Interval type-2 polynomial fuzzy fault detection scheme with a multi-order homogeneous polynomial Lyapunov functions considering unmeasurable premise variables

A polynomial fuzzy fault detection scheme for sampled-output-measurements-based interval type-2 (IT2) polynomial-fuzzy-model-based (PFMB) systems is investigated in this paper, where the uncertainties in the premise variables (PVs) and membership functions (MFs) are described by IT2 fuzzy sets. Fully or partially unmeasurable PVs cause the parameter matrices of the polynomial fuzzy fault detection observer (PFFDO) to rely on the estimated states and the corresponding mismatching problems are further considered. Lyapunov stability theory is carried out with a novel multi-order homogenous polynomial Lyapunov functions (MHPLF) to introduce more information of the states when eliminating the partial derivatives, and the time-delays introduced by sampled-output measurements are handled by L-K functions. Unlike the membership-function-independent (MFI) approaches, the membership-function-dependent (MFD) approaches carry the information of the MFs for the relaxation of the stability constraints. Corresponding stable constraints in sum-of-squares (SOS) form are given to hold the asymptotic stability of the fault detection system with H∞ performance γ. A numerical example with many cases illustrates the effectiveness of the proposed techniques in uncertainty handling and conservativeness reduction, while an inverted pendulum example verifies the feasibility of the method on physical systems.