Recent designs for brake-by-wire systems use 'resolvers' to provide accurate and continuous measurements for the absolute position and speed of the rotor of the electric actuators in brake callipers (permanent magnet DC motors). Resolvers are absolute-angle transducers that are integrated with estimator modules called 'angle tracking observer' and together they provide position and speed measurements. Current designs for angle-tracking observers are unstable in applications with high acceleration and/or speed. In this paper, we introduce a new angle-tracking observer in which a closed-loop linear time-invariant (LTI) observer is integrated with a quadrature encoder. Finite-gain stability of the proposed design and its robustness to three different kinds of parameter variations are proven based on theorems of input-output stability in nonlinear control theory. In our experiments, we examined the performance of our observer and two other methods (a well-known LTI observer and an extended Kalman filter) to estimate the position and speed of a brake-by-wire actuator. The results show that because of the very high speed and acceleration of the actuator in this application, the LTI observer and Kalman filter cannot track the rotor position and diverge. In contrast, with a properly designed open-loop transfer function and selecting a suitable switching threshold, our proposed angle-tracking observer is stable and highly accurate in a brake-by-wire application.