Robust Observer Based on Fixed-Time Sliding Mode Control of Position/Velocity for a T-S Fuzzy MEMS Gyroscope

This study focused on a control system of the nonlinear micro-electro-mechanical systems (MEMS) gyroscope. First, sector nonlinearity was used to model a MEMS gyroscope in the Takagi-Sugeno (T-S) fuzzy system. Second, a state observer was designed based on linear matrix inequality (LMI) to identify the optimal eigenvalues of the state tracking error function. Then, full-state fixed-time sliding mode control (FTSMC) was constructed to control the system. Third, a case study of a harmonic disturbance observer was used to address the unknown disturbance of the system. A disturbance observer (DOB) was simply designed based on the error signals of the system outputs and observer outputs. The output signals precisely converged to the predefined trajectories in a very short time, with no overshoots and small of steady-state errors. Moreover, the estimated output states were precisely tracked by the system outputs. These important factors were used to confirm that the control of the T-S fuzzy MEMS was effective and easy to achieve. The study used MATLAB simulation to archive the verification. The maximum of tracking error was <inline-formula> <tex-math notation="LaTeX">$e_{4} \in [-4.657:5.565]\times 10^{-11}$ </tex-math></inline-formula>, and the maximum settling time was <inline-formula> <tex-math notation="LaTeX">$T_{e3} \sim 0.144$ </tex-math></inline-formula> for the error of the <inline-formula> <tex-math notation="LaTeX">$\dot {y}-$ </tex-math></inline-formula> axis and the settling time of the <inline-formula> <tex-math notation="LaTeX">$\dot {x}-$ </tex-math></inline-formula> axis, respectively.