Bode's Sensitivity Integral Constraints: The Waterbed Effect Revisited

Abstract: Bode's sensitivity integral constraints define a fundamental rule about the limitations of feedback and is referred to as the waterbed effect. We take a fresh look at this problem and reveal an elegant and fundamental result that has been seemingly masked by previous derivations. The main result is that the sensitivity integral constraint is crucially related to the difference in speed of the closed-loop system as compared to that of the open-loop system. This makes much intuitive sense. Similar results are also derived for the complementary sensitivity function. In that case the integral constraint is related to the sum of the differences of the reciprocal of the transmission zeros and the closed-loop poles of the system. Hence all performance limitations are inherently related to the locations of the open-loop and closed-loop poles, and the transmission zeros. A number of illustrative examples are presented.