The proliferation of renewable energy sources has placed grid-connected inverters at the forefront of modern power electronics research. These critical interfaces between distributed generation resources and the utility grid are tasked with delivering high-quality power. This necessitates that the output current injected into the network accurately tracks a specified sinusoidal reference signal in terms of magnitude, phase, and frequency, despite inherent system challenges. A primary challenge stems from the widespread use of LCL filters, favored for their superior high-frequency harmonic attenuation compared to simple L filters. However, the LCL filter introduces a resonant pole, potentially leading to system instability. While passive damping methods exist, they often incur additional power losses and increased physical volume. Consequently, advanced active damping control strategies, implemented within the control algorithm itself, are essential for ensuring stable and high-performance operation of the grid connected inverter.
Traditional control methods like Proportional-Integral (PI) in the synchronous reference frame or Proportional-Resonant (PR) in the stationary frame are commonplace. More advanced techniques, including Model Predictive Control (MPC), Sliding Mode Control (SMC), and H∞ control, have been investigated to improve dynamics, constraint handling, and robustness. However, many approaches either lack explicit robustness guarantees against parameter variations and grid disturbances or rely on high-gain designs or exact model knowledge. This article addresses the robust current tracking problem for a three-phase grid connected inverter with an LCL filter by employing the output regulation theory and the internal model principle. This framework systematically handles known external disturbances (like grid voltage variations) and provides inherent robustness to parameter uncertainties within a known range, offering a structured design procedure for the controller gains.
The control objective for the grid connected inverter is formalized as an output regulation problem. The goal is to design a controller such that the grid current asymptotically tracks a sinusoidal reference while rejecting a sinusoidal grid voltage disturbance, and all internal system states remain bounded. The reference and disturbance are modeled as outputs of an exogenous system, or exosystem. The core theoretical tool is the internal model principle, which states that a controller must incorporate a model of the dynamics generating the reference and disturbance signals to achieve asymptotic tracking and rejection. The solution involves solving the regulator equations to find the desired steady-state behavior of the system and then constructing a dynamic controller that embeds this internal model and stabilizes the overall closed-loop system.
System Modeling and Problem Formulation
The topology of a single-phase equivalent of a three-phase grid connected inverter with an LCL filter is considered. The dynamics are described by the following differential equations, where system parameters are subject to variation around their nominal values.
$$
\begin{aligned}
V_I &= I_L R_1 + L_1 \frac{dI_L}{dt} + V_C, \\
I_L &= I_O + C \frac{dV_C}{dt}, \\
V_C &= I_O R_2 + L_2 \frac{dI_O}{dt} + V_G,
\end{aligned}
$$
Let the state variables be defined as $x_1 = I_O$ (grid current), $x_2 = V_C$ (capacitor voltage), and $x_3 = I_L$ (inverter-side current). The control input is $u = V_I$ (inverter bridge output voltage). The controlled output is $y = x_1$. The grid voltage $V_G(t)$ is considered a disturbance, and $y_r(t)$ is the reference current. Both are sinusoidal: $y_r(t) = A_r \sin(\omega t + \phi_r)$ and $V_G(t) = A_d \sin(\omega t + \phi_d)$. The parameter vector $\Omega = (R_1, R_2, L_1, L_2, C)$ is expressed as its nominal value $\bar{\Omega}$ plus an uncertainty $w \in \mathcal{W} \subset \mathbb{R}^5$, i.e., $\Omega = \bar{\Omega} + w$.
The exosystem generating the reference and disturbance signals is defined as:
$$
\dot{v} = S v = \begin{bmatrix}
0 & \omega & 0 & 0 \\
-\omega & 0 & 0 & 0 \\
0 & 0 & 0 & \omega \\
0 & 0 & -\omega & 0
\end{bmatrix} v,
$$
with $y_r(t) = v_1(t)$ and $V_G(t) = v_3(t)$. The tracking error is $e = y – y_r = x_1 – v_1$.
Thus, the composite system for the grid connected inverter control problem is:
$$
\begin{aligned}
\dot{x}_1 &= -\frac{R_2}{L_2}x_1 + \frac{1}{L_2}x_2 – \frac{1}{L_2}v_3, \\
\dot{x}_2 &= -\frac{1}{C}x_1 + \frac{1}{C}x_3, \\
\dot{x}_3 &= -\frac{1}{L_1}x_2 – \frac{R_1}{L_1}x_3 + \frac{1}{L_1}u, \\
\dot{v} &= S v, \\
e &= x_1 – v_1.
\end{aligned}
$$
Control Problem: Design a dynamic state feedback control law $u$ for the system above such that, for all initial conditions and for all admissible parameter variations $w \in \mathcal{W}$, the closed-loop system satisfies:
1. All trajectories $(x(t), v(t), \eta(t))$ are bounded.
2. The tracking error converges to zero asymptotically: $\lim_{t \to \infty} e(t) = 0$.
Controller Design via Output Regulation Framework
The solution follows a structured procedure based on output regulation theory for nonlinear systems.
Step 1: Solving the Regulator Equations
The first step is to find the steady-state manifold and input. This involves solving the regulator equations, which define the state and input trajectories $(x(v,w), u(v,w))$ that yield perfect tracking ($e=0$) when the system is precisely on this manifold. For the grid connected inverter model, these equations are:
$$
\begin{aligned}
\frac{\partial x_1(v,w)}{\partial v} S v &= -\frac{R_2}{L_2}x_1(v,w) + \frac{1}{L_2}x_2(v,w) – \frac{1}{L_2}v_3, \\
\frac{\partial x_2(v,w)}{\partial v} S v &= -\frac{1}{C}x_1(v,w) + \frac{1}{C}x_3(v,w), \\
\frac{\partial x_3(v,w)}{\partial v} S v &= -\frac{1}{L_1}x_2(v,w) – \frac{R_1}{L_1}x_3(v,w) + \frac{1}{L_1}u(v,w), \\
0 &= x_1(v,w) – v_1.
\end{aligned}
$$
The solution is obtained as:
$$
\begin{aligned}
x_1(v,w) &= v_1, \\
x_2(v,w) &= R_2 v_1 + L_2 \omega v_2 + v_3, \\
x_3(v,w) &= (1 – L_2 C \omega^2) v_1 + R_2 C \omega v_2 + C \omega v_4, \\
u(v,w) &= a_1 v_1 + a_2 v_2 + a_3 v_3 + a_4 v_4,
\end{aligned}
$$
where the coefficients $a_1, a_2, a_3, a_4$ are functions of the system parameters $(R_1, R_2, L_1, L_2, C)$ and the frequency $\omega$.
Step 2: Constructing the Internal Model
The steady-state input $u(v,w)$ is a linear combination of the exosystem states $v$. Since $v$ is unknown, we cannot implement $u(v,w)$ directly. The internal model principle resolves this by constructing a dynamical system (the internal model) that can asymptotically generate this required steady-state input. Observing that $x_2(v,w)$, $x_3(v,w)$, and $u(v,w)$ are all linear functions of $v$, they can be viewed as outputs of linear systems driven by $v$.
Define $\tau_i(v,w) = [x_{i+1}(v,w), \dot{x}_{i+1}(v,w)]^T$ for $i=1,2,3$ (with $x_4 \triangleq u$). It can be verified that $\frac{\partial \tau_i}{\partial v} S v = \Phi_i \tau_i$ and $x_{i+1} = \Gamma_i \tau_i$, where $\Phi_i = \begin{bmatrix} 0 & 1 \\ -\omega^2 & 0 \end{bmatrix}$ and $\Gamma_i = [1, 0]$.
For each $i$, we choose a controllable pair $(M_i, N_i)$ with $M_i$ being a Hurwitz matrix. Solving the Sylvester equation $T_i \Phi_i – M_i T_i = N_i \Gamma_i$ yields a non-singular matrix $T_i$. Defining $\Psi_i = \Gamma_i T_i^{-1}$ allows us to establish the following steady-state generator:
$$
\begin{aligned}
\frac{\partial \theta_i(v,w)}{\partial v} S v &= (M_i + N_i \Psi_i) \theta_i(v,w), \\
x_{i+1}(v,w) &= \Psi_i \theta_i(v,w),
\end{aligned}
$$
where $\theta_i(v,w) = T_i \tau_i(v,w)$.
Consequently, the internal model for the grid connected inverter is constructed as the following dynamical system with $x_2$, $x_3$, and $u$ as its inputs:
$$
\dot{\eta}_i = M_i \eta_i + N_i x_{i+1}, \quad \text{for } i = 1, 2, 3.
$$
Step 3: Coordinate Transformation and Stabilization
With the internal model in place, the original tracking problem is transformed into a stabilization problem. Define new error variables:
$$
\begin{aligned}
\tilde{x}_1 &= x_1 – x_1(v,w) = e, \\
\tilde{x}_{i+1} &= x_{i+1} – \Psi_i \eta_i, \quad \text{for } i=1,2, \\
\bar{\eta}_i &= \eta_i – \theta_i(v,w) – N_i b_i^{-1} \tilde{x}_i, \quad \text{for } i=1,2,3,
\end{aligned}
$$
where $b_1 = 1/L_2$, $b_2 = 1/C$, $b_3 = 1/L_1$, and $\tilde{x}_4 \triangleq \bar{u} = u – \Psi_3 \eta_3$ is a new control input.
In these new coordinates, the system takes a lower-triangular form suitable for backstepping or recursive stabilization:
$$
\begin{aligned}
\dot{\bar{\eta}}_i &= M_i \bar{\eta}_i + \Theta_i(\bar{\eta}_1,\dots,\bar{\eta}_{i-1}, \tilde{x}_1,\dots,\tilde{x}_i, w), \\
\dot{\tilde{x}}_i &= \Upsilon_i(\bar{\eta}_1,\dots,\bar{\eta}_i, \tilde{x}_1,\dots,\tilde{x}_i, w) + b_i \tilde{x}_{i+1},
\end{aligned}
$$
for $i=1,2,3$, with $\tilde{x}_4 = \bar{u}$. The functions $\Theta_i$ and $\Upsilon_i$ vanish at the origin for all $w \in \mathcal{W}$.
The stabilization problem is then solved by the following virtual and actual control laws using a backstepping-like procedure:
$$
\begin{aligned}
\bar{x}_1 &= \tilde{x}_1, \\
\bar{x}_{i+1} &= \tilde{x}_{i+1} + k_i \bar{x}_i, \quad \text{for } i=1,2, \\
\bar{u} &= -k_3 \bar{x}_3,
\end{aligned}
$$
where $k_1, k_2, k_3 > 0$ are controller gains to be chosen sufficiently large to dominate the uncertain terms arising from $\Theta_i$ and $\Upsilon_i$. A Lyapunov analysis proves that with appropriate gain selection, the origin $(\bar{\eta}, \bar{x}) = 0$ is asymptotically stable for all $w \in \mathcal{W}$.
Step 4: Final Dynamic State Feedback Controller
Reverting to the original coordinates, the final robust current tracking controller for the three-phase grid connected inverter is given by the following set of dynamic equations:
$$
\begin{aligned}
u &= -k_3 \bar{x}_3 + \Psi_3 \eta_3, \\
\dot{\eta}_i &= M_i \eta_i + N_i x_{i+1}, \quad i = 1, 2, 3, \\
\bar{x}_1 &= e = x_1 – v_1, \\
\bar{x}_{i+1} &= x_{i+1} – \Psi_i \eta_i + k_i \bar{x}_i, \quad i = 1, 2.
\end{aligned}
$$
This controller consists of the internal model states $\eta_i$ and the stabilizing feedback loops on the error variables $\bar{x}_i$. It guarantees that for the grid connected inverter system with uncertain LCL filter parameters and under sinusoidal grid voltage disturbance, the grid current $x_1$ asymptotically tracks the reference $v_1$, and all internal signals remain bounded.
Simulation Verification and Comparative Analysis
The performance of the proposed output regulation-based controller is validated via simulation of a three-phase grid connected inverter system and compared against a standard feedforward controller derived from the ideal regulator equations.
Simulation Setup
A detailed simulation model is built in the MATLAB/Simulink environment. The nominal system parameters and their variation ranges are listed in the table below. The Space Vector Pulse Width Modulation (SVPWM) technique is used to generate the gate signals for the inverter bridge from the control signal $u$.
| Parameter | Description | Nominal Value / Range |
|---|---|---|
| $V_{DC}$ | DC Link Voltage | 600 V |
| $L_1$ | Inverter-side Inductance | 600 µH (450 – 900 µH) |
| $R_1$ | Inverter-side Resistance | 1.8 Ω (0.1 – 3.5 Ω) |
| $C$ | Filter Capacitance | 15 µF (12 – 15 µF) |
| $L_2$ | Grid-side Inductance | 700 µH (450 – 980 µH) |
| $R_2$ | Grid-side Resistance | 1.4 Ω (0.01 – 2.81 Ω) |
| $V_G$ | Grid Voltage (RMS) | 220 V, 50 Hz |
| $f_s$ | Switching Frequency | 100 kHz |
The control objective is to track a reference current $y_r(t) = 10\sin(100\pi t)$ A while rejecting a grid voltage disturbance $V_G(t) = 220\sqrt{2}\sin(100\pi t + \pi/6)$ V. Two scenarios are tested:
Case 1 (Nominal): Parameters at their nominal values.
Case 2 (Uncertain): Parameters perturbed to $L_1=450 \mu H$, $R_1=2.5 \Omega$, $C=12 \mu F$, $L_2=450 \mu H$, $R_2=0.8 \Omega$.
Proposed Controller Performance
The proposed dynamic controller is implemented with gains $k_1=12$, $k_2=0.035$, $k_3=340$. The internal model matrices $(M_i, N_i)$ are designed as described. The tracking errors for all three phases ($e_A, e_B, e_C$) under both nominal and uncertain parameter cases are shown to converge rapidly to a very small neighborhood of zero. The small residual error is due to the SVPWM harmonics and non-ideal switching. The Total Harmonic Distortion (THD) of the output current for Phase A is analyzed:
| Case | THD of Output Current (Phase A) |
|---|---|
| Nominal (Case 1) | 0.85% |
| Uncertain (Case 2) | 1.14% |
Both THD values are well below the typical 5% limit, confirming high power quality. Crucially, the performance degradation from Case 1 to Case 2 is minimal, demonstrating the robustness of the proposed controller to significant parameter variations in the LCL filter of the grid connected inverter.
Comparison with Feedforward Control
For comparison, a feedforward-plus-state-feedback controller is designed based on the perfect knowledge solution from the regulator equations:
$$
u_{ff} = K_1 x + K_2 v,
$$
where $K_1$ is chosen to place closed-loop poles for stability, and $K_2 = U – K_1 X$ is derived from the ideal feedforward gain. This controller assumes precise knowledge of both the system states $x$ and the exosystem states $v$.
The simulation results under the same two cases reveal a stark contrast:
| Controller | Case (Parameters) | Tracking Error | THD (Phase A) | Robustness Observation |
|---|---|---|---|---|
| Proposed (Output Regulation) | Nominal | Converges ~0 | 0.85% | High performance maintained. |
| Uncertain | Converges ~0 | 1.14% | ||
| Feedforward ($u_{ff}$) | Nominal | Small steady-state error | 0.46% | Performance severely degrades with parameter drift. |
| Uncertain | Large steady-state error & oscillation | 2.38% |
The feedforward controller performs adequately under nominal conditions, albeit with a slight steady-state error due to the lack of an integral action equivalent. However, under the uncertain parameter case (Case 2), its performance deteriorates significantly, exhibiting larger steady-state error and increased oscillation. This is because the feedforward gain $K_2$ is calculated based on nominal parameters; when the actual parameters change, the perfect cancellation condition is violated. In contrast, the proposed output regulation-based controller, with its embedded internal model and robust stabilization loop, adapts to the changing dynamics, maintaining excellent tracking and low THD. This comparison conclusively validates the superior robustness of the internal model-based approach for controlling a grid connected inverter subject to real-world parameter uncertainties.
Discussion and Implementation Aspects
The presented control strategy offers a systematic and theoretically grounded solution for the grid connected inverter. The internal model, designed for a specific frequency $\omega$, guarantees zero asymptotic error for the fundamental component. To address harmonic disturbances (e.g., grid voltage harmonics), the exosystem model and the internal model can be extended to include multiple harmonic frequencies (e.g., $5\omega$, $7\omega$, etc.), following the same principle. This would make the controller a generalized proportional-resonant (PR) controller with inherent robustness properties.
The design requires the selection of Hurwitz matrices $M_i$ for the internal model and the stabilization gains $k_i$. The matrices $M_i$ determine the convergence speed of the internal model states and can be tuned for desired dynamics. The gains $k_i$ must be chosen large enough to satisfy the conditions derived from the Lyapunov analysis, ensuring stability for all $w \in \mathcal{W}$. This provides a clear, albeit conservative, tuning guideline. In practice, these gains can be adjusted within these bounds to optimize transient response, such as settling time and overshoot, for the specific grid connected inverter hardware.
An important practical consideration is the required measurements. The proposed controller as formulated uses full state feedback ($x_1, x_2, x_3$). In many industrial applications, to reduce cost and complexity, only the grid current $x_1$ and possibly the capacitor voltage $x_2$ are measured. Future work naturally extends this design to an output-feedback framework by incorporating state observers. The separation principle typically holds in such linear-like error systems, allowing the combination of the internal model-based controller with a Luenberger or $H_\infty$ observer to estimate the unmeasured states (like the inverter-side current $x_3$), leading to a practical, sensor-reduced implementation without sacrificing the core robustness benefits.
Conclusion
This article has presented a robust current tracking control solution for three-phase grid connected inverters equipped with LCL filters. By formulating the problem within the output regulation framework, the design systematically addresses the dual challenge of asymptotic reference tracking and sinusoidal disturbance rejection. The core of the solution is the incorporation of an internal model that learns the necessary steady-state input signal corresponding to perfect tracking. Through a series of coordinate transformations, the complex tracking problem is converted into a robust stabilization problem for an interconnected lower-triangular system, which is solved using a dynamic state feedback control law.
The key advantages of this approach for the grid connected inverter are its explicit robustness to parameter uncertainties within a known compact set and its structured, step-by-step design procedure. Unlike feedforward strategies or controllers reliant on exact feedback linearization, this method does not require precise real-time knowledge of the disturbance or exact system parameters to achieve zero steady-state error in principle. Simulations confirmed the effectiveness and robustness of the controller, showing minimal performance degradation under significant LCL filter parameter variations and maintaining output current THD below stringent limits. The comparative analysis with a model-based feedforward controller highlighted the critical importance of robustness in practical inverter applications where parameters can drift. The proposed method provides a powerful and theoretically sound alternative for designing high-performance, reliable controllers for modern renewable energy integration systems.