A Novel Finite-Frequency Optimization Fault Detection for Fuzzy Systems by Membership Functions Iterative Learning

This article presents the finite-frequency optimization fault detection (FD) strategy for Takagi-Sugeno (T-S) fuzzy systems. Under the imperfect premise matching (IPM) policy, a weighted fuzzy FD observer (WFFDO) with the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L_{\infty }/L_{2}$ </tex-math></inline-formula> robustness performance and the finite-frequency <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$ H_{-}$ </tex-math></inline-formula> fault sensitivity performance is first proposed, which signifies the residual signal is robust to the external interference and sensitive to potential faults. Some parameters and slack matrices are introduced to obtain more relaxed conditions of designing the WFFDO with mixed performance. Afterward, a new online membership functions (MFs) iterative learning algorithm with the exponential decay learning rate is proposed for the sake of updating the observer MFs in real-time such that optimal <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L_{\infty }/L_{2}$ </tex-math></inline-formula> performance can be achieved in this article. In addition, sufficient criterion is established so as to ensure the convergence of the structured mean squared error cost function by means of Lyapunov stability theory. Eventually, two simulation examples are given for illustrating the feasibility and superiority of the developed optimization FD technique.