Research on Sliding Mode Control of Dual Active Bridge Converter Based on Linear Extended State Observer in Distributed Electric Propulsion System

Abstract

This paper focuses on the high-performance bidirectional DC-DC converter required in distributed electric propulsion (DEP) systems, with the dual active bridge (DAB) converter chosen as the subject of study. To achieve the goal of stabilizing the output voltage while improving the converter’s anti-interference ability and dynamic performance, this paper proposes a novel strategy. In particular, it combines the Linear Extended State Observer (LESO) with a sliding mode control (SMC), proposing a sliding mode control strategy based on the Linear Extended State Observer (LESO-SMC). Notably, this control strategy not only retains the fast dynamic performance of Linear Active Disturbance Rejection Control (LADRC) and the robustness of SMC but also addresses the significant chattering issue inherent in traditional SMC. Comparing the traditional PI, LADRC, and SMC strategies, the results show that when the load changes, the voltage fluctuation of the LESO-SMC strategy proposed in this paper is 0.165 V (0.25 V) in the Matlab/Simulink and RT-Lab platforms, and the average adjustment time is 4 ms (3.5 ms). In contrast, the average voltage fluctuations of PI and LADRC strategies were 3.7 V (4.9 V) and 0.55 V (1.35 V), and the average adjustment times were 99.5 ms (201 ms) and 71.5 ms (77.5 ms), respectively. When the input voltage changes, the proposed LESO-SMC strategy adjusts faster and has almost no voltage fluctuations, while the average voltage fluctuations of the PI and LADRC strategies in the simulation are 0.5 V and 0.1 V, and the average adjustment times are 89.5 ms and 35 ms, and the change in the input voltage in the RT-Lab platform has very little effect on the output voltage. Compared with SMC, the LESO-SMC strategy has no chattering problem. In summary, compared to the other three control strategies, the LESO-SMC strategy proposed in this paper exhibits superior performance in terms of voltage fluctuation and adjustment time during load changes and input voltage changes. It shows a robust anti-interference ability and a rapid dynamic response performance.

1. Introduction

With the continuous rise in global carbon dioxide emissions, countries increasingly focus on energy conservation, emission reduction, and green, sustainable development. In recent years, the new energy industry entered a phase of rapid development, with various advanced energy conversion systems being proposed. Among these, distributed electric propulsion (DEP) systems, known for their low loss and high-quality power, received special attention [1,2]. As shown in Figure 1, the DEP system obviously requires the DC-DC converter to adjust the voltage to meet different load requirements. Moreover, it also serves as a bridge between the energy storage system and the DC bus, which can be used to balance power fluctuations [3]. Therefore, a converter with good performance is needed. Enhancing the anti-interference ability and the dynamic response speed of the DC-DC converter is also a key factor when studying the characteristics of such converters.

The DAB converter, known for its key features such as high power transmission efficiency, electrical isolation, and bidirectional energy flow, attracts widespread attention in the new energy industry [4].

The efficiency and performance of a converter are closely related to its mathematical modeling and control strategy. Various modeling methods and control strategies are proposed by scholars to enhance the disturbance suppression ability and dynamic performance of the DAB converter. Existing research primarily includes three types of mathematical modeling: reduced-order models, generalized averaging models, and discrete-time models [5,6,7,8]. Based on these, various advanced control strategies are proposed. For example, Reference [9] borrows the idea of power control in motor control and proposes a direct power control (DPC) strategy. Building on this, reference [10] introduces a virtual power control (VPC). However, the former is easily influenced by system internal parameters, and the latter has poor robustness and dynamic performance. Reference [11] applies load current feedforward to improve the converter’s dynamic performance to load changes and reduce the use of sensors, but it does not deeply analyze its stability, and its performance is poor when the parameters are inaccurate. References [12,13,14] apply the model predictive control (MPC) to the converter, but MPC requires strong computing power, which might lead to an increase in hardware costs, and the derivation process of the control method is complex. To address the instability problem caused by constant power loads, References [15,16] introduce Linear Active Disturbance Rejection Control (LADRC) into the converter, achieving fast dynamic performance while maintaining system stability. Sliding mode control (SMC) has strong robustness against system parameter changes and disturbances and has been validated in many industrial fields. References [17,18] employ SMC to control DAB converters, but traditional SMC suffers from the problem of chattering, which can cause serious issues during converter control.

Observer control technology is widely studied over the past few decades due to its ability to estimate and compensate for disturbances [19]. Reference [20], based on load current feedforward and combined with the Nonlinear Disturbance Observer (NOD), proposes a control strategy without a current sensor. By using NOD to estimate the load current, it regulates the bus voltage without the use of current sensors. Reference [21] combines the backstepping control with the NOD to achieve accurate voltage tracking under large signal disturbances. Reference [22] proposes a robust voltage control strategy based on the Uncertainty and Disturbance Estimator (UDE) to deal with internal and external disturbances and uncertainties, thereby enhancing the stability of the converter system. In summary, observer technology can effectively enhance the disturbance suppression ability and improve the dynamic performance of converters. However, the observer control strategies proposed in the above articles are overly complex and not easily understandable or transferable.

Compared with the discrete model and the generalized average model, the reduced-order model of the DAB converter has excellent small signal accuracy while maintaining low complexity [23]. Therefore, this paper chooses the reduced-order model, combines the Linear Extended State Observer (LESO) in LADRC with SMC, and proposes a sliding mode control strategy based on the Linear Extended State Observer (LESO-SMC). Through the LESO, the system’s internal and external disturbances and model parameter uncertainties are treated as total disturbances, and the LESO is used for precise estimation and compensation of these total disturbances. To further improve the control effect and achieve an accurate tracking of the reference output voltage, an equivalent control law is derived by constructing an error integral sliding surface in combination with SMC. Then, an improved exponential reaching law is chosen as the switching control term to ensure the global asymptotic stability of the system, and finally, an error feedback law is designed using the two observer output values. Comparative experiment results show that the LESO-SMC strategy proposed in this paper is simple and can significantly improve the anti-interference ability and dynamic performance of the DAB converter.

The organization of this paper is as follows: Section 2 introduces and analyzes the working principle of the DAB converter and carries out average dynamic modeling. Section 3 outlines the design of the converter’s controller, where Section 3.1 describes the design of the LADRC strategy, Section 3.2 combines the core LESO of LADRC with SMC, proposing the control strategy suggested in this paper. Section 4 is the simulation experiments and comparative analysis of various control strategy. Section 5 presents the overall conclusion of this paper.

2. Analysis of the Working Principle of DAB Converter

The circuit structure of the DAB converter is shown in Figure 2. H1 and H2 represent the symmetrical switching bridges on the primary and secondary sides of the transformer T, respectively, which include a total of eight switching tubes, ($S_1-S_8$). $U_i$ and $U_o$, respectively, represent the input and output voltages, $C_1$ and $C_2$ are support capacitors, and L represents the equivalent inductance of the leakage inductance of the transformer and the sum of the auxiliary inductances for power transmission. $U_{ab}$ and $U_{cd}$ each represent the square wave voltage on the primary and secondary sides, respectively.

In order to control the DAB converter simply and effectively, the control method adopted in this paper is a single-phase-shift (SPS) control. It controls the phase shift between the H1 and H2 bridges, denoted as D, which represents the phase difference between the square wave voltages $U_{ab}$ and $U_{cd}$. D is defined as the ratio of the phase shift angle to 180°. When $D>0$, the power is output forward ($U_{ab}$’s phase is ahead of $U_{cd}$), and when $D<0$, the power is transmitted in reverse. To facilitate the analysis, only the case of forward power transmission is considered, and the main waveform diagram of the DAB converter under its control is shown in Figure 3. The rules for turning on and off the eight switching tubes under SPS control are as follows: within one switch period $T_s$ ($t_0-t_4$), the drive signals of the diagonal switching tubes in the bridges H1 and H2 are the same, and the upper and lower switching tubes in the same bridge arm are complementary. There is a certain phase difference between switching tube $S_1$ ($S_4$) of the H1 bridge and switching tube $S_5$ ($S_8$) of the H2 bridge (the phase difference between $U_{ab}$ and $U_{cd}$), and all drive signals have a duty cycle of 50%.

As shown in Figure 3, the switching function of the DAB converter under SPS control can be defined as follows:

$$\left\{\begin{array}{c}\hfill S_a=\left\{\begin{array}{c}\hfill 1,S_1\mathrm{and}{S}_4\mathrm{ON},S_2\mathrm{and}{S}_3\mathrm{OFF}\hfill \\ \hfill -1,S_1\mathrm{and}{S}_4\mathrm{OFF},S_2\mathrm{and}{S}_3\mathrm{ON}\hfill \end{array}\right.\hfill \\ \hfill S_b=\left\{\begin{array}{c}\hfill 1,S_5\mathrm{and}{S}_8\mathrm{ON},S_6\mathrm{and}{S}_7\mathrm{OFF}\hfill \\ \hfill -1,S_5\mathrm{and}{S}_8\mathrm{OFF},S_6\mathrm{and}{S}_7\mathrm{ON}\hfill \end{array}\right.\hfill \end{array}\right.$$

(1)

Then, the state equation of the DAB converter can be expressed as follows:

$$\left\{\begin{array}{c}\hfill \frac{d{i}_L}{dt}=\frac{S_a{U}_i}{L}-\frac{n{S}_b{U}_o}{L}\hfill \\ \hfill \frac{d{U}_o}{dt}=\frac{n{S}_b{i}_L}{C_2}-\frac{U_o}{R{C}_2}\hfill \end{array}\right.$$

(2)

The waveform of inductive current $i_L$ splits into four operational phases: $t_0-t_1$, $t_1-t_2$, $t_2-t_3$, and $t_3-t_4$. By leveraging the waveform’s symmetry for analytical simplicity, a half cycle from $t_0$ to $t_2$ is selected. According to Formulas (1) and (2), the expressions for inductive current during the periods $t_0-t_1$ and $t_1-t_2$ are as follows:

$$i_L\left(t\right)=\left\{\begin{array}{c}\hfill i_L\left(t_0\right)+\frac{U_{in}+n{U}_o}{L}(t-t_o)t_0\le t\le t_1\hfill \\ \hfill i_L\left(t_1\right)+\frac{U_{in}-n{U}_o}{L}(t-t_1)t_1\le t\le t_2\hfill \end{array}\right.$$

(3)

In the Formula (3), $i_L\left(t_0\right)$ and $i_L\left(t_1\right)$ represent the inductor currents at the respective moments $t_0$ and $t_1$, and due to the symmetry of the waveform of the inductance current $i_L$, it follows that $i_L\left(t_0\right)=-i_L\left(t_2\right)$, $i_L\left(t_1\right)=-i_L\left(t_3\right)$.

Neglecting system losses and combining Formula (3) and Figure 3, the power transmission $P_o$ of the DAB converter under SPS control is defined as follows:

$$P_o=\frac{T_s}{2}{\int }_{t_0}^{t_2}{U}_i{i}_L\left(t\right)dt=\frac{n{U}_o{U}_i}{2L{f}_s}D(1-D)$$

(4)

$f_s$ in the Formula (4) is the switching frequency, and $T_s=1/f_s$.

Based on Formula (4), the average output current on the secondary side can be calculated, and a low-frequency small signal disturbance is further introduced. By eliminating the direct current and disturbances of the secondary and higher small signals, the small signal value of the output current on the secondary side can be obtained as follows:

$${\widehat{i}}_2=\frac{n{U}_i}{2f_{s}L}(1-2D)\widehat{d}$$

(5)

In Formula (5), $\widehat{d}$ is the small signal value of the shift phase ratio D, and only the output side is concerned. At this time, the corresponding small signal model of the DAB converter is shown in Figure 4:

From Figure 4 and Formula (5), according to Kirchhoff’s current law, the dynamic equation of the output voltage can be obtained as follows:

$$\frac{d{\widehat{U}}_o}{dt}=\frac{n{U}_i}{2L{C}_2{f}_s}(1-2D)\widehat{d}-\frac{{\widehat{U}}_o}{R{C}_2}$$

(6)

In Formula (6), ${\widehat{U}}_o$ represents the small signal value of the output voltage $U_o$.

Formula (6) can be further simplified as follows:

$$\dot{y}=b_{0}u+f$$

(7)

In Formula (7), f represents the total disturbance, which includes both the unknown disturbances from the system and external sources as well as the uncertain parameters of the converter. $b_o$ represents the known part of the input control gain.

3. Controller Design of DAB Converter

The working performance of the DAB converter is closely related to its control strategy. The traditional PI control of the DAB converter has problems such as weak anti-interference ability and slow response speed. Therefore, this chapter combines LESO with SMC, significantly improving the anti-interference ability and dynamic response speed of the converter.

3.1. Design of Linear Active Disturbance Rejection Control

LADRC is a control strategy with good dynamic performance and certain anti-interference ability. Applying the LADRC to the DAB converter can improve its dynamic performance and anti-interference ability. The framework of the LADRC strategy is shown in Figure 5. Its LESO is the core part of LADRC. It is a dynamic observer. Its main function is to estimate the system state variables. It can extend the disturbances inside and outside the system into new state variables and compensate them to the control signal. Through the LESO, the overall dynamic performance and stability of the system can be improved [24].

The selection of state variables is given as $x={\left[\begin{array}{cc}{x}_1& x_2\end{array}\right]}^T={\left[\begin{array}{cc}y& f\end{array}\right]}^T$, turning Formula (7) into a state-space equation description [25] as follows:

$$\left\{\begin{array}{c}y=x_1\hfill \\ {\dot{x}}_1=b_{0}u+x_2\hfill \\ {\dot{x}}_2=\dot{f}\hfill \end{array}\right.$$

(8)

In Formula (8), y denotes the system’s output (output voltage $U_o$), u stands for the system’s control variable (shift phase ratio D), and $x_2$ signifies the total disturbance extended into a state variable. Building on Formula (8), the following depicts the matrix equation for constructing the LESO as follows:

$$\left[\begin{array}{c}\hfill {\dot{z}}_1\hfill \\ \hfill {\dot{z}}_2\hfill \end{array}\right]=\left[\begin{array}{cc}-{\beta }_1& 1\\ -{\beta }_2& 0\end{array}\right]\left[\begin{array}{c}\hfill z_1\hfill \\ \hfill z_2\hfill \end{array}\right]+\left[\begin{array}{cc}{b}_0& {\beta }_1\\ 0& {\beta }_2\end{array}\right]\left[\begin{array}{c}u\\ y\end{array}\right]$$

(9)

In Formula (9), $z_1$ and $z_2$ are the observed values of state variables y and f, respectively. ${\beta }_1$ and ${\beta }_2$ are the gain parameters of the observer. Adjusting these two gain parameters can enable the observer to estimate state variables well. To simplify the selection of parameters, the gain parameters can be chosen using the bandwidth method [26] as follows:

$$\left\{\begin{array}{c}\hfill {\beta }_1=2{\omega }_0\hfill \\ \hfill {\beta }_2={\omega }_0^2\hfill \end{array}\right.$$

(10)

According to the Hurwitz stability criterion, the selection of parameters should satisfy the characteristic polynomial equation $s^2+{\beta }_{1}s+{\beta }_2={(s+{\omega }_0)}^2$ to ensure the convergence of the LESO.

For the LESO, the input signals are output voltage $U_o$ and shift phase ratio D, while the output signals $z_1$ is the estimated value of the output voltage $U_o$, and $z_2$ is the estimated value of the total disturbance f. By substituting Formula (10) into Formula (9) and performing Laplace transform, we have the transfer function $G_1$ as follows:

$$G_1=\frac{z_2\left(s\right)}{U_o\left(s\right)}=\frac{{\omega }_0^{2}s}{s^2+2{\omega }_{0}s+{\omega }_0^2}$$

(11)

Furthermore, the transfer function $G_2$ between the disturbance observation $z_2$ and the total disturbance f can be derived as follows:

$$G_2=\frac{z_2\left(s\right)}{f\left(s\right)}=\frac{{\omega }_0^2}{s^2+2{\omega }_{0}s+{\omega }_0^2}$$

(12)

The Bode plots of the transfer functions in LESO under different bandwidths ${\omega }_0$ are shown in Figure 6. The tracking performance of LESO is better when the bandwidth ${\omega }_0$ is larger. However, as ${\omega }_0$ increases, the noise immunity of LESO weakens, creating an inevitable trade-off between noise immunity and disturbance rejection capabilities. Therefore, when choosing ${\omega }_0$, a balance should be struck between noise immunity and disturbance rejection capabilities. After consideration, ${\omega }_0=1600$ rad/s is chosen in this paper.

When the gain parameters for LESO are appropriately chosen, $z_1$ tends to the output voltage $U_o$ and $z_2$ tends to total disturbance f. After compensation, the system is simplified to a single integral series system. Good control results can be achieved with just a linear $K_p$ proportional control, enhancing the system’s dynamic response performance.

In this case, the control signal u can be expressed as follows:

$$u=\frac{k_p(U_{ref}-U_o)-f}{b_0}$$

(13)

In Formula (13), $k_p(U_{ref}-U_o)/b_0$ represents the feedback control component, and $-f/b_0$ represents the disturbance compensation component.

3.2. Design of Sliding Mode Control Based on Linear Extended State Observer

While the aforementioned LADRC strategy improves the dynamic performance and anti-interference ability of the converter, considering the non-linear characteristics of the DAB converter, the linear feedback control law might not be suitable. The introduction of SMC into the LADRC structure and the design of non-linear state error feedback law can further improve the converter’s ability to suppress disturbances. However, in traditional SMC, there exists a sign function sgn(s). When the system trajectory reaches the sliding mode surface, the sgn(s) function causes the input control law to switch at high frequencies continuously. The inertia of the movement makes it difficult for the system state to slide along the sliding mode surface towards the desired equilibrium point, resulting in the state trajectory crossing back and forth on both sides of the sliding mode surface, thereby causing chattering. By combining SMC with the LESO from the LADRC, the LESO estimates and compensates for the total disturbance in a feedforward manner. The SMC feedback control law is designed using the observation values $z_1$ and $z_2$ and continuous functions. This reduces the switching gain in SMC, which can effectively reduce the chattering problem while improving the system response speed.

Compared with the non-integral SM control, the integral SM control can eliminate the steady-state error more effectively [27]. Therefore, in order to stabilize the output voltage while reducing the steady-state error, the constructed integral sliding surface s is expressed as follows:

$$s=k_{1}e+k_2\int edt$$

(14)

In the Formula (14), $k_1$ and $k_2$ are adjustable parameters, both greater than 0, $e=U_{ref}-U_o$ is the tracking error of the output voltage, where $U_{ref}$ is the reference value of the voltage.

Differentiating the sliding surface s and combining it with Formula (7), the formula can be obtained:

$$\dot{s}=k_1\dot{e}+k_{2}e=-k_1(b_{0}u+f)+k_{2}e$$

(15)

According to the equivalent control approach, the control law of the system can be designed as $u=u_{eq}+u_{sw}$, where $u_{eq}$ represents the equivalent control signal, and $u_{sw}$ represents the switching control term. Setting Formula (15) equal to 0, the equivalent control law $u_{eq}$ can be solved as follows:

$$u_{eq}=-\frac{1}{b_0}(f-\frac{k_2}{k_1}e)$$

(16)

To ensure that the system reaches its equilibrium point within a finite time, the exponential convergence law can be chosen as the switching control term $u_{sw}$. Based on Formula (16), the final system control law u can be designed as follows:

$$u=-\frac{1}{b_0}(f-\frac{k_2}{k_1}e+k_{3}s+\epsilon \mathrm{sgn}\left(s\right))$$

(17)

The Lyapunov function can be used to analyze whether a system is stable. For the system to reach a stable state, the Lyapunov function must always be greater than zero throughout the system’s entire state space, and its derivative must be less than zero. Therefore, the chosen Lyapunov function is expressed as follows:

$$V=\frac{1}{2}{s}^2$$

(18)

To differentiate Formula (18) and substitute Formula (15) into it, the result can be obtained as follows:

$$\dot{V}=s\dot{s}=s(k_1\dot{e}+k_{2}e)=s(-k_1(b_{0}u+f)+k_{2}e)$$

(19)

Substituting Formula (17) into Formula (19), the result can be obtained as follows:

$$\dot{V}=s(-k_1(-k_{3}s-\epsilon \mathrm{sgn}\left(s\right)))=k_1{k}_3{s}^2+k_1\epsilon \left|s\right|$$

(20)

It is known that the known coefficient $k_1$ is greater than 0 from Formulas (18) and (20). When the selected adjustable coefficients $k_3$ and $\epsilon$ are both less than 0 and $s\ne 0$, there exists $V>0$ and $\dot{V}=s\dot{s}<0$, satisfying the basis for Lyapunov stability judgment. In SM control, it manifests as the initial state outside the sliding mode surface $s=0$, which satisfies the condition to reach the sliding mode surface, i.e., any initial state where $s\ne 0$ can tend toward the equilibrium point in finite time, thereby achieving a stable state for the system.

As shown in Formula (17), the control law u (the phase shift ratio D) of the system contains a high-frequency sign function $\mathrm{sgn}\left(s\right)$. Its presence can cause the high-frequency switching of the phase shift ratio D, resulting in vibrations. For the DAB converter, the high-frequency chattering of the phase shift ratio D can cause huge fluctuations in power transmission. In addition, it may lead to an increase in the transformer current, causing core saturation, thereby increasing energy consumption and affecting the dynamic performance of the converter [28]. In order to reduce the vibration of the transient switching of the phase shift ratio D, the observed values $z_1$, $z_2$ and the continuous function $sat\left(s\right)$ are used to optimize the sign function to smooth the change in the control law. In Figure 7, (a) and (b) are comparisons of the phase shift ratio D before and after the replacement of the continuous function, and it is clear that the control law after the replacement with the continuous function is smoother.

The chosen continuous function is expressed as follows:

$$sat\left(s\right)=\frac{s}{\left|s\right|+\eta }$$

(21)

In Equation (21), the term $\eta$ is the anti-chattering factor, and $\eta >0$.

Combining Equations (9), (17), and (21), the SMC law based on the LESO and designed using the observed values $z_1$ and $z_2$ can be expressed as follows:

$$u=-\frac{1}{b_0}(z_2-\frac{k_2}{k_1}\widehat{e}+k_3\widehat{s}+\epsilon sat\left(\widehat{s}\right))$$

(22)

In Equation (22), $\widehat{e}=U_{ref}-z_1$, $\widehat{s}=k_1\widehat{e}+k_2\int \widehat{e}dt$.

The design of LESO-SMC strategy is depicted in Figure 8, as mentioned in this article.

After collecting $U_o$ and D to LESO, LESO outputs the estimated value $z_1$ of $U_o$ and the estimated value $z_2$ of total disturbance. Using the errors between $U_{ref}$ and $z_1$ and $z_2$, the phase shift ratio D is output using Formula (22), and finally, eight switches are driven to work. It should be noted that D generally ranges from $-0.5$ to $+0.5$, so the saturation module should be added in front of the output control signal.

4. Analysis of Simulation Experiment Results

To validate the superiority and feasibility of the proposed LESO-SMC strategy, a simulation model of the DAB converter is constructed using the Matlab/Simulink platform, as shown in Figure 9. The main circuit parameters are shown in

Table 1. DAB converter circuit parameters.

Main Circuit Parameters	Value
Input voltage $U_i$/V	100
Output voltage $U_o$/V	60
Transformer ratio n	1:1
switching frequency $f_s$/Khz	10
Equivalent inductance L/μH	200
Input side capacitance $C_1$/μF	2000
output side capacitance $C_2$/μF	2000
load resistance R/$\mathsf{\Omega }$	30/15

.

This paper compares the dynamic performance and disturbance resistance of the converter’s output voltage under load change and input voltage change, using four different control strategies: traditional single closed-loop voltage PI control; the LADRC; the SMC presented in reference [29]; and the LESO-SMC strategy. The parameter selection for the four control strategies are shown in

Table 2. The parameter values of the proposed control method.

Control Methods	Parameters	Value
PI	$k_p$	$0.05$
	$k_i$	$1.5$
	$b_0$	2000
LADRC	${\omega }_0$	1600
	$k_p$	50
	$k_1$	1000
SMC	$k_2$	10
	$k_3$	40
	$\epsilon$	40
	$k_1$	1000
	$k_2$	10
LESO-SMC	$b_0$	2000
	${\omega }_0$	1600
	$k_3$	40
	$\epsilon$	40

.

4.1. Load Change Experiment

Figure 10 presents the output voltage dynamic performance waveform under the four control strategies during the load change. Figure 10 contains four subfigures, where (a) represents PI control, (b) shows LADRC, (c) denotes SMC, and (d) illustrates the LESO-SMC strategy proposed in this paper. The system experiences output voltage fluctuations due to the sudden load change and, after an adjustment period, recovers to the reference voltage. From the four subfigures in Figure 10, it can be noted that, when the load abruptly changes from $30\mathsf{\Omega }$ to $15\mathsf{\Omega }$ at 0.3 s, the output voltage of the PI control drops the most, about 2.9 V, and has the longest adjustment time, approximately 104 ms. The LADRC strategy experiences a smaller voltage drop of about 1.1 V, but the adjustment time is still relatively long, requiring about 78 ms. Traditional SMC strategy can rapidly regain stability after the load change with virtually no dynamic process and voltage fluctuation, but, as shown in Figure 10c, there is also notable chattering in the SMC. Additionally, the back-and-forth crossing of the control law over the sliding surface results in pronounced voltage ripples. The LESO-SMC strategy proposed in this paper requires a shorter adjustment time, recovering in approximately 3 ms, with the voltage dropping only about 0.13 V. More importantly, it resolves the issue of excessive chattering.

At 0.5 s, when the load changes from $15\mathsf{\Omega }$ back to $30\mathsf{\Omega }$, under traditional PI control, the adjustment time is the longest, requiring 95 ms, and the output voltage increase is the largest, about 4.5 V. The LADRC strategy experiences a voltage rise of about 0.6 V, with a longer adjustment time still needed, around 65 ms. Traditional SMC strategy can quickly transition back to the original output voltage without a noticeable voltage increase, but it still presents a significant chattering phenomenon. The LESO-SMC strategy proposed in this paper requires a shorter adjustment time, needing only about 5 ms, and a smaller voltage increase of approximately 0.3 V.

4.2. Input Voltage Change Experiment

Figure 11 and Figure 12 show the output voltage dynamic response waveforms under the four control methods when the input voltage jumps at 0.4 s. The former depicts the input voltage rising from 110 V to 115 V, and the latter represents the input voltage falling from 110 V to 85 V. In both figures, (a) represents PI control, (b) shows LADRC, (c) denotes SMC, and (d) illustrates the LESO-SMC strategy. Simulation results reveal that the system’s output voltage fluctuates with the jump in input voltage. After a certain period, the output voltage will recover to a stable level. As observed in the four subfigures in Figure 11, when the input voltage increases, the output voltage under traditional PI control rises the most, about 0.6 V, and requires the longest adjustment time, around 87 ms. The LADRC strategy experiences a smaller voltage rise, only about 0.1 V, but still requires a long adjustment time of approximately 30 ms. The SMC strategy has virtually no dynamic adjustment process and voltage increase, but from the magnified part of the subfigure Figure 11c, it can be seen that traditional SMC has significant chattering. The control strategy proposed in this paper features an almost negligible voltage rise and virtually no dynamic adjustment time. In the case of a decrease in input voltage, as seen in the four subfigures in Figure 12, the output voltage under PI control drops the most, approximately 0.6 V, with the longest adjustment time of around 92 ms. Under LADRC strategy, the output voltage drop is smaller, only about 0.1 V, but the adjustment time is still long, around 40 ms. Both SMC and the LESO-SMC strategy proposed in this paper exhibit rapid dynamic responses and stability, but SMC has notable chattering. However, as shown in Figure 12c, there is also notable chattering in the SMC.

4.3. Semi-Physical Experimental Verification

In order to further prove the feasibility and advanced nature of the control strategy mentioned, it is verified on the RT-LAB platform. Figure 13 shows the RT-LAB platform. The lower machine contains multiple real-time simulators that transmit data over the PCIe bus. The selected CPU simulation step is 10 μs.

Figure 14 shows the load transient waveforms of the four control strategies under the operation of the RT-Lab platform.

From Figure 14, it can be observed that under the PI control, when the load changes from full load to half load, the output voltage fluctuates by 2.6 V, and the settling time is 297 ms. When the load is restored, the voltage fluctuation is 7.2 V, and the settling time is 105 ms. Under LADRC strategy, when the load changes from full load to half load, the voltage fluctuates by 1.2 V, and the settling time is 80 ms. When the load is restored, the voltage fluctuation is 1.5 V, and the settling time is 75 ms; Under SMC and LESO-SMC strategies, the response speed is very fast, and the voltage shows minimal fluctuations. However, it is evident that the LESO-SMC strategy does not exhibit noticeable chattering.

Figure 15 and Figure 16 show the output voltage waveforms under the four control strategies on the RT-Lab platform during input voltage rising and falling, respectively.

From Figure 15 and Figure 16, it can be observed that the variation in the input voltage on the RT-Lab platform has a minor impact on the output voltage. Under PI control and LADRC strategy, the voltage slightly increases or decreases. Under SMC and LESO-SMC strategies, the voltage shows almost no fluctuations. However, it is worth noting that, compared to SMC, the LESO-SMC strategy exhibits no significant chattering.

Table 3. Performance comparison of four control strategies for load change.

Conditions	Control Methods	Voltages Fluctuation	Adjustment Times	Ripples Level
	PI	$-2.9$ V/$-2.6$ V	104 ms/297 ms	+/+
Form $30\mathsf{\Omega }$ to $15\mathsf{\Omega }$	LADRC	$-0.5$ V/$-1.2$ V	78 ms/80 ms	++/+++
	SMC	-/-	-/-	+++/+++
	LESO-SMC	$-0.13$ V/$-0.2$ V	3 ms/3 ms	+/+
	PI	$+4.5$ V/$+7.2$	95 ms/105 ms	+/+
Form $15\mathsf{\Omega }$ to $30\mathsf{\Omega }$	LADRC	$+0.6$ V/$+1.5$ V	65 ms/75 ms	++/+++
	SMC	-/-	-/-	+++/+++
	LESO-SMC	$+0.2$ V/$+0.3$ V	5 ms/4 ms	+/+

Note: + = Low; ++ = Medium; +++ = High; The slash/behind is the data under the RT-Lab platform.

and

Table 4. Performance comparison of four control strategies for input voltage changes.

Conditions	Control Methods	Voltages Fluctuation	Adjustment Times	Ripples Level
	PI	$+0.6$ V/$+0.15$ V	87 ms/-	++/+
Form 100 V to 115 V	LADRC	$+0.1$ V/$+0.25$ V	30 ms/-	++/+++
	SMC	-/-	-/-	+++/+++
	LESO-SMC	-/-	-/-	+/+
	PI	$-0.4$ V/$-0.15$ V	92 ms/-	++/+
Form 100 V to 85 V	LADRC	$-0.1$ V/$-0.3$ V	40 ms/-	++/+++
	SMC	-/-	-/-	+++/+++
	LESO-SMC	-/-	-/-	+/+

Note: + = Low; ++ = Medium; +++ = High; The slash/behind is the data under the RT-Lab platform.

, respectively, compare the performance of the DAB converter under four control strategies, specifically, the voltage fluctuation, adjustment time, and ripple level during sudden load changes and input voltage variations. The comparison results indicate that under the two aforementioned disturbance scenarios, the LESO-SMC strategy proposed in this paper exhibits smaller output voltage fluctuation and superior dynamic response performance compared to PI control and LADRC strategy. Compared to traditional SMC, it does not present obvious chattering issues, making the system more stable. In summary, the LESO-SMC strategy can significantly enhance the system’s dynamic performance and anti-interference ability.

5. Conclusions

This paper focuses on the DC-DC converter with good performance required in the distributed electric propulsion system, and selects the DAB converter as the research object. In order to improve the anti-disturbance ability and dynamic performance of the converter, this paper proposes a LESO-SMC strategy as follows:

(1)

Firstly, the principle of SPS control is elaborated in detail, and the reduced order dynamic model is derived. Then, based on this model, the LADRC strategy is designed, and the ability of the LESO to compensate for the system disturbance is used to improve the anti-interference ability and dynamic performance of the converter.

(2)

In order to further improve the performance of the converter, the LESO is combined with SMC, and the nonlinear error feedback law constructed by the observer’s value and the switching function can better adapt to the nonlinear characteristics of the converter, which is no serious chattering phenomenon.

(3)

Finally, the strategy is verified on the Matlab/Simulink and RT-Lab platform. The theoretical and experimental results show that the LESO-SMC strategy proposed in this paper has stronger robustness and better dynamic performance under the conditions of load mutation and input voltage change and can achieve a stable tracking of the output reference voltage.