The paper deals with a class of non linear systems consisting of the cascade of a static nonlinear map depending on the state and the input and a dynamic input affine system. The static nonlinearity is approximatively inverted and the approximation error is treated as an input disturbance acting in the same direction as the control input. For this class of systems it is possible to compute a priori an upper bound for the H∞ attenuation level achieved by the optimal L2 controller and the suboptimal H ∞ central controller. Notably, these bounds depend only on the function mapping the control input to the performance variable.