Necessary and sufficient conditions for robust stability of a class of nonlinear systems

Input-output stability results for feedback systems are developed. Robust stability conditions are presented for nonlinear systems with nonlinear uncertainty defined by some function (with argument equal to the norm of the input) that bounds its output norm. A sufficient small gain theorem for a class of these systems is presented. Then it is also shown that, for the vector spaces (l∞, ||·||∞) and (l2, ||·||2), the sufficient conditions are also necessary with some additional assumptions on the systems. These results capture the conservatism of the small gain theorem as it is applied to systems that do not need to have linear gain.