Fault detection for T–S fuzzy systems with unmeasurable premise variables based on a two-step interval estimation method

This paper proposes a fault detection strategy based on a two-step interval estimation method for T–S fuzzy systems with unmeasurable premise variables. First, an $L_{\infty }$ observer is designed to achieve robust point estimation under Lipschitz conditions. Then, the estimated error bounds are analyzed and optimized using the $L_{\infty }$ performance conditions to enable interval estimation. Furthermore, the residual threshold is derived from the interval estimation to achieve robust fault detection. Finally, an activated sludge process in a wastewater treatment is considered to validate the proposed method. Simulation results demonstrate that the proposed approach can provide more accurate state interval estimation and outperforms standard $L_{\infty }$ observer design methods in addressing fault detection problems compared with existing methods.