Methods from Fuzzy Logic since the end of the 80th were the sources for remarkable applications of computer modelling in fields which before looked essentially inaccessible. The main tool for that, the fuzzy controllers - a method of rule based rough modelling using fuzzy information - is presented in this book and investigated from a mathematical point of view. The basic notions from fuzzy set theory and many-valued logic are explained in detail, and a theory of fuzzy equations and systems of them is developed and applied to fuzzy controllers. The final chapter discussed methodological issues arising out of the process of developing and evaluating fuzzy models. Methoden der Fuzzy-Logik haben seit dem Ende der 80er Jahre zu bemerkenswerten Automatisierungslösungen in Bereichen geführt, die zuvor dem Computereinsatz weitgehend verschlossen schienen. Die dabei vor allem benutzten unscharfen Regler, eine Methode regelbasierter Grobmodellierungen mit Hilfe unscharfer Informationen, werden in diesem Buch dargestellt und mathematisch untersucht. Die dazu nötigen Grundlagen aus der Theorie der fuzzy sets und der mehrwertigen Logik werden ausgiebig erörtert, und es wird eine Theorie unscharfer Gleichungssysteme und ihrer Lösbarkeit entwickelt und auf unscharfe Regler angewendet. Ein Kapitel zu methodologischen Problemen der Bildung und Bewertung unscharfer Modelle beschließt das Werk, das als Standardwerk Theoretikern und Praktikern empfohlen ist.
Inhaltsverzeichnis
1 Logical Preliminaries. - 1. 1 Basic notions. - 1. 2 t-norms and ? -operators. - 1. 3 t-norm based connectives. - 2 Basic fuzzy set theory. - 2. 1 Set algebra for fuzzy sets. - 2. 2 Fuzzy relations. - 2. 3 The full image under a relation. - 2. 4 Special types of fuzzy relations. - 2. 5 Graded properties of fuzzy relations. - 3 Set equations with fuzzy sets. - 3. 1 Fuzzy equations and some of their applications. - 3. 2 Solvability of fuzzy relational equations. - 3. 3 Solvability of fuzzy arithmetical equations. - 3. 4 Solvability of systems of fuzzy equations. - 3. 5 Solvability degrees and approximate solutions. - 3. 6 Towards more difficult equations. - 4 Fuzzy controllers. - 4. 1 The construction of fuzzy controllers. - 4. 2 The problem of interaction. - 4. 3 Manipulation of fuzzy data. - 5 Methodological issues. - 5. 1 Comparison of fuzzy sets. - 5. 2 Approximate solutions. - 5. 3 Fuzzy equations for processing fuzzy data. - 5. 4 Evaluation of fuzzy models. - 5. 5 Controllability and predictability.
