Ensuring zero steady state error in interval type-2 fuzzy PI control system with non-symmetrical fuzzy sets

Interval type-2 fuzzy logic controllers (IT2 FLCs) have been treated as a black box in most control applications because the input-output relationship is not fully understood. Reason is that the input-output mapping is very difficult to be expressed in closed-form, therefore, most fuzzy control designers use evolutionary computation methods such as genetic algorithm and big bang-big crunch optimisation to optimise the parameters for specific control applications. However, this is not the case for classical PI controllers, where the input-output mappings are always known. Apart from using optimisation method, symmetrical antecedent fuzzy sets are usually employed to construct IT2 FLCs. It will be interesting to see how the control performance can be further improved by introducing non-symmetrical fuzzy sets in one of the input domain. In this paper, we analyse an IT2 fuzzy PI control system with non-symmetrical IT2 antecedent sets, symmetrical consequent sets and a symmetrical rule base. Results show that, contrary to conventional knowledge, a fuzzy PI controller with non-symmetrical IT2 antecedent sets may not be able to ensure zero steady-state error. Thus, this paper presents guideline for designing IT2 fuzzy PI control system with the aforementioned structure that has zero steady state error. Two conditions and 4 inequality equations are derived to act as constraints on the firing strengths and fuzzy set parameters, thereby ensuring zero steady-state error when non-symmetrical fuzzy sets are introduced into one of the two input variable domains. Simulation results are presented to demonstrate that application of the proposed conditions ensure zero steady-state error in a fuzzy system with non-symmetrical fuzzy sets.