Mathematical modeling and control of micro electromechanical gyroscopes

Micro electromechanical gyroscopes serve as inertial sensors for measuring the angular rate with respect to an inertial reference frame. They are utilized in an increasing number of modern technical applications, in particular in mass products with high demands on cost efficiency and miniaturization, and have been extensively treated in the literature in recent years. Many contributions are dealing in particular with the design and modeling of micro machined gyroscopes. Thereby, the demands on the mathematical models strongly vary with the intended applications. In order to understand the basic principle of operation, typically lumped parameter models are used, which in general are insufficient in terms of accuracy. More accurate models, especially of the mechanical structure, are required for the overall design process of micro electromechanical gyroscopes. Finite element methods are preferably utilized providing numerical models at a high level of detail, even if the geometry of the device is complex. These methods, however, have the disadvantage that the resulting system of differential equations is very large and can neither be used for analysis and control design purposes nor for transient simulations within reasonable time. Therefore, one major motivation for this work is to provide an analytical model with a reasonable number of degrees-of-freedom and sufficient accuracy while preserving the physical properties of the electromechanical system. Different from various order reduction techniques applied to the complex finite element models the presented approach is based on a classification of the device into functional components well-defined by their physical properties (e. g., rigid bodies, elastic bodies, lumped capacitors, etc.). Thereby, the actual model simplification is achieved by applying physically motivated assumptions to the mathematical representations of the individual functional components. This allows for the derivation of dynamical models with a suitable level of detail for the respective application.