Theoretical analysis of a derivative free control based continuation algorithm with path following capability for autonomous systems

TL;DR

This paper introduces a control-based continuation algorithm with path following capabilities for autonomous systems, combining derivative feedback, phase, and arclength controllers, supported by theoretical analysis for parameter tuning.

Contribution

It presents a minimal control-based continuation method with theoretical tuning rules, applicable to autonomous systems for tracking limit cycle branches.

Findings

The controller stabilizes and tracks limit cycles effectively.

Tuning rules are system-independent, derived via averaging method.

The approach enables systematic branch tracking in autonomous systems.

Abstract

We present a minimal control-based continuation algorithm designed to track branches of limit cycles in autonomous systems. The controller can be viewed as three sub-controllers: (i) a derivative feedback controller that is used to stabilize the limit cycle, (ii) an integral phase controller, used to synthesize the unknown phase of the limit cycle and (iii) an integral arclength controller, used to track branches of limit cycles. The controlled system is analyzed theoretically, using the averaging method, allowing us to express tuning rules for the different parameters of the controller. Remarkably, theses tuning rules are independent of the studied system.