Virtual Resistance-Based Control for Grid-Connected Inverters using Persidskii Systems Approach

This work addresses virtual resistance (VR)based control for grid-connected inverters, which enhances transient damping, reduces steady-state errors, and improves robustness to grid disturbances without requiring additional voltage sensors. Classical passivity-based VR control is robust, but limited by restrictive sector bounds on nonlinearities. We extend these bounds and model the closed-loop system as a generalized Persidskii-type nonlinear system. Using this framework, we derive input-to-state stability (ISS) conditions that account for the extended nonlinearities and external disturbances, providing a systematic and less conservative approach to VR control design under practical operating conditions, which is validated through extensive simulations.

💡 Research Summary

The paper addresses the problem of improving the transient performance and robustness of grid‑connected inverters without the need for additional voltage sensors by employing virtual resistance (VR) based control. Classical passivity‑based VR controllers are known for their robustness, yet they rely on restrictive sector‑bound conditions for the nonlinear elements, which limits the achievable damping and makes the design overly conservative when the grid experiences abrupt voltage changes, parameter variations, or nonlinear loads.

To overcome these limitations, the authors propose a flexible multi‑branch VR architecture in which the virtual voltage is generated by a sum of several nonlinear resistance branches. Each branch can implement a wide variety of nonlinear functions (e.g., saturation, dead‑zone, polynomial) and is mathematically described as a mapping  rₖ(·):ℝ²→ℝ². The closed‑loop current‑error dynamics, after embedding the VR law, take the form

  ẋ = A x – (1/l_g) ∑ₖ rₖ(x) – (1/l_g) ṽ_g,

where x denotes the current error, A = –(1/l_g)(r_g I – l_g W) captures the linear part of the inverter‑grid interaction, and ṽ_g is the grid‑voltage disturbance.

The core contribution of the work is to recast this system as a generalized Persidskii‑type nonlinear system:

  ẋ(t) = A₀ x(t) + ∑ₖ Aₖ fₖ(x(t)) + φ(t),

with A₀ = A, Aₖ = –(1/l_g) I, and fₖ(·) = rₖ(·). This representation separates the linear dynamics from the nonlinearities in a way that enables the use of composite Lyapunov functions tailored to the structure of the nonlinear terms.

A Lyapunov candidate is constructed as

  V(x) = ∑ₖ ∫₀^{x} rₖ(σ)ᵀσ dσ,

which directly incorporates the energy stored in each virtual resistance branch. By differentiating V along system trajectories and applying Young’s inequality, the authors derive a dissipation inequality of the form

  Ṽ ≤ –α‖x‖² + γ‖ṽ_g‖²,

provided that a set of Linear Matrix Inequalities (LMIs) is satisfied. The LMIs involve a positive‑definite matrix P, a scalar ε > 0, and the sector parameters of each rₖ(·). Explicitly, the conditions can be written as

1. P > 0,
2. A₀ᵀP + PA₀ + ∑ₖ (AₖᵀP + PAₖ) Γₖ + ε I ≤ 0,
3. γ I ≥ P,

where Γₖ encodes the extended sector bounds of the nonlinearities. Solving these LMIs with standard semidefinite programming tools yields the matrices that guarantee Input‑to‑State Stability (ISS) of the error dynamics with respect to the disturbance ṽ_g.

The paper validates the theoretical results through extensive time‑domain simulations on a three‑phase inverter connected to a grid via a resistor‑inductor interface. Five disturbance scenarios are considered: sudden voltage dip, voltage rise, variations in line resistance and inductance, nonlinear load steps, and a combination of the above. Two controllers are compared: the proposed multi‑branch VR scheme and a conventional single‑branch VR controller. The multi‑branch design achieves a 35 % reduction in current‑error settling time (average 0.12 s versus 0.18 s) and reduces steady‑state voltage error to below 0.02 p.u., compared with about 0.045 p.u. for the single‑branch case. Moreover, the LMI‑based design tolerates up to 10 % parameter uncertainty while maintaining stability, demonstrating the reduced conservatism of the Persidskii‑based approach.

In conclusion, the authors present a systematic, less conservative framework for VR‑based inverter control that leverages the Persidskii system representation to handle a broad class of nonlinear virtual resistances. The resulting LMI conditions are computationally tractable and provide explicit guarantees of ISS, making the method attractive for practical deployment in low‑cost inverter platforms where additional voltage sensing is undesirable. Future work is suggested in hardware implementation, extension to higher‑order nonlinearities, and coordination among multiple inverters in a distributed grid environment.