of the response, and the cells forwards in time, which are called the future of the response.

The fuzzy model compares the past of the actual response with the past of a number of responses in the knowledge structure. The responses which resemble the actual response are used in the calculation of the output of the fuzzy model.

The fuzzy controller not only compares the past of the actual response with the past of a number of responses in the knowledge structure. It also compares the future of these responses with a desired future (which is called the reference). The responses with both similar pasts and similar futures are used in the calculation of the output of the fuzzy controller.

A decision table is used to record the results of the comparisons of the responses. After all the responses have been compared, the output of the fuzzy model and the output of the fuzzy controller are calculated on the basis of this decision table.

Both the fuzzy model and the fuzzy controller generate a reliability measure, which is their judgement about their own behavior.

The fuzzy model and the fuzzy controller have been applied to a number of processes in simulations. The results are promising. The thesis ends with conclusions and suggestions.

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Contents in

SYMBOLS

List of frequently used symbols

Cells and their values:

C BJ CB:_U CBj_y CL, CSm

cs

m

_d

basic cell with number j u value stored in basic cell CB: y value stored in basic cell CB: list cell with number 1

search cell with number m discriminant of search cell CSm

Operators on the cells:

CB PC PL PT SC SL ST S U 1+ , SU2+ SUi + S U r , S U 2 SUi"

sx

+

sx-sx°

SYI+ , SY: SYi+ SY1- , SY2-+ .

basic cell operator

class-predecessor operator list-predecessor operator time-predecessor operator class-successor operator list-successor operator time-successor operator

search operator, to obtain a basic value of u

search operator, to obtain a basic value of u, for search depth i

search operator, to obtain a basic value of u

search operator, to obtain a basic value of u, for search depth i

search cell operator, to obtain a aspect with a higher discriminant search cell operator, to obtain a aspect with a lower discriminant search cell operator, to obtain a aspect, or to obtain a basic cell

search operator, to obtain a basic value of y

search operator, to obtain a basic value of y, for search depth i

search operator, to obtain a basic value of y

cell with a higher class cell with a higher class cell with a lower class cell with a lower class search cell of the same search cell of the same search cell of the next cell with a higher class cell with a higher class cell with a lower class

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X Fuzzy modeling and control based on cell structures

SYi search operator, to obtain a basic cell with a lower class value of y, for search depth i

Fuzzy sets: f _c,u,e f 1c,y f _c,y,e fd «in

a fuzzy comparison set a fuzzy comparison set on u

a fuzzy comparison set on y for the evaluation of the future an extended fuzzy comparison set on u

a fuzzy comparison set on y

an extended fuzzy comparison set on y fuzzy time distance set

fuzzy linearity set fuzzy order set

Similarity measures: shc(e) shm(e) shp(hp,e) smf(hf,d,e) , shf(hf,d',e) "smin,db(hP>u(k),l(k),CBj,g) + smi_db(hf,r + (k),STd(CBi),g) "smindb(hf>y.+ (k),CBj,g)

successive similarity measures for increasing depths of the past, and as a function of the extension, used in the fuzzy controller

successive similarity measures for increasing depths of the past, and as a function of the extension, used in the fuzzy model

similarity measures for the past similarity measures for the future

similarity between the past of an actual response and the past of a response in the knowledge structure

similarity between the reference and the future of a response in the knowledge structure

similarity between the desired future of an actual response and the future of a response in the knowledge structure

Components of the decision table Dm(h~e) of the fuzzy model:

Dm(hp,e)_g the factor of linearity of the response with the

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Symbols XI

Dm(hp,e)_maxsm : the maximum of the similarity measures of the rules

Dm(rL,e)_nr : the number of the basic cell of the response with the

greatest similarity measure

Dm(rL))e)_ssm : the sum of the similarity measures of the rules

Dm(hp,e)_sysm : the sum of the products of the similarity measure and

the result of a rule

Components of the decision table Dc(h„h^d,e) of the fuzzy controller:

Dc(hp,hf,d,e)_maxp : the maximum of the similarity measures of the past of

the rules

Dc(hp,hf)d,e)_maxf : the maximum of the similarity measures of the future of

the rules

Dc(hp,hf,d)e)_nr : the number of the basic cell óf the response with the

greatest intersection of the similarity measure of the past and the future

Other symbols:

class operator

two dimensional class of the (u,y) plane operator to obtain the depth of a substructure

operator on a search cell,. to obtain the maximum value of the discriminants of a substructure

operator on a search cell, to obtain the minimum value of the discriminants of a substructure

operator to obtain the number of cells of a substructure time distance between the past and the future

maximum time distance minimum time distance resulting distance

the opposite of the class function UD the opposite of the class function YD extension of the fuzzy comparison set

maximum extension of the fuzzy comparison set resulting extension of the fuzzy comparison set

minimum value for the resulting maximum of the similarity of the fuzzy model as a function of the extension

minimum value for the resulting maximum of the similarity of the fuzzy model as a function of the order

factor of linearity

bound of the factor of linearity (gb > 1) C CSDEPTH CSMAX CSMIN CSNRC d amax ''min ds DU DY e emax es feb fhb g gb

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Fuzzy modeling and control based on cell structures

maximum intensification factor of the factor of linearity resulting factor of linearity

evaluation depth of the future

minimum for the evaluation depth of the future for short time distances

resulting evalution depth of the future evaluation depth of the past

resulting evalution depth of the past transfer function of a process number of time responses

interpretation bound for the decision table of the fuzzy controller

minimum value of the interpretation bound for the decision table of the fuzzy controller

interpretation coefficient for the decision table of the fuzzy controller

resulting rule

maximum value of a class value of u, and equal to the number of classes on u minus one

maximum value of a class value of y, and equal to the number of classes on y minus one

similarity measure of the resulting rule

minimum of the number of cells in a substructure and of the number of classes

number of samples stored in the knowledge structure

number of cells of a substructure as a function of the depth v maximum number of samples of the future which are taken into account

maximum number of samples of the past which are taken into account

reliability measure reference

reference vector at the time kTs

root of the knowledge structure, concerning the search procedure

time operator

number of times that a response passes the reference root of the knowledge structure, concerning the time sampling period

input of a process boundary values of u

input of a process at the time kTs

input vector of the past at the time kTs

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Symbols X l l l UD uc Uf us w wc,max w _c,ey,max w c,u,max w _c,y,max wu wy y . yP y " . y+ y(k) I(k)

x+00

ycmax ya YD ym ys ZH(CBj) a

P

6u óy 6ey £ Xc Xm

5 n

E class function of u

output of a non-fuzzy controller output of the fuzzy controller shift of the u axis

maximum width of the fuzzy comparison set fc

maximum width of the fuzzy comparison set on y for evaluation of the future

maximum width of the fuzzy comparison set on u for evaluation of the past

maximum width of the fuzzy comparison set on y for evaluation of the past

maximum width of the extended fuzzy comparison set on u the evaluation of the past

maximum width of the extended fuzzy comparison set on y the evaluation of the past

output of a process boundary values of y

output of a process at time kTs

output vector of the past output vector of the future evaluation width

class value of y class function of y

output of the fuzzy model shift of the y axis

position of cell CB: in its class list

the extent to which a PI(D) controller is used ratio between the widths of two comparison sets

unit of the fuzzy comparison set on u for the evaluation the past

unit of the fuzzy comparison set on y for the evaluation the past

unit of the fuzzy comparison set on y for the evaluation the future

difference between the reference and the output of process

criterion for the fuzzy controller criterion for the fuzzy model

£TS is a virtual time of a response in the knowledge structure

set of the natural numbers set of the real numbers

the the the for for of of of the

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1.1 Introduction 1

1 A concept for fuzzy modeling and control

1.1 Introduction

A human being is a creature of enormous complexity. Man is born a helpless being, crying for milk and affection. It is difficult to imagine that somebody at an age of, for instance, 30 years, occupied in talking, walking, laughing, swimming, working or whatever he may be doing was once a child. When we think about it, we are astonished by the great mental flexibility and learning capacity of man. Obviously his brain is structured very flexibly and has a very great capacity, so that many different types of knowledge and behavior can be learned and successfully applied in many different situations.

It is a great challenge to try to create artificial beings that ever more closely approximate the intelligent behavior of mankind. This field of research is generally called "artificial intelligence" (A.I.). However, there is no generally accepted definition of "intelligence" or "artificial intelligence". The field of A.I. covers many subjects, for example, expert systems, robots, vision, the understanding of natural language, logical reasoning, problem solving, and so on.

The research carried out in A.I. can be roughly divided into two approaches: - the research on the organization of the brain of humans and animals, and

the realization of similar kinds of cell structures with computers. The structures can help to understand some aspects of the functioning of the brain, which can lead to new methods to apply to vision, reasoning, and so on.

- the research on computer programs which in some way reflect a behavior which would generally be recognized as intelligent behavior. However, a program may produce "intelligent behavior", while most experts on A.I. do not accept it as an "intelligent program" (for example chess programs based on brute force). If programs are based on principles of which it is believed that human beings also apply these principles, the programs are more generally accepted as being "intelligent programs".

The purpose of this thesis is the development of a concept for modeling and control, based on a fuzzy reasoning method. Because of the enormous mental flexibility of man, the fuzzy reasoning method is based on some ideas of human brain function. These ideas are illustrated in this chapter by means of some examples. The concept is related tó the second approach and includes the organization of knowledge in a structure built up with simple basic units which are called cells. These cells are far simpler than the neurons of our brain. The algorithms of the model and the controller will be based on a simultaneous

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usage of the contents of these cells. This can be compared with the parallel functioning of the neurons of the brain. The thesis concludes with a suggestion as to how the concept could be extended to obtain a fuzzy model and a fuzzy controller based on a concept of communicating cells. If these cells could be considered as being a very simple kind of a neuron or group of neurons, the concept is also related to the first approach. Perhaps neuro-scientists could combine some of their results with this approach.

A vital element of the concept is a learning phase, in which the knowledge which is necessary in order to be able to function as a model or a controller is obtained. In the following section some examples of learning processes in human life will be given.

1.2 Examples of learning processes in human life. 1.2.1 The process of learning to swim

How does a man learn to swim?

One may think that each movement of the arms and legs (the "output" of the brain, the input of the "swimming process") is registered in the brain, together with the resulting behavior of the body in the water (the "input" of the brain, the output of the "swimming process"). It seems that some kind of model is built up that represents the behavior of the body in the water as a function of the movement of the arms and legs. In my opinion during swimming (and similar things hold, for instance, for driving and the other examples) searches are performed in the model, that consists of all prior experiences, in order to find the best transitions from the actual behavior of the body to a desired behavior of the body. Of course the decision about the best transitions to take is reviewed again and again. The decision about the actual movement of the arms and legs is based on these transitions so that according to the experiences of the swimmer the behavior of the body is as good as possible.

In the beginning an effective movement can hardly be found in the model. This means that the resulting behavior of the body may be far from effective, or, in other words, if somebody tried to learn to swim in deep water, he would probably drown. During the learning time the model of experiences is extended by new experiences of the behavior of the body in the water, and it will become more and more easy to find a good transition. It can be thought that a good swimmer has built up a base of experiences with many good transitions which are close together, and which model the body behavior accurately for all the situations which occur when he swims. The better he has learned to swim the longer will be the transitions, so that he swims more fluently than

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1.2 Examples of learning processes in human life. 3

somebody who has not yet had so much experience. (It can be viewed as a kind of forward control.)

It often happens that somebody receives instruction on how to swim. The man who receives instruction relies at first more on the instruction than on his data base. In this way his data base may be faster filled with useful experience, and he will be able to swim sooner than if he had to learn completely by self-discovering. However, the learning process itself is the same and as time goes by the swimmer relies more and more on his data base: swimming becomes an unconscious process.

Notes:

- A similar sort of learning process has been observed for driving.

Experienced drivers seem to display a more forward control pattern, while inexperienced and fatigued drivers seem to display a more backwards control pattern. Probably the same phenomenon (experienced versus inexperienced; fatigue is somewhat difficult) will appear in a configuration for an adaptive fuzzy controller which will be given in section 8.2.3.

- A whole class of examples like the process of how to learn to swim exists, for example how to learn to walk, how to learn to play tennis, and so on. - In the process of swimming, walking, playing tennis and so on, feedback

also plays some role. The thesis deals mainly with the forward control. However it could be advantageous to investigate the incorporation of the use of feedback in the concept.

1.2.2 The process of learning to speak

In the author's view the process by which a human being learns to speak is

governed by the same principles as the process by which a man learns to swim. However, the process by which a man learns to speak is probably much more complicated. One may imagine that a large data base is formed, containing both his own experiences (movements of mouth, tongue and throat and the produced sound), as well as the sounds of the people around the child. At the age at which a child is learning to speak it is inherent in him to imitate those around him. The gibberish that is uttered by a child that still cannot speak properly is an attempt to imitate the speech of man. Although perhaps people cannot recognize it, the 'data base' of the child is enriched with new experiences and in this way the child acquires a data base that increasingly models its own behavior. In this view a child has learned to speak if the search for the movements of mouth, tongue and throat to imitate the sounds of surrounding people results in recognizable words. The problem of

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how a child learns the meaning of the words is not dealt with, that is outside the scope of this example.

Notes:

- A child often begins to utter sounds when the surrounding people stop speaking. The child uses these quiet periods to listen to itself, and by imitating other people the child builds a data base to model its own speech behavior.

- Children who are born deaf cannot speak well, they have problems in building a correct and sufficient data base.

- This view is supported by the observation of some researchers that in the utterances of children who are learning to speak, in some way sounds of the spoken sentences are recognizable. A child does not learn single words at first, but imitates complete sentences.

- This view does not only agree with (as of course it should) the observation that children learn the language of their own mother (father), but also explains two other things:

- The ease (in respect to the scope of this example) with which a man can learn other languages with the same phonemes. A man has ngl_ learned which movements to make in order to speak some word, but he has learned the sound as a result of the movements!

- The relatively greater difficulty of learning a language which is based on other phonemes: for such languages the sound, as a result of the movements of mouth, tongue and throat, is poorly present in the data base.

13 Knowledge through experience and knowledge through understanding

Let us imagine a factory in which the parameters of some controller have to be set to correct values. There may be an engineer who can calculate the settings of the parameters of the controller in order to obtain a good system behavior. There may also be an operator, who cannot explain why, but who is able to set the parameters correctly through experience. One could summarize this by saying that, roughly speaking, the engineer depends more on knowledge through understanding while the operator depends more on knowledge through experience.

Knowledge through understanding can, for example, be found in books, be taught at schools, and so on. Most expert systems try to capture knowledge through understanding in their knowledge bases. The fuzzy model and controller which will be described in this thesis try to capture knowledge through experience. This knowledge will be processed by a special-purpose

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1.4 The concept 5

expert system. Whereas the knowledge of an expert system dealing with knowledge by understanding can be taught by an expert, or found in books, the knowledge which is captured in the fuzzy model and controller is obtained in a learning phase. In this learning phase the behavior of a process is stored in a knowledge structure. This learning process can be compared with that of acquiring experiences.

Note:

The distinction between knowledge through experience and knowledge through understanding does not exclude the fact that somebody can try to explain the knowledge of the operator. Somebody can perhaps give a theoretical basis for the behavior of the operator, or he can try, for example, to capture the knowledge of the operator in rules. Such an approach may also result in an expert system.

1.4 The concept

The above ideas have been condensed into a system which can function both as a model and as a controller. In its initial state this system contains no information. First the system enters a learning phase in which the system learns the process behavior by recording time responses of a process in a knowledge structure. In this way the system gets a lot of 'experience' of the behavior of the process. After the learning phase the system enters another phase in which it can be used to act as a model of the process, or to control the process. This phase will be called the application phase. In this application phase the system searches in the knowledge structure to find responses which can be used by a pattern recognition method to yield, for example, information on how to act either as a model or as a controller. The pattern recognition method is based on the fuzzy set theory. If, in the application phase, it appears that the system did not learn enough, the learning phase can be entered again to learn more about the process behavior. .

Four central points can be distinguished:

- the learning phase, in which the knowledge is learned, - the knowledge structure and its organization,

- the application phase, in which the knowledge is used, and

- the search in order to find the parts of the knowledge which will be applied in relation to the structure of the knowledge structure.

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Notes:

- In many control algorithms the calculation of the inverse of the relation

between input and output signals is required. This may be difficult if, for example, the relation is not completely known, or if the relation contains nonlinearities. In this concept no calculation of an inverse relationship takes place, instead a search method is used.

- In this setup the relation between the input and the output is not given by an explicit function, but is given implicitly by the knowledge obtained in the learning phase.

It may be advantageous that neither a mathematical relationship has to be derived nor an inverse relationship has to be calculated. If the system, for example, does not control accurately enough, it can simply be subjected to more learning. A disadvantage may be the length of the search time and the amount of knowledge which needs to be stored. It could be said that research into an efficient calculation of an inverse relationship is replaced by, among others, research into efficient search algorithms and an efficient application of the stored knowledge.

1.5 The learning phase

The learning phase plays a vital role in the concept. Without the learning phase the fuzzy system does not have any knowledge of a process. Throughout the learning phase a process is activated by a learning signal and a knowledge structure of the process is built up. During this time the process input and output are sampled with a constant sample frequency and each sample is added to the knowledge structure. The set of samples which belongs to one period of time is called a response. The knowledge structure can contain one or more responses of variable length. No assumptions about how to use the learned knowledge are made during the learning phase. Every preprocessing of the data (for example the averaging of responses) can be delayed till the application phase, so that assumptions about the application of the knowledge can be made flexibly, suited to the actual situations which occur in the application phase.

Note:

The algorithms of the learning phase and of the application phase are strictly separated, but these phases need not be separated in time. The system can be extended so that it can be simultaneous in the learning phase and in the application phase, while the learned knowledge immediately influences the behavior in the application phase, and conversely, the application phase could dictate whether a time response has to be learned or not.

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1.7 The application phase 7

1.6 The knowledge structure and its organization

The knowledge will be stored in cells, which are the basic memory units of the knowledge structure. Each cell represents the data at one sampling period. A cell is connected with other cells. These connections represent relations between cells. Thus the knowledge structure is a structure of cells.

Different cells may contain the same values for the process input and output. (It is assumed that the process has one input and one output.) So the same values for the input and output can be at physically different locations (cells) in the knowledge structure. However, because each cell has its own connections with other cells, the represented knowledge does not need to be the same.

If the system has learned two time responses which are the same, the resulting knowledge structure will contain this time response twice, in two physically separated strings of cells. It is up to the user to prevent this inefficient use of the memory space in the knowledge structure.

The knowledge structure is organized according to two fundamentally different types of relations:

- relations in time, and - relations in value.

These relations (connections between the cells) are built up in the learning phase. In the application phase they are used:

- The relations in time together with the values in the cells reflect information about the process behavior. They are used in the pattern recognition algorithms..

- The relations in value reflect information about similar responses. They are used to search for the responses which are used in the pattern recognition algorithms.

1.7 The application phase

In the application phase the system possesses a knowledge structure filled with knowledge about the process behavior. The whole concept does not only deal with answers to questions like "What will be the next output value of the process?", or "How must the process be controlled?", but covers also answers to questions like "Is the knowledge structure filled with consistent data?", "What is the complexity of the process?", and "Can something be said about the extent of the nonlinearity of the process?".

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The answers to such questions are given by algorithms which are used in pattern recognition methods which apply the information contained in the knowledge structure. These algorithms are based on the first type of relations of the knowledge structure, the relations in time. In the thesis an algorithm for a fuzzy model and an algorithm for a fuzzy controller is treated.

The algorithms have in common that they have to evaluate (examine) time responses in the knowledge structure.

1.8 Search in the knowledge structure

In the application phase the system possesses a knowledge structure filled with knowledge about the process behavior. However, not all knowledge is relevant in every situation. Which knowledge is relevant and which knowledge is not relevant is dependent on the typical situation. The second type of relations of the knowledge structure, the relations in value, are used to search the responses which are relevant for the application algorithms. The search algorithm selects parts of the responses which the application algorithm evaluates. The application algorithms are essentially parallel algorithms.

However, the results have been obtained with a sequential computer. The search algorithm is dependent on the cell structure. A different cell structure usually means a different search algorithm. (Although the reverse is also true: the cell structure has been set up according to a desired search algorithm.) From this one could imagine that it would be possible to define the fuzzy model and the fuzzy controller in terms of communicating cells. This will be discussed in section 9.12.

1.9 The contents of the thesis

In chapter 2 a start is made on developing the cell structure. There a

two-dimensional structure is designed. Besides the application of this structure, it can serve well as a basic example of the type of cell structures which are designed in other chapters. It will be of great help in the understanding of the more complex cell structures if the two-dimensional structure of chapter 2 is kept in mind. Before the structures are further developed, it is shown how these structures are applied. Chapter 3 contains some preparations for the usage. Chapter 4 treats the application as a fuzzy model. Chapter 5 treats the application as a fuzzy controller. Hereafter in chapter 6 it is shown how the cell structure can be used in a search algorithm, the structure is extended to a more general n-dimensional cell structure in chapter 7. In chapter 8 it is described how the value of various parameters in the algorithms can be

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1.9 The contents of the thesis 9

chosen, and experimental results are given. Chapter 9 suggests, among others things, the extension of the concept for a multi- variable process, a measure for the consistency of the knowledge structure, an, adaptive fuzzy controller, and how the cell structure can be extended to a structure of communicating cells which act as a fuzzy controller or as a fuzzy model. This chapter also ends with some conclusions.

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2.1 Introduction 11

2 The knowledge representation

2.1 Introduction

Knowledge representation is a very important aspect of many problems. Often there are various possibilities to represent the knowledge in a formal way. In some representations it may be hardly possible to find a solution for a particular problem, while in other representations the relations between the "pieces of knowledge" may be so transparent that a solution may easily be found. Once a suitable knowledge representation has been found, an important part of the work has been done already. This chapter deals with the representation of the knowledge in the fuzzy system. The basic structure of the knowledge is described. In chapter 7 knowledge structures are derived, which are more suited to be used in practice.

The way in which the knowledge is structured can be regarded in the context of a general principle, which is in general called the association principle.

This principle had already been formulated by Aristotle [Ref 4]:

"Aristotle stated a set of observations on human memory which were later compiled as the Classical Laws of Association. The conventional way for their expression is:

Mental items (ideas, perceptions, sensations, or feelings) are connected in memory under the following conditions:

1) If they occur simultaneously ('spatial contact'). 2) If they occur in close succession ('temporal contact'). 3) If they are similar.

4) If they are contrary."

Two items are connected if one condition is satisfied. The first three conditions are used in the composition of a knowledge representation of a process. The relation between these conditions and the concept (see chapter 1.4 and 1.6) is as follows:

- The simultaneous occurrence of the input and the output of a process is taken into account by the recording of the values of the input and the output at the same point in time in one cell.

- The successive occurrence of the input and the output of a process at successive points of time will be reflected in the relations in time.

- The occurrence of similar inputs and outputs of a process will be reflected in the relations in value.

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22 Cells, operators and roots

Cells, operators and roots are treated in this section only in respect to the cells and the roots used in this thesis, and the operators which operate on these cells and roots.

A cell is a piece of the memory space in a computer which the designer of the fuzzy system declares to be a cell. It will be assumed that each cell can be indicated by its own unique number, which must be an element of the set of natural numbers, excluding zero, which will indicated by W+.

The piece of the memory space that forms a cell is divided into a number of parts. Some parts may be reserved for data, while each of the other parts is reserved to store the number of another cell. In this way a cell has one or more references to one or more other cells. While the knowledge structure is being built up, these parts of the memory space of a cell are filled with data. The knowledge structure changes if these parts of the memory space of a cell are changed.

The symbol "_" will be used to refer to data which are stored in a memory place within a cell. So for example data d of a cell X: (j is the number of cell X:) will be indicated by X j d .

Operators are introduced which, applied to a cell, will yield another cell indicated by the number which is stored in the memory space of the first cell.

It may happen that in the memory place of a cell which is reserved for the number of another cell actually no number is stored. In this case the memory place will be assumed to contain the value of zero. This will be expressed by the notation that the result of the concerning operator when applied to that cell does not yield an existing cell, or shorter: the concerning operator applied to that cell results zero.

The cells with the same arrangement belong to the same type of cells. It will be assumed that the memory space of a cell is large enough to contain all the data and operators which are attributed to that cell.

Each cell will be graphically represented as a circle, and their operators as arrows (insofar as they are relevant to the discussion).

A root is a memory place (variable) in a computer, which the designer of the fuzzy system declares to be a root. It is reserved to store the number of one cell. Because such a memory place is comparable with the memory places within the cells which are reserved to store the number of a cell, these roots

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2.3 The basic cells 13

are formalized in the same way by means of an operator. The concerning operator applied to a root will yield a cell, indicated by the number which is stored in the memory place of the root. The roots themselves will be referred to as dummy objects.

Note:

- To connect the concept of a cell with a computer implementation: a cell

can be defined as a "structure" in C, or as a "record" in PASCAL.

2 3 The basic cells

A basic cell is a cell in which the process data at one sampling period is stored. A basic cell is denoted by CB:, where jeW+ is the number of the cell. It

will be assumed that the process has one input and one output. Thus each basic cell CB: has two values: CB- u and CB:_y, which represent respectively the input u and the output y of a process at one sampling period. In fig. 2.1 a basic cell CB: is shown.

-O

Fig. 2.1 A basic cell

CB-A cell without relations to other cells has no part in the knowledge structure. In the following sections the basic cells will be extended with relations in time and relations in value.

Note:

It is possible that the same values for u and y occur at different sampling periods. These values are stored in different basic cells because the values represent different samples. Because each cell has its own unique set of relations, the values may also be used in a different way.

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2.4 Relations in time

2.4.1 The time-successor and the time-predecessor operator

A basic cell is only useful when it is part of a larger portion: a time response. In order to be able to construct a time response with basic cells each basic cell is extended with relations in time. Two types of operators are used as relations in time: a time-successor operator ST yields the basic cell at the next sampling period of a response, and a time-predecessor operator PT yields the basic cell at the previous sampling period. A basic cell CB: extended with its relations in time is shown in fig. 2.2.

PT _ ^ ST

Fig. 2.2 A basic cell CB- with relations in time

The range of these operators is the set of basic cells, while the image of these operators is also the set of basic cells.

Let us suppose the existence of the basic cell CB:, with the values of the input and the output respectively CB:_u and CB:_y. The input and the output of the next basic cell are then given by (ST CBj)_u and (ST CB:)_y while the input and the output of the previous basic cell are given by (PT CB:)_u and (PT CBj)_y, when these cells exist.

The basic cell, following the next basic cell, can be obtained by applying the successor operator twice, so this cell is given by (ST (ST CB:)). This will also be written as (ST2 CBj). In general: (ST (ST1 CBj)) = (S'H + 1 CBj). Similar

expressions hold for the predecessor operator PT, which is the inverse of the successor operator ST. So we have (PT (ST CBj)) = CBj, and (ST CBj) = -(PT-1 CBj).

Each basic cell can have at maximum only one time-predecessor operator, and one time-successor operator, because for each sample it holds that there can only be one sample immediately before and immediately after that sample.

Note:

The number of each basic cell is unique, but no ordering has been assumed. Let us consider a basic cell with, for example, number 121. The application of the time-successor operator may yield a basic cell with, for example, number

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2.4 Relations in time 15

322, while the application of the time-predecessor operator may yield a basic cell with, for example, number 279. These numbers may seem to be quite random!

So the following relation (ST CB .) = CB. ,

J J + l is in general false.

Nomenclature:

Let us consider a basic cell CB:, which forms part of a response. With respect to this cell CB: can be defined:

- the past of the response:

{ ( ( P T1 C B . ^ u . C P T1 C B . ) _ y ) , ieW } , with

(PT° CB.) = C B . , and - the future of the response:

{ ( ( S T1 C B . ^ u . t S T1 C B . ) _ y ) , ieW+ } .

Note that the past and the future of a response are defined with respect to a reference basic cell.

2.42 The representation of time responses

The basic cells and their time operators are used to construct strings of cells. Each string contains the data of a time response. This is more briefly expressed as: a string of cells connected with the time operators is a time response. The number of cells, and their contents, is determined by the user of the fuzzy system. If he wants, for example, to store 1000 samples of a response of a process, 1000 cells are used. The user is free to choose the beginning and the end of the time response. A time response is shown in fig. 2.3.

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^~*. ST ^~*. ST ^-^ ST ^ - ^

000= O

_{^ " " ^ PT ^ - ^ PT ^ " ^ PT ^ " ^}

Fig. 2.3 A time response

The user may wish to store more than one time response. (He could, for example, activate the process with a series of step functions, and store it as one time response, and activate the process with a series of sines, and store it as another time response.) Therefore the knowledge structure must be organized so that multiple time responses are stored in a well-organized way. To this end a new type of cell is introduced: list cells. These list cells must be suited to construct a list of time responses. Therefore a cell which contains, the following three operators is taken as a list cell (which will be denoted by CL|):

- an operator CB which yields the first basic cell of a time response,

- an operator PL (list-predecessor operator) which yields the previous list cell, and

an operator SL (list-successor operator) which yields the next list cell. A list cell CL, is shown in fig. 2.4.

CB SL

Fig. 2.4 A list cell CL{

At this stage in the development of a representation for time responses the situation is as follows: it has been defined, how strings of basic cells may represent time responses, and it has been defined, how a string of list cells may represent a list of time responses. If it would be possible for us to see all the cells and their contents, we would know which time responses had been stored, but how can the system "know" which cells belong to a time response and which cells do not? The system must have some point from which it can reach the cells which belong to the knowledge structure. In general such a point is called a "root". Let us assume therefore for the formal description

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2.4 Relations in time 17

that there is an object Tr o o t, which points to the first list cell. (In a

computer this object is a variable, in which the number of a list cell can be stored.) All the basic cells of the time responses spring from this root as follows:

The list cells are found by applying the list-successor operator:

(VCLL) (3ie/V) [ CL1= ( S L1 Tr o Q t) ] . (2.1)

Let us assume that the list-successor operator yields zero when it is applied more than I times:

(SL1 T ) = 0 , for i > I, (2.2)

r o o t

and that the list-successor operator yields a list cell when it is applied I times or less (I > 0). This means that I responses have been stored in the knowledge structure.

(SL Tr o o t) = 0 means that no samples have yet been stored in the knowledge

structure.

A cell of a time response is given by applying operator CB of a list cell to find the first cell of the time response, and successively applying the time-successor operator:

( V C B , ) ( 3 £ e l 0 ( 3 i e { l , I } ) [CB = (ST^(CB ( S L1 Tr o o t> ) ) \ l - (2-V

The end of a time response is reached when the time-successor operator yields zero instead of a basic cell.

An example with three time responses is shown in fig. 2.5.

2.43 The time of a basic cell

When the knowledge structure is applied, it is sometimes useful if the system displays information about what is happening. In this case the user does not want to know the number of a basic cell (which may seem to be random to the user), but rather a point in time which corresponds with the basic cell. For this purpose a time operator T is defined as follows:

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Fig. 2.5 A knowledge structure that contains three time responses

T(CB.) = ( i . S T ) , such that (2.4)

j s

CB. = (ST^ (CB ( S L1 T ^ ) ) ) , and

j r o o t in which Ts is the sampling period.

Because each cell has a unique number, a unique combination (i,£Ts) is

assigned to each basic cell. This combination can be regarded as an extended time, and consists of two elements:

- i is the number of the time response, and - £TS is a point in time.

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2.5 Relations in value 19

Notes:

The relation is most useful if the knowledge does not change when it is applied. In this case, before the application, a table can be set up, in which each basic cell is related to a point in time. A convenient way to construct this table is to start from the root Tr 0 0 t, and to assign a point

in time to each basic cell found in the time responses instead of, as according to the definition, starting from the basic cell and finding the point in time.

2.5 Relations in value

2.5.1 The class-successor and the class-predecessor operator

Just as a relation in time is either a successor operator or a time-predecessor operator, a relation in value is either a class-successor operator SC or a class-predecessor operator PC, with which a basic cell is extended. These operators enable the construction of strings of basic cells with about the same u and about the same y. A region with about the same u and about the same y is called a class and is worked out in the following section. A string of basic cells constructed with the class operators SC and PC is called a class list. The class-successor operator SC yields the next cell of the class list, and the class-predecessor operator PC yields the previous cell of the class list. A basic cell CB; with both time operators and class operators is shown in fig. 2.6.

PT

SC

O

ST

t PC

Fig. 2.6 A basic cell CB-. with both relations in time and relations in value

The range of all these operators is the set of basic cells, while the image of these operators is also the set of basic cells.

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2.5.2 The division into classes

The division into classes is based on a nonfuzzy division of u and of y. This division has nothing to do with the analog to digital (AD) conversion, which is a discretisation. It is assumed that this AD conversion is sufficiently accurate. The division into classes means that both u as well as y are divided into a number of parts, while the cartesian product of these parts yields the similar classes: the classes with about the same u and the same y. This division is two-dimensional. Chapter 7 contains a further treatment of the knowledge structure, in which a more dimensional division will be used. Let us suppose that u and y are bounded between respectively u", u + , y" and y + , with u ' < u +

and y"<y + . Let us further suppose 1 + LU classes on u and 1+LY classes on y,

with LUeW and LYeW. LU and LY are arbitrarily chosen.

First the u and y values are rescaled respectively to the ranges [0,LU] and [0,LY] by the monotonous rescaling functions RU and RY, for which the following linear functions have been chosen:

(u - u ) . LU ( u+ - u " )

RU(u) = , u e [ u " , u+] , RU(u) e [0.LU] (2.5)

(y - y") • LY

R Y( y ) = , y e [ y- , y ] , RY( y ) g [ 0 , LY ] (y - y )

Note:

The rescaling functions RU and RY are linear. Theoretically there is an infinity of possibilities to define a rescaling function. Practical investigation could be done on the behavior of the application algorithms with nonlinear rescaling functions, for example logarithmic.

Each class will be indicated by a number. These numbers will be called class values. Let us suppose that ud and yd are class values which are associated

with u and y, defined by the two class functions UD and YD:

u = UD(u) = e n t i e r ((RU(u) + 0 . 5 ) , u e { 0 , L U } , and (2.6) yd = YD(y) = e n t i e r ( ( R Y ( y ) + 0 . 5 ) , yd e { 0 . L Y } , .

Because the rescaling functions are monotonous the range of ud and yd follows

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2.5 Relations in value 21 u = u ~ u , = e n t i e r ( 0 . 5 ) = 0 , and d u = u * u , = e n t i e r (LU + 0 . 5 ) = LU . d If u" = -u + then: LU u = 0 - u , = e n t i e r ( — + 0 . 5 ) . d 2

For example, if u" = -10, u+= 1 0 and LU = 20, the class values for -10, 0 and 10

are respectively 0, 10 and 20.

The relations, which have the character of an inverse, can be derived from equations 2.5 and 2.6: let us suppose a class value ud for which it holds

ud = UD(u). Using equation 2.6 yields:

u , <: RU(u) + 0 . 5 < u , + 1 . (2.7)

a a

Substituting equation 2.5 yields the interval for which it holds ud = UD(u):

(u - 0 . 5 ) ( u++ u ~ ) (u + 0 . 5 ) ( u +u~)

u e [ u~ + — - , u " + —2 : ) (2.8)

LU LU It is chosen that the relation DU yields the middle of the interval:

u (u - u " )

u = DU ( u . ) = u " + — , u , e {0.LU} . (2.9) d u

LU

Similar relations hold for y, so DY is defined by:

yri (y - y")

y = DY (y ) = y " + — , y e {0.LY} . (2.10) LY

By means of the rescaling functions RU and RY the (u,y) plane between u", u + , y and y+ is divided into (1 + LU).(1 + LY) classes:

C = { ( u . y ) | RU(u) e [ p - 0 . . 5 , p + 0 . 5 ) A (2.11) P i H

RY(y) e [ q - 0 . 5 , q + 0 . 5 ) } , where p = UD(u) and q = YD(y).

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A division of u and y is shown in fig. 2.7 in the (u,y) plane.

Fig. 2.7 Division of the (u,y) plane

2.53 The class of a basic cell

The class of a basic cell can be defined by a class operator C, which operates on a basic cell CB: and yields the class as follows:

C(CB.) = C , with (2.12) j p . q

p = UD (CB. u ) a n d q = YD (CB. y) .

J - J

By this operator C each cell CB: is assigned to an existent class, because it will be assumed that

u ^ CB. u ^ u , and y " < C Bj _ y < y •

2.5.4 The class lists

By means of the class operator each basic cell belongs to a class. Thus each class Cp q can be related to a list of the basic cells which belong to that

class. Let us consider Cp q as a matrix of (1 + LU).(1 + LY) elements, on which

the class-successor operator can be defined. In each element the number of a basic cell can be stored. Thus each element can be viewed as a root according to section 2.4.2. The cells of the list of class Cp q can be given by:

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2.5 Relations in value 23

(SCh C ) , w i t h h e W+, a n d (S(^ C ) > 0 . (2.13)

p . q p . q The class-predecessor operator PC yields zero for the first cell of a class list, so if (PC CBj)=0 then cell CBj is the first cell of a class list. In fig. 2.8 an example of a (u,y) plane with some class lists is shown.

Fig. 2.8 A (u,y) plane with relations in value

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Fig. 2.9 A (u,y) plane with relations in time and relations in value

In fig. 2.10 the time response from which the cell structure has been derived is shown in the time domain.

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2.6 Visualization of the cell structure 25

u.y

Note:

Fig. 2.10 The time response belonging to fig. 2.9

The class lists are used by starting from the classes in the (u,y) plane, and following the class-successor operators. Thus the class-predecessor operators are not necessary for the application. However, if it would happen that a basic cell must be deleted (and the knowledge structure repaired again), it is practical that also class-predecessor operators exist.

2.6 Visualization of the cell structure

2.6.1 Representation of the cell structure in a three-dimensional space

In fig. 2.8 and fig. 2.9 it has already been shown that the knowledge structure

which has been developed so far can be represented in a three-dimensional space. In this subsection some notes about the representation will be made.

- The three-dimensional space will have the axis u, y and z, in which z is the "height".

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- The (u,y) plane is the base of the representation. Of course a different plane is also possible, but this convention seems convenient.

- The position of a cell CB: in the (u,y,z) space is given by three coordinates. The coordinates in the (u,y) plane are CB:_u and CB:_y. The third coordinate of the position of a cell will be denoted by ZH(CB:), which will represent the place of cell CB: in its class list. Application ZH(CBj) times of the class-successor operator to the root of a class will yield the cell, hence (see section 2.5.4):

ZH(CB.)

(SC J C ) = CB. . (2.14)

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2.6 Visualization of the cell structure 27

An example of the representation of a knowledge structure with about 200 samples is shown in fig. 2.11. Whereas in this figure the relations in value can be still distinguished, in fig. 2.12 a knowledge structure with about 1000 samples is shown, in which the cells have been drawn so close together, that the relations in value could not be shown.

Note:

If the relations in time are disregarded, the resulting visualization of the knowledge structure is a histogram. This is perhaps most clear if the knowledge structure is viewed from almost straight above the (u,y) plane, as in fig. 2.13, in which the knowledge structure of fig. 2.12 is seen from a different direction.

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Fig. 2.13 The knowledge structure of fig. 2.12 viewed from above

2.62 Number of possible three-dimensional visualizations of the knowledge structure

The three-dimensional representation of the knowledge described up to now is not unique. Besides a different choice of axes, the order of cells in a class list is arbitrary, and this implies that the height ZH(CBj) of a cell CBj in the visualization is rather arbitrary.

Each class list of n elements can be ordered in n! ways. Let us assume a knowledge structure containing a class list with 20, a class list with 18, and a

Fuzzy modeling and control based on cell structures

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SUMMARY

CONTENTS

SYMBOLS

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x+00

P

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1 A concept for fuzzy modeling and control

2 The knowledge representation

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000= O

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