Fault Detection for Discrete-Time Takagi-Sugeno Fuzzy Systems With Unmeasurable Premise Variable With <i>L</i>₂ – <i>L</i>∞/<i>H</i>∞ Mixed Observer and Zonotopic Analysis

For complex fuzzy nonlinear systems, the set membership estimation technique is often applied to fault detection or safety monitoring by giving a guaranteed estimation of the state. The main difficulty affecting the accuracy of existing set membership estimation methods for fuzzy system is the inability to obtain accurate model information online due to the unmeasurable premise variables. Therefore, a fault detection method based on membership function dependent (MFD) <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathcal {L}}_{2}-{\mathcal {L}}_{\infty }/{\mathcal {H}}_{\infty }$ </tex-math></inline-formula> mixed performance observer and zonotopic analysis is proposed for discrete fuzzy systems with unmeasurable premise variables. First, a novel MFD <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathcal {L}}_{2}-{{\mathcal {L}}}_{\infty }/{{\mathcal {H}}}_{\infty }$ </tex-math></inline-formula> performance is proposed, which reduces the conservatism of the traditional approach and provides more freedom to design. Second, on the basis of the proposed performance, the design conditions for the fault detection observer adopting the T-N-L structure are given using fuzzy basis-dependent Lyapunov functions, taking into account the effect of imprecise premise variables. Further, the effects caused by disturbances in the error dynamics of the observer as well as imprecise premise variables are handled using zonotopic analysis. The estimation results of the states and outputs in zonotopic and interval forms are given and applied to fault detection. Finally, the simulation shows that the proposed method provides guaranteed estimation in the absence of system faults and facilitates rapid fault detection when the system is faulty.