Nonlinear Observability Analysis of Micro-machined Electrostatic Actuators Using Self-Sensing

Micro-machined electrostatic actuators (MEA) like parallel plate actuators (PPAs) and comb drive actuators (GCAs) are commonly used in many applications, including gyroscopes, resonators and RF switches. The detection of an actuators' mechanical motion is desired when they are combined with feedback control techniques, especially when the application requires high performance or is affected by disturbances. The motion can be detected by a variety of sensing techniques, including capacitive, piezoresistive and optical. Electrostatic parallel plate actuators can be modeled as a type of variable capacitor, which depends on the gap between a fixed electrode and a movable electrode. Thus, the displacement of the actuator can be obtained by measuring the capacitance. However, this practical method often requires high frequency excitation signal sources or additional sensing structures. The excitation power source not only affects the performance in the actuator's steady state, but it may also generate harmonics that distort the measurement signals due to the nonlinear characteristics of the actuator. In addition, the information about velocity may not be obtained without specific sensing structures. The additional structures occupy more space in each die, which could increase the cost and size, or decrease the performance. In this study, an estimator with a series resistor configuration is proposed. The estimator can estimate the displacement and velocity by measuring the voltage of the power supply and the voltage across the actuator itself. To evaluate the feasibility, a nonlinear observability analysis is applied. The analysis shows the observability index among different system states. A simulation study verified the proposed theories.