Focus-Center-Limit Cycle Bifurcation in a Bidirectional Buck Converter with PI Controller

This work analyzes the stability and bifurcation behavior of a bidirectional DC–DC converter operating in Buck mode and regulated by a Proportional–Integral (PI) controller subject to saturation. The dynamic behavior is modeled by a three-dimensional continuous piecewise-linear system with three zones and a unique equilibrium point. Explicit conditions on the control parameters are derived to ensure local stability of the equilibrium and to characterize the emergence of a stable limit cycle via the Focus-Center-Limit Cycle bifurcation. These results provide practical guidelines for tuning PI controllers, helping designers avoid unwanted oscillations and improve robustness against load disturbances. Numerical simulations and experimental results support the theoretical analysis.