Enhanced Three-Phase Inverter Control: Robust Sliding Mode Control with Washout Filter for Low Harmonics

Abstract

This paper presents a robust control strategy for three-phase inverters that combines Sliding Mode Control with a Washout Filter (SMC-w) to achieve low harmonic distortion and high dynamic stability. The proposed approach addresses the critical challenge of maintaining the stability of a high-quality output signal while ensuring robustness against disturbances and adaptability under variable, unbalanced, and nonlinear loads. The proposed hybrid controller integrates the fast response and disturbance rejection capability of SMC with the filtering properties of the washout stage, effectively mitigating low-frequency chattering and steady-state offsets. A detailed stability analysis is provided to ensure the closed-loop convergence of the SMC–w. Simulation results obtained in MATLAB–Simulink demonstrate significant improvements in transient response, total harmonic distortion, and robustness under unbalanced and nonlinear load conditions compared to conventional control methods. The inverter demonstrated rapid tracking of the reference signals with a minimal error margin of 3%, effective frequency regulation with a low steady-state error, and resilience to input disturbances and load variations. For instance, under a load variation from 20 Ω to 5 Ω, the system maintained the output voltage accuracy within a 3% error threshold. In addition, the input perturbations and frequency shifts in the reference signals were effectively rejected, confirming the robustness of the control strategy. Furthermore, the integration of the SMC proved to be highly effective in reducing harmonic distortion and delivering a stable and high-quality sinusoidal output. The integration of the washout filter minimized the chattering phenomenon typically associated with the SMC, further enhancing the smooth response and reliability of the system. This study highlights the potential of SMC–w to optimize power quality and operational stability. This study offers significant insights into the development of advanced inverter systems that can operate in dynamic and challenging environments.

1. Introduction

Three-phase inverters are important components for electrical energy conversion, particularly in industrial and renewable energy applications. They regulate the output voltage of each phase by applying pulse-width modulation (PWM) to each bridge branch, thereby enabling precise control of the generated waveform [1]. The quality of the output signal is crucial because a pure sine waveform minimizes harmonic distortion, which is vital for the proper operation of the connected loads [2,3]. However, designing a three-phase inverter that operates under variable, unbalanced, and nonlinear loads while maintaining a pure sinusoidal waveform with a low harmonic content is a technical challenge [4,5].

Harmonic distortion can negatively affect the performance of electrical equipment, causing overheating, reduced efficiency, and shorter lifespans of connected devices [6]. Three-phase inverters can produce zero-sequence currents under unbalanced load conditions or in the presence of nonlinear loads, further complicating the system control and stability [6]. This situation underscores the need for advanced control strategies to maintain output signal quality and adapt to load variations and external disturbances.

Conventional controllers based on proportional–integral–derivative (PID) can be designed to minimize the total harmonic distortion (THD) of the output waveform by carefully tuning the gains [7,8]. For example, in [9], a new approach was proposed for the direct torque control of a doubly fed induction motor using a PID controller optimized by a genetic algorithm (GA). The proposed method achieved better speed control, response time, rejection time, overshoot, and THD reduction. However, the study did not consider other types of load and did not present details related to other tests for complete three-phase systems with different loads.

Recent research on three-phase inverter control has explored hybrid strategies and fault-tolerant schemes that improve dynamic stability and harmonic suppression. For instance, Liu et al. [10] proposed an LCC-type DC grid-forming method and a fault ride-through strategy based on fault current limiters, demonstrating effective current limitation and grid fault resilience. This approach provides useful insights for developing robust inverter controls under grid disturbances.

Sinusoidal pulse-width modulation (SPWM) is a widely used modulation technique for inverters, because it can produce output waveforms with low harmonic content [11]. For example, a study proposed a harmonic suppression strategy for single-phase sinusoidal pulse width modulation (SPWM) inverters using mixed integer nonlinear programming [12]. The method reduced harmonic distortion in the output voltage of SPWM inverters and investigated the relationship between amplitude modulation (AM) depth and harmonic components, determining an optimal AM depth to efficiently reduce the harmonics. Another study focused on the selection method of the modulation index (ma) and frequency ratio (mf) to obtain the minimum harmonic distortion in a single-phase SPWM inverter [13], showing minimum harmonics of 11.48% before filtering and 2.45% after filtering. Another study focused on the analysis and suppression of harmonics in SPWM inverters [14], concluding that selecting an appropriate carrier frequency and injecting suitable 3rd harmonic components are effective and practical strategies for harmonic suppression in SPWM inverters. Additionally, a study used a modified Twelve-Sector Space Vector Pulse Width Modulation (TS-SVPWM) technique to reconstruct three-phase currents using a single current sensor [15]; they achieved a reduction in phase current harmonics. However, practical implementations still face challenges in achieving pure sinusoidal waveforms and harmonic suppression.

Multilevel inverter topologies, such as three-, five-, and cascaded H-bridges, can significantly reduce the harmonic content in the output waveform compared to traditional two-level inverters. For example, in ref. [16], a novel three-phase six-level multilevel inverter (MLI) topology was proposed and analyzed. The proposed MLI utilizes fewer power components than conventional and recently developed MLIs, while still generating a high number of voltage levels, producing a low-current total harmonic distortion (THD) of only 5.8%, making it suitable for various grid applications. Another example is the application of a single-phase five-level multilevel inverter to a grid-connected single-stage solar photovoltaic system [17], where the output voltage total harmonic distortion obtained was only 1.18%. Additionally, another study focused on comparing different optimization algorithms for Selective Harmonic Elimination Pulse Width Modulation (SHEPWM) in a five-phase multilevel inverter [18], eliminating the targeted lower-order harmonics (3rd and 7th) in the seven-level five-phase multilevel inverter. Although these studies offer advanced harmonic elimination, these methods increase the complexity and cost of implementation and cause reliability issues because they require more components to form topologies.

Another study focused on the design and implementation of a robust artificial neural network-based proportional integral derivative (ANN-PID) controller to improve the quality of photovoltaic (PV) energy injected into the grid [19]. The ANN-PID controller demonstrated superior performance compared to the conventional PID controller in terms of voltage regulation, with better tracking of the DC-link voltage reference and a significantly lower maximum overshoot. This method ensures better robustness and injects a current with a lower THD than the classical PID regulator. In addition, this method enhances the performance of three-phase inverters by addressing issues related to the intermittent nature of photovoltaic sources and the challenges in maintaining a constant DC bus voltage. Although these techniques are useful, they require various system parameters for proper operation and are difficult to implement. In addition, this method requires a longer time to reach a steady state than a conventional controller, resulting in a longer transient period to follow the reference.

Another study proposed a novel deep deterministic policy gradient (DDPG)-assisted integral reinforcement learning (IRL)-based control algorithm for three-phase DC/AC inverters with RL loads [20]. The proposed controller can autonomously update its control gains online without requiring knowledge of the system model. It achieves excellent steady-state and dynamic system responses with a reasonably low computational cost. Unlike most existing approaches, the proposed method is independent of the system model used for online control. It realizes autonomous online control gain adjustment by learning the solution to the H∞ control problem in real time. The DDPG technique was used to systematically determine the initial stabilizing gain for the IRL, addressing the limitations of previous studies. The proposed algorithm reduces the computational burden by eliminating the disturbance updating process, making it easier to implement in real-time applications. However, the implementation of deep learning and reinforcement learning techniques is more complex to implement than traditional or simple methods, as it requires training, computational resources, and tuning challenges.

Another study proposed an improved whale optimization algorithm (IWOA) to minimize the THD in MLIs [21]. The IWOA incorporates new features, such as a diffusion process, ranking system, and position updating techniques, to improve performance. The proposed IWOA outperformed nine other metaheuristic algorithms in minimizing THD for 5-level and 7-level MLIs across various performance parameters. The proposed method is effective in minimizing the THD across the full range of the modulation index from 0 to 1, which is an improvement over existing techniques. However, it requires the IWOA to introduce additional components, such as a novel diffusion process and new position updating techniques, which may increase the algorithmic complexity compared to simpler methods. Although the paper claims better computational efficiency, the added features of the IWOA might require more computational resources than some simpler algorithms, especially for higher-level MLIs. Minimal tuning of the algorithm parameters is claimed in the paper, but some level of tuning might still be required for optimal performance in different scenarios. Although the authors claimed that their method is applicable to higher-level MLIs and can be extended to three-phase systems, further tests are required to evaluate these assumptions.

Sliding mode control (SMC) is a robust technique that has proven to be effective in regulating nonlinear systems [22,23]. SMC can provide a better transient response, faster convergence, and higher accuracy than classical linear controllers, particularly in the presence of parameter uncertainties, external disturbances, and model nonlinearities [24,25,26]. For instance, a study based on grid-connected voltage source inverters showed that SMC controllers can improve system stability under variable load conditions [27]. An adaptive nonsingular terminal sliding mode control technique using a barrier function was designed for nonlinear systems with outdoor disturbances to improve performance and robust stability [28]. A study presented the use of SMC of a single-phase VSI with an inductor current observer for voltage reference tracking and THD removal against linear and nonlinear load disturbances [29]. In ref. [30], a study highlighted the effectiveness of second-order SMC in managing disturbances in various parts of a microgrid system, including the main DC bus, AC bus, and constant power loads. In ref. [31], SMC was used for a four-leg inverter in an uninterruptible power supply, reducing chattering, low total harmonic distortion (THD) values, fast dynamic response, and high robustness against parameter mismatches and model uncertainties. Thus, the SMC can handle disturbances and load variations [32], ensuring that the output signal remains within the desired range. Therefore, SMC offers a promising solution because it is robust against uncertainties and disturbances [27,33,34]. However, more research is needed to prove that this controller is useful for improving the operation under different load conditions and simplifying the construction of three-phase inverters.

In summary, the literature shows that PID requires additional techniques to improve harmonics, and obtaining a better sinusoidal signal requires more advanced techniques. Conventional control must handle chartering, subharmonics, and infinite-frequency commutation. Conventional techniques manage analog signals that create delays, which can be improved using digital control. SPWM is a better technique for improving the signal; however, practical implementations still face challenges in achieving a pure sinusoidal waveform, and better techniques are required to suppress the harmonics in three-phase systems. In addition, SPWM may result in higher switching losses and increased electromagnetic interference than advanced PWM schemes [35]. Furthermore, other techniques based on multilevel and artificial intelligence have increased complexity compared to conventional methods, which may necessitate higher-performance control hardware. Moreover, the multilevel and modulation techniques used to minimize harmonics [36,37,38] require several adjustments to provide good results and switches with different voltages, which makes them more complex to implement.

Therefore, this paper presents a three-phase inverter controlled by an SMC with a washout filter (SMC-w) to generate a pure sinusoidal waveform with low harmonics. This type of control offers a more stable and reliable operation of the inverter [39]. The SMC-w is designed to maintain a pure sinusoidal waveform while efficiently responding to disturbances. In addition, it is designed to mitigate the chattering phenomenon that is common in SMC controllers, resulting in a smoother response and fewer oscillations in the inverter output, thereby improving the signal quality [40]. The proposed design allows the inverter to adapt to varying load conditions and external disturbances, which is essential in practical applications where the operating conditions can fluctuate significantly [41]. The proposed system shows high robustness not only under variable RL loads but also under unbalanced and nonlinear loads, ensuring stability and waveform quality even under adverse conditions. Furthermore, the controller demonstrates resilience to electrical faults, such as ground faults and short circuits, while maintaining safe operation. Another relevant contribution is that the control strategy is independent of the inverter and load parameters, requires only one tuning constant, and is simple to implement in current digital electronics (DSPs or microcontrollers of any technology), without the need for high computational resources. This makes the proposed method practical, low-cost, and suitable for real-time applications. This technique has previously demonstrated to provide better voltage regulation, load disturbance rejection, and parameter variation handling than traditional PI or PWM control methods [42,43]. In addition, the harmonic distortion can be reduced to ensure that the output waveform satisfies the required standards [42].

The main contributions of this study are the following:

• The study introduces a hybrid control structure that integrates Sliding Mode Control (SMC) with a Washout Filter to improve the performance of three-phase inverters. This method offers enhanced robustness against disturbances and load variations, and achieves lower harmonic distortion compared to traditional SMC and PID control methods. The approach is particularly effective for inverters operating under variable RL, unbalanced, and nonlinear loads, producing a pure sine waveform with reduced harmonics, an area not widely covered in existing literature.
• The proposed controller requires only one parameter for implementation, simplifying use while achieving performance similar to PID control. It is independent of system parameters, unlike other controllers. Simulations show it is robust to load parameter variations, with no need for new settings.
• The proposed method offers robustness to external disturbances such as short circuits and power fluctuations by maintaining stable inverter output. A systematic methodology for tuning the control gain and filter cutoff frequency is established to balance transient speed and steady-state smoothness. A formal Lyapunov-based stability analysis is provided for the integrated control law.

The proposed method can inspire new control techniques that are beneficial for various electrical and electronic systems, improving system efficiency and stability [44]. The research and development of this type of inverter adds to the existing body of knowledge on electrical system control, offering new perspectives and methodologies for future studies [45]. This project represents a step forward in a three-phase inverter design, proposing an approach that combines robustness, adaptability, and efficiency, and setting a potential model for future developments in the industry [46].

The following sections are organized as follows. Section 2 describes the materials and methods, including the design of the three-phase inverter, SMC, and the implementation of the washout filter. Section 3 presents the results obtained from the simulation, evaluating the performance of the system in terms of the output signal stability, robustness against load variations, and harmonic distortion reduction. Section 4 provides conclusions, summarizing the key findings, highlighting the contributions of the study, and suggesting potential directions for future research on optimizing three-phase inverters in power systems.

2. Materials and Methods

2.1. Three-Phase DC-AC Inverter

The three-phase inverter is shown at the top of Figure 1, which is the main component responsible for converting the direct current (DC) from the voltage source $\mathrm{E}$ into alternating current (AC) in three phases, each 120° out-of-phase. This inverter uses six transistors, labeled $\mathrm{T}1 \mathrm{t}\mathrm{h}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h} \mathrm{T}6$, which are alternately activated to generate output signals in each phase (A, B, and C). The switching of the transistors is carried out through the control signals $u_1,u_2,\mathrm{a}\mathrm{n}\mathrm{d} u_3$, and the negated control signals ${\overline{u}}_1,{\overline{u}}_2,\mathrm{a}\mathrm{n}\mathrm{d} {\overline{u}}_3$, which are activated following a specific pattern to produce AC voltage waves in each phase. These signals can be generated using techniques such as pulse-width modulation (PWM) and SMC-w control, which allow the output voltage to be regulated according to the desired reference values.

Each output phase (A, B, and C) of the inverter is connected to an LC filter, which consists of an inductor and capacitor. These filters smoothen the waveform generated by the inverter and reduce the harmonic content of the output signal. In practice, the output signal of an inverter is usually a square wave owing to transistor switching, and LC filters convert this signal into a cleaner sinusoidal wave, which is ideal for powering sensitive loads. In each phase, inductors $L_{an}, L_{bn}, \mathrm{a}\mathrm{n}\mathrm{d} L_{cn}$ represent the inductances in phases A, B, and C. These inductors limit sudden changes in the current and eliminate unwanted high-frequency signals. The terms $C_{an}, C_{bn}, \mathrm{a}\mathrm{n}\mathrm{d} C_{cn}$ represent the capacitance in each phase, respectively. These capacitors helped stabilize the signal by smoothing the voltage variations. After the LC filters, each phase was connected to an RL load connected to phases A, B, and C.

The inverter converts the DC voltage from source $\mathrm{E}$ into three AC signals, each 120° out of phase, through the control of transistors $\mathrm{T}1 \mathrm{t}\mathrm{o} \mathrm{T}6$. The alternating signals generated in each phase pass through an LC filter, which smooths the signal and reduces high-frequency noise, resulting in a signal that is closer to a sinusoidal wave. Finally, the filtered signal was delivered to the load for each phase. The load consumed power from each phase in a balanced manner. A three-phase DC-AC inverter is an electronic circuit consisting of six transistors that allows efficient control of alternating current (AC) in three phases. This type of inverter is derived from three-phase buck converters [18]. The circuit shown in Figure 1 enables the generation of an AC signal in each output phase, supplying voltages ${\upsilon }_{an}\left(t\right),{\upsilon }_{bn}\left(t\right),{\upsilon }_{cn}\left(t\right)$ to the RL load connected to each phase. Using pulse-width modulation (PWM) and SMC-w control techniques, precise voltage regulation can be achieved for the load by controlling the six transistors in the converter. This control system allows the adjustment of the reference signals for each phase ${\upsilon }_{an}ref\left(t\right),{\upsilon }_{bn}ref\left(t\right),{\upsilon }_{cn}ref\left(t\right)$, thereby optimizing the voltage supply to the load under variable operating conditions.

2.2. Washout Filter

In this setup, the phase currents ${iL}_{an}\left(t\right)$, ${iL}_{bn}\left(t\right)$, and ${iL}_{cn}\left(t\right)$ are processed using washout filters as part of the three-phase control system. The washout filter applies the transfer functions $G_f\left(s\right)$, $G_{af}\left(t\right)$, $G_{bf}\left(t\right)$, and $G_{cf}\left(t\right)$, to generate the filtered signals $I_F\left(s\right): I_{Fan}\left(t\right)$, $I_{Fbn}\left(t\right)$, and $I_{Fcn}\left(t\right)$ [47,48]:

$$G_f\left(s\right)={\displaystyle \frac{I_F\left(s\right)}{iL\left(s\right)}}={\displaystyle \frac{s}{s+w}}=1-{\displaystyle \frac{w}{s+w}}$$

(1)

In this expression $iL\left(s\right)$ represents the phase currents, ${iL}_{an}\left(t\right)$, ${iL}_{bn}\left(t\right)$, and ${iL}_{cn}\left(t\right)$, s represents the Laplace variable, and $w$ is the cutoff frequency of the high-pass filter. After filtering each phase current, a differential equation is incorporated for each filter in the system to capture the transient response (see Equation (2)) [47,49].

$${\displaystyle \frac{dz}{dt}}=w\left(iL-z\right)$$

(2)

where $z$ represents the low-frequency component of the current signal. By integrating both sides of Equation (2), we obtain [47,49]

$$z=⎰w\left(iL-z\right)dt$$

(3)

This filter configuration allows the controller to focus on transient deviations in the phase currents, providing smoother and more responsive control within the three-phase system, as shown in Figure 1.

2.3. Sliding Mode Control

The circuit in Figure 1 represents a three-phase SMC, where each single-phase output of the three-phase inverter receives two signals: one voltage ${\upsilon }_{an}\left(t\right), {\upsilon }_{bn}\left(t\right) or {\upsilon }_{cn}\left(t\right)$, and one current signal per phase ${iL}_{an}\left(t\right), {iL}_{bn}\left(t\right) or {iL}_{cn}\left(t\right)$. For phase $a$, a sliding surface is defined as [42,48,49]:

$$h_a\left(x\right)={\upsilon }_{an}\left(t\right)-{\upsilon }_{an}ref\left(t\right)+k({iL}_{an}\left(t\right)-z_a)=0$$

(4)

The term ${\upsilon }_{an}\left(t\right)$ is the phase $a$ voltage at the load, ${\upsilon }_{an}ref\left(t\right)$ is the desired reference voltage for phase $a$, ${iL}_{an}\left(t\right)$ is the phase $a$ current passing through inductor $L_{an}$, and $z_a$ can be obtained from Equation (9), as shown in (10). Term $k$ is a control parameter that multiplies the filtered current and takes values greater than zero ($k$ > 0). Parameter $k$ can be adjusted in the controller to achieve different system responses, and it will be varied to study the system dynamics.

$${\displaystyle \frac{d{z}_a}{dt}}=w({iL}_{an}-z_a)$$

(5)

$$z_a=⎰w\left({iL}_{an}-z_a\right)dt$$

(6)

For phase $b$, the sliding surface is defined as [42,48,49]

$$h_b\left(x\right)={\upsilon }_{bn}\left(t\right)-{\upsilon }_{bn}ref\left(t\right)+k({iL}_{bn}\left(t\right)-z_b)=0$$

(7)

The term ${\upsilon }_{bn}\left(t\right)$ is the phase $b$ voltage at the load, ${\upsilon }_{bn}ref\left(t\right)$ is the desired reference voltage for phase $b$, ${iL}_{bn}\left(t\right)$ is the phase $b$ current passing through inductor $L_{bn}$, and $z_b$ is obtained by integrating Equation (8), and the resulting term is shown in Equation (9).

$${\displaystyle \frac{d{z}_b}{dt}}=w({iL}_{bn}-z_b)$$

(8)

$$z_b=⎰w\left({iL}_{bn}-z_b\right)dt$$

(9)

For phase $c$, the sliding surface is defined as [42,48,49]

$$h_c\left(x\right)={\upsilon }_{cn}\left(t\right)-{\upsilon }_{cn}ref\left(t\right)+k({iL}_{cn}\left(t\right)-z_c)=0$$

(10)

The term ${\upsilon }_{cn}\left(t\right)$ is the phase $c$ voltage at the load, ${\upsilon }_{cn}ref\left(t\right)$ is the desired reference voltage for phase $c$, ${iL}_{cn}\left(t\right)$ is the phase $c$ current passing through the inductor $L_{cn}$, $z_c$ is obtained by integrating Equation (11), and the resulting term is presented in Equation (12).

$${\displaystyle \frac{d{z}_c}{dt}}=w({iL}_{cn}-z_c)$$

(11)

$$z_c=⎰w\left({iL}_{cn}-z_c\right)dt$$

(12)

A control law is then applied to determine the final output signal depending on the two switching states, as shown in Equation (13). Here, $u$ is a scalar that depends on $h_a\left(x\right),$ which can take the value $u=0$ if or $u=1$ if $h_a\left(x\right)<0$ [48]:

$$h_{a}u=\left\{\begin{array}{c}0\\ 1\end{array}\right.\begin{array}{ll}& \end{array}\begin{array}{l}if\\ if\end{array}\begin{array}{ll}& \end{array}\begin{array}{l}h\left(x\right)>0\\ h\left(x\right)<0\end{array} \left(x\right)>0$$

(13)

For phase $a$, the control input $u_1=u$ is required for transistor T1, located at the top of the first arm of the inverter, whereas ${\overline{u}}_1=\overline{u}$ is the control input for the same arm but at the bottom, corresponding to transistor T4. The same configuration applies to phases $b$ and $c$: $u_2$ is the control input for transistor T2 and ${\overline{u}}_2$ for transistor T5, $u_3$ is the control input for transistor T3 and ${\overline{u}}_3$ for transistor T6. A complete explanation is provided in Figure 1. Finally, in all the results shown later, the percentage error was calculated using Equation (14) for phase a, and the same was applied to phases b and c.

$$Error={\displaystyle \frac{{\upsilon }_{an}ref\left(t\right)-{\upsilon }_{an}\left(t\right)}{{\upsilon }_{an}ref\left(t\right)}}\times 100\%$$

(14)

2.4. Parameters and Signals

The control performance of the SMC–washout system depends primarily on two parameters: (1) the sliding gain $k$, and (2) the cutoff frequency of the washout filter ${\omega }_c$.

2.4.1. Selection of the Sliding Gain ($k$)

The parameter $k$ determines the slope of the sliding surface and directly affects the system’s convergence rate and chattering amplitude. A higher $k$ increases the control effort and accelerates convergence, while a smaller $k$ results in smoother behavior but slower dynamics.

To ensure system stability and convergence, $k$ must satisfy the sliding condition:

$$k>{\displaystyle \frac{\left|L\left(f\left(x,t\right)\right)\right|}{\left|g(x,t)\right|}}$$

(15)

where $f(x,t)$ represents the system dynamics and $g(x,t)$ is the control input gain.

The appropriate value of $k$ was obtained experimentally by analyzing the transient response of the inverter. The settling time $t_s$ was found to vary approximately according to:

$$t_s\propto {\displaystyle \frac{1}{k}}$$

(16)

Based on this relationship, three representative values were tested ($k$ = 0.1, 1, 4). Results show that $k$ = 0.1 provides a smooth transient with minimal chattering, whereas $k$ = 4 yields a faster response but increases switching activity. Thus, $k$ = 1 represents a compromise between stability and response speed.

2.4.2. Influence of the Washout Filter Frequency (${\mathit{\omega }}_{\mathit{c}}$)

The cutoff frequency ${\omega }_c$ defines the transient sensitivity of the washout filter, which acts as a high-pass element:

$$H\left(s\right)={\displaystyle \frac{s}{s+{\omega }_c}}$$

(17)

This filter removes low-frequency steady-state components of the current and allows rapid detection of transient variations. Simulation results showed that increasing ${\omega }_c$ (e.g., from 10,000 to 26,563 rad/s) enhances the dynamic tracking capability but can slightly increase noise sensitivity. Conversely, smaller ${\omega }_c$ values improve steady-state smoothness but slow down transient response.

Therefore, the combination of $k$ = 1 and ${\omega }_c$ = 26,563 rad/s was selected as optimal, ensuring a balance between dynamic speed, robustness, and signal smoothness. A sensitivity analysis confirmed that variations in $k$ and ${\omega }_c$ proportionally affect the system’s transient response and harmonic suppression capability, validating the proposed selection criteria.

3. Results

3.1. Parameters for the Tests

Table 1. Parameters of the elements used for the simulation tests.

Parameter	Description	Value
Fc	Commutation frequency	40 kHz
E	Input voltage	400 VDC
${iL}_{an}\left(t\right)$	Inductor current phase A	Variable (A)
${iL}_{bn}\left(t\right)$	Inductor current phase B	Variable (A)
${iL}_{cn}\left(t\right)$	Inductor current phase C	Variable (A)
${\upsilon }_{an}\left(t\right)$	Voltage output phase A	Variable (V)
${\upsilon }_{bn}\left(t\right)$	Voltage output phase B	Variable (V)
${\upsilon }_{cn}\left(t\right)$	Voltage output phase C	Variable (V)
${\upsilon }_{an}ref\left(t\right)$	Reference voltage phase A	Variable (V)
${\upsilon }_{bn}ref\left(t\right)$	Reference voltage phase B	Variable (V)
${\upsilon }_{cn}ref\left(t\right)$	Reference voltage phase C	Variable (V)
$L_{an}, L_{bn}, L_{cn}$	Inductance of each phase	62.5 ($\mathsf{\mu }$H)
$C_{an}, C_{bn}, C_{cn}$	Capacitance of each phase	11.11 ($\mathsf{\mu }$F)
$R_{an}, R_{bn}, R_{cn}$	Resistance of each phase	$20 \Omega$$\mathrm{or} 10 \Omega$$\mathrm{or} 5 \Omega$
$u_1, u_2, u_3,$	Control signals for switches	0 or 1
${\overline{u}}_1, {\overline{u}}_2,{\overline{u}}_3$	Complementary control signals	1 or 0
$w$	Cut-off frequency for a high-pass filter	$w=0.6{\displaystyle \frac{1}{\sqrt{LC}}}=$ 26,563
$k$	Control parameter	0.1, 1 or 4

lists the parameters required for the circuit and the description and values used in the simulation tests. These parameters were obtained from real experiments; therefore, they are a close representation of a real circuit. This table shows the values of the input voltage and current and voltage outputs per phase of the system. Other parameters, such as the impedance and control signals, were included. With the data provided in Table 1, the parameters of the LC filter can be calculated using Equations (18) and (19) for each phase, where $R_{an}$ is the load connected to each phase of the inverter, and $∆\mathrm{v}$ represents the desired ripple at the output as a percentage. Then, the inductance is L = 62.5 µH and the capacitance is 11.11 µF. The inductor current ($iL$) in all results shows a relatively high ripple, which is due to the inherent abrupt changes in the inductor current influenced by the switching frequency and inductance value. This behavior is expected in this type of converter.

$$L_{an}={\displaystyle \frac{1.5\times \mathrm{E}}{8\times \mathrm{F}\mathrm{c}\times \frac{{\upsilon }_{an}}{R_{an}}}}$$

(18)

$$C_{an}={\displaystyle \frac{0.8\times \mathrm{E}}{32\times ∆\mathrm{v}\times L_{an}\times {\mathrm{F}\mathrm{c}}^2}}$$

(19)

All simulations were developed using MATLAB–Simulink 2018a with the Simscape Electrical Toolbox, which enables the inclusion of realistic non-ideal effects in the inverter model. The simulation environment accounts for switching device nonlinearities, parasitic inductances and capacitances, sensor quantization, and numerical noise. The control loops also include discrete sampling and computation delays, reflecting practical digital implementation aspects. These inherent non-idealities make the simulation results highly representative of the system’s real behavior. Previous studies have experimentally validated the strong correlation between Simulink-based results and real hardware measurements [50,51,52], confirming that the proposed control maintains stability and robustness even under parameter variations and noisy conditions. Appendix A provides an evaluation of the SMC-w in the frequency domain. Appendix B presents an assessment of the stability of the same controller using the direct Lyapunov method. Finally, Appendix C demonstrates how SMC-w mitigates chattering and subharmonics, while also exhibiting a lower steady-state error compared to traditional SMC.

Figure 2 shows the injection voltage signal for all tests performed with the system. This figure shows a square signal used as the power source or pulsating wave, with a vertical axis representing the magnitude E(t) [V], and a horizontal axis representing the time in seconds t [s]. The amplitude changes between 350 V and 450 V. Each wave cycle is formed by a switching signal with a pulse width modulation (PWM) system. This source is pulsed to demonstrate that the system is robust against disturbances in the input power source. The disturbance of 100 volts at the input is performed to show the robustness to large changes in the power supply. This last test evaluates if the controller tracks the reference signals even in the presence of these strong perturbations.

Table 2. Perturbation events applied to the system.

No. Event	Perturbation	From	To	Time [s]
1	Load change	R = 20 ohm	R = 5 ohm	0.0225
2	Frequency change	f = 60 Hz	f = 180 hz	0.015
3	Voltage amplitude change	120 Vrms	60 Vrms	0.015

lists the events applied in the test to evaluate the proposed method. This table shows the three events and a description of the perturbation applied to the system. In addition, the initial value of the variable in the system and the final value were obtained after perturbation. This perturbation was applied at different times, as defined in the same table.

3.2. Load Change

Figure 3 shows the three-phase voltages of the system represented in milliseconds in the range of 0–0.045 s. The figure represents a voltage signal that varies between −$120\sqrt{2}=-169.7$ V to +$120\sqrt{2}=169.7$ V. The frequency is 60 Hz, and the signals were represented according to a three-phase balance circuit with an angle separation of 120°, and k = 4. This figure shows the perturbation created at 0.225 s when the load was changed from R = 20 to 5 Ω. The event creates a variation in the signals noted in phases $a$ and $c$. In addition, a slight variation was observed in phase b. However, the variation created a temporal transient that was restored to the original signal without changing the voltage magnitude and frequency of the signals. Figure 3 shows that the system is robust to load variations. As observed, the voltage experienced a brief disturbance but quickly stabilized back to the reference value of 120 V RMS in all three phases.

Figure 4 shows the currents of the three-phase system. These currents correspond to the phases A, B and C and they are represented as ${iL}_{an}\left(t\right)$ in black, ${iL}_{bn}\left(t\right)$ in blue, and ${iL}_{cn}\left(t\right)$ in red. The currents are expressed in amperes, and the time is expressed in seconds. The figure shows that before the event, the current varies from −9.8 A to 9.8 A. When the event occurs at t = 0.015 s, the signals present a transient behavior that ends quickly in new sinusoidal signals with amplitudes close to 32 A. It shows that the system reaches a steady state after the event. The steady state is reached approximately 0.0225 s, after which the currents become sinusoidal and symmetrical, characteristics of a balanced three-phase system under stable conditions. The symmetry in the amplitude and phase shift is maintained in the system after the event, which is desirable in most power applications to avoid significant variations.

Figure 5 shows the percentage error in the phase voltages of the three-phase system. The errors are denoted as $Error{\upsilon }_{an}\left(t\right)$ in black, $Error{\upsilon }_{bn}\left(t\right)$ in blue, and $Error{\upsilon }_{cn}\left(t\right)$ in red, corresponding to phases A, B, and C. The graph represents the error as a function of time in seconds, and the error is expressed as a percentage [%].

The results show that the percentage error ranged between −2% and 2%. There are some peaks during the transient owing to the load change, which is controlled by the inverter with the SMC-w. These transients affect the phase voltages, resulting in a temporary increase in the percentage error. The observed transients can have significant implications for the power quality and performance of devices connected to the system. However, the results show that the errors remained low after the event, suggesting that the technique is appropriate for regulating signals.

3.3. Frequency Change in the Reference Signal

The frequency of the reference signal was varied from 60 to 180 Hz. The simulation time was set to 0.03 s, the load was 10 Ω, and k = 4. Figure 6 shows the voltage signal corresponding to a balanced three-phase system. The maximum amplitude of each voltage signal was approximately ±$120\sqrt{2}=169.7$ V. There was an amplitude change at t = 0.015 s during the event, but the system regulated the voltage and changed the frequency to 180 Hz. This indicates modulation during transient changes in the system. The regulation of the signal shows that the system responds appropriately to frequency changes.

Figure 7 shows the current signals for the three phases of the system. The initial signal represents the current of a balanced three-phase system. The initial frequency of the system was 60 Hz before the event. After t = 0.015 s, the signals exhibit a transient response during the frequency change. After this time, the system immediately changes the frequency, and the inverter with the controller helps establish a new frequency of 180 Hz. All signals have small variations or “noise” in their waveforms before and after the events created by the system with the power inverters. Subsequently, the system continued as a balanced three-phase system with a new frequency. The characterization of these phenomena can contribute to improvements in the design of power control systems and the mitigation of harmonics to increase the efficiency and stability of electrical systems. The results show that the controller is effective in regulating the signals when the frequency changes.

Figure 8 shows the relative errors of the three signals as a function of time represented in percentages, as $Error{\upsilon }_{an}\left(t\right), Error{\upsilon }_{bn}\left(t\right),$ and $Error{\upsilon }_{cn}\left(t\right)$. The three signals are shown in black (phase A), blue (phase B), and red (phase C). The relative errors for the three signals remained mostly within the range of ±3%, which is low. The error amplitude increases at t = 0.015 s, due to the frequency change. At t = 0.015 s, a pronounced peak was observed in the three signals, particularly in the red vertical line, which marks the beginning of a transient change. This transient suggests a disturbance that momentarily affects the precision of the system, followed by stabilization with more regular oscillations. The subsequent recovery indicates that the system tends to return to a balanced state after the disturbance. After the transient, there was a slight increase in the error, but the value was maintained for the following seconds, suggesting a stable condition. Therefore, the inverter with the proposed controller was effective in maintaining a stable signal with low error. This analysis is relevant for evaluating the response of the inverter to the proposed controller by representing the errors in the signals.

3.4. Voltage Amplitude Change

A third event was created to study the system response. The voltage is varied from 120 V_RMS to 60 V_RMS at t = 0.015 s, and k = 4. Figure 9 shows the three-phase voltage signals ${\upsilon }_{an}\left(t\right)$ ${\upsilon }_{bn}\left(t\right)$, and ${\upsilon }_{cn}\left(t\right)$, as a function of time, each represented in distinct colors (black, blue, and red, respectively). The initial signal maintains its magnitude before the event, and no variations are observed. At t = 0.015 s, the amplitude changes, and the magnitude varies from 120 V to 60 V, creating a transient event that is quickly regulated. After the transient event, the frequency of the oscillations was not significantly altered, although the signals gradually recovered toward stable operation. Therefore, with the amplitude change, the system remained stable, and the controller was effective in maintaining the signals at the new target.

Figure 10 shows the three currents of the system, ${iL}_{an}\left(t\right)$ ${iL}_{bn}\left(t\right)$, and ${iL}_{cn}\left(t\right)$ as a function of time, represented by black, blue, and red colors, respectively. Before the event, the signals maintained a sinusoidal shape, indicating normal operating conditions in the three-phase system. At approximately t = 0.015 s, a clear transient is observed in the three signals, indicating a response to the voltage amplitude change. The immediate response in each current shows an alteration in the amplitude and waveform, which suggests recovery of the system after the disturbance. The maximum amplitude was initially approximately ±28 A, and after the event, it changed to ±9 A. After the event, the signals continuously oscillated with small variations in their amplitude and waveforms, indicating the stabilization of the new value. This study contributes to the understanding of the dynamics of three-phase electrical systems and the implementation of mitigation strategies for transient events using the proposed controller.

Figure 11 shows the relative errors [%] of the three voltage signals, represented as $Error{\upsilon }_{an}\left(t\right)$, $Error{\upsilon }_{bn}\left(t\right)$, and $Error{\upsilon }_{cn}\left(t\right)$. The signs are colored black (phase A), blue (phase B), and red (phase C). The presence of a notable transient around t = 0.015 s is due to the change in voltage amplitude. The amplitude of the errors varied between −3% and 3%, which is low. Before the transient event, the errors in the three signals showed a gradual trend, reaching peaks of approximately ± 2%. A notable transient event is observed in all signals around t = 0.015 s, highlighted by a vertical line red on the graph, generating a significant disturbance in the signal errors. After the event, the error signals tended to stabilize. The signal error was reduced and maintained for the following seconds. Therefore, the controller is effective at regulating signals with an amplitude change, representing a good option for installation in power systems that are subject to voltage variations.

3.5. Comparison of Performance Between SMC-w and PID-Controlled Systems Under Load Variations (RL)

The performance of the SMC-w was compared with the PID control. The impedance of the load in the range 0 < t < 0.015 s is Z = 10 + j10, composed of a 10-ohm resistor in series with an inductance L1 = 26.5 mH. The load was changed at t = 0.015 s, obtaining a new value in the range 0.015 s < t < 0.03 s with an impedance equivalent to Z = (10 + j10) in parallel with a 5 ohms resistor, resulting in an equivalent impedance of Z = 3.85 + j0.77 Ohms. The test was performed with a switching frequency of Fc = 20 kHz, and an impedance for the filter L = 62.5 mH and C = 44.4 µF. The parameters for the SMC were w = 13282 and k = 0.1. For PID control, the parameters Kp = 150, Ki = 100, and Kd = 0.05 were calculated, as described in [50,51,52].

As shown in Figure 12a,b, both controllers (PID and SMC) respond well to the changes in the load. In addition, the PID control stabilizes at t = 0.75 ms and the SMC at t = 0.55 ms, demonstrating that the system with the SMC stabilizes faster than the system with the PID. It is important to note that, in the load change t = 150 ms, no differences are observed in the behavior of the two controllers, and both the PID and SMC work very well and follow the reference signals.

Figure 13 shows the differential error between the controlled and reference signals for each phase of the three-phase inverter controlled by each PID controller (Figure 13a) and the SMC (Figure 13b). In the case of PID, the disturbance in the load is instantly rejected, and no difference is observed in the error before and after the change. Similarly, the SMC presents a small increase in the error at the instant of the load change at t = 15 ms. The most notable difference was found in the differential error, as the SMC presents a steady-state differential error approximately between +0.56% and −0.56%, while for the PID the differential error is between +1.10% and −1.14%. Therefore, the steady-state differential error is lower for the SMC.

Figure 14 shows the behavior of the line current ($i_L$) under load changes when the voltages in the loads are controlled. In general, the currents in Figure 14a,b exhibit the same behavior for the two types of controllers because these signals are not controlled, and it is observed that the load increases for time t = 15 ms, where it is observed that the currents go from an approximate value of 10 to 44 amps. The currents at the beginning at t = 0 s reach values close to 100 A for both controllers. In general, the fundamental difference in this comparison is observed in the differential error because the differential error is 50% smaller in the SMC. Another advantage the SMC has is that the only control parameter is k1, so it can easily be adjusted to the value desired by the user depending on the operating conditions.

The proposed SMC–washout (SMC-w) control strategy shows a significant reduction in chattering effects compared with the conventional PID controller. The inclusion of the washout filter introduces a dynamic compensation that removes low-frequency components from the sliding surface, which smooths the control signal and minimizes high-frequency switching oscillations.

As shown in the comparative results, the SMC-w controller provides a smoother transient response and greater stability under RL load variations, maintaining the robustness and fast convergence characteristics of SMC while avoiding the excessive control effort and oscillations typically present in PID-based systems.

3.6. Computational Complexity and Implementation Feasibility (SMC–Washout vs. PID)

The proposed SMC–washout (SMC-w) controller is computationally light because it relies on a simple sliding-surface evaluation, a sign-based switching rule, and a single first-order washout filter per phase. In contrast, a conventional three-phase PID requires proportional, integral, and derivative paths (often with derivative filtering) and anti-windup logic.

3.6.1. Operation Count at 20 kHz

For SMC-w, the per-phase per-sample load is ≈10 primitive operations (additions/multiplications/comparison): washout update (first-order, one state) and sliding-surface computation plus sign decision. This yields ~30 ops per sampling step for the three phases, equivalent to

$$\approx 3.0\times {10}^{4}FLOPS (20 \mathrm{kHz})$$

For a three-phase PID with derivative filtering and anti-windup, the per-phase cost is ≈20 operations (error, P/I/D paths, filter updates, summation, saturation checks), i.e., ~60 ops per sample for the three phases, or

$$\approx 6.0\times {10}^{4}FLOPS (20 \mathrm{kHz})$$

Therefore, under identical sampling conditions, SMC-w requires roughly half the computational load of a conventional PID.

3.6.2. Memory Footprint and Latency

SMC-w stores one washout state per phase and a few scalar parameters ($k,w_c)$, resulting in a compact memory footprint (<15 kB for code and data on typical DSP/MCU targets). A full three-phase PID with filtered derivative and anti-windup typically remains <25 kB. Both are easily accommodated by low-cost processors (e.g., TI F28335, STM32F4), but the smaller state set of SMC-w eases integration and verification. The absence of iterative steps or matrix operations yields deterministic, low-latency execution compatible with 20 kHz real-time control.

3.6.3. Practical Implications

Besides the lower computational cost, SMC-w requires fewer tuning parameters (mainly $k$ and $w_c$) than PID (three gains per phase plus filter/anti-windup settings). This reduces commissioning effort while preserving robustness to uncertainties and load variations documented in this work. In summary, SMC-w offers a favorable trade-off between dynamic performance, robustness, and implementation effort compared with conventional PID at a 20 kHz switching frequency.

3.7. Behavior of the System Controlled with SMC Under Short Circuits

The following test evaluates the behavior of a system controlled by SMC under short-circuit conditions. Initially, a load of R = 20 was considered between 0 and t < 0.015 s. At t = 0.015 s, a three-phase ground fault was simulated with an impedance of R = 0.5 Ω. The system parameters are listed in Table 1; the control parameter is k = 0.1, and the switching frequency Fc = 20 kHz. Figure 15a shows that for 15 ms, a 0.5 ohms ground fault is applied, and the system suffers the disturbance, but the SMC controller rejects the change and keeps the system controlled, regulating the voltage at the output even with the ground fault.

Figure 15b illustrates the current waveform before and after a ground fault. Prior to the fault, the system exhibited a peak current of 11 A, whereas after the fault, the maximum current reached 341 A. Nevertheless, as observed in Figure 15a,b, the system maintains proper operation, with all voltage and current signals preserving equal magnitudes and a 120° phase displacement. Figure 15c depicts the differential error that appeared in the three phases before the fault. In this figure, the maximum steady-state error ranges between −1.9 V and 2 V, and after the fault, the differential error varies between −2.92 V and 3.88 V. These magnitudes are relatively small and confirm that the system governed by the SMC remains robust under this type of fault.

3.8. SMC-w Controlled System Response to Unbalanced Loads in a Three-Phase System

Figure 16 illustrates the behavior of an SMC-w-controlled system when subjected to unbalanced loads, with $k=4$. The reference signals were 120 VRMS at 60 Hz, with a 120-degree phase shift, following the abc sequence. From t = 0 s to t = 30 ms, the load was balanced, with values $z_a=z_b=z_c=20 \Omega$. However, between t = 30 ms and t = 60 ms, the load changed abruptly to $z_a=10 \Omega$, $z_b=5 \Omega$ y $z_c=2.5 \Omega$. In Figure 16a, the voltages of the three phases are shown as ${\upsilon }_{an}\left(t\right)$ ${\upsilon }_{bn}\left(t\right)$, and ${\upsilon }_{cn}\left(t\right)$. During the first segment, up to t = 30 ms, the system maintained identical steady-state currents (Figure 16c), and the voltages remained at the reference value of 120 VRMS at 60 Hz in all three phases because the load was balanced. The currents reach a peak value of approximately 9.1 A in all phases, which are very similar, as shown in Figure 16c.

After t = 30 ms, when the load becomes unbalanced, the voltages ${\upsilon }_{an}\left(t\right)$ ${\upsilon }_{bn}\left(t\right)$, and ${\upsilon }_{cn}\left(t\right)$ remain unaffected, remaining at 120 VRMS at 60 Hz across all three phases with the same 120-degree phase shift and 60 Hz frequency. Despite the abrupt shift to unbalanced loads, the three-phase voltage remains constant. At t = 30 ms, a small disturbance was observed; however, it was quickly corrected by the control system. Figure 16b shows the percentage error of the output voltages with respect to the reference voltage for each phase. The error remained below 3% even after the load was changed to an unbalanced value. In Figure 16c, for t > 30 ms, the currents in each phase, ${iL}_{an}\left(t\right)$, ${iL}_{bn}\left(t\right)$, and ${iL}_{cn}\left(t\right)$ are shown, which have different magnitudes owing to the unbalanced nature of the loads. However, despite this difference in current, the voltages in all three phases remain at the reference value, demonstrating that the system is robust and effectively controls the voltage in all phases.

3.9. SMC-w Controlled System Response to Unbalanced Nonlinear Load Transitions in a Three-Phase System

Figure 17 shows the behavior of the SMC-w-controlled system when unbalanced nonlinear loads were applied, with $k=4$. The reference signals are the same as those in the previous section, with a value of 120 VRMS at 60 Hz, shifted by 120°, and following the abc sequence. For the interval between t = 0 s and t = 30 ms, the load was balanced with the values $z_a=z_b=z_c=20 \Omega$. However, from t = 30 ms to t = 60 ms, the load abruptly changed to three unbalanced nonlinear loads. To construct these loads, diodes were placed in series. Thus, the loads were $z_a=10 \Omega$, $z_b=5 \Omega$ y $z_c=2.5 \Omega$. The diodes in series with the $z_a=10 \Omega$ and $z_c=2.5 \Omega$ loads were connected in the same direction, while the diode in series with the $z_b=5 \Omega$ load was placed in the opposite direction.

In Figure 17a, for t > 30 ms, when the load becomes unbalanced and nonlinear, it is observed that the voltage waveforms ${\upsilon }_{an}\left(t\right)$ ${\upsilon }_{bn}\left(t\right)$, and ${\upsilon }_{cn}\left(t\right)$ remain unchanged, remaining at 120 VRMS at 60 Hz in all three phases with a 120-degree phase shift and maintaining a frequency of 60 Hz. Despite the abrupt transition to unbalanced nonlinear loads in the second segment, three-phase voltages were not altered. Figure 17b shows the percentage error of the output voltages with respect to the reference voltage for each phase. As shown, the error remained below 3% even after the transition to unbalanced nonlinear loads. Finally, in Figure 17c, for t > 30 ms, the currents in each phase, ${iL}_{an}\left(t\right)$, ${iL}_{bn}\left(t\right)$, and ${iL}_{cn}\left(t\right)$ are shown. These currents display different magnitudes and waveforms owing to unbalanced nonlinear loads; however, despite these variations, the voltage in all three phases remains at the value set by the reference signals. This demonstrates that the system is robust and maintains efficient voltage control across all three phases.

4. Conclusions

This work presented the design of a robust three-phase inverter controlled by an SMC-w, achieving a pure sinusoidal output with low harmonic content under RL load conditions. The following conclusions were obtained from the results.

• The SMC controller demonstrated superior robustness compared to the PID controller, particularly under abrupt inductive load variations. This enhanced performance is attributed to its ability to adapt rapidly to system changes without depending on the exact mathematical model.
• During the ground fault tests, the SMC-w effectively maintained the overall system stability, mitigated adverse effects, and quickly restored normal operating conditions. This capability underscores its suitability for applications requiring resilience to severe faults, where minimal disruption to the system performance is critical.
• A comparative analysis of the SMC-w and PID controllers highlighted the advantages of the proposed approach. The PID controller required precise tuning and exhibited longer settling times under rapid load variations, and the SMC-w provided faster stabilization and superior steady-state performance with fewer control parameters. The robustness of the SMC also enables it to handle complex disturbances, such as ground faults and input voltage perturbations, with minimal impact on the system output.
• The results showed that the SMC-w maintained stable performance under unbalanced nonlinear loads. Despite the abrupt load change, the voltages remained constant in all three phases at 120 VRMS and 60 Hz, with a voltage error of less than 3%. Although the currents exhibited different magnitudes and waveforms because of the unbalanced and nonlinear nature of the loads, the voltages remained stable. This shows that the system is robust and effective in controlling the voltages. This highlights the capability of the SMC-w to maintain efficient and stable control under complex and dynamic load conditions.
• The SMC-w control strategy not only ensured stability under RL load conditions but also demonstrated outstanding performance under more demanding scenarios. This capability guarantees the applicability of the system in industrial and renewable energy environments, where load conditions are rarely ideal, thereby strengthening the practical relevance of the proposed approach. The controller design, which requires only a reduced number of parameters and is oriented toward current digital technologies, is easily implementable in microprocessors or DSPs of any architecture. This reduces implementation complexity, minimizes hardware costs, and accelerates the transfer of the system from the simulation environment to practical industrial applications.
• The Lyapunov-based analysis confirmed that the SMC–washout filter combination guarantees asymptotic stability of the controlled inverter, ensuring convergence of the sliding surface and robustness under disturbances.

These contributions demonstrate the feasibility and advantages of integrating the SMC-w for three-phase inverters and establish a robust framework for designing advanced inverter systems capable of achieving high performance in dynamic and challenging environments. The simulation models included realistic representations of components such as MOSFETs, diodes, RL, and nonlinear loads. However, one limitation of this study is the lack of experimental implementation on a physical three-phase inverter platform, and switching losses are not considered.

Future work will focus on experimentally validating the robustness of the proposed inverter when connected to the power grid, assessing its behavior under unbalanced and time-varying load conditions, and extending the study to other power converter topologies. In parallel, the development of a hardware prototype will be carried out to verify the controller’s real-time performance and implementation feasibility. Furthermore, future research will explore advanced optimization methods and adaptive control algorithms, including machine learning (ML) techniques such as adaptive neuro-fuzzy inference systems (ANFIS) and reinforcement learning, to enable real-time adjustment of control parameters ($k$ and ${\omega }_c$). These intelligent approaches are expected to enhance the controller’s adaptability, minimize total harmonic distortion, and maintain optimal performance under nonideal and dynamic operating conditions.