This paper presents a state and fault observer design for a class of Takagi-Sugeno implicit models (TSIMs) with unmeasurable premise variables satisfying the Lipschitz constraints. The fault variable is constituted by the actuator and sensor faults. The actuator fault affects the state and the sensor fault affects the output of the system. The approach is based on the separation between dynamic and static relations in the TSIM. Firstly, the method begins by decomposing the dynamic equations of the algebraic equations. Secondly, the fuzzy observer design that satisfies the Lipschitz conditions and permits to estimate simultaneously the unknown states, actuator and sensor faults is developed. The aim of this approach for the observer design is to construct an augmented model where the fault variable is added to the state vector. The exponential convergence of the state estimation error is studied by using the Lyapunov theory and the stability condition is given in term of only one linear matrix inequality (LMI). Finally, numerical simulation results are given to highlight the performances of the proposed method by using a TSIM of a single-link flexible joint robot.
In this study, we develop a fuzzy observer for a class of discrete-time nonlinear implicit models that are described by the Takagi-Sugeno structure and affected by actuator and sensor faults with unmeasurable premise variables satisfying the Lipschitz constraints. This study is based on separating dynamic and static equations in discrete-time Takagi-Sugeno implicit models. The design of a fuzzy observer is proposed to estimate unknown states, actuators, and sensor faults simultaneously. It is designed by considering the fault variables constituted by the actuator and sensor faults as auxiliary state variables. The observer gain is calculated by studying the exponential convergence of the state estimation error using the Lyapunov theory and the stability condition given as a linear matrix inequality. Simulation results demonstrated the effectiveness and validity of the proposed method.
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