Lyapunov-Based Partial Practical Generalized Mittag–Leffler Stability and Polynomial Fuzzy Observer Design for Tempered Fractional-Order Systems

This paper develops a Lyapunov-based framework for the analysis of partial practical stability in tempered fractional-order systems described by the tempered ζ-Caputo derivative. In contrast with classical full-state stability concepts, the proposed approach focuses on practical generalized Mittag–Leffler stability with respect to a prescribed component of the state vector. Sufficient conditions are established in terms of a Lyapunov function and suitable comparison estimates, leading to a notion of practical uniform generalized Mittag–Leffler stability for substates. The theoretical results are then applied to the observer design problem for a class of tempered fractional-order polynomial fuzzy systems. By exploiting a block decomposition of the estimation error and using polynomial fuzzy observer gains, new sufficient conditions are derived to guarantee practical generalized Mittag–Leffler stability of the estimation error with respect to its first component. The observer synthesis is formulated through polynomial matrix inequalities and sum-of-squares conditions on an admissible operating region. A numerical example involving a 3-dimensional tempered fractional-order polynomial fuzzy system is provided to illustrate the proposed methodology. The simulations confirm the effectiveness of the developed observer and show that the error component of interest converges inside a small neighborhood of the origin in agreement with the theoretical estimates. Quantitatively, the first error component is driven to |e1(10)|≈1.22×10−3, two orders of magnitude below the theoretical envelope (1.07×10−1), while the kernel-boundedness diagnostic remains finite (supςK(ς)≈1.59). In contrast with existing full-state Mittag–Leffler observer designs for fractional and tempered fuzzy systems, the proposed partial formulation certifies convergence of only the substate of interest and therefore requires the synthesis of a single observer-gain block. The obtained results contribute to the stability theory of tempered fractional-order systems and provide a constructive framework for partial-state observer design in nonlinear fuzzy settings.