The simplest fuzzy PID controllers: mathematical models and stability analysis

Abstract

This paper reveals mathematical models for the simplest fuzzy PID controllers which employ two fuzzy sets for each of the three input variables and four fuzzy sets for the output variable. Mathematical models are derived via left and right trapezoidal membership functions for each input, singleton or triangular membership functions for output, algebraic product triangular norm, different combinations of triangular co-norms and inference methods, and center of sums (COS) defuzzification method. Properties of these structures are studied to examine their suitability for control application. For the structure which is suitable for control, bounded-input bounded-output (BIBO) stability proof is presented. An approach to design fuzzy PID controllers is given. Finally, some numerical examples along with their simulation results are included to demonstrate the effectiveness of the simplest fuzzy PID controllers.

Fuzzy control of a heat exchanger

“…• use a rule base that covers all possible combinations of fuzzy sets for the two input variables [2,3,4,7,9].…”

“…Design procedure for a linear fuzzy system Having designed the conventional discrete time controller, an equivalent FLC for the plant can be obtained, by replacing in equation (5) the PD component (6) with a linear fuzzy system [3,4,9].…”

Gain Scheduling of PID Controller Based on Fuzzy Systems

“…proportional gain, integral time constant and derivative time constant [1]. Due to their linear structure, the conventional PID controllers are usually not effective if the processes involved are higher order and time-delay systems, nonlinear systems, complex and vague systems without precise mathematical models, and systems with uncertainties [2]. So for the overall improved performance, fuzzy PID controllers are preferred.…”

Modelling of the simplest fuzzy PID controller of Takagi-Sugeno type using algebraic product t-norm and Maximum t-co-norm

“…Some analytical structures for Mamdani type PID controllers [3][4][5][6]8,9] are available, but analytical structures for TS type PID controllers [7,11] are rather limited. Some design methods of Mamdani type [10] and TS type [13] PID controllers have been reported.…”