This study introduces the notion of interval of confidence (IoC) for the modeling of fuzzy controllers and presents three novel mathematical models of interval type-2 fuzzy proportional-integral-derivative (IT2FPID) controllers by utilizing the concept of IoC and various types of footprints of uncertainty (FoUs) having parallelogram, bottom-wide triangle, and top-wide triangle shapes. The properties and computational aspects of the proposed controllers are analyzed mainly in terms of IoCs and FoUs. All the developed controllers with various types of FoUs are plant-model-free nonlinear controllers with variable gains and structures. It is proved that more generalized models of IT2FPID controllers can be attained by replacing the fuzzy singletons with IoCs. The suitability of introducing IoC in IT2FPID controllers is verified by showing that IoC does not play any role in the mathematical modeling of type-1 (T1) fuzzy PID controllers. To demonstrate the effectiveness of the proposed controllers, extensive simulation studies and numerical and hardware experiments are conducted. The simulation and experimental results affirm the proposed controllers' superiority over conventional and fuzzy PID ones, opening up new possibilities for improved performance in real-world control applications.