Observer Design for Takagi–Sugeno Fuzzy Systems with Unmeasurable Premise Variables Based on Differential Mean Value Theorem

In this work, we present the design of an observer for Takagi–Sugeno fuzzy systems with unmeasurable premise variables. Moving away from Lipschitz-based and L2 attenuation-based methods—which fall short in eliminating the mismatching terms in the estimation error dynamics—we leverage the differential mean value theorem. This approach not only removes these terms but also streamlines the factorization of the estimation error dynamics, making it directly proportional to the estimation error. To ensure the asymptotic convergence of the estimation error, we apply the second Lyapunov theorem, which provides sufficient stability conditions described as linear matrix inequalities. A numerical example applied on a three-tank hydraulic system is presented to demonstrate the observer’s effectiveness.