Necessary 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 dened by some function (with argument equal to the norm of the input) that bounds its output norm. A suÆcient small gain theorem for a class of these systems is known. Here, necessary conditions are presented for the vector space (` 1; k k 1). These results capture the conservatism of the small gain theorem as it is applied to systems that do not have linear gain. The theory is also developed for the case of 2 signal norms, indicating some diÆculties which make this case less natural than