Systematic design of a stable type-2 fuzzy logic controller

The concept of a type-2 fuzzy set was introduced by Prof. Zadeh, as an extension of the concept of an ordinary fuzzy set. A Fuzzy Logic System (FLS) described using at least one type-2 fuzzy set is called a type-2 FLS. Type-1 FLSs are unable to directly handle rule uncertainties, because they use type-1 fuzzy sets that are certain. On the other hand, type-2 FLSs, are very useful in circumstances where it is difficult to determine an exact membership function and the measurement of uncertainties is difficult or even impossible. Similar to a type-1 FLS, a type-2 FLS includes type-2 fuzzyfier, rule-base, inference engine and substitutes the defuzzifier by the output processor. The output processor includes a type-reducer and a type-2 defuzzyfier; it generates a type-1 fuzzy set output (from the type reducer) or a crisp number (from the defuzzyfier). A type-2 FLS is again characterized by IF-THEN rules, but its antecedent and/or consequent fuzzy sets are now of type-2. Type-2 FLSs, can be used when the circumstances are too uncertain to determine exact membership grades. For our description we are going to consider the well-known problem of designing a stabilizing controller for the inverted pendulum system. The state-variables are θ = 1 x the pendulum’s angle, and θ = 2 x its angular velocity. The system’s actual dynamical equation, which we will assume unknown, are shown in (1)-(3): u x x g x x f x x x ) , ( ) , ( 2 1 2 1 2 2 1 + = = (1)