Fractional order Thau-Luenberger observer for fractional order Takagi-Sugeno dynamical systems under uncertain nonmeasurable variables and unknown inputs

This paper proposed a novel state estimation framework for nonlinear systems described by fractional-order (FO) and Takagi-Sugeno (TS) fuzzy models, targeting critical challenges associated with large modeling uncertainties. The key innovation was in the development of two complementary observer structures that estimated system states when premise variables were either measurable or non-measurable, and in the presence of unknown inputs and uncertain dynamics. The proposed methodology addressed uncertainties affecting system matrices, input matrices, and unknown input transmission matrices. It introduced a fractional-order Thau-Luenberger observer (FO-TLO) for systems with measurable premise variables, and a dedicated observer adapted to the case of non-measurable premise variables. Both configurations utilized Lyapunov-based stability theory and linear matrix inequality (LMI) methods to ensure robust, asymptotic, and theoretically guaranteed convergence of the state estimation error in the presence of time-varying and bounded uncertainties. The framework extended the observer design to a broader class of FO-TS systems, and it offered effective tools for fault diagnosis, system monitoring, and robust control in the presence of uncertain environments.