A fundamental performance limit is derived for a class of positive nonlinear systems. The performance limit describes the achievable output response in the presence of a positive disturbance and subject to a sign constraint on the allowable input. An explicit optimal input is derived which minimises the maximum output response whilst ensuring that the minimum output response does not fall below a pre-specified lower bound. The result provides a fundamental performance standard against which all control policies, including closed loop schemes, can be compared. Implications of the result are examined in the context of blood glucose regulation for Type 1 Diabetes. (C) 2018 Elsevier Ltd. All rights reserved.