Fuzzy logic models with improved accuracy and continuous differentiability

One method of obtaining a fuzzy logic model (FLM) for use in process optimization and control involves the fuzzy weighting of many simple, often linear, functions. The method of identifying the fuzzy weighting scheme typically makes use of triangular membership functions and the minimum operator, both of which result in a model with discontinuous derivatives. Alternate methods of model identification that make use of continuously differentiable membership functions and the product operator will be discussed here. These methods result in models that have improved predictive capabilities for smooth, complex processes typical of those found in the semiconductor manufacturing industry. In addition, with continuously differentiable membership functions and the product operator, the FLM will have continuous derivatives. As a result, the operation of algorithms that rely on derivatives obtained from the model will be enhanced. Three types of FLM´s and an artificial neural network (ANN) model were obtained for eight different sets of data representing three processes, and the resulting models were compared. The predictive capability of each model was estimated by computing the multiple correlation coefficient, R_predict², over a large number of points distributed throughout the input space. For one of the most complex examples studied, a simulated six-input chemical vapor deposition process, the traditional FLM using the minimum operator and triangular membership functions had R_predict²=-0.43, indicating very poor predictive capability. The fuzzy logic model using the product operator and triangular membership functions had R_predict²=0.87. The fuzzy logic model using the product operator and piece wise defined quadratic membership functions with continuous first derivatives had R_predict²=0.92. The ANN model for the same data had R_predict²=0.73. Similar results were observed for each example presented, indicating appropriately defined FLM´s can meet or exceed the modeling capabilities of artificial neural networks for nonlinear, multi-dimensional problems