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Observability and detectability analyses are often used to guide the measurement setup and select the estimation models used in dynamic state estimation (DSE). Yet, marginally observable states of a synchronous machine prevent the direct application of conventional observability and detectability analyses in determining the existence of a DSE observer. To address this issue, the authors propose to identify the marginally observable states and their associate eigenvalues by finding the smallest perturbation matrices that make the system unobservable. The proposed method extends the observability and detectability analyses to marginally observable estimation models, often encountered in the DSE of a synchronous machine. The effectiveness and application of the proposed method are illustrated on the IEEE 10-machine 39-bus system, verified using the unscented Kalman filter and the extended Kalman filter, and compared with conventional methods. The proposed analysis method can be applied to guide the selection of estimation models and measurements to determine the existence of a DSE observer in power-system planning.