We investigate theoretically the stabilization of a fixed point of a discrete one-dimensional nonlinear map by applying small perturbations to an accessible system parameter or variable. The size of the perturbations is determined in real time using feedback schemes incorporating only the dynamical state of the system and its state at previous iterates without making a comparison to a reference state. In particular, we compare and contrast two algorithms: extended time-delay autosynchronization, which uses an infinite series of past iterates with weights that decay by a factor of [formula presented] with each time step, and [formula presented]-time-delay autosynchronization, which uses an average of [formula presented] past iterates with equal weights. The range of feedback parameters that successfully stabilize the fixed point and the robustness of the schemes to noise are determined. It is found that the domain of control for the two schemes is similar for appropriately matched values of [formula presented] and [formula presented], and that [formula presented]-time-delay autosynchronization tends to be less sensitive to noise. © 1998 The American Physical Society.