Constructing a Fuzzy Controller from Data
Frank Klawonn and Rudolf Kruse
Fuzzy control at the executive level can be interpreted as an
approximation technique for a control function based on typical,
imprecisely specified input-output tuples that are represented by
fuzzy sets. The imprecision is characterized by similarity relations
that are induced by transformations of the canonical distance function
between real numbers. Taking this interpretation of fuzzy controllers
into account, in order to derive a fuzzy controller from observed data
typical input-output tuples have to be identified. In addition, a
concept of similarity based on a transformations of the canonical
distance is needed in order to characterize the typical input-output
tuples by suitable fuzzy sets.
A variety of fuzzy clustering algorithms exists that are exactly
working in this spirit: They identify prototypes and assign fuzzy sets
to the prototypes on the basis of a suitable transformed distance. In
this paper we discuss how such fuzzy clustering techniques can be
applied to construct a fuzzy controller from data and introduce
special clustering algorithms that are tailored for this problem.
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The Relation between Inference and Interpolation in the Framework ofFuzzy Systems
Frank Klawonn and Vilem Novak
This papers aims at clarifying the meaning of different
interpretations of the Max-Min or, more generally, the Max-t-norm rule
in fuzzy systems. It turns out that basically two distinct approaches
play an important role in fuzzy logic and its applications: fuzzy
interpolation on the basis of an imprecisely known function and
logical inference in the presence of fuzzy information.
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Fuzzy Control on the Basis of Equality Relations - with an Example from Idle Speed Control
F. Klawonn, J. Gebhardt, R. Kruse
The way engineers use fuzzy control in real world applications is
often not coherent with an understanding of the control rules as
logical statements or implications. In most cases fuzzy control can be
seen as an interpolation of a partially specified control function in
a vague environment, which reflects the indistinguishability of
measurements or control values.
In this paper we show that equality relations turn out to be the
natural way to represent such vague environments and we develop
suitable interpolation methods to obtain a control function. \par
As a special case of our approach we obtain Mamdani's model and can
justify the inference mechanism in this model and the use of
triangular membership functions not only for the reason of simplified
computations, and we can explain why typical fuzzy partitions are
preferred. We also obtain a criterion for reasonable defuzzification
strategies.
The fuzzy control methodology introduced in this paper has been
applied successfully in a case study of engine idle speed control for
the Volkswagen Golf GTI.
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Similarity in Fuzzy Reasoning
Frank Klawonn, J. L. Castro
Fuzzy set theory is based on a `fuzzification' of the predicate $\in$
(element of), the concept of membership degrees is considered as
fundamental. In this paper we elucidate the connection between
indistinguishability modelled by fuzzy equivalence relations and fuzzy
sets. We show that the indistinguishability inherent to fuzzy sets can
be computed and that this indistinguishability cannot be overcome in
approximate reasoning.
For our investigations we generalize from the unit interval as the
basis for fuzzy sets, to the framework of GL-monoids that can be
understood as a generalization of MV-algebras. Residuation is a basic
concept in GL-monoids and many proofs can be formulated in a simple
and clear way instead of using special properties of the unit
interval.
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Fuzzy Sets and Vague Environments
Frank Klawonn
In this paper we propose a natural approach to handle imprecise
numbers as they arise for example from measurements. Fuzzy sets turn
out to be a canonical representation for such imprecise numbers that
are induced by taking different tolerance or error bounds into
account. Fuzzy sets are induced by scaling factors that describe the
magnitude of the imprecision. On the other, the scaling factors can
be derived from given fuzzy sets so that we have a correspondence
between scaling factors and fuzzy sets.
When these concepts are applied to control problems, the max-min rule
is rediscovered as an interpolations technique. Viewing fuzzy control
as an interpolation technique in vague environments enables us to
validate various concepts for the design and tuning of fuzzy
controllers and suggests new also new methods based on clear
semantics.
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A Lukasiewicz Logic Based Prolog
Frank Klawonn and Rudolf Kruse
Prolog is a programming language based on a restricted subset of
classical first order predicate logic. In order to overcome some
problems of classical logic to handle imperfect human knowledge, we
provide a formal framework for a Lukasiewicz logic based Prolog
system. The use of Lukasiewicz logic with its connection to Ulam games
enables us to deal with partial inconsistencies by interpreting the
truth values as relative distance to contradiction.
We also present the software tool LULOG which is based on the
theoretical results of this paper and can be seen as a Prolog system
for many-valued logic. Applications of LULOG to an Ulam game and an
example of reasoning with imperfect knowledge are also discussed.
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Equality Relations as a Basis for Fuzzy Control
F. Klawonn, R. Kruse
The aim of this paper is to introduce a fuzzy control model with
well-founded semantics in order to explain the concepts applied in
fuzzy control. Assuming that the domains of the input- and output
variables for the process are endowed with equality relations, that
reflect the indistinguishability of values lying closely together, the
use of triangular and trapezoidal membership functions can be
justified and max-$\sqcap$ inference where $\sqcap$ is a t-norm turns
out to be a consequence of our model. Distinguishing between a
functional and a relational view of the control rules it is possible
to explain when defuzzification strategies like MOM or COA are
appropriate or lead to undesired results.
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Techniques and Applications of Control Systems
Based on Knowledge-Based Interpolation
F. Klawonn, R. Kruse
Fuzzy control was established as an alternative control method when it is
difficult to develop a suitable mathematical model of the process, but
expert knowledge in the form of vague rule is available. Although the
principal idea of a fuzzy set as a model of a vague linguistic expression is
very appealing, a naive approach to fuzzy sets can cause tedious
problems. Without a concrete semantics for fuzzy sets the design of a fuzzy
controller can end up in a trial and error experiment with a large number of
parameters and options.
In this paper we review an approach to fuzzy control that interprets fuzzy
sets as vague values in a vague environment. The vague environment is
characterised by a scaling function that describes how sensitive the process
reacts when a certain value is slightly changed. We also discuss a
regression technique based on the concept of vague environments, enabling to
construct a fuzzy controller from data.
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The Role of Similarity in Fuzzy Reasonin
F. Klawonn
Fuzzy reasoning mechanisms are designed to cope with vague and
uncertain knowledge and information. In this paper we demonstrate that
from the vagueness inherent in a fuzzy system a canonical
indistinguishability of objects can be derived which cannot be
overcome by the standard reasoning schemes. We discuss also the
consequences for fuzzy logic in the narrow sense.
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Fuzzy Shell Cluster Analysis
F. Klawonn, R. Kruse and H. Timm
In this paper we survey the main approaches to fuzzy shell cluster
analysis which is simply a generalization of fuzzy cluster analysis to
shell like clusters, i.e. clusters that lie in nonlinear subspaces.
Therefore we introduce the main principles of fuzzy cluster analysis
first. In the following we present some fuzzy shell clustering
algorithms. In many applications it is necessary to determine the
number of clusters as well as the classification of the data set.
Subsequently therefore we review the main ideas of unsupervised fuzzy
shell cluster analysis. Finally we present an application of
unsupervised fuzzy shell cluster analysis in computer vision.
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Prolog Extensions to Many--Valued Logics
F. Klawonn
The aim of this paper is to show that a restriction of a logical
language to clauses like Horn clauses, as they are used in Prolog,
applied to [0,1]-valued logics leads to calculi with a sound and
complete proof theory. In opposition to other models where generally
the set of axioms as well as the deduction schemata are enriched we
restrict ourselves to a simple modification of the deduction rules of
classical logic without adding new axioms.
In our model the truth values from the unit interval can be
interpreted in a probabilistic sense, so that a value between 0 and 1
is not just intuitively interpreted as a `degree of truth'.
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Learning the Rule Base of a Fuzzy Controller by a Genetic Algorithm
J. Hopf and F. Klawonn
For the design of a fuzzy controller it is necessary to choose,
besides other parameters, suitable membership functions for the
linguistic terms and to determine a rule base.
This paper deals with the problem of finding a good rule base - the
basis of a fuzzy controller. Consulting experts still is the usual but
time-consuming and therefore rather expensive method. Besides, after
having designed the controller, one cannot be sure that the rule base
will lead to near optimal control. This paper shows how to reduce
significantly the period of development (and the costs) of fuzzy
controllers with the help of genetic algorithms and, above all, how to
engender a rule base which is very close to an optimum solution.
The example of the inverted pendulum is used to demonstrate how a
genetic algorithm can be designed for an automatic construction of a
rule base. So this paper does not deal with the tuning of an existing
fuzzy controller but with the genetic (re-)production of rules, even
without the need for experts. Thus, a program is engendered,
consisting of simple IF...THEN instructions.
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Context Sensitive Fuzzy Clustering
Annete Keller, Frank Klawonn
We introduce an objective function-based fuzzy clustering technique that incorporates linear combinations of attributes in the distance function. The main application field of our method is image processing where a a comparison pixel by pixel is usually not adequate, but the environment of a pixel or a groupd of pixels characterize important properties of an image or parts of it. In addition, our approach can be seen as a generalization of other fuzzy clustering techniques like the axes-parallel version of the Gustafson-Kessel algorithm.
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Fuzzy Clustering Based on Modified Distance Measures
Frank Klawonn, Annete Keller
The well-known fuzzy c-means algorithm is an objective function based fuzzy
clustering technique that extends the classical k-means method to fuzzy
partitions. By replacing the Euclidean distance in the objective function
other cluster shapes than the simple (hyper-)spheres of the fuzzy c-means
algorithm can be detected, for instance ellipsoids, lines or shells of
circles and ellipses. We propose a modified distance function that is based
on the scalar product and allows to detect a new kind of cluster shape and
also lines and (hyper-)planes.
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Fuzzy Clustering with Weighting of Data Variables
Annete Keller, Frank Klawonn
We introduce an objective function-based fuzzy clustering technique that assigns one influence parameter to each single data variable for each cluster. Our method is not only suited to detect structures or groups in unevenly over the structure's single domains distributed data, but gives also information about the influence of individual variables on the detected groups. In addition, our approach can be seen as as generalization of the well-known fuzzy c-means clustering algorithm.
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Mathematical Analysis of Fuzzy Classifiers
Frank Klawonn and Erich-Peter Klement
We examine the principle capabilities and limits of fuzzy classifiers
that are based on a finite set of fuzzy if-then rules like they are
used for fuzzy controllers, except that the conclusion of a rule
specifies a discrete class instead of a (fuzzy) real output value. Our
results show that in the two-dimensional case, for classification
problems whose solutions can only be solved approximately by crisp
classification rules, very simple fuzzy rules provide an exact
solution. However, in the multi-dimensional case, even for linear
separable problems, max-min rules are not sufficient.
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Fuzzy Clustering and Fuzzy Rules
Frank Klawonn and Annette Keller
Fuzzy clustering offers various possibilities for learning fuzzy
if-then rules from data for classification tasks as well as for
function approximation problems like in fuzzy control. In this paper
we review approaches for deriving rules from data by fuzzy clustering
and discuss some of their common problems. As a consequence, we
propose a new method which is specifically tailored for the task of
learning rules.
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Fuzzy Clustering with Evolutionary Algorithms
Frank Klawonn
Abstract
Objective function based fuzzy clustering aims at finding a fuzzy
partition by optimizing a function evaluating a (fuzzy) assignment of
a given data set to clusters, that are characterized by a set of
parameters, the so-called prototypes. The iterative optimization
technique usually requires the objective function not only to be
differentiable, but prefers also an analytical solution for the
equations of necessary conditions for local optima. Evolutionary
algorithms are known to be an alternative robust optimization
technique which are applicable to quite general forms of objective
functions. We investigate the possibility of making use of
evolutionary algorithms in fuzzy clustering. Our experiments and
theoretical investigations show that the application of evolutionary
algorithms to shell clustering, where the clusters are in the form of
geometric contours, is not very promising due to the shape of the
objective function, whereas they can be helpful in finding solid
clusters that are not smooth, for example rectangles or cubes. These
types of clusters play an important role for fuzzy rule extraction
from data.
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Fuzzy Cluster Analysis for Identification of Gene Regulating Regions
Lars Pickert, Frank Klawonn, Edgar Wingender
The main approach of this work is the implementation of a cluster
analysis program for identification of regulatory regions in
genomes. These regions are important parts of the genetic pool in
higher developed organisms. They are composed of several basic
elements, so called transcription factor sites, which can be
identified by special analysis tools more or less vaguely. The
program we have developed is able to search for two-dimensional
clusters in the results of such analysis tools to give hints on gene
regulatory regions. For this purpose two fuzzy clustering algorithms
have been implemented: The fuzzy c-means (FCM) and the Gath and Geva
fuzzy clustering algorithm (GG) with two conventional cluster validity
methods and one which has been developed especially for this
application. All results of the cluster analysis program can be
visualized and documented automatically.
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Neuro-Fuzzy Classification Initialized by Fuzzy Clustering
D. Nauck, F. Klawonn
In this paper we discuss how a neuro-fuzzy classifier can be
initialized by rules generated by fuzzy clustering. The neuro-fuzzy
classifier NEFCLASS can learn fuzzy classification rules completely
from data. The learning algorithm for fuzzy sets can be constrained in
order to obtain interpretable classifiers. However, fuzzy clustering
provides more sophisticated rule learning procedures. We show that the
learning process of NEFCLASS produces better results, if it is
initialized by fuzzy clustering.
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Derivation of Fuzzy Classification Rules from Multidimensional Data
F. Klawonn and R. Kruse
A
This paper describes techniques for deriving fuzzy classification
rules based on special modified fuzzy clustering algorithms. The basic
idea is that each fuzzy cluster induces a fuzzy classification rule.
The fuzzy sets appearing in a rule associated with a fuzzy cluster are
obtained by projecting the cluster to the one-dimensional coordinate
spaces. In order to allow clusters of varying shape and size we derive
special fuzzy clustering algorithms which are searching for clusters
in the form of axes-parallel hyper-ellipsoids. Our method can be
applied to classification tasks where the classification of the sample
data is known as well as when it is not known.
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Clustering Methods in Fuzzy Control
F. Klawonn, R. Kruse
Fuzzy controllers can be interpreted as an interpolation technique on
the basis of fuzzy clusters of input/output pairs. It is therefore
obvious that fuzzy clustering algorithms are a promising tool for
supporting the design of a fuzzy controller when data of the process
to be controlled are available.
This paper discusses the possibilities and limitations of fuzzy
clustering for fuzzy control.
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Similarity Based Reasoning
Frank Klawonn
This paper is devoted to the duality between fuzzy sets and equality
relations. It comprises various results that allow to interchange
from one framework to the other. Finally it is shown that in fuzzy
reasoning the inherent similarity characterized by equality relations
cannot be avoided.
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Modifications of Genetic Algorithms for Designing and Optimizing FuzzyControllers
J. Kinzel, F. Klawonn, R. Kruse
This paper investigates the possibilities for applications of genetic
algorithms to tuning and optimizing fuzzy controllers, or even to
generate fuzzy controllers automatically. There are various ad-hoc
approaches to use genetic algorithms for the design of fuzzy
controllers, which already indicated good results. However, there is a
need for systematic techniques that take the properties of fuzzy
controllers and genetic algorithm into account in order to obtain fast
convergence and to be able to tackle more complex control problems.
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Fuzzy Clustering with Evolutionary Algorithms
Frank Klawonn and Annette Keller
Objective function based fuzzy clustering aims at finding a fuzzy
partition by optimizing a function evaluating a (fuzzy) assignment of
a given data set to clusters, that are characterized by a set of
parameters, the so-called prototypes. The iterative optimization
technique usually requires the objective function not only to be
differentiable, but prefers also an analytical solution for the
equations of necessary conditions for local optima. Evolutionary
algorithms are known to be an alternative robust optimization
technique which are applicable to quite general forms of objective
functions. We investigate the possibility of making use of
evolutionary algorithms in fuzzy clustering. Our experiments and
theoretical investigations show that the application of evolutionary
algorithms to shell clustering, where the clusters are in the form of
geometric contours, is not very promising due to the shape of the
objective function, whereas they can be helpful in finding solid
clusters that are not smooth, for example rectangles or cubes. These
types of clusters play an important role for fuzzy rule extraction
from data.
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Fuzzy Max-Min Classifiers Decide locally on the Basis of Two Attributes
Birka von Schmidt, Frank Klawonn
Fuzzy classification systems differ from fuzzy controllers in the form
of their outputs. For classification problems a decision between a
finite number of discrete classes has to be made, whereas in fuzzy
control the output domain is usually continuous, i.e. a real interval.
In this paper we consider fuzzy classification systems using the
max-min inference scheme and classifying an unknown datum on the basis
of maximum matching, i.e. assigning it to the class appearing in the
consequent of the rule whose premise fits best. We basically show that
this inference scheme locally takes only two attributes (variables)
into account for the classification decision.
Letzte
Änderung am
07.09.2006
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