# Control and Observability Aspects of Phase Synchronization

## TL;DR

This paper investigates control and observability issues in phase synchronization of oscillators, proposing a feedback control framework, analyzing coordinate choices, and addressing practical challenges in phase-based control.

## Contribution

It introduces a novel control framework for phase synchronization, compares coordinate effects, and proposes a method to handle indirect phase error control.

## Key findings

- Observability is less critical for phase synchronization than for complete synchronization.
- Coordinate choice impacts the control design but not the core synchronization ability.
- A practical method is proposed to control phase when phase error isn't directly used in control.

## Abstract

This paper addresses important control and observability aspects of the phase synchronization of two oscillators. To this aim a feedback control framework is proposed based on which issues related to master-slave synchronization are analyzed. Comparing results using Cartesian and cylindrical coordinates in the context of the proposed framework it is argued that: i)~observability does not play a significant role in phase synchronization, although it is granted that it might be relevant for complete synchronization; and ii)~a practical difficulty is faced when phase synchronization is aimed at but the control action is not a direct function of the phase error. A procedure for overcoming such a problem is proposed. The only assumption made is that the phase can be estimated using the arctangent function. The main aspects of the paper are illustrated using the Poincar\'e equations, van der Pol and R\"ossler oscillators in dynamical regimes for which the phase is well defined.