The nonlinear robust stability theory of Georgiou and Smith [IEEE Trans. Automat. Control, 42 (1997), pp. 1200-1229] is generalized to the case of notions of stability with bias terms. An example from adaptive control illustrates nontrivial robust stability certificates for systems which the previous unbiased theory could not establish a nonzero robust stability margin. This treatment also shows that the bounded-input bounded-output robust stability results for adaptive controllers in French [IEEE Trans. Automat. Control, 53 (2008), pp. 461-478] can be refined to show preservation of biased forms of stability under gap perturbations. In the nonlinear setting, it also is shown that in contrast to linear time invariant systems, the problem of optimizing nominal performance is not equivalent to maximizing the robust stability margin.