Perturbed versions of 1D simple maps such as logistic, tent, cubic ones (close in uniform norm but with strongly varying derivatives) can exhibit completely different behavior. We encounter such situation when dealing with chaos stabilization via small control. In this paper we present analytical investigation of this effect for one particular case - piecewise linear perturbation of the tent map. Surprisingly, iterations of this map converge to the unique fixed point very fast for all initial points. The result is in sharp contrast with iterations of the original tent map but explains fast stabilization of unstable periodic orbits by predictive control.