Fuzzy two-term controllers with multi-fuzzy sets: mathematical models and analysis

This paper first presents mathematical models for fuzzy two-term (PI/PD) controllers
which employ N 1 (≥3) number of symmetric fuzzy sets for the input variable displacement,
N 2 (≥3) number of symmetric fuzzy sets for the input variable velocity and N 1 + N 2 – 1 number of
symmetric fuzzy sets for the output variable. These models are derived via triangular
membership functions for fuzzification of the inputs and output variables, linear control rules,
minimum/algebraic product triangular norm, different triangular conorms, different inference
methods and centre of sums (COS) defuzzification method. Properties of such models are
investigated. Using the well-known small-gain theorem bounded-input bounded-output (BIBO)
stability analysis of feedback systems involving fuzzy PD controller as a subsystem is presented.
Next, mathematical models with N 1 and N 2 number of asymmetric input fuzzy sets and
N 1 + N 2 – 1 number of asymmetric output fuzzy sets (both triangular and trapezoidal) are also
presented. Finally, some numerical examples along with their simulation results are included to
demonstrate the effectiveness of the fuzzy two-term controllers.