Sensor Fault Estimation via Polynomial Observers for T–S Fuzzy Caputo–Hadamard Fractional-Order Systems with Monotone Nonlinearities

In this paper, the issue of robust sensor fault estimation for Takagi–Sugeno (T–S) fuzzy systems with Caputo–Hadamard fractional-order dynamics subject to monotone nonlinearities is addressed. An adaptive observer is designed for the joint estimation of the system state and a globally constant sensor bias fault. The Caputo–Hadamard operator is used to handle logarithmic memory effects, and the T–S fuzzy representation is used for multi-regime nonlinear dynamics through a convex interpolation structure. Sufficient linear matrix inequality (LMI) conditions are obtained to ensure generalized Mittag–Leffler stability of the augmented estimation error system under a constant-fault assumption, by combining a sector inequality for strongly monotone nonlinearities with a fractional Lyapunov approach. The stability conditions are directly posed in the decision variables and the observer gains are recovered through a standard change of variables. To broaden the engineering applicability of the result, a finite-horizon practical Mittag–Leffler stability theorem is also derived for absolutely-continuous time-varying sensor faults whose Caputo–Hadamard derivative is bounded on the operating horizon [t0,T], in which the augmented estimation error remains in a residual ball whose radius is proportional to that bound. An alternative design, called a polynomial gain-scheduled observer, is also developed to reduce the conservatism of the constant-gain design, with observer gains given as polynomials of a measurable, fault-free scheduling vector. Quantitative root-mean-square performance metrics, LMI feasibility margins and an adaptation-gain sensitivity study are reported, and the polynomial matrix inequality is certified both by a dense grid check and by a sum-of-squares (SOS) feasibility argument so that the polynomial design is supported by a constructive certificate over the admissible scheduling set. Three numerical scenarios with fractional order 0.8 are provided: a strict constant-bias scenario that exactly validates the LMI theorem, a bounded-derivative ramp scenario that validates the practical Mittag–Leffler theorem, and a polynomial gain-scheduled scenario that validates the polynomial observer.