Membership function modification of fuzzy logic controllers with histogram equalization

In most fuzzy logic controllers (FLCs), initial membership functions (MFs) are normally laid evenly all across the universes of discourse (UD) that represent fuzzy control inputs. However, for evenly distributed MFs, there exists a potential problem that may adversely affect the control performance; that is, if the actual inputs are not equally distributed, but instead concentrate within a certain interval that is only part of the entire input area, this will result in two negative effects. On one hand, the MFs staying in the dense-input area will not be sufficient to react precisely to the inputs, because these inputs are too close to each other compared to the MFs in this area. The same fuzzy control output could be triggered for several different inputs. On the other hand, some of the MFs assigned for the sparse-input area are “wasted”. In this paper we argue that, if we arrange the placement of these MFs according to a statistical study of feedback errors in a closed-loop system, we can expect a better control performance. To this end, we introduce a new mechanism to modify the evenly distributed MFs with the help of a technique termed histogram equalization. The histogram of the errors is actually the spatial distribution of real-time errors of the control system. To illustrate the proposed MF modification approach, a computer simulation of a simple system that has a known mathematical model is first analyzed, leading to our understanding of how this histogram-based modification mechanism functions. We then apply this method to an experimental laser tracking system to demonstrate that in real-world applications, a better control performance can he obtained by using this proposed technique