Bode's Sensitivity Integral Constraints: The Waterbed Effect Revisited

TL;DR

This paper revisits Bode's sensitivity integral constraints, revealing their fundamental link to the difference in speed between open-loop and closed-loop systems, and clarifies the inherent limitations imposed by system pole and zero locations.

Contribution

The paper provides a new, elegant derivation of Bode's sensitivity integral constraints, emphasizing their relation to system speed differences and transmission zeros, enhancing understanding of fundamental performance limits.

Findings

Sensitivity integral constraints relate to the speed difference between open and closed-loop systems.

The integral constraint for the complementary sensitivity function involves transmission zeros and closed-loop poles.

Performance limitations are inherently tied to the locations of poles and zeros in the system.

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.