Relaxed LMI-based stability conditions for fuzzy-model-based control systems under imperfect premise matching: Approximated membership function approach

This paper investigates the stability of the fuzzy-model-based control systems under imperfect premise matching that both fuzzy model and fuzzy controller are not required to share the same number of fuzzy rules and the same set of premise membership functions. Under the case of imperfect premise matching, it allows a greater design flexibility for fuzzy controller and is able to lower the implementation complexity when a less number of fuzzy rules and/or some simple membership functions are employed. However, due to the mismatch of the number of fuzzy rules and/or the membership functions, the existing analysis techniques with the parallel distributed compensation (PDC) cannot be applied to deal with the cross term of the membership functions and thus it leads to comparatively conservative stability conditions. In order to relax the stability conditions, we approximate the multiplication of the membership functions and the approximated membership functions exhibit some nice properties in favour of the stability analysis. Through the approximated membership functions, the information of the original membership functions is brought to the stability conditions. As a result, the proposed stability conditions are applied to a specified nonlinear plant characterized by the approximated membership functions rather than a family. A simulation example is given to demonstrate the effectiveness of the proposed approach.