In the context of advancing renewable energy integration, the control of single-phase inverters has become a critical area of research. These inverters are essential for converting DC power from sources like solar panels into AC power for grid connection or standalone applications. However, achieving high-quality output voltage with low total harmonic distortion (THD) and fast dynamic response remains challenging, especially under nonlinear loads. Traditional control methods, such as proportional-integral (PI) control, often struggle with zero steady-state error tracking for sinusoidal signals, while other strategies like repetitive control or deadbeat control may suffer from computational complexity or sensitivity to model inaccuracies. In this work, we propose a novel approach combining state feedback control with proportional resonant (PR) control to enhance the performance of single-phase inverters. This method leverages state feedback for rapid dynamic response and PR control for precise steady-state tracking, resulting in improved output voltage quality. We begin by deriving a mathematical model of the single-phase inverter, followed by detailed controller design and experimental validation. The results demonstrate that our approach effectively reduces THD and ensures accurate voltage tracking, making it suitable for real-world applications in power electronics.
The single-phase inverter topology considered in this study consists of a full-bridge configuration using MOSFETs and an LC filter to smooth the output waveform. Key parameters include the DC bus voltage, filter inductance and capacitance, and load resistance. The system operates by switching the MOSFETs to generate an AC voltage from the DC input, with the LC filter attenuating high-frequency harmonics. To analyze and control this system, we first develop a state-space model that captures the dynamics of the inverter output voltage and current. This model forms the foundation for designing our control strategy, which aims to achieve robust performance under various load conditions.
The state-space representation of the single-phase inverter is derived from the differential equations governing the circuit. For the LC filter and load, the equations are based on Kirchhoff’s laws. The continuous-time state-space model is given by:
$$ \dot{x} = A x + B u $$
where \( x = [u_c, \dot{u}_c]^T \) represents the state vector (output voltage and its derivative), \( u \) is the control input (inverter-side voltage), and the matrices \( A \) and \( B \) are defined as:
$$ A = \begin{bmatrix} 0 & 1 \\ -\frac{1}{2LC} & -\frac{1}{RC} \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ \frac{1}{2LC} \end{bmatrix} $$
Here, \( L \) and \( C \) are the filter inductance and capacitance, respectively, and \( R \) is the load resistance. This model assumes a linear load, but we extend it to include nonlinear loads later. To implement digital control, we discretize the system using a sampling period \( T \), resulting in the discrete-time state-space model:
$$ x(k+1) = \Phi x(k) + \Gamma u(k) $$
where \( \Phi \) and \( \Gamma \) are the discrete-time system matrices. For instance, the elements of \( \Phi \) and \( \Gamma \) can be computed as:
$$ \Phi = \begin{bmatrix} \alpha_{11} & \alpha_{12} \\ \alpha_{21} & \alpha_{22} \end{bmatrix}, \quad \Gamma = \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} $$
with:
$$ \alpha_{11} = 1 – \frac{T^2}{4LC}, \quad \alpha_{12} = T – \frac{T^2}{2RC}, \quad \alpha_{21} = \frac{T^2}{4RLC^2} – \frac{T}{2LC}, \quad \alpha_{22} = 1 + \frac{T^2}{(RC)^2} – \frac{T^2}{4LC} – \frac{T}{RC}, \quad b_1 = \frac{T^2}{4LC}, \quad b_2 = \frac{T}{2LC} – \frac{T^2}{4RLC^2} $$
This discrete model allows us to design a state feedback controller in the z-domain, which is crucial for real-time implementation on digital signal processors. The control law for state feedback is expressed as:
$$ u(k) = -K x(k) + k_{\text{ref}} u^*(k) $$
where \( K = [k_1, k_2] \) is the state feedback gain vector, \( u^*(k) \) is the reference voltage, and \( k_{\text{ref}} \) is a scaling factor. The gains \( k_1 \) and \( k_2 \) are determined by pole placement, where we aim to position the closed-loop poles within the unit circle to ensure stability and desired dynamic response. For example, by setting the poles at \( 0.3 \pm j0.6 \) in the z-plane, we solve for \( k_1 \) and \( k_2 \) using algebraic methods or numerical tools like MATLAB. This results in fast response times, as evidenced by the unit step response of the closed-loop system, which shows minimal overshoot and rapid settling.
To further improve steady-state accuracy, we incorporate a proportional resonant (PR) controller into the system. The PR controller is designed based on the internal model principle, which ensures zero steady-state error for sinusoidal signals by including a model of the reference frequency in the controller. The transfer function of the PR controller is:
$$ G_{\text{PR}}(s) = K_P + \frac{2K_R s}{s^2 + \omega_0^2} $$
where \( K_P \) is the proportional gain, \( K_R \) is the resonant gain, and \( \omega_0 \) is the fundamental angular frequency (e.g., 50 Hz or 60 Hz). The PR controller provides infinite gain at the resonant frequency, enabling perfect tracking of AC signals. However, the parameters \( K_P \) and \( K_R \) must be carefully selected to balance performance and stability. We use root locus analysis to tune these parameters. First, we set \( K_R = 0 \) to determine \( K_P \) based on the desired crossover frequency. For a sampling frequency of 18 kHz, we choose the voltage loop crossover frequency as one-sixth of this value, leading to:
$$ \left| \frac{K_P k_{\text{pwm}}}{2LC s^2 + k_{\text{pwm}} C s + 1} \right| = 1 \quad \text{at} \quad s = j \cdot 6000\pi $$
Solving this equation yields \( K_P = 0.95 \). Next, we introduce the resonant term and plot the root locus with respect to \( K_R \). The characteristic equation of the closed-loop system becomes:
$$ 1 + \frac{2 k_{\text{pwm}} K_R s}{(2LC s^2 + k_{\text{pwm}} C s + 1 + k_{\text{pwm}} K_P)(s^2 + \omega_0^2)} = 0 $$
Analysis shows that the system remains stable for \( K_R < 1.22 \times 10^4 \), and we select \( K_R = 400 \) to achieve optimal dynamic performance without compromising stability. This combination of state feedback and PR control ensures that the single-phase inverter can handle both linear and nonlinear loads effectively.
The overall control system structure integrates state feedback and PR control in a cascaded manner. The state feedback controller provides rapid disturbance rejection and setpoint tracking, while the PR controller eliminates steady-state errors at the fundamental frequency and its harmonics. This is particularly important for nonlinear loads, such as rectifier circuits, which introduce harmonic distortions. The output voltage equation under this combined control strategy can be expressed as:
$$ y(s) = \frac{[G_c(s) + G_{\text{PR}}(s)] G_p(s)}{1 + [G_c(s) + G_{\text{PR}}(s)] G_p(s)} r(s) + \frac{1}{1 + [G_c(s) + G_{\text{PR}}(s)] G_p(s)} d(s) $$
where \( G_c(s) \) represents the state feedback controller, \( G_p(s) \) is the plant transfer function, \( r(s) \) is the reference signal, and \( d(s) \) is the disturbance. At the resonant frequency, \( G_{\text{PR}}(s) \to \infty \), so \( y(s) \to r(s) \), indicating perfect tracking.
To validate our approach, we conducted experiments using a PTS-1000 power electronics platform. The system parameters are summarized in Table 1 below.
| Parameter | Value |
|---|---|
| DC Bus Voltage (U_dc) | 70 V |
| Filter Inductance (L) | 0.665 mH |
| Filter Capacitance (C) | 10 µF |
| Load Resistance (R) | 42 Ω |
| Sampling Frequency | 18 kHz |
| Reference Output Voltage | 40 V |
We tested the single-phase inverter under both linear and nonlinear load conditions. For linear loads, the output voltage with state feedback alone showed a significant error (34.1 V vs. 40 V reference) and a THD of 2.37%. In contrast, with the combined state feedback and PR control, the output voltage reached 39.9 V with a THD of 1.67%. Under nonlinear loads, such as a rectifier with capacitive filtering, the state feedback controller resulted in a distorted output voltage of 33.9 V and a THD of 7.76%. However, with the addition of PR control, the output voltage improved to 39.8 V, and THD reduced to 3.40%, meeting the standard requirement of below 5%. These results highlight the effectiveness of our method in enhancing both dynamic and steady-state performance for single-phase inverters.
In terms of implementation, the control algorithm was programmed on a digital signal processor, with the state feedback gains and PR parameters calculated offline. The discrete control law was executed at each sampling interval, ensuring real-time responsiveness. We also analyzed the sensitivity of the system to parameter variations, such as changes in load resistance or filter components. The state feedback design provides inherent robustness due to pole placement, while the PR controller adapts to frequency variations through its resonant structure. For practical applications, this makes the single-phase inverter control system reliable in varying operating conditions.
Further, we explored the impact of different PR controller parameters on system stability using Bode plots and Nyquist criteria. The gain margin and phase margin were evaluated to ensure adequate stability margins. For instance, with \( K_P = 0.95 \) and \( K_R = 400 \), the phase margin was above 45 degrees, indicating good relative stability. Additionally, we considered the effect of multiple resonant controllers for harmonic compensation, but for simplicity, we focused on the fundamental frequency in this study. Future work could extend this to include selective harmonic compensation for improved performance under highly distorted loads.
The mathematical derivation of the state feedback controller involves solving the discrete-time algebraic Riccati equation for optimal control, but we used pole placement for simplicity. The closed-loop system matrix with state feedback is given by:
$$ \Phi_{\text{cl}} = \Phi – \Gamma K $$
and the characteristic equation is:
$$ \det(zI – \Phi_{\text{cl}}) = 0 $$
By specifying desired poles, we compute \( K \) such that the eigenvalues of \( \Phi_{\text{cl}} \) match these poles. For the PR controller, the discrete-time equivalent was implemented using the Tustin transformation to maintain accuracy. The discrete PR controller transfer function is:
$$ G_{\text{PR}}(z) = K_P + \frac{2K_R \omega_0 T (z – 1)}{(z^2 – 2\cos(\omega_0 T) z + 1)} $$
This allows seamless integration with the state feedback controller in the digital domain.
In conclusion, our research demonstrates that the combination of state feedback and proportional resonant control offers a robust solution for single-phase inverter applications. This approach addresses key challenges in power electronics, such as fast dynamic response and low THD, making it suitable for renewable energy systems. The experimental results confirm that the proposed control strategy outperforms traditional methods, providing high-quality output voltage under various load conditions. As the demand for efficient power conversion grows, further refinements in control algorithms will continue to enhance the performance of single-phase inverters in modern grid applications.
To summarize the key equations and parameters, we provide Table 2 below, which outlines the main formulas used in the controller design.
| Component | Equation |
|---|---|
| State-Space Model (Continuous) | $$ \dot{x} = A x + B u $$ with $$ A = \begin{bmatrix} 0 & 1 \\ -\frac{1}{2LC} & -\frac{1}{RC} \end{bmatrix}, B = \begin{bmatrix} 0 \\ \frac{1}{2LC} \end{bmatrix} $$ |
| Discrete Model | $$ x(k+1) = \Phi x(k) + \Gamma u(k) $$ with $$ \Phi = \begin{bmatrix} \alpha_{11} & \alpha_{12} \\ \alpha_{21} & \alpha_{22} \end{bmatrix}, \Gamma = \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} $$ |
| State Feedback Control | $$ u(k) = -K x(k) + k_{\text{ref}} u^*(k) $$ |
| PR Controller | $$ G_{\text{PR}}(s) = K_P + \frac{2K_R s}{s^2 + \omega_0^2} $$ |
| Closed-Loop Transfer Function | $$ y(s) = \frac{[G_c(s) + G_{\text{PR}}(s)] G_p(s)}{1 + [G_c(s) + G_{\text{PR}}(s)] G_p(s)} r(s) + \frac{1}{1 + [G_c(s) + G_{\text{PR}}(s)] G_p(s)} d(s) $$ |
Overall, this work underscores the importance of advanced control techniques in power electronics and provides a practical framework for implementing state feedback and PR control in single-phase inverters. The methods described here can be adapted to other inverter topologies and applications, contributing to the ongoing development of efficient and reliable energy conversion systems.