During my time as a research assistant in the Laboratory for Control Theory and Robotics at the Technische Universität Darmstadt, I developed a theory for recurrent fuzzy systems. But what are fuzzy systems and especially recurrent fuzzy systems? A vague answer to this question is provided below. Some more information can be found in the two short popular scientific articles "Rekurrente Fuzzy-Systeme" and "Modellierung ökologischer Systeme mit rekurrenten Fuzzy-Systemen" (in German). For a deep and broad discussion of recurrent fuzzy systems refer to my German Ph.D. thesis and the English journal articles, which are listed in the Publications section.

## What are Fuzzy Systems?

In the classical logic, as it is used as a fundamental basis of mathematics, there are only two categories: "true" or "false". There is no "in between". In every days life however, this hard discrimination in these categories are often not respected or are even not appropriate. The transitions between "true" and "false" are often blurred and fuzzy. E.g., if you want to describe at which temperatures the tab water for a bath is pleasant, you would not heartily affirm this for 24 degrees centigrade and, at the same time, downright negate it for a temperature of 23.9 degrees centigrade. While 26 degrees centigrade are certainly pleasant and 18 degrees centigrade are unpleasant, there is a gradual transition between these two ratings.

Logic may be described by numbers. Normally, true statement are assigned a truth value of one and false statements are assigned a truth value of zero. When using fuzzy logic, the fuzzy transitions between true and false statements are described by functions that may not only take on the (truth) values of one oder zero, but all values in between.

The big advantage in assigning truth values to statements is to be able to systematically derive the truth values of relations and conclusions containing those statements without referring to the nature of the statement themselves. E.g., if the two statements "The water temperature is pleasant" and "If the water temperature is pleasant, I can stay long in the bathtub" are both true (i.e. have a truth value of one), one may derive the the conclusion "I can stay long in the bathtub" is also true (i.e. has also a truth value of one). There is an analogous mechanism for the fuzzy logic as well. A mathematical description for determining the truth values of conclusions from the truth values of the premises of rules are called fuzzy systems. E.g., if I rate the water temperature only as partially pleasant (i.e. with a truth value of 0.8), as the temperature is already a little low, I may regard the conclusion of the rule above also as only partially true; well, I won't stay in the tub that long.

This shallow example of the tub water temperature is there to give a vague feeling for the mechanisms of fuzzy systems. The property of fuzzy logic and fuzzy systems to handle fuzzy terms, such as "pleasant", "long", "many", "fast", etc., enables us to mathematically evaluate vague rules containing such fuzzy terms. With such a mathematically description of rules, it is possible to use the knowledge contained in such rules also in technical devices. E.g., for automatic quality assurance, control of machines, diagnosis of faults in machines or evaluation of data series.

## What are Recurrent Fuzzy Systems?

The fuzzy systems sketched above are normally restricted to rules that describe a static relationship between the objects in the premise and those in the conclusion. However, rules may also describe the change of the value of an object. E.g., the change in the water temperature may be described by rules such as the following: "If the tub water is to cold and I add hot water, then the tub water will be pleasant." This change in the value of the water temperature is not static, but dynamic, as the premise of the relevant rule changes with the evaluation of the rule. Some time later, the appropriate rule may be: "If the tub water is pleasant and I add hot water, then the tub water will be too hot." This additional temporal component changes the (static) fuzzy system to a (dynamic) recurrent fuzzy system. The attribute "recurrent" points to the fact that rules are recurrently evaluated with new emerging values for the premises.

There is a general question that needs to be answered when using fuzzy systems: Does the mathematical model represent the linguistical knowledge adequately? If this is not the case the rules have normally to be changed slightly. In the case of recurrent fuzzy systems, not only a single evaluation of the rules has to match the linguistic interpretation of the rules, but the whole temporal evolution of the system has to. In order to be able to test this matching of a liguistical model and its recurrent fuzzy system, the dependencies of the rules and the temporal evaluation of the system has to be investigated. The first steps in this direction can be found in my Ph.D.thesis (ref. Publications).

## Applications for Recurrent Fuzzy Systems

Recurrent fuzzy systems may not only be investigated rigorously as a nice mathematical modelling technique and be used to develop a theory for their temporal behaviour. No, recurrent fuzzy systems may also be used as a problem solver in various ways. With the help of some students, I set up a recurrent fuzzy system to model the behaviour of car drivers on a motorway. The accuracy of the model, who's rules were set up entirely subjective, is astounding. The following movie (1.6 MB) gives an idea of the model behaviour. It shows car drivers following mathematically interpreted linguistical rules to take over a breakdown car, which is symbolised by a fivepointed star. The different driver characters are depicted by different symbols and the speed is decoded by the colour of the symbols:

Recurrent fuzzy systems are also used in industry: As a fault diagnosis system in continuous casting plants. They evaluate measured data series. From the shape of the signal it derives a danger level for the occurrence of a certain error. The danger level on the other hand influences the interpretation of the measured data. Thus the danger level does not only occur in the consequence but also in the premises of the rules and its new value is recurrently reused to evaluate the signal. By this means, the measured signal is investigated for a specific series of features that is characteristic for the occurrence of a dangerous fault in the machine.