# Phase limitations of Zames-Falb multipliers

## TL;DR

This paper investigates phase limitations of Zames-Falb multipliers in continuous and discrete time, relating these limitations to the Kalman conjecture, and demonstrates their implications through classical and novel counterexamples.

## Contribution

It generalizes a known phase limitation for continuous-time multipliers, introduces a new phase limitation for discrete-time multipliers, and explores their impact on stability criteria and the Kalman conjecture.

## Key findings

- Existence of fourth-order plants without suitable Zames-Falb multipliers showing instability.
- A new discrete-time phase limitation for Zames-Falb multipliers is established.
- Discrete-time limitations imply no direct off-axis circle criterion counterpart.

## Abstract

Phase limitations of both continuous-time and discrete-time Zames-Falb multipliers and their relation with the Kalman conjecture are analysed. A phase limitation for continuous-time multipliers given by Megretski is generalised and its applicability is clarified; its relation to the Kalman conjecture is illustrated with a classical example from the literature. It is demonstrated that there exist fourth-order plants where the existence of a suitable Zames-Falb multiplier can be discarded and for which simulations show unstable behavior. A novel phase-limitation for discrete-time Zames-Falb multipliers is developed. Its application is demonstrated with a second-order counterexample to the Kalman conjecture. Finally, the discrete-time limitation is used to show that there can be no direct counterpart of the off-axis circle criterion in the discrete-time domain.