One-dimensional input space modelling of a simplified general type-2 Mamdani and Takagi-Sugeno Fuzzy Proportional Integral Derivative controller

A wide variety of fuzzy controllers have evolved over several decades. Several mathematical models of type-1 and interval type-2 fuzzy controllers have been explored. Most of these modelling approaches involved two/three- dimensional input space. In this work, we have presented a simplified modelling approach fora General Type-2 (GT2) Mamdani and Takagi-Sugeno (TS) fuzzy Proportional Integral Derivative (PID) controllers involving input space of one-dimension. The fuzzy PID controller's structure is the parallel combination of the Fuzzy Proportional (FP) plus the Fuzzy Integral (FI) plus the Fuzzy Derivative (FD) control actions. Having a parallel PID control structure simplifies the fuzzy 'IF - THEN' rules by eliminating the role of 'AND' (triangular norms) and 'OR' (triangular co-norms) operators. This decoupled rule base aids in decreasing the computing complexity of the GT2 fuzzy PID controller. Owing to the one-dimensional input space, the number of tuneable parameters for fuzzy controllers reduces significantly when compared to twoor three-dimensional input spaces. It is also demonstrated that the type-1 (T1) and interval type-2 (IT2) fuzzy controllers are variations of the GT2 fuzzy controller. In order to assess the controller models, we simulate two systems: the unstable first-order system with dead time and the Continuously Stirred Tank Reactor (CSTR). These models, nevertheless, may also be applied to other dynamic processes and systems.