In the context of hybrid anti-lock brake systems (ABS), a closed-loop wheel-acceleration controller based on the observation of the extended braking stiffness (XBS) is provided. Its objective is to improve the system's robustness with respect to changes in the environment (as changes in road conditions, brake properties, etc.). The observer design is based on Burckhardt's tyre model, which provides a wheel acceleration dynamics that is linear up to time-scaling. The XBS is one of the state variables of this model. The paper's main result is an observer that estimates this unmeasured variable. When the road conditions are known, a three-dimensional observer solves the problem. But, for unknown road conditions, a more complex four-dimensional observer must be used instead. In both cases, the observer's convergence is analysed using tools for switched linear systems that ensure uniform exponential stability (provided that a dwell-time condition is satisfied). Both experiments and simulations confirm the convergence properties predicted by the theoretical analysis.