In this paper, state estimation of a single degree of freedom flexible cable-actuated system governed by nonlinear equations of motion is considered. State estimation formulations are developed in discrete time using an extended Kalman filter (EKF) and an unscented Kalman filter (UKF). Measurements of the payload acceleration and either the angular position of one or both winches are obtained by an accelerometer mounted on the payload and angular winch encoders. These sensors are relatively small and inexpensive, and allow for operation in many different environments. This is an improvement over current estimation techniques for cable-actuated systems that typically rely on a position measurements of the payload that are obtained by large and expensive sensors that often only operate in a particular environment (e.g., motion capture system, GPS receiver). An EKF and UKF are implemented to estimate the states of the system using a nonlinear dynamic model. A proportional-derivative-based controller from the literature is used for control. Estimation of the tension in each cable is also performed, which demonstrates that additional sensors (e.g., load cells) are not needed to obtain an accurate estimate of the cable tensions. Consideration of sensors with different sampling rates is also included in the EKF and UKF formulations. Simulation results using the EKF and UKF along with the proportional-derivative controller in discrete time are presented.