The design of output feedback controllers that eliminate uncertainties caused by critical races in asynchronous sequential machines is considered. The objective is to build controllers that drive a race-afflicted machine so as to match a prescribed deterministic model. Necessary and sufficient conditions for the existence of such controllers are presented in terms of a numerical matrix derived from the given machine. When controllers exist, an algorithm for their construction is also provided. The discussion depends on the novel notion of 'generalised state', which helps represent the uncertainty created by critical races and facilitates the construction of controllers.