Effects of Increasing the Footprints of Uncertainty on Analytical Structure of the Classes of Interval Type-2 Mamdani and TS Fuzzy Controllers

The fundamental difference between an interval type-2 (IT2) fuzzy controller and a type-1 fuzzy controller is the footprint of uncertainty (FOU) of an IT2 fuzzy set. In this paper, we study how FOUs affect the analytical structure (i.e., the input-output mathematical relationship) of a broad class of IT2 Mamdani and takagi-sugeno (TS) controllers. The controllers employ arbitrary fuzzy rules, the Karnik-Mendel (KM) or Enhanced KM type-reducer, the minimum AND operator, and the centroid defuzzifier. The controllers utilize commonly used IT2 fuzzy sets for their input variables and any kind of type-2 fuzzy sets for their output variable. We prove that, with increase of FOUs of the input fuzzy sets, the Mamdani controllers approach constant controllers, and the TS controllers approach piecewise linear controllers. The resemblance to the constant or pricewise linear controllers increases as the FOUs increase. When all the FOUs are at their maximum (to reflect the highest level of uncertainties), the Mamdani and TS controllers become the constant controllers and piecewise linear controllers, respectively. We investigate how change in the resemblance takes place progressively as FOUs increase. We also show that an increase in the resemblance narrows control gain variations for part of the IT2 controllers, which can worsen control performance. These findings implicit controller design-too large FOUs are generally undesirable for the input fuzzy sets because they can make an IT2 controller behave like a constant or piecewise linear controller. Real-time control experiment results are provided to illustrate the theoretical analysis.

I. Introduction

Developed from interval type-2 (IT2) fuzzy logic [13], the IT2 fuzzy control is a relatively new field of interest and research because IT2 fuzzy controllers [5], [7], [10], [11] have the potential to outperform their type-1 (T1) counterparts [1], [2], [4], [6], [9], [22], [30]. An IT2 controller uses IT2 fuzzy sets for its input and/or output variables. Hence, the IT2 fuzzy set is the fundamental factor differentiating an IT2 controller from a T1 controller. A unique feature of an IT2 fuzzy set is a footprint of uncertainty (FOU) that is used to characterize additional uncertainties beyond what a T1 fuzzy set can capture [15], [16], [18], [20], [22], [30].