Observer design for Takagi-Sugeno fuzzy systems with unmeasured premise variables: Conservatism reduction using line integral Lyapunov function.

In this paper, a non-quadratic Lyapunov function is employed to reduce conservatism in a nonlinear observer designed for a class of continuous-time Takagi-Sugeno fuzzy systems with unmeasurable premise variables. This structure presents greater challenges compared to systems with measured variables. To overcome this issue, we utilize the mean value theorem and the sector nonlinearity transformation to convert the nonlinear error dynamics into a linear parameter-varying system. Moreover, we introduce the line integral Lyapunov function, which based on the integral of the membership functions, in order to ensure the global stability of the fuzzy systems under consideration. The use of this function offers several notable advantages over conventional quadratic forms, including a reduction in conservatism. Additionally, this type of functions constructed in a manner that eliminates the need for generating time derivatives of the membership functions, thereby simplifying calculations and analysis in comparison to other nonquadratic functions. Furthermore, it also enables capturing the system's behavior along a trajectory. The stability conditions are more relaxed and expressed as linear matrix inequalities, which can be solved using a linear programming approach through specialized software tools. To validate the effectiveness of the proposed methodology, we conducted a hardware-in-the-loop test using a flexible joint robot. The obtained results clearly underscore the success of the proposed approach.