Proportional integral observer design for T‐S fuzzy fractional‐order systems with multiple faults

Abstract This article proposes a fuzzy fractional‐order proportional integral observer (FFOPIO) design strategy for T‐S fuzzy fractional‐order systems. Compared with the previous studies, the T‐S fuzzy model is used to characterize nonlinear fractional‐order systems, which eliminates the constraint of the Lipschitz condition for general nonlinear fractional‐order systems. Meanwhile, the difficulty of applying fuzzy Lyapunov functions to analyze the stability of estimation errors is solved by introducing a complex fractional derivative rule. The application of fuzzy Lyapunov functions effectively reduces conservatism in observer design. Based on these, an FFOPIO is designed for the transformed system. Subsequently, by combining the fractional derivative rule and linear matrix inequality (LMI) technique, the stability criterion of observer synthesis is provided. It deserves mentioning that the FFOPIO achieves simultaneous estimation of system states and multiple faults. Finally, the validity of the FFOPIO is demonstrated through a fractional‐order permanent magnet synchronous motor (PMSM) model.