# Bode's Sensitivity Integral Constraints: The Waterbed Effect in Discrete Time

## TL;DR

This paper explores the fundamental limitations of feedback control in discrete-time systems, revealing how the sensitivity integral constraints are intrinsically linked to unstable poles and zeros outside the unit circle, highlighting the waterbed effect.

## Contribution

It extends Bode's sensitivity integral constraints to discrete-time systems, emphasizing the role of unstable poles and transmission zeros outside the unit circle in performance limitations.

## Key findings

- Sensitivity integral constraints relate to unstable open-loop poles.
- Complementary sensitivity constraints relate to zeros outside the unit circle.
- Illustrative examples demonstrate the theoretical results.

## Abstract

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.