A biased approach to nonlinear robust stability with applications in adaptive control

The nonlinear robust stability theory of Georgiou and Smith (IEEE Trans. Auto. Control, 42(9):1200-1229, 1997) is generalized to the case of notions of stability with bias terms. An example from adaptive control illustrates non trivial robust stability certificates for systems which the previous unbiased theory could not establish a non-zero robust stability margin. This treatment also shows that the BIBO robust stability results for adaptive controllers in French (IEEE Trans. Auto. Control, 53(2):461-478, 2008) 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 LTI systems, the problem of minimizing nominal performance is not equivalent to maximizing the robust stability margin.