A quasi-linear feedback compensator is one in which its poles depend in an appropriate way on its gain. The reason for introducing this new concept was the desire to remove the limitation to performance imposed by a plant with more than one pole in excess of its zeros. In this article it is shown that this objective is realized for plants with zeros in the left half of the complex plane. The consequences are surprising. In time domain it is possible to track arbitrarily fast a class of reference inputs despite a large class of disturbances and uncertainty in plant parameters. The response is non-oscillatory for high enough compensator gains, which is explained by the automatic adaptation of the closed loop poles to stability and stability margins for such gains. And in frequency domain the phase margin tends to 90degrees while the gain margin and crossover frequency become unlimited. Technically the design procedure of quasi-linear compensators presented here is based on our theoretical result concerning the asymptotic behaviour of the roots of certain polynomials in a complex variable which depend also on a large positive parameter. We also show how to implement such quasi-linear compensators in practical feedback control schemes, and their use at lower gains which is the case of most industrial applications.