Frequency–amplitude double closed-loop control of weakly coupled resonators based on energy conservation

The mode localization phenomenon of the weakly coupled resonators (WCRs), which is revealed as a small asymmetric perturbation introduced in the symmetric weakly coupled resonant system will lead to a drastic variation in the system energy distribution and amplitude ratios of the resonators, has been widely used in the topology design of micro-electro-mechanical system (MEMS) sensors. In order to improve the noise characteristics of mode-localized sensors, this paper presents an optimized measurement and control system, including the frequency closed-loop system based on phase-locked loop (PLL) and the amplitude closed-loop system based on energy conservation. The vibration energy of a weakly coupled resonant system can be represented by the sum of the squares of the amplitude of each resonant unit, and the total energy of resonant units is stabilized at a constant value by the amplitude closed-loop system based on energy conservation. The proposed measurement and control system is optimized by Simulink simulation software and finally verified with the fabricated prototype of a mode-localized airflow sensor. Experiment results demonstrate that compared with the amplitude open-loop detection method, the closed-loop control scheme based on energy conservation has better bias stability, which indicates that our proposal can improve the long-term stability of output signals of the weakly coupled resonant system.