On comparing non-singleton type-1 and singleton type-2 fuzzy controllers for a nonlinear servo system

Uncertainty handling is a major issue for the control of real-world systems. Traditional singleton type-1 Fuzzy Logic Controllers (FLCs) with crisp inputs and precise fuzzy sets cannot fully cope with the high levels of uncertainties present in real world environments (e.g. sensor noise, environmental impacts, etc.). While non-singleton type-1 fuzzy systems can provide an additional degree of freedom through non-singleton fuzzification of the inputs, it is unclear how this capability relates to singleton type-1 and specifically interval type-2 FLCs in terms of control performance (also because the application of non-singleton type-1 FLCs is quite rare in the literature). In recent years interval type-2 FLCs employing type-2 fuzzy sets with a Footprint of Uncertainty (FOU) have become increasingly popular. This FOU provides an additional degree of freedom that can enable type-2 FLCs to handle the uncertainties associated with the inputs and the outputs of the FLCs. One of the main criticisms of singleton type-2 FLCs is that they outperform (the usually singleton-) type-1 FLCs because they - respectively their type-2 fuzzy sets, employ extra parameters, thus making improved performance an obvious result. In order to address this criticism, we have implemented a non-singleton type-1 FLC which allows a more direct comparison between the non-singleton type-1 FLC and singleton interval type-2 FLC as the number of parameters for both controllers is very similar. The paper details the implementation details of the FLCs for the application of a nonlinear servo system and provides the experimental simulation results which were performed to study the effect of increasing levels of uncertainty (in the form of input noise) and the capability of the individual FLCs to cope with them. We conclude by providing our interpretation of the results and highlighting the essential differences in the uncertainty handling between the (non-) singleton type-1 and singleton interval type-2 FLCs.