Numerical estimation of the lock-in domain of a DC/AC inverter

We estimate the lock-in domain of the origin of a current control system which is used in common DC/AC inverter designs. The system is a cascade connection of a 4-dimensional linear system (current controller, CC) followed by a two-dimensional nonlinear system (phase-locked loop, PLL). For the PLL, we construct a Lyapunov function via numerical approximation of its level curves. In combination with the quadratic Lyapunov function of the CC, it forms a vector Lyapunov function (VLF) for the overall system. A forward-invariant set of the VLF is found via numerical application of the comparison principle. By LaSalle’s invariance principle, convergence to the origin is established.

💡 Research Summary

The paper addresses the problem of estimating the lock‑in domain of a DC/AC inverter current‑control loop that consists of a four‑dimensional linear current controller (CC) cascaded with a two‑dimensional nonlinear phase‑locked loop (PLL). The lock‑in domain is defined as the set of initial conditions from which the closed‑loop trajectories converge to the origin without the phase‑error Δθ ever crossing the lines Δθ = ±π, i.e., without a 2π “cycle slip”.

System model.
The CC is modeled as (\dot x = A x) with A Hurwitz, guaranteeing exponential decay of the state vector x∈ℝ⁴. The PLL dynamics are written as

\