340 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 10, NO. 3, JUNE 2002 Fuzzy Controllers With Conditionally Firing Rules Bernhard Moser and Mirko Navara Abstract—Mamdani controller was successfully used in many applications. One of its interpretations is that it uses a fuzzy rela- tion as an approximation of the desirable input–output correspon- dence. We analyze mathematical properties of Mamdani controller and notice that it has lower computational complexity when com- pared to the residuum-based controller. However, we show that in standard situations, both these fuzzy controllers do not repre- sent the rule base properly in the sense of finding a solution to the related system of fuzzy relational equations. First, we consider the premises and consequents as typical inputs and outputs and we want their correspondence to be kept. Second, we require that each normal input produces an output that bears nontrivial in- formation. These two conditions appear to be almost contradic- tory for the previous controllers. We suggest a generalization of Mamdani controller which allows us to satisfy these requirements. Theory and experiments suggest that it performs better without any change of rule base and without a substantial increase of com- plexity. Index Terms—Fuzzy control, fuzzy interpolation, fuzzy rela- tional equation. I. MOTIVATION T HE concept of approximate reasoning as it was conceived by Zadeh [12], [13] provides a framework which allows us to model and process vague linguistic information. The idea is to model linguistic terms by fuzzy sets, their (logical) relation- ship by fuzzy relations and their composition by the so-called compositional rule of inference [12]. As an important field of applications we refer to control processes for which linguistic information of a human expert about the required input–output behavior of the controller is available (for an introduction see, e.g., [4]). Let and denote the input and the output space, respec- tively. The spaces and are supposed to be convex sub- sets of finite-dimensional real vector spaces. Then usually the expert’s knowledge can be expressed by means of a rule base of if–then rules having the form where and are fuzzy subsets of , respectively. For a universe of discourse , let denote the set of all fuzzy subsets of . According to the paradigm of Manuscript received August 29, 2000; revised July 26, 2001 and October 23, 2001. This work was supported by the Czech Ministry of Education under Re- search Programme MSM 212300013 “Decision Making and Control in Manu- facturing,” Projects Aktion Österreich - Tschechien 16p12 and 23p16, and Grant 201/97/0437 of the Grant Agency of the Czech Republic. B. Moser is with CYBERhouse, GmbH, A-4020 Linz, Austria (e-mail: moser@cyberhouse.at). M. Navara is with the Center for Machine Perception, Department of Cyber- netics, Faculty of Electrical Engineering, Czech Technical University, CZ-166, Czech Republic (e-mail: navara@cmp.felk.cvut.cz). Publisher Item Identifier S 1063-6706(02)04834-8. approximate reasoning, the knowledge from the rule base can be represented by a fuzzy relation . Applying the compositional rule of inference to a fuzzy input and the relation , a fuzzy output is derived via (1) i.e., where is a t-norm modeling a fuzzy conjunction [3]. It is interesting to point out that the first successful prac- tical applications of fuzzy sets were realized by means of the so-called Mamdani (or Mamdani–Assilian) inference [9] which results from (1) for (2) We denote the induced mapping by . Its explicit expression is (3) Note that Mamdani’s approach is not fully coherent with the paradigm of approximate reasoning. For example, by employing a fuzzy conjunction instead of a fuzzy implication, Mamdani ’s approach does not represent the logical meaning of a linguistic if–then rule. A fuzzy set is called convex if all its -cuts are convex sets. For fuzzy subsets of the same universe, , their convex hull is the smallest (w.r.t. the pointwise ordering) convex fuzzy set satisfying for all , . A fuzzy set is called normal if it attains the value 1 at some point. The support of a fuzzy set is . For , let denote a Singleton, i.e., if if for all . In this paper, we are interested in Mamdani’s formula and other inference methods as fuzzy interpolation with respect to the rule base . The fol- lowing axioms will be considered. Int1) If the input coincides with one of the premises, then the resulting output coincides with the corresponding consequent, i.e., . Int2) For each normal input , the output is not contained in all consequents, i.e., there is an index with . Int3) The output belongs to the convex hull of , , where . 1063-6706/02$17.00 © 2002 IEEE