Stability bounds for fuzzy estimation and control – Part I: State estimation

Analysis and observer design for nonlinear systems have long been investigated, but no generally applicable methods exist as yet. A large class of nonlinear systems can be well approximated by Takagi-Sugeno fuzzy models, for which methods and algorithms have been developed to analyze their stability and to design observers. However, results obtained for Takagi-Sugeno fuzzy models are in general not directly applicable to the original nonlinear system. In this paper, we investigate what conclusions can be drawn and what guarantees can be expected when an observer is designed based on an approximate fuzzy model and applied to the original nonlinear system. It is shown that in general, exponential stability of the estimation error dynamics cannot be obtained. However, the estimation error is bounded. This bound is computed based on the approximation error and the Lyapunov function used. The results are illustrated using simulation examples.