Approaches to frequency tracking and vibration control

^ H I S thesis is concerned with the development and analysis of algorithms for fre­ quency tracking and estimation. The frequency of a sinusoidal signal embedded in noise can carry important information in many areas of signal processing and control. This makes the estimation of a constant frequency or the tracking of a time-varying frequency important issues. Two problems in this area are examined. The first is the tracking of a time-varying frequency in open-loop using the extended Kalman filter (EKF). The second is the estimation of a constant frequency for the purposes of vibration control. The extended Kalman filter was chosen as a frequency tracker because of its widespread use as a method of deriving filters for nonlinear systems. However, a thorough under­ standing of its behaviour and modes of failure was not available. Accordingly the stability of the EKF as an observer for nonlinear systems is examined. A new result giv­ ing sufficient conditions for bounded-input bounded-output stability of the EKF when applied to stochastic, discrete-time systems is presented. This extends previous results which were available only for continuous time and deterministic systems. The result also allows the development of theoretically supported design guidelines. Following the stability analysis of the EKF, design guidelines for constructing EKF-based observers are presented. This section collects previously known results, as well as the new guidelines which can be derived from the new stability result. These guidelines are used to construct EKF-based frequency trackers for high strength signals as well as weak, narrowband signals. This work also illustrates the flexibility of the EKF approach to nonlinear observer design by demonstrating how the particular features of a problem

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