Stability of interpolative fuzzy KH controllers

The classical approaches in fuzzy control (Zadeh and Mamdani) deal with dense rule bases. When this is not the case, i.e. in sparse rule bases, one has to choose another method. Fuzzy rule interpolation [proposed first by Kóczy and Hirota, Internat. J. Approx. Reason. 9 (1993) 197-225] offers a possibility to construct fuzzy controllers (KH controllers) under such conditions. The main result of this paper shows that the KH interpolation method is stable. It also contributes to the application oriented use of Balázs-Shepard interpolation operators investigated extensively by researchers in approximation theory. The numerical analysis aspect of the result contributes to the well-known problem of finding a stable interpolation method in the following sense. We would like to approximate a function that is only known at distinct points (e.g. where it can be measured). Since measurement errors cannot be eliminated in practice, it is a natural requirement that the interpolation method should be stable independently from the selection of measurement points. The classical interpolation methods generally do not fulfil this condition, only with certain strong restrictions concerning the measurement points. If we neglect the classical form of the approximating function (polynomial and trigonometrical), we can produce better behaving approximation. The KH controller approximating function, being a simple fractional function, fulfils the stability condition. This can also be interpreted as KH controllers are universal approximators in the space of continuous functions (with respect to, e.g., the supremum norm).