Phase limitations of multipliers at harmonics

TL;DR

This paper introduces a phase condition that determines the impossibility of finding suitable multipliers for certain continuous-time plants, with applications to system analysis and counterexamples to the Kalman Conjecture.

Contribution

It provides a new, easily testable phase condition derived from duality and frequency interval approaches, improving upon previous results.

Findings

The phase condition can be tested efficiently.

Numerical examples demonstrate significant improvements.

A third order system with delay is shown as a counterexample to the Kalman Conjecture.

Abstract

We present a phase condition under which there is no suitable multiplier for a given continuous-time plant. The condition can be derived from either the duality approach or from the frequency interval approach. The condition has a simple graphical interpretation, can be tested in a numerically efficient manner and may be applied systematically. Numerical examples show significant improvement over existing results in the literature. The condition is used to demonstrate a third order system with delay that is a counterexample to the Kalman Conjecture.