Realizing Interval Type-2 Fuzzy Systems with Type-1 Fuzzy Systems

In this chapter, the authors have realized Interval Type-2 Fuzzy Logic Systems (IT2 FLSs) with the average of two Type-1 Fuzzy Logic Systems (T1 FLSs). The authors have presented two case studies by applying this realization methodology on (i) an arbitrary system, where an IT2 FLS is considered, in which its footprint of uncertainty (FOU) is expressed using Principal T1 FS+FOU approach, and the second (ii) the Mackey-Glass time-series forecasting. In the second case study, T1 FLS is evolved using Particle Swarm Optimization (PSO) algorithm for the Mackey-Glass time-series data with added noise, and is then upgraded to IT2 FLS by adding FOU. Further, four experiments are conducted in this case study for four different noise levels. For each case study, a comparative study of the results of the average of two T1 FLSs and the corresponding IT2 FLS, obtained through computer simulations in MATLAB environment, is presented to demonstrate the effectiveness of the realization approach. Very low values of Mean Square Error (MSE) and Root Mean Square Error (RMSE) demonstrate that IT2 FLS performance is equivalent to the average of two T1 FLSs. This approach is helpful in the absence of the availability of development tools for T2 FLSs or because of complexity and difficulty in understanding T2 FLSs that makes the implementation difficult. It provides an easy route to the simulation/realization of IT2 FLSs and by following this approach, all existing tools/methodologies for the design, simulation and realization of T1 FLSs can be directly extended to T2 FLSs.