We address in this lecture several concepts of robustness in the design of fuzzy logic controllers. By the very nature of fuzziness, membership functions of linguistic labels are only "nominal" so that variations are possible. Thus the advantageous, but delicate art of fuzzy designs (of fuzzy controllers as well as of fuzzy plant dynamics) calls for some uncertainty reduction techniques in the choice of logical components of inference procedures. With respect to robust stability of fuzzy systems, we emphasize analytic framework and tools.

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The aim of this paper is to clarify the properties of fuzzy arithmetic based on a generalized extension principle and consider the possibility of its applications. The generalized extension principle corresponding to sup-(t-norm) convolution enables us to control the growth of fuzziness in calculation by choosing an adequate t-norm. In this paper, it was made clear that the family of fuzzy numbers forms commutative monoids concerning addition and multiplication, respectively. Also, the authors found some new properties on the distribution of multiplication on addition, and showed their differences from those of the ordinary fuzzy arithmetic based on sup-min convolution.

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This paper proposes some basic operations on fuzzy sets and a generalizing of t-norms fuzzy operations on fuzzy numbers by using interval representation. This algorithm presented is less calculating, more convenient, more flexible and more practicable than another one(e.g.L-R function, Digital Representation etc.) as for the fuzzy numbers operations. Firstly, we describe some basic operations on fuzzy sets using the Interval Representation instead of using the usual point function. All algorithms on the fuzzy numbers operations are belonging to the extension principle proposed by L.A.ZADEH in 1975. Furthermore, we devote to generalize the algorithm which a min fuzzy operations of t-norms on fuzzy numbers defined by possibility distribution on real line are performed by using the Interval Representation. In order to accomplish this purpose we introduce idea of the α-contour line from meaning t-norms operations to these calculations.

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In conventional fuzzy retrieval systems, retrieved results do not always satisfy users' requests since only some operators for 'and' and 'or' in retreival queries are defined. In this paper, we propose new fuzzy connectives which can express 'and' and 'or' operators that provide users with results satisfactory to their request. The fuzzy connectives are defined using a convex combination with a weighting function of parametric t-norm and t-conorm proposed by Schweizer. When users' estimate for retrieved results are given, parameters of the fuzzy connectives are adjusted to minimize the square of errors between users' estimate and output of the fuzzy connectives by steepest descent method. The fuzzy connectives are called "fuzzy connectives with learning function." The fuzzy connectives with learning function mean any operators between drastic product and drastic sum, and can also represent linear mean operator between t-norm and t-conorm by using convex combination. Reported here is the formulation of fuzzy connectives with learing function, and a retrieval experiment employing the fuzzy connectives to show its usefulness.

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The fuzzy relation equations that were proposed by Sanchez in 1976,are very useful, when we want to deal with fuzzy relations between human behaviors and subjective evaluations. This paper presents a study on the method of finding whether or not the fuzzy relation equations have solutions. This method can be applied not only the above-mentioned fuzzy relation equations but also be extended to fuzzy relation equations such as fuzzy inequality and fuzzy relation equations whose elements are indicated by interval values between O and 1. This method consists of "the Signs Matrix Technique" that was proposed. "Signs Matrix" is the martix whose elements are a greater/less then comparison sign between 2 matrixes. This technique enables us to find the existence of solutions of extended fuzzy relation equations very easily.

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Fuzzy modeling is a method to describe the characteristics of complex systems using fuzzy inference. The method has a distinguishing feature in that it can express complex nonlinear systems linguistically. However in the case where the system has multi-inputs, the acquired fuzzy rules through some sorts of learning are hard to understand. And when it is difficult to obtain sufficient input-output data from the system with multi-inputs, the fuzzy model cannot be made precise. This paper presents a hierarchical fuzzy modeling method. The proposed fuzzy modeling method uses fuzzy neural networks(FNNs). The identified fuzzy rules with the new method are easy to understand. The proposed method enables to obtain a precise fuzzy model even with limited number of input-output data.

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A personal computer has developed as a tool which is effectively used by designers and engineers. In recent years, it has required to read a drawing by machine automatically. It has become very important to develop the method which recognizes automatically the segment of a line. Many kinds of method for the curve recognition, such as the segment chain encoding and vector method, have developed. The curve recognition has many cases that the human vagueness are treated. The author doesn't use the method of the old recognition techniques and has a try to do the curve recognition by using the approximate reasoning. In this study, it is convenient to apply the elliptic curve that the radius of curvature is changed by the length of a major or minor axis. As the result of the test, a method of curve recognition by fuzzy reasoning can be obtained sufficiently recognition for the curved lines that a major and minor axis in the elliptic form is the ratio 4.6 : 1.0. And it became clear that this method is useful.

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With the progress of the research of fuzzy theory and technology, it gets to be desired to develop computers and various logic circuits treatable fuzzy information directly. However at present, such circuit design method and the characteristic are not clarified like it is so in the classics two-valued logic. Fuzzy flip-flops will be basic sequential circuits for the research of fuzzy computer as flip-flops are so in two-valued computers, where fuzzy flip-flops is a flip-flops expanded into fuzzy. A usual flip-flops uses 0 and 1 as the inputs. However, fuzzy flip-flops permits not only 0 or 1 but also any value in [0,l] as the inputs. Fuzzy flip-flops is a typical example of fuzzy sequential circuits. By now, some researches have proposed basic circuits of fuzzy computers. In these researches fuzzy flip-flops has been investigated by Mr. K.Hirota, Mr. K.Ozawa, Mr. W.Pedrycz, and Mr. L.T.Koczy, etc. until now. In those researches, a fuzzy logic function representing fuzzy JK flip-flops has two expressions for set and reset respectively. These two expressions are switched if necessary for the use. In this paper, JK flip-flops is taken up also as a typical fuzzy circuit expressible by a fuzzy logical expression. Karnaugh graph for basic circuit design had been paid to attention before thinking of the logical expression. Here, we expand it into ternary logic case which satisfies the same axiomatic system of fuzzy logic. Then characteristics of fuzzy logic function was paid to attention. All V2-equivalent (equivalent in two-valued sense) fuzzy logic functions are derived from the ternary Karnaugh graph. As a result, we can derived one logical expression for the fuzzy JK flip-flops which we need not to switch two expression.

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In Tanaka-Sugeno's stability theorem for fuzzy systems, the existence of a symmetric positive definite matrix P satisfying the Lyapunov inequalities plays an important role. The aim of this paper is to propose the P-region method for finding a set of P(2×2), and to generalize it into an extended case P(n×n). Also, a corollary for the theorem is given, and it is shown that for applications of Tanaka-Sugeno's stability theorem to the design of fuzzy control, an extension based on the proposed method and corollary will present a more actual and helpful tool.

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The conventional control theories can be divided into two classes, that is, classical control theory whose objective is to minimize the control error and modern control theories whose objective is to obtain the dynamic control which optimize the integral type performance index. The both class of control theories are different in design concept and structure. On the other hand, control action of most of the fuzzy control is a fuzzified PD or PI control action used in classical control theory. There are many studies on fuzzy dynamic programming. But it is hard to apply fuzzy dynamic programming to control , because it is basically considered as a feedforward control. Since fuzzy control rules can be chosen more freely in order to achieve a higher goal, it should be possible to design fuzzy control system which has both advantages of classical control theory and modern control theory. This paper proposes the new control strategy that the fuzzy optimal control rules are selected when the state variables are apart from the goal and it is switched to ordinary fuzzy PD control rules automatically near the goal in order to solve the problem of stability and off-set. For the optimal control, phase plane is digitized and the optimal output of the controller is obtained by applying fuzzy dynamic programming which has linguistically explained goals and constraints to each state on the phase plane. Quasi-optimal feedback control rules are obtained by describing the optimal output of controller for each state as fuzzy rules whose inputs are state variables. The effectiveness of the system designed with this method is verified by simulations and it was also found that the characteristic of the system is similar to that of fuzzy sliding mode control.

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In self-learning, most of the learners improve gradually their understanding through trial-and-error learning process. It is difficult for the traditional frame type CAL (Computer Assisted Learning) to assist a learner in his trial-and-error learning due to lack of its assisting ability. In this paper, the authors design the trial-and-error learning model for a frame type CAL with fuzzy grades of learner's understanding in fuzzy set theory. Our CAL model fulfils successful assist in trial-and-error learning.

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Structural modeling of learning tasks is an important process of designing classes. Designing classes is divided into three phases. The first phase is to determine a structural model which shows how the learning tasks are related. This is called the structural modeling of learning tasks. The second phase is to determine the sequence of learning tasks using the structural model obtained in the first phase. The third phase is to formulate an actual curriculum using the sequence decided on. In this paper, a method is proposed to obtain the structural model of learning tasks when a number of teachers plan the lessons. Let L_p be the set of learning tasks. Each teacher evaluates numerically the difficulty of each learning task. Structural model, that is, pseudo-ordered relation R on L_p, of each teacher can be calculated by applying a fuzzy implication function to the data. Teachers are assigned into clusters based on the evaluation for each learning task. Finally, FISM procedure is introduced to develop a consensus model between clusters that will help to determine the best curriculum structure.

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In order to reduce the workload of designers and to help them stimulate ideas, several support systems for car styling design have been investigated. These systems are aimed at changing the designer's thinking process into objective models and then reasoning other cases from this model. However, many of these models have two problems; firstly, variables of the formative styling elements combine subjective scales and measurable scales, and secondly the reasoning process lacks the flexibility of that of the designer. In this paper, therefore, in order to solve these problems, the authors propose a more flexible model which improves the formative element variables, changing these variables from measurable scales to subjective scales by the use of fuzzy sets. Additionally, by the use of fuzzy inverse reasoning, broad styling solutions can be produced. A problem of the design process is that the fuzzy relational matrix contains ambiguities, and therefore a reasoning method using a neural network was introduced into the system. The process of evolving the styling from conceptual images of cars was used as a sample study in order to demonstrate the design support system developed in this paper, and by the use of practical data, the efficiency of this model was confirmed.

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