Excellent and comprehensive guide to Fuzzy Modeling and Control. This book successfully bridges theory with practice by including many concrete examples. It provides detailed examples of Fuzzy Logic Control (FLC) systems. And it provides the theoretical background behind many fundamental aspects of the subject, including the theory of Approximate Reasoning (AR).
Here is the Table of Contents annotated with key points from the book
Chapter 1 - Basic Concepts of Fuzzy Set Theory
Chapter 2 - Aggregation Operations on Fuzzy Sets
Intersection and union operations can be viewed in a more general setting as an aggregation of fuzzy subsets.
Many applications of fuzzy sets involve aggregations.
In this chapter we provide a number of different methods for aggregating fuzzy subsets.
We first look at the intersection and the union operation.
We introduce two classes of aggregation operators, the t-norm and the t-conorm, which generalize the intersection and union operations.
We then turn to the issue of weighted intersections and unions.
In some applications in artificial intelligence, a need arises for the ability to reason with default knowledge.
We then turn from intersection and union to other aggregation operations which lie between them, mean operators.
Then we turn to a specific type of mean operator, ordered weighted aggregation (OWA).
We then use OWA operators to help make a connection between natural language stipulations of aggregations
in terms of linguistic quantifiers and the mathematical realization of the stipulated aggregation.
Last, fuzzy measure and fuzzy integral are introduced.
Chapter 3 - Theory of Approximate Reasoning (AR)
Lotfi Zadeh introduced the concept of Approximate Reasoning in 1979.
Based upon fuzzy sets, it provides a powerful framework for reasoning in the face of uncertain information.
Central to this theory is the representation of propositions as statements assigning fuzzy sets as values to variables.
The formal AR framework is provided in this chapter.
The AR framework provides a mechanism for modeling and making inferences from imprecise functional relationships.
This mechanism forms the basis of the fuzzy systems modeling technique.
Definitions: atomic variables, universe of discourse, joint variable, canonical statement, possibility, tautology,
conjoin, cylindrical extension, proposition equivalence.
Translation rules: proposition negation, proposition Anding, proposition Oring, Plying operation
Definitions: deduction, Inference rules 1 and 2
Theorems: monotonicity
Definitions: projection, minimal value,
Theorems: 6 theorems
Binary propositional logic as a special case of AR
Extension of the AR inference mechanism to (non-binary) functional relationships.
Chapter 4 - introduction to Fuzzy Logic Control
The best chapter of the book.
Introduction to conventional system control; feedback.
Definitions
C: controller
S: system
y: output
w: set point
u: controller action
e: error, w-y
f: non-linear function
Conventional controller algorithms are expressed via analytic algorithms based upon conventional set theory.
Fuzzy logic controllers (FLC) are expressed via knowledge-based algorithms based upon fuzzy set theory.
FLCs are expressed via a linguistic description, and a predicate rule-based inference algorithm.
The general form of the conventional logic controller is
(1) u(k) = f(e(k),e(k-1),...,e(k-v),u(k-1),u(k-2),...,u(k-v))
A FLC is a knowledge=based system consisting of IF ... THEN rules with vague predicates and a fuzzy logic inference mechanism.
The general form of the fuzzy logic controller is
(2) u(k) = F(e(k),e(k-1),...,e(k-v),u(k-1),u(k-2),...,u(k-v))
where the function F is described by a rule-base.
An example using 5 predicate rules and linguistic labels represented as fuzzy sets.
The linguistic labels & predicate logic are what allow a heuristic implementation.
These are based upon human knowledge of the particular system.
Very important section ... and difficult to capture without a technical document editor (LaTex here we come):
The phrase 'reasoning' is used to indicate 'the process' in which the given input values and antecedent variables are used in conjunction with the knowledge-base to obtain the output value, the consequent. This is based upon the theory of Approximate Reasoning (AR).
The procedure for obtaining the fuzzy output value is:
1 - find the firing level of each of the rules
2 - find the output of each of the rules
3 - aggregate the individual rule outputs to obtain the overall system output.
The degree (level) of firing (DOF) for the ith is shown to be
DOF = B1(x1*) ^ B2(x2*)
for antecedent linguistic labels B1, B2 and inputs x1, x2
the ith output fuzzy set F is obtained by anding the DOF and the consequent set D:
F = DOF ^ D
Similarly, aggregation across ALL i individual rule outputs F gives the fuzzy output F
F(y) = V F(y) = V ( DOF ^ D )
This approach is known as the (constructive) Mamdani method
The procedure for obtaining the de-fuzzified output value is further detailed as follows:
1 - calculate the DOF of the rules for crisp inputs or for fuzzified noisy inputs
2 - find the membership function of output fuzzy set Fi inferred by the ith rule, for all i
3 - form the membership function of output fuzzy set F, inferred by the rule-base of FLC by aggregating the Fi
4 - calculate the crisp output of FLC y* by defuzzification of fuzzy set F using the center of area (COA) or mean of maxima (MOM) method
Extended example included
How Mamdani-type FLCs can be regarded as PI, PD and PID-like control algorithms
Now, the next two sections address 2 of the most important FLC implementation challenges
Determination of the linguistic values, the membership functions of the associated reference fuzzy sets, can be a challenge.
This is a surprisingly complex topic.
A complex example of reference fuzzy set membership functions in analytic form is provided.
Creating the membership functions requires pairing skills and experience in both FL methodology and the particular problem domain.
Construction of the knowledge base is the most crucial and difficult part of FLC design. In the past, knowledge base construction was an ad hoc process. FLC has been criticized for a lack of a general systematic tool for creating the knowledge base. However, there are a variety of (non-systematic) approaches. Building the FLC knowledge base requires both a system operator expert, as well as an FLC system specialist.
One more formal approach utilizes the concept of a knowledge template rule-base.
The MacVicar-Whelan template rule-base is described in this section.
Summary of practical design rules in FLC design
step 1 - determine the input and output variables of the FLC, and any PI, PD, or PID-like similarities
step 2 - define the FLC parameters - scaling factors, normalized universe of discourse, linguistic variables, and membership functions
step 3 - determine the FLC rule-base
step 4 - computational realization of the FLC
Originally, the Mamdani FLC was considered a knowledge-base alternative to conventional system control methods.
The concept of fuzzy control can be applied (with appropriate rules) not only to calculating the control variable, but also for updating the control strategy itself.
For example, modifying the setpoint.
An example of this is provided, for modifying the setpoint of a non-linear, non-stationary biochemical process.
Chapter 5 - Fuzzy System Models
The complexity of the real world is most dramatically reflected in "soft" area such as social, economic, ecological, and biological systems.
Multi-dimensionality, hierarchical structures, mutual interactions, internal feedback mechanisms, and unpredictable dynamics are only a part of the characteristics of such complex systems.
This complexity accounts for some of the reasons for the difficulties in the attempts to transfer the powerful systems and control techniques to these disciplines.
The weakness of the traditional quantitative techniques to adequately describe complex phenomena was summarized in the well-known principle of incompatibility, formulated by L. Zadeh. This principle states that "as the complexity of a system increases, our ability to make precise and yet significant statements about its behaviors diminishes, until a threshold is reached beyond which precision and significance or relevance) become almost mutually exclusive characteristics."
Fuzzy system models fall into 2 categories
(1) Linguistic Models (LM) based upon collections of IF-THEN rules with fuzzy predicates, fuzzy quantities, and natural language system description
(2) Takagi-Sugeno-Kang (TSK) models, formed by logical rules with a fuzzy antecedent and a functional consequent.
TSK models combine fuzzy and non-fuzzy models.
[[ note that this 60-page chapter devotes only 13 pages to TSK models ]]
According to the theory of approximate reasoning, each model rule: IF U is B THEN V is D
can be translated into a canonical proposition of the form (U,V) is R
where R is a fuzzy relationship defined on the Cartesian product universe X x Y
There are 2 types of Linguistic models
(1) Mamdani-type (constructive) LM
(2) Logical-type (destructive) LM
Mamdani-type models are built by constructing the union of all the individual fuzzy relations.
Under this approach the aggregation of individual rules (ALSO connectives) is accomplished by the fuzzy intersection.
[28] Box, Jenkins, "Time Series Analysis, Forcasting and Control", Holden Day 1970
Chapter 6 - Developing Fuzzy Models
Three fuzzy modeling principals according to Zadeh for complex/ill-defined systems
1 - linguistic variables
2 - conditional fuzzy statements
3 - fuzzy algorithms
an important issue in model design is ... how to build the model, the methodology. Here are two approaches.
*1* (direct approach) early examples inspired by expert systems (ES) extracted the model from the operator expert describe the system linguistically using natural language, then translate into formal structure using approximate reasoning (AR)
*2*(system identification approach) inspired by classical systems theory, and more recent neural network development, this is a two phase approach.
phase 1: structure identification
phase 2: parameter identification
p205
Structure identification ... template based models combine expert knowledge the system expert provides template linguistic values which allow us to partition the input-output (I/O) space
fuzzy sets are given a priori the template values are used to define 'potential' rules; I/O data is used to generate weights or probabilities
If template linguistic values are not available, then the system structure can be obtained by clustering the IO space. Mountain-clustering provides a systematic approach to identifying the most important rules from the IO data.
Parameter identification is closely related to member function estimation. Real success in parameter id was achieved with the TSK method, which combines Kalman filtering. The TSK method also demonstrates a way of developing fuzzy models though a simplification of the Mamdani reasoning.
The coupling of FL and NN is a new and powerful tool for parameter identification of FL models with the use of back-propagation
[35] Aoki, "Optimization of Stochastic Systems", 1967
Chapter 7 - Theoretic Analysis of FLC
p265 FLC does not require a precise mathematical model of the controller object because it models the experience & knowledge of the operator.
But, thee is more ... recent developments into FLC self tuning
Comparison to (classical) PD controllers, like a virtual non-linear PD controller ...
Lots of discussion of sliding mode ... complex ...
p311 references include [9], [10] design of self-tuning fuzzy controllers.
Chapter 8 - Defuzzification Problem
p311 Compare & contrast the various defuzzification methods
p315 Defuzzification is a "selection" problem. First, a fuzzy set must be converted into a "probability" distribution. Then, a crisp value is selected either:
(a) by random experiment, or
(b) by expected value
Chapter 9 - The Flexible Structure of Fuzzy Systems
p357 introduce a more powerful & flexible parameterized fuzzy reasoning framework that unifies both Mamdani-type and logical-types models.
More powerful ... how ?
