Identification of fuzzy relational models from unevenly distributed data using optimization methods

A wide range of algorithms for Fuzzy Relational Model (FRM) identification from process data have been developed over the last decade. It has been shown that the use of multi-variable optimization methods to establish the strengths of fuzzy relationships results in accurate models. Accuracy means here, however, the fit between training data and model predictions. If the available training data is unevenly distributed then optimization results in good model fit in regions where training data is clustered, but the regions where training data is sparse are less well modelled. The problem is frequently encountered as little influence is often possible on how training data is acquired. In this paper identification methods are proposed which take into account the distribution of training data. To enhance the model quality over the entire model range the input space is divided into regions in which the model error is considered separately during optimization. It is shown that the division of the input space by fuzzy partitioning is not sufficient. Problems which can occur due to unevenly distributed data are discussed and, based on the fuzzy partitioning, a new division of the input space is derived. These new input regions are described using rectangular or triangular fuzzy sets. It is shown that both shapes of fuzzy regions result in FRM with better predictive capabilities on unseen process data. An example for the application of the FRM in a Model Predictive Controller (MPC) is also given