Stability analysis in state space for non-driven MEMS gyro

The non-driven MEMS gyro is a new kind of micromechanical vibratory gyro, which has no a driving structure itself. The gyro is installed on a rotating aircraft and utilizes the spinning of the carrier to obtain an angular momentum. When the carrier produces a transverse rotation, a periodic Coriolis force acts on the sensitive mass of the MEMS gyro to sense the transverse input angular velocity of the rotating carrier. In applications, we found that the MEMS gyro is subjected to a high shock when the carrier begins to launch. If the sensitive mass cannot return to balance, the gyro will not work properly. So the stability of the gyro is the key issue on whether it can properly work. In this paper, we have analyzed the stability of the MEMS gyro in details by using Lyapunov stability principle for the first time. Firstly, based on the designed structural principle of the MEMS gyro, by using Euler dynamic equation of a rigid body rotating around a fixed point, we have described the angular vibration of the sensitive mass of the gyro and obtained its motion equation. The motion is the second order system. Then, we have chosen an appropriate state vector and established a state space model in state space for describing the motion of the sensitive mass. In order to research the stability of the designed MEMS gyro by using Lyapunov stability principle, a Lyapunov function needs to be found. Therefore, we have built a quadratic function and proved that its Lyapunov matrix equation has a solution. The matrix solution is symmetric and positive definite. Thus, the found quadratic function is a Lyapunov function. According to Lyapunov stability principle, the designed MEMS gyro is asymptotically stable. Next, utilizing numerical calculation, we have done the simulation of the unit-impulse response. The response curve has shown that the system of the designed MEMS gyro can come back to the balance after 160 ms. Finally, for further verification, the MEMS gyro is fixed on the shock table to test. The shock wave is a half-sine with the strength of 60 g and the impulse width of 80 ms. The tested result has demonstrated that the output signal of the designed MEMS gyro can again come back to zero state position after 150 ms under shock disturbance.