Fuzzy Sets and Fuzzy Logic is a true magnum opus. An enlargement of Fuzzy Sets, Uncertainty,
and Information—an earlier work of Professor Klir and Tina Folger—Fuzzy Sets and Fuzzy Logic
addresses practically every significant topic in the broad expanse of the union of fuzzy set theory
and fuzzy logic. To me Fuzzy Sets and Fuzzy Logic is a remarkable achievement; it covers its vast
territory with impeccable authority, deep insight and a meticulous attention to detail.
To view Fuzzy Sets and Fuzzy Logic in a proper perspective, it is necessary to clarify a point
of semantics which relates to the meanings of fuzzy sets and fuzzy logic.
A frequent source of misunderstanding fias to do with the interpretation of fuzzy logic. The
problem is that the term fuzzy logic has two different meanings. More specifically, in a narrow
sense, fuzzy logic, FLn, is a logical system which may be viewed as an extension and generalization
of classical multivalued logics. But in a wider sense, fuzzy logic, FL^ is almost synonymous
with the theory of fuzzy sets. In this context, what is important to recognize is that: (a) FLW is much
broader than FLn and subsumes FLn as one of its branches; (b) the agenda of FLn is very different
from the agendas of classical multivalued logics; and (c) at this juncture, the term fuzzy logic is
usually used in its wide rather than narrow sense, effectively equating fuzzy logic with FLW
In Fuzzy Sets and Fuzzy Logic, fuzzy logic is interpreted in a sense that is close to FLW. However,
to avoid misunderstanding, the title refers to both fuzzy sets and fuzzy logic.
Underlying the organization of Fuzzy Sets and Fuzzy Logic is a fundamental fact, namely,
that any field X and any theory Y can be fuzzified by replacing the concept of a crisp set in X and Y
by that of a fuzzy set. In application to basic fields such as arithmetic, topology, graph theory, probability
theory and logic, fuzzification leads to fuzzy arithmetic, fuzzy topology, fuzzy graph theory,
fuzzy probability theory and FLn. Similarly, hi application to applied fields such as neural network
theory, stability theory, pattern recognition and mathematical programming, fuzzification leads to
fuzzy neural network theory, fuzzy stability theory, fuzzy pattern recognition and fuzzy mathematical
programming. What is gained through fuzzification is greater generality, higher expressive
power, an enhanced ability to model real-world problems and, most importantly, a methodology for
exploiting the tolerance for imprecision—a methodology which serves to achieve tractability,

Fuzzy Sets and Fuzzy Logic: Theory and Applications

Conventional fuzzy control can be considered mainly composed of fuzzy two-term control and fuzzy three-term control. In this paper, more systematic analysis and design are given for the conventional fuzzy control. A general robust rule base is proposed for fuzzy two-term control, leaving the optimum tuning to the scaling gains, which greatly reduces the difficulties of design and tuning. The digital implementation of fuzzy control is also presented for avoiding the influence of the sampling time. Based on the results of previous fuzzy two-term controllers, a simplified fuzzy three-term controller is proposed to enhance performance. A two-level tuning strategy is also planned, which first tries to set up the relationship between fuzzy proportional/integral/derivative gain and scaling gains at the high level, and optionally tunes the control resolution at low level. Simulation of different order models show the characteristics of fuzzy control, effectiveness of the new design methodologies, and advantages of the enhanced fuzzy three-term control.

Conventional fuzzy control and its enhancement

A new methodology is proposed for designing a fuzzy logic controller (FLC). A phase plane is used to bridge the gap between the time-response and rule base. The rule base can be easily built using the general dynamics of the process, and then readily updated to contain the delayed information for reducing the deadtime effects of the process. An adaptive gain method is also proposed to help the database design and the controller tuning. Much of the FLC design can be shifted to the design and tuning of gain. A good performance can be achieved both in transient state and steady state without use of multidecision tables. Application of FLC with these new methodologies is presented for a thermal process with a varying deadtime to show the robust performance of FLC and the effectiveness of these methodologies. >

A new methodology for designing a fuzzy logic controller

Points out how the nonlinearities involved in multivariable Takagi-Sugeno (T-S) fuzzy control systems could originate complex behavior phenomena, such as multiple equilibrium points or limit cycles, that cannot be detected using conventional stability analysis techniques. In the paper, the application of MIMO frequency-domain methods to predict the existence of multiple equilibria and of limit cycles are presented. The proposed method is based on the formulation of a Lur'e problem from the original structure of a T-S fuzzy system with a fuzzy controller. Furthermore, this technique makes straightforward the application of input-output stability techniques such as the multivariable circle criterion, also called the conicity criterion, and the harmonic balance method. Moreover, in the paper, the application of the harmonic balance method has been generalized to the case of a MIMO fuzzy system with asymmetric nonlinearities and improved by the decreasing conservatism. A new and more general stability index which could be used to perform a bifurcation analysis of fuzzy control systems is presented. The paper includes a collection of examples where the advantages of the proposed approach are made explicit comparing it to the input-output conicity criterion and the Lyapunov direct method.

Stability analysis of nonlinear multivariable Takagi-Sugeno fuzzy control systems

The paper proposes a complete design method for an online self-organizing fuzzy logic controller without using any plant model. By mimicking the human learning process, the control algorithm finds control rules of a system for which little knowledge has been known. In a conventional fuzzy logic control, knowledge on the system supplied by an expert is required in developing control rules, however, the proposed new fuzzy logic controller needs no expert in making control rules, Instead, rules are generated using the history of input-output pairs, and new inference and defuzzification methods are developed. The generated rules are stored in the fuzzy rule space and updated online by a self-organizing procedure. The validity of the proposed fuzzy logic control method has been demonstrated numerically in controlling an inverted pendulum. >

A self-organizing fuzzy logic controller for dynamic systems using a fuzzy auto-regressive moving average (FARMA) model

A complete design procedure for a fuzzy three-term PID controller is presented. A plant model is not required to achieve this design. A reduced look-up table containing the rules along with the quantization and tuning procedures by means of input and output mapping factors are introduced. The scaling factors of the output error time sequences are preselected arbitrarily and the search for an optimal input-to-output mapping factors ratio is performed through a phase diagram analysis. the applicability of the design procedure is demonstrated through computer simulations. >

Design of a fuzzy controller using input and output mapping factors

The steady-state response of a fuzzy logic controller that uses fuzzy addition as the basis for its decision making process is analyzed mathematically. The nature of the look-up table constructed from fuzzy addition enables the decomposition of the fuzzy controller into three one-dimensional-multilevel relays. Unlike the sinusoidal-input describing function technique, the steady-state response of the closed-loop system is analyzed. The sufficient conditions derived depend on the parameters of the controller and can be used as guidelines to achieve a good step response at steady-state, one that does not lend itself to sustained oscillations or limit cycles. >

Steady-state analysis of a three-term fuzzy controller

The exponential-input describing function technique is used to investigate the transient response of a fuzzy controller. The rule-base of this fuzzy controller is generated by fuzzy addition and the partitioning of the universes of discourse of its input and output variables is based on an exponential mapping scheme. By using the symmetric features introduced by fuzzy addition in the decision table of the fuzzy control algorithm and the new mapping scheme, the effects of the output error time sequences can be represented by independent multilevel relays and a multiplication junction. Computer simulations on a second-order system confirm the behaviour of this analysis technique. >

G. Abdelnour

Papers

Design of a fuzzy controller using input and output mapping factors

Steady-state analysis of a three-term fuzzy controller

Application of describing functions in the transient response analysis of a three-term fuzzy controller