We propose a framework for disturbance attenuation in case of nonlinear systems affine in a multivariable input. Roughly speaking, the underlying idea of our approach exploits the nature of nonlinear systems with a multivariable input. In particular, we extend the system under investigation by additional control possibilities, which were formed out by the so-called mixed Lie brackets. It is shown that this extension is reasonable and may allow to establish special dissipation inequalities by using a certain type of discontinuous feedback law. We illustrate this analysis technique by examples and suggest a synthesis framework for disturbance attenuation which fits very well into this concept.