Specification and optimization of fuzzy systems using convolution techniques. (Q1575334)

Fuzzy logic is now widely accepted as a formal tool for describing control or decision-making processes that are based on incomplete, vague or uncertain information. This paper presents a new approach for the description and efficient computation of fuzzy rules based on fuzzy logic and also on convolution techniques. The fuzzy rules \(R_{r}\) of the MISO considered are an extension of the zero-order Takagi-Sugeno fuzzy model and are given in the form of \(R_{r}\): If \(X_{1}\) is \(A_{r1}\) and \(\ldots\) and \(X_{N}\) is \(A_{rN}\) then \(z\) is \(c_{r}\), where \(X_{j}\) are fuzzified input variables, \(A_{rj}\) are fuzzy numbers which belong to the corresponding partition of unity \(A_{rj}\) and \(c_{r}\) is a nonfuzzy singleton term of the output variable \(z\). The global consideration of the fuzziness (imprecision) of each input \(X_{j}\) and the fuzziness (vagueness) of the corresponding fuzzy partition \(A_{rj}\) greatly simplifies the corresponding specification process and the involved matching computation. Two general fuzzy partitions are introduced: F-splines and \(\phi\)-splines. Both capture the fuzziness of input variables and fuzzy terms considered, as well the smoothness constraints of outputs. As an application, the control of an inverted pendulum is analyzed.