An online adjustment mechanism for membership functions of single input interval type-2 fuzzy PID controller

Abstract

The characteristics of the footprint of uncertainty (FOU) in interval type-2 membership functions (IT2-MFs) are crucial for the performance and robustness of interval type-2 fuzzy controllers (IT2 FCs). However, existing IT2-FC designs mostly use fixed FOU structures. This study proposes an online adjustment mechanism for membership functions of single input interval type-2 fuzzy PID controller (SIT2-FPID) by adjusting the footprint of uncertainty (FOU) and the weights of the antecedent and consequent membership functions (MFs) respectively to achieve high performance and robustness. The proposed online adjustment mechanism consists of two main parts: relative rate observer (RRO) and adjustment mechanism which has two inputs "error" and "normalized acceleration (Rv)", whereas the "normalized acceleration" provides relative information about the fastness or slowness of the system response. Meta-rules for the modification of the output of online adjustment mechanism (γ) are derived according to the error value and the relative information on the fastness or slowness of the system response and by analyzing the transient phase of the unit step response of the closed-loop system. The output of online adjustment mechanism (γ) in the proposed online tuning method is used as a tuning variable for the footprint of uncertainty (FOU) of the antecedent interval type-2 membership functions and the weights of the consequent crisp membership functions. This provides a dynamic membership functions (MFs) structure, where the heights of the Lower MFs (LMFs) or Upper MFs (UMFs) of each IT2 fuzzy set and the weights of the crisp output are defined as functions of the output of online adjustment mechanism (γ). By doing so, the method accomplishes the task of an online adjustment of the FOU and the weights of the antecedent and consequent membership functions respectively. The single input interval type-2 fuzzy PID controller (SIT2-FPID) with the proposed membership function adjustment mechanism was compared with the conventional PID controller and single input interval type-2 fuzzy PID controller with fixed membership functions through simulations. Throughout the simulation studies seven different performance measures are considered, three of them classical transient system response criteria: settling time (Ts), overshoot (%OS), and rise time (Tr) and the other performance measures are considered as: Integral Absolute Error (IAE), Integral Square Error (ISE), Integral Time Squared Error (ITSE) and Integral Time Absolute Error (ITAE). In addition, a step input and output disturbances have been employed to observe the disturbance rejection performance of the proposed method. The proposed online adjustment mechanism for membership functions method is demonstrated to be effective in linear and non-linear systems through simulations, and to be efficient in compensation of input and output disturbances in a short period of time.