In the context of hybrid anti-lock brake systems (ABS), we provide a closed-loop wheel-acceleration controller based on the observation of the extended braking stiffness (XBS). Its objective is to improve the system's robustness with respect to changes in the environment (as changes in road conditions, brake properties, etc.). In our design, we take advantage of Burckhardt's tyre model, in order to obtain a wheel acceleration dynamics that is linear up to time-scaling. The XBS is one of the state variables of this model. Our main result is an observer that estimates this unmeasured variable. The observer's convergence analysis is established using tools for switched linear systems that allow us to ensure its uniform exponential stability (provided that a dwell-time condition is satisfied). Our simulation results confirm the convergence properties predicted by our main theorem.