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

TL;DR

This work addresses tracking branches of limit cycles in autonomous systems with a derivative-free control-based continuation approach. It proposes a minimal controller architecture composed of three sub-controllers: a derivative feedback loop for stabilization, a phase-locked loop to synthesize the unknown limit-cycle phase, and an arclength controller to enable path-following along bifurcation diagrams; the dynamics are analyzed via averaging to derive generic tuning rules that are largely system-independent. Theoretical results establish fixed-point conditions and stability criteria, and numerical experiments on a generalized Van der Pol oscillator demonstrate non-invasive tracking and the ability to pass turning points; the approach is limited by mono-harmonic control but can be extended with adaptive filtering. Overall, the paper provides a practical, tunable framework for experimental continuation of autonomous oscillations, enabling robust exploration of limit-cycle branches without Jacobian estimates.

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.

Figures (6)

• Figure 1: Graphical interpretation of the stability condition given in Eq. (\ref{['eq:36']}). Blue and red squares correspond to stable and unstable fixed point, respectively.
• Figure 2: Bifurcation diagram of the uncontrolled system Eq. (\ref{['eq:37']}) with $\beta=1$. plain and dotted lines represent the amplitude of stable and unstable limit cycle, respectively. The dot corresponds to fold bifurcation.
• Figure 3: Result of numerical integration for parameters in Table \ref{['tab:1']}.
• Figure 4: Result of numerical integration for the bifurcation parameters $\mu$ and $G$ plotted on the bifurcation diagram of the uncontrolled system for $K_{p3}=0.1$ (blue) and $K_{p3}=-0.1$ (red). The other parameters are given in Table \ref{['tab:1']}. The grey dot represents the starting point $(\mu_0,G_0)=(0,0.3)$ and the dotted grey line, the circle of radius $\Delta=0.1$.
• Figure 5: Result of numerical integration of the controlled system displayed on the bifurcation diagram of the uncontrolled system for $K_{p3}=0.1$ (blue) and $K_{p3}=-0.1$ (red). (a) $K_{d1}=0.1$, (b) $K_{d1}=0.2$, the other parameters are given in Table \ref{['tab:1']}. The grey dot represents the starting point on the fold bifurcation point.