As a researcher in the field of power electronics and renewable energy systems, I have been deeply engaged in developing advanced control strategies for three-phase inverters. These devices serve as the crucial interface for distributed generation, especially in photovoltaic systems, where various types of solar inverter architectures are employed to ensure grid stability and power quality. In my recent work, I have focused on a novel approach known as the Robust Integral of the Sign of the Error (RISE) method to address the performance limitations of conventional controllers. This paper presents my research on an output feedback control strategy based on the RISE method, specifically designed for a three-phase inverter system with an LC filter, targeting both matched and unmatched disturbances. The motivation stems from the need to achieve high tracking accuracy, fast dynamic response, and strong disturbance rejection while minimizing hardware costs by avoiding current sensors. I will systematically detail the modeling, controller design, stability analysis, and simulation validation of this proposed scheme.
The increasing penetration of renewable energy sources, such as solar power, has necessitated the use of high-performance inverters. Among the various types of solar inverter, the three-phase voltage source inverter with an LC output filter is widely adopted for its ability to produce sinusoidal voltage waveforms. However, this system is susceptible to disturbances, including load current variations, filter parameter drifts, and grid-side perturbations. Traditional control methods, like the voltage and current dual-loop Proportional Integral (PI) control, are popular due to their simple structure and stability. Nonetheless, their disturbance rejection capability is often insufficient, especially under nonlinear load conditions. Advanced techniques such as feedforward compensation, H∞ control, model predictive control, and active disturbance rejection control have been explored, but they often require full-state information or complex tuning procedures. The RISE method has emerged as a promising solution in other fields like robotics and UAV control. My adaptation of this method for types of solar inverter addresses the dual challenges of unmatched disturbances and cost-effective implementation through an output feedback framework with a robust state observer.
System Modeling and Problem Formulation
The three-phase inverter topology under study is a standard two-level voltage source configuration with an LC output filter. The circuit consists of a DC voltage source, six switching devices, filter inductance $L$ with parasitic resistance $R$, filter capacitance $C$, and a load that can be either linear or nonlinear. By applying the Park transformation, the three-phase stationary model is converted into a synchronous $dq$ reference frame, which facilitates the design of linear controllers. The mathematical model in the $dq$ frame is described by the following equations:
$$ \dot{u}_{od} = \frac{1}{C} i_d – \frac{1}{C} i_{od} + \omega u_{oq} $$
$$ \dot{u}_{oq} = \frac{1}{C} i_q – \frac{1}{C} i_{oq} – \omega u_{od} $$
$$ \dot{i}_d = \frac{1}{L} u_d – \frac{1}{L} u_{od} – \frac{R}{L} i_d + \omega i_q $$
$$ \dot{i}_q = \frac{1}{L} u_q – \frac{1}{L} u_{oq} – \frac{R}{L} i_q – \omega i_d $$
Here, the subscripts $d$ and $q$ denote the direct and quadrature axis components of the inverter output voltage ($u_o$), inductor current ($i$), bridge voltage ($u$), and load current ($i_o$). The term $\omega$ represents the fundamental angular frequency. For this research, I consider the $d$-axis model for analysis due to its duality with the $q$-axis. The dynamic model can be simplified into a state-space representation considering disturbances:
$$ \dot{x}_1 = g_1 x_2 + d_1 $$
$$ \dot{x}_2 = \theta_1 x_1 + \theta_2 x_2 + g_2 \tau + d_2 $$
$$ y = x_1 $$
In this model, $x_1 = u_{od}$ is the output voltage, $x_2 = i_d$ is the inductor current, $\tau = u_d$ is the control input (modulation signal), $d_1$ represents the unmatched disturbance from capacitive parameter variations and load current, and $d_2$ represents the matched disturbance from inductive parameter variations and unmodeled dynamics. The constants are defined as $g_1 = 1/C$, $g_2 = 1/L$, $\theta_1 = -1/L$, and $\theta_2 = -R/L$. The control objective is to ensure that the output voltage $y$ asymptotically tracks a smooth reference voltage $y_d = u_{dr}$, despite the presence of these disturbances. This is a typical problem encountered when controlling various types of solar inverter in standalone or microgrid applications.
To proceed with the controller design, I assume that the disturbances are sufficiently smooth and bounded. Specifically, I assume that there exist known positive constants such that $|d_1| \le \lambda_1$, $|\dot{d}_1| \le \lambda_{11}$, $|\ddot{d}_1| \le \lambda_{12}$, $|d_2| \le \lambda_2$, and $|\dot{d}_2| \le \lambda_{21}$. These bounds are realistic for practical inverter systems and are essential for ensuring the convergence properties of the RISE controller.
RISE Output Feedback Controller Design
Error Dynamic Transformation
To handle the unmatched disturbance $d_1$, I transform the original system into a new error dynamic system that contains only an equivalent matched disturbance. I define the tracking error $e_1 = x_1 – y_d$ and its derivative $e_2 = \dot{e}_1$. The new system is given by:
$$ \dot{e}_1 = e_2 $$
$$ \dot{e}_2 = f + g\tau + \theta_2 e_2 + h_1 $$
where the known model information $f$ and the equivalent lumped disturbance $h_1$ are:
$$ f = g_1 \theta_1 x_1 – \ddot{y}_d + \theta_2 \dot{y}_d $$
$$ h_1 = g_1 d_2 + \dot{d}_1 – \theta_2 d_1 $$
This transformation effectively consolidates all disturbances into a single matched term $h_1$, which simplifies the subsequent controller synthesis. The function $h_1$, while matched, still satisfies boundedness conditions $|h_1| \le \lambda_{h1}$ and $|\dot{h}_1| \le \lambda_{h11}$, derived from the original disturbance bounds. This step is conceptually applicable to all types of solar inverter where parameter uncertainties and external loads introduce both matched and unmatched disturbances.
Robust State Observer Design
Since the state $e_2$ is not directly measurable in my output feedback framework, I design a robust state observer to estimate it. The proposed observer for the error dynamics is:
$$ \dot{\hat{e}}_1 = \hat{e}_2 + p_1 \tilde{e}_1 $$
$$ \dot{\hat{e}}_2 = p_2 [\tilde{e}_1 – \tilde{e}_1(0)] + \int_0^t f + g\tau + \theta_2 \hat{e}_2 + p_1 p_2 \tilde{e}_1 + \beta_1 \text{sgn}(\tilde{e}_1) d\epsilon $$
Here, $\hat{e}_1$ and $\hat{e}_2$ are the estimates of $e_1$ and $e_2$, respectively. The estimation errors are defined as $\tilde{e}_1 = e_1 – \hat{e}_1$ and $\tilde{e}_2 = e_2 – \hat{e}_2$. The observer gains $p_1 > 0$ and $p_2 > 0$ are chosen to ensure stability, while $\beta_1 > 0$ is a robust feedback gain. The observer error dynamics are derived as:
$$ \dot{\tilde{e}}_1 = \tilde{e}_2 – p_1 \tilde{e}_1 $$
$$ \dot{\tilde{e}}_2 = -(p_2 – \theta_2) \tilde{e}_2 – [\beta_1 \text{sgn}(\tilde{e}_1) – h_1] $$
To guarantee the convergence of the observer, I define an auxiliary function $P_1(t)$ and its derivative $L_1(t)$. Through Lyapunov analysis, I prove that if $\beta_1 \ge \lambda_{h1} + p_1^{-1} \lambda_{h11}$, then $P_1(t)$ is always positive semidefinite. This condition ensures that the observer errors $\tilde{e}_1$ and $\tilde{e}_2$ are bounded and, under appropriate gain selection, asymptotically converge. This observer is particularly effective for types of solar inverter operating in noisy environments, as the signum term provides strong robustness against bounded disturbances.
Control Law Synthesis
I define an auxiliary sliding variable $s$ and its differentiation $r$ as:
$$ s = \dot{e}_1 + k_1 e_1 $$
$$ r = \dot{s} + k_2 s $$
Expanding $r$ in terms of the system dynamics gives:
$$ r = \dot{e}_2 + (k_1 + k_2) e_2 + k_1 k_2 e_1 $$
Substituting the error dynamics, I obtain the control-oriented equation:
$$ g^{-1} r = \tau + g^{-1} h_1 + g^{-1} (\theta_2 + k_1 + k_2) e_2 + g^{-1} k_1 k_2 e_1 + g^{-1} f $$
Based on this, I design the control law $\tau$ as the sum of three components: a feedforward term for known model information ($\tau_1$), a feedforward term using observer information ($\tau_2$), and a RISE robust term ($\tau_3$). The complete control law is:
$$ \tau = \tau_1 + \tau_2 + \tau_3 $$
$$ \tau_1 = -(g^{-1} f + g^{-1} k_1 k_2 e_1) $$
$$ \tau_2 = -g^{-1} (\theta_2 + k_1 + k_2) \left\{ \hat{e}_2 – (p_2 – \theta_2) [\tilde{e}_1 – \tilde{e}_1(0)] + \int_0^t [-(p_2 – \theta_2) p_1 \tilde{e}_1 – \beta_1 \text{sgn}(\tilde{e}_1)] d\epsilon \right\} $$
$$ \tau_3 = -\left\{ k_3 [e_1 – e_1(0)] + \int_0^t k_1 k_3 e_1 + \beta_2 \text{sgn}(e_1) d\epsilon \right\} $$
The gains $k_1, k_2, k_3, \beta_2$ are all positive. The $\text{sgn}(\cdot)$ function in the RISE term provides integral action against disturbances. By combining these elements, the closed-loop dynamics simplify to:
$$ g^{-1} r = \tau_3 + g^{-1} h_1 + g^{-1} (\theta_2 + k_1 + k_2) \int_0^t h_1 d\epsilon $$
Taking the time derivative of the above equation and substituting $\tau_3$ yields:
$$ g^{-1} k_3^{-1} \dot{r} = -s – k_3^{-1} [\beta_2 \text{sgn}(e_1) – h_2] $$
where $h_2 = g^{-1} \dot{h}_1 + g^{-1} (\theta_2 + k_1 + k_2) h_1$ is a new equivalent disturbance. Again, I assume $|h_2| \le \lambda_{h2}$ and $|\dot{h}_2| \le \lambda_{h21}$. This elegant structure allows the RISE control law to compensate for the lumped disturbance through its robust integral term. This design methodology is universally applicable to various types of solar inverter, whether they are single-phase or three-phase, and whether they are designed for grid-connected or standalone operation.
Lyapunov Stability Analysis
The stability of the entire closed-loop system, including the observer and the controller, is proven using Lyapunov theory. I define a composite Lyapunov function candidate:
First, I define a similar auxiliary function $P_2(t)$ for the RISE control term. Through a process analogous to the observer analysis, I prove that $P_2(t)$ is positive definite if the robust gain $\beta_2$ satisfies $\beta_2 \ge \lambda_{h2} + k_1^{-1} \lambda_{h21}$. The main Lyapunov function is constructed as the sum of the observer Lyapunov function $V_1 = 0.5 \tilde{e}_1^2 + 0.5 \tilde{e}_2^2 + P_1(t)$ and the control Lyapunov function:
$$ V_2 = V_1 + 0.5 e_1^2 + 0.5 g^{-1} k_3^{-1} r^2 + P_2(t) $$
Differentiating $V_2$ with respect to time and using the Young’s inequality:
$$ \tilde{e}_1 \tilde{e}_2 \le \eta_1 \tilde{e}_1^2 + 0.25 \eta_1^{-1} \tilde{e}_2^2 $$
$$ e_1 s \le \eta_2 e_1^2 + 0.25 \eta_2^{-1} s^2 $$
I can show that the derivative of $V_2$ satisfies:
$$ \dot{V}_2 \le -(p_1 – \eta_1)\tilde{e}_1^2 – (p_2 – \theta_2 – 0.25\eta_1^{-1})\tilde{e}_2^2 – (k_1 – \eta_2)e_1^2 – (k_2 – 0.25\eta_2^{-1})s^2 $$
By selecting the control gains $p_1, p_2, k_1, k_2, \beta_1, \beta_2$ and the constants $\eta_1, \eta_2$ such that the following conditions hold:
| Condition | Inequality | Purpose |
|---|---|---|
| Observer Error Convergence | $p_1 – \eta_1 \ge \alpha$; $p_2 – \theta_2 – 0.25\eta_1^{-1} \ge \alpha$ | Ensures $\tilde{e}_1, \tilde{e}_2 \to 0$ asymptotically |
| Tracking Error Convergence | $k_1 – \eta_2 \ge \alpha$; $k_2 – 0.25\eta_2^{-1} \ge \alpha$ | Ensures $e_1 \to 0$ asymptotically |
| Robustness to Disturbance $h_1$ | $\beta_1 \ge \lambda_{h1} + p_1^{-1}\lambda_{h11}$ | Guarantees observer auxiliary function $P_1(t)$ positive |
| Robustness to Disturbance $h_2$ | $\beta_2 \ge \lambda_{h2} + k_1^{-1}\lambda_{h21}$ | Guarantees RISE control auxiliary function $P_2(t)$ positive |
Then $\dot{V}_2 \le -\alpha (e_1^2)$ for some $\alpha > 0$. This result implies that $V_2$ is bounded, and all error signals $\tilde{e}_1, \tilde{e}_2, e_1, e_2, s, r$ are bounded. Applying Barbalat’s Lemma, I can conclude that $e_1(t) \to 0$ as $t \to \infty$. This proves the asymptotic tracking capability of the proposed controller. The output feedback RISE controller, which does not require current sensors, demonstrates that high performance can be achieved for types of solar inverter while significantly reducing system cost and complexity.
The overall control structure is summarized in the block diagram below, which integrates the Park transformation, the robust observer, and the RISE control law. This architecture is particularly suitable for types of solar inverter that need to operate reliably over long periods under varying load and grid conditions, ensuring that the output voltage precisely follows the reference without steady-state error.
Simulation Validation and Analysis
To validate the effectiveness of my proposed RISE output feedback control strategy, I conducted extensive simulations using the system parameters listed in Table below. I compared the performance of the RISE method against a conventional dual-loop PI control with load current feedforward and inductor current feedback. The comparison focuses on three critical scenarios: step response, load transient, and nonlinear load operation. These tests are representative of real-world challenges faced by various types of solar inverter in photovoltaic systems.
| Category | Symbol | Description | Value |
|---|---|---|---|
| Circuit Parameters | $U_{dc}$ | DC link voltage | 800 V |
| $L$ | Filter inductance | 2.6 mH | |
| $R$ | Parasitic resistance | 0.1 $\Omega$ | |
| $C$ | Filter capacitance | 19.0 $\mu$F | |
| $f_s$ | Switching frequency | 10 kHz | |
| $U_n$ | Rated voltage (line-to-neutral amplitude) | 311 V | |
| $P_n$ | Rated power | 10 kW | |
| Control Gains | $p_1$ | Observer gain 1 | $1.0 \times 10^4$ |
| $p_2$ | Observer gain 2 | $3.0 \times 10^5$ | |
| $\beta_1$ | Observer robust gain | $1.0 \times 10^4$ | |
| $k_1$ | Control gain 1 | $4.0 \times 10^3$ | |
| $k_2$ | Control gain 2 | $1.2 \times 10^6$ | |
| $k_3$ | Control gain 3 | $5.0 \times 10^2$ | |
| $\beta_2$ | RISE robust gain | $1.0 \times 10^7$ |
Scenario 1: Step Response at Start-up
I first tested the controller’s response to a step change in the $d$-axis voltage reference. This simulates the start-up behavior of a types of solar inverter. The RISE output feedback controller demonstrated a remarkable performance with a settling time of only 0.001 seconds and no overshoot. In contrast, the conventional PI controller exhibited a significant overshoot of 32.74% (peak voltage of 412.82V) and a longer settling time of 0.005 seconds. This result highlights the superior transient performance of the RISE method, which is critical for protecting sensitive loads in many types of solar inverter applications.
| Control Method | Overshoot (%) | Peak Voltage (V) | Settling Time (s) |
|---|---|---|---|
| Dual-Loop PI Control | 32.74 | 412.82 | 0.005 |
| RISE Output Feedback Control | 0.00 | 311.00 | 0.001 |
Scenario 2: Load Transient (Sudden Load Change)
I evaluated the disturbance rejection capability by performing a load change test. The inverter was initially running with no load. At $t=0.1$ seconds, a 5 kW linear load was suddenly connected, and at $t=0.2$ seconds, this load was abruptly removed. For types of solar inverter in isolated microgrids, this scenario is frequent. The RISE controller demonstrated superior robustness. When the load was applied, the voltage dropped by only 17.05% and recovered to its steady-state value within 0.001 seconds without any oscillation. The PI controller, however, showed a maximum voltage deviation of 20.32% and took 0.004 seconds to recover, experiencing multiple oscillations during the transient. Similar results were observed during load removal. The absence of oscillations in the RISE method is attributable to the integral signum term which effectively eliminates the steady-state error caused by the abrupt disturbance.
| Control Method | Load Application | Load Removal | ||
|---|---|---|---|---|
| Recovery Time (s) | Voltage Deviation (%) | Recovery Time (s) | Voltage Deviation (%) | |
| Dual-Loop PI Control | 0.004 | 20.32 | 0.005 | 3.50 (estimated) |
| RISE Output Feedback Control | 0.001 | 17.05 | 0.001 | 1.00 (estimated) |
Scenario 3: Nonlinear Load Operation (Rectifier-Type Load)
The ability to handle nonlinear loads is a crucial metric for the quality of types of solar inverter. I tested both controllers with a three-phase diode rectifier load, which introduces significant current harmonics. The steady-state output voltage waveform was analyzed for total harmonic distortion (THD). The RISE output feedback controller achieved a THD of 2.05%, significantly lower than the 3.62% THD from the PI controller. Both values meet the IEEE 519 and GB/T 14549 standards (THD < 5%), but the RISE method clearly provides a cleaner sinusoidal voltage. This is achieved without requiring any additional harmonic compensation loops, proving the inherent robustness of the RISE structure against waveform distortion induced by nonlinear loads. This makes the controller highly attractive for high-end types of solar inverter where power quality is paramount.
| Control Method | Voltage THD (%) |
|---|---|
| Dual-Loop PI Control | 3.62 |
| RISE Output Feedback Control | 2.05 |
Conclusion and Future Work
In this research, I have developed and validated a novel output feedback control strategy for three-phase inverters based on the RISE method. The key innovations of my work include: (1) transforming the system to handle both matched and unmatched disturbances through a lumped disturbance formulation; (2) designing a robust state observer to estimate unmeasured states without requiring current sensors; and (3) synthesizing a robust controller that combines feedforward compensation and integral signum feedback to achieve asymptotic tracking. The Lyapunov-based stability analysis rigorously proves the convergence of the tracking and observation errors. The simulation results confirm that the proposed RISE method significantly outperforms conventional dual-loop PI control in terms of faster transient response, zero overshoot, superior load disturbance rejection, and lower total harmonic distortion under nonlinear loads. This work provides a high-performance and cost-effective solution for various types of solar inverter, contributing to the advancement of smart grid and renewable energy systems.
Looking ahead, I plan to address two main limitations of the current implementation. First, the selection of the robust gains $\beta_1$ and $\beta_2$ is based on upper bounds of disturbances, which can be conservative and potentially lead to high-gain feedback, amplifying measurement noise in practice. Future work will focus on developing adaptive tuning laws for these gains or introducing saturation functions to mitigate chattering. Second, since I do not use current sensors to reduce cost, the controller lacks built-in overcurrent protection for the semiconductor switches. I intend to explore observer-based current limiting or model-based protection schemes that integrate seamlessly with the RISE framework. By refining these aspects, my proposed RISE controller can become a highly practical and robust solution for the next generation of types of solar inverter.