A robust stabilizing controller for a class of fuzzy systems

This paper discusses the construction of a robust stabilizing controller for a Takagi-Sugeno class of fuzzy systems. It should be clear at the outset that the fuzzy system is inherently uncertain due to the vagueness of the associated linguistic terms, even though crisp and useful results are obtained via inferencing and defuzzification. This vagueness uncertainty is the essential fuzziness of the system as introduced via the membership functions. However, the fuzzy system considered here is uncertain for two other reasons. One, the parameterized membership functions are not completely known but the bounds of the characterizing parameters are known. Two, the mathematical model itself is uncertain via parameters whose bounds are known. The problem is cast within a robust bounded parameter design which then allows analysis within that framework. The parameters of the membership functions considered are those that determine its spread, and therefore the fuzziness, in the universe of discourse. It is shown that by knowing the bounds of the parametric uncertainties in both the system model and the membership functions, and using a matching condition, the local feedback controller gains can be selected so that the overall control law stabilizes the system. Furthermore, it is shown that the resulting Lyapunov equation that needs to be solved for global stability is indeed dependent on the fuzzy firing order which has not been the case with previous results