Stability multipliers for memoryless positive nonlinearities: parameterizations based on the nonlinearity graph

We establish that the class of all multipliers that preserve the positivity of a given memoryless positive nonlinearity can be derived by parameterizing the Zames-Falb multipliers, provided certain quantitative information concerning the nonlinearity graph is available. Further, we extend this result to derive the class of all positivity preserving multipliers for a repeated memoryless positive nonlinearity when such information concerning its graph is available. These results are useful in reducing the conservatism inherent in the prevalent multiplier theoretic stability analysis techniques