Abbas Emami-Naeini

Abbas Emami-Naeini, Dick de Roover

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.

Abbas Emami-Naeini, Dick de Roover

Bode's sensitivity integral constraints define a fundamental rule about the limitations of feedback and is referred to as the waterbed effect. In a companion paper, we took a fresh look at this problem using a direct approach to derive our results. In this paper, we will address the same problem, but now in discrete time. Although similar to the continuous case, the discrete-time case poses its own peculiarities and subtleties. The main result is that the sensitivity integral constraint is crucially related to the locations of the unstable open-loop poles of the 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 locations of the transmission zeros outside the unit circle. Hence all performance limitations are inherently related to the open-loop poles and the transmission zeros outside the unit circle. A number of illustrative examples are presented.