Fractional-Order Linear Active Disturbance Rejection Control Strategy for DC-DC BUCK Converters

Abstract

This paper explores the problems of slow response speed, poor anti-interference performance, and low control accuracy that exist in traditional Active Disturbance Rejection Control methods in Buck-type DC/DC converters. To address these issues, a fractional-order Active Disturbance Rejection Control (FO-LADRC) controller is proposed to enhance the dynamic characteristics and anti-interference ability of Buck-type DC/DC converters, while expanding the control range and flexibility of traditional linear Active Disturbance Rejection Control (LADRC). Firstly, the mathematical model of the Buck-type DC/DC converter is established. Secondly, based on Active Disturbance Rejection Control, a fractional-order linear Extended State Observer (FO-LESO) is constructed to estimate the model error and external disturbance of the system. Then, the stability of the system is studied through transfer function and error analysis. Finally, the effectiveness of the FO-LADRC controller method is verified through simulation. The simulation and experiment results show that the proposed FO-LADRC method outperforms traditional PI and LADRC methods in terms of dynamic performance. It can effectively improve the dynamic characteristics of the system and enhance the anti-interference ability of the system.

1. Introduction

The Buck converter, which is widely employed in electric vehicles [1,2,3], microgrids [4], and energy storage systems [5,6], is a common type of DC/DC circuit. However, due to its inherent characteristics as a time-varying nonlinear system, the Buck converter is particularly sensitive to variations in internal parameters, load disturbances, and fluctuations in input voltage under real-world operating conditions. These factors frequently result in issues such as unstable output voltage and diminished control accuracy. As a result, numerous scholars have undertaken extensive research into innovative control strategies aimed at improving the output voltage control performance and enhancing the anti-interference capability of Buck converters.

For the Buck DC/DC converters, numerous researchers have proposed a series of control methods. Reference [7] developed a self-PID control strategy for the DC/DC converter using an FPGA-based experimental platform. This approach enhanced the system’s response speed to some extent; however, it was associated with a high computational complexity issue. Reference [8] presents the design of a sliding mode control (SMC) for a DC/DC converter, achieving integral sliding mode control through the implementation of a nonlinear control law. Reference [9] proposes a voltage-ripple (VR)-based dynamic freewheeling (DF) control technique for a hybrid conduction mode (HCM) single-inductor dual-output (SIDO) Buck converter to address the key limitations of Pseudocontinuous Conduction Mode (PCCM) SIDO converters. Reference [10] addresses the noise issue in DC/DC converters by employing Extended State Observer (ESO) cascading and Active Disturbance Rejection Control (ADRC). This approach not only mitigates the noise problem, but also enhances the stability of the system. For the parameter design in the DC/DC converter, reference [11]. Six discrete iterative models with potential evolutionary capabilities were established based on the power characteristics of the Buck converter operating in Continuous Conduction Mode (CCM). Furthermore, the parameter dynamic behavior of PWM modulation under varying pulse widths was elucidated via bifurcation diagrams. This analysis was extended to derive the boundary equations for the closed-loop stability of the system, thereby providing theoretical guidance for ensuring stable system control. As an alternative control strategy to the PI controller, Active Disturbance Rejection Control (ADRC) demonstrates strong robustness and high control accuracy. This can effectively address the trade-off between “rapidity” and “overshoot” associated with the PI controller. ADRC has been widely implemented in practical applications [12,13,14]. The Buck converter is a nonlinear time-varying system. In certain complex operating environments, the traditional integer PI control method exhibits low precision and a limited range. To compensate for the limitations of linear system control methods, active disturbance rejection can be applied to the Buck converter. However, LADRC is typically designed based on a simplified mathematical model. Consequently, when the input voltage or load undergoes rapid changes, the controller, limited by the model simplification, fails to estimate the system state and disturbances accurately and promptly, which in turn compromises the control performance. Additionally, the effectiveness of LADRC heavily relies on the appropriate tuning of parameters such as observer bandwidth and controller gain. Nevertheless, in practical applications, owing to the complex dynamic behavior of Buck-type DC/DC converters and their sensitivity to varying operating conditions, it is challenging to identify a fixed set of parameters that can accommodate all operational scenarios while ensuring the system’s rapid response, high precision, and robust anti-interference capabilities.

Reference [15] proposed a fractional-order controller tailored for the frequency response model of a servo system. The open-loop system incorporating this fractional-order controller exhibits a flat phase characteristic near the gain crossover frequency, which significantly enhances the robustness against loop gain variations. When compared with the optimized traditional integer-order controller, this controller demonstrates superior control performance. Reference [16] focuses on the topic of fractional control, with particular emphasis on fractional PID controllers. This study provides a comprehensive review of the current automatic tuning methods for fractional-order PID controllers. The primary focus is on the latest survey results. Comparisons among several methods are presented for various types of processes. Numerical examples are provided to demonstrate the practical applicability of these methods, which can be extended to simple industrial processes. Fractional-order linear Active Disturbance Rejection Control is an innovative controller that combines the merits of linear Active Disturbance Rejection Control with fractional-order theory, while incorporating targeted improvements. This approach can, to a certain extent, enhance the dynamic performance of the system. Reference [17] investigated a novel fractional-order Active Disturbance Rejection Control (FOADRC) method to address the issue of noise sensitivity in the Extended State Observer (ESO) within ADRC systems. The fractional-order Extended State Observer was designed by leveraging available information from the controlled system. Under a specified observer bandwidth, the performance of perturbation estimation was enhanced without increasing the noise sensitivity. Moreover, the fractional-order Extended State Observer transformed the conventional second-order controlled system into a fractional-order dual integrator model, thereby reducing the phase lag to less than 180 degrees. Reference [18] investigates the issue wherein delays in power systems can degrade the dynamic performance of load frequency control, potentially resulting in unstable system frequencies. This study proposes a fractionally ordered sliding mode controller tailored for interconnected power systems with time-varying delays and performs stability analysis using the Lyapunov direct method.

Therefore, because of the problems such as the slow response speed, poor anti-interference ability, and low control accuracy of the traditional Active Disturbance Rejection Control method in the Buck DC/DC converter [19], this paper proposes a novel fractional-order Active Disturbance Rejection Control (FO-ADRC) method for Buck converters. While the application of fractional-order ADRC is already relatively widespread [20], this paper will focus on the innovative aspect of the fractional-order ADRC proposed in this paper, in terms of the novelty in its practical applications. This paper has made improvements and optimizations, systematically improving and optimizing the FO-LADRC method for the special requirements of Buck converters, introducing new control structures and parameter adjustment strategies. Firstly, the Buck DC/DC converter is modeled. Then, considering the system disturbance, the FO-LADRC controller of the Buck converter is designed based on the existing ADRC, and the parameters of the system are determined. Next, the stability of the system is verified using two methods: system transfer function analysis and error analysis, and the Bode diagram of the system is drawn. Finally, the system control model is built in MATLAB/Simulink, and the superiority and effectiveness of the proposed FO-LADRC strategy are verified through simulation.

2. Modeling and Analysis of Buck DC/DC Converter

The typical topology of a PWM-based Buck converter [21] is illustrated in Figure 1.

As shown in Figure 1, the average state equation of the Buck converter can be derived as follows:

$$\left[\begin{array}{l}{\displaystyle \frac{\mathrm{d}{i}_L(t)}{\mathrm{d}t}}\\ {\displaystyle \frac{\mathrm{d}{V}_o(t)}{\mathrm{d}t}}\end{array}\right]=\left[\begin{array}{cc}0& -{\displaystyle \frac{1}{L}}\\ {\displaystyle \frac{1}{C}}& -{\displaystyle \frac{1}{RC}}\end{array}\right]\left[\begin{array}{c}{i}_L(t)\\ V_o(t)\end{array}\right]+\left[\begin{array}{c}{\displaystyle \frac{D}{L}}\\ 0\end{array}\right]\left[V_{in}(t)\right]$$

(1)

where i_L is the inductor current. Vo is the output voltage. V_in is the input voltage. D is the duty cycle.

Based on the average state Equation (1) of the Buck converter, the control input transfer function G_vd(s) can be derived as follows:

$$G_{\mathrm{vd}}\left(s\right)=\frac{V_{in}}{LC{s}^2+\frac{L}{R}s+1}$$

(2)

3. Design of FO-ADRC Algorithm for Buck Converter

Active Disturbance Rejection Control (ADRC) [21] is mainly composed of three parts: a Tracking Differentiator (TD), Extended State Observer (ESO), and Nonlinear State Error Feedback (NLSEF). The ADRC system structure is shown in Figure 2.

The primary distinction of ADRC from other control methodologies lies in its operation without requiring detailed mathematical model information on the controlled object as a prerequisite. This feature renders it suitable for complex systems, even in the presence of various uncertainties such as parameter perturbations. The robust applicability of ADRC is attributed to its core component, the Extended State Observer (ESO). Within ADRC theory, both internal uncertainties and external disturbances are treated as aggregate disturbances, which are estimated and compensated for in real time by the ESO.

For a common second-order nonlinear system:

$$\left\{\begin{array}{l}{\dot{x}}_1=x_2\\ {\dot{x}}_2=f(x_1,x_2,g(t),t)+bu\\ y=x_1\end{array}\right.$$

(3)

In the equation, g(t) denotes the external disturbances acting on the system, while f represents various internal uncertainties. Consequently, when the system dynamics are effectively controlled, the entire controlled system can function properly.

Let f = x₃, then the system can be written as follows:

$$\left\{\begin{array}{l}{\dot{x}}_1=x_2\\ {\dot{x}}_2=x_3+bu\\ {\dot{x}}_3=\dot{f}\\ y=x_1\end{array}\right.$$

(4)

Based on Equation (4), an observer can be designed. Given that it incorporates a new state variable x₃, this observer is referred to as an (ESO).

$$\left\{\begin{array}{l}e=z_1-x_1\\ {\dot{z}}_1=z_2-{\beta }_{1}e\\ {\dot{z}}_2=z_3-{\beta }_{2}e+bu\\ {\dot{z}}_3=-{\beta }_{3}e\end{array}\right.$$

(5)

where z₁, z₂, and z₃ denote the observed values of the system state variables, β₁, β₂, and β₃ represent the observer parameters, b signifies the system gain, and u is the input variable of the system.

To improve the control accuracy of the Active Disturbance Rejection Control (ADRC), nonlinear functions were employed as the parameters of the observer and controller. Although this led to an improvement in the engineering control quality, it also gave rise to problems such as a complex structure, difficult parameter adjustment, and challenging theoretical analysis. In addressing these issues, scholars such as Gao Zhiqiang, after conducting an in-depth analysis of the control principle and design method of ADRC, proposed a simplified form of the Linear Active Disturbance Rejection Controller (LADRC). While the LADRC retains the merits of the original ADRC, it reduces the utilization of nonlinear functions. Through the linearization treatment of the Extended State Observer (ESO) and Nonlinear State Error Feedback (NLSEF), the control system structure of the LADRC is significantly simplified, and it also brings considerable convenience in parameter tuning and theoretical analysis. The control structure of LADRC is illustrated in Figure 3.

4. Design of FO-ADRC for Buck Converter

In recent years, fractional-order calculus has been adopted in control theory along with the continuous refinement of control system design. Since the beginning of this century, research scholars have discovered that, compared with the traditional integer-order control scenarios, the controlled systems in fractional-order control possess superior dynamic performance and robustness, which has significantly facilitated the application of fractional-order theory in control engineering domains such as motors and aircraft. In the realm of DC/DC converters, the fractional-order control of the single-voltage closed loop has been accomplished, and better effects have been obtained compared to integer-order control. To further enhance and improve the stability and dynamic characteristics of the Buck converter, in this section, based on the merits of the LADRC and the advantages of fractional-order theory, a fractional-order linear Active Disturbance Rejection Control (FO-LADRC)method based on the output voltage is designed, to elevate the dynamic performance and anti-interference capability of the system.

4.1. Fractional-Order Linear Active Disturbance Rejection Control

Fractional-order linear Active Disturbance Rejection Control (FO-LADRC) is an extension of linear Active Disturbance Rejection Control (LADRC), which itself is based on integer-order control theory and comprises an error feedback mechanism and an Extended State Observer (ESO). The control structure of FO-LADRC is illustrated in Figure 4.

To retain the advantages of both PID and LADRC, the FO-LADRC scheme typically incorporates a PD controller as its error feedback mechanism. The key component of FO-LADRC is the fractional-order Extended State Observer (FO-ESO), whose primary function is to estimate and compensate for system disturbances in real time, thereby transforming the system into an integral series structure. This transformation introduces an integral term to the controlled object without explicitly adding an integrator to the system, enabling it to achieve static error-free output. The structural diagram of the FO-LESO is shown in Figure 5.

The expression for the fractional-order linear Extended State Observer (FO-LESO) can be formulated as follows:

$$\left\{\begin{array}{l}{e}_1=V_{ref}-z_1\\ D^{\alpha }{z}_1=z_2+{\beta }_1{e}_1\\ D^{\alpha }{z}_2=z_3+{\beta }_2{e}_1+b_{0}u\\ D^{\alpha }{z}_3={\beta }_3{e}_1\end{array}\right.$$

(6)

where e₁ is the system error. z₃ is the estimate of the total disturbance f of the system. D is the fractional differential operator. α is the differential order. β₁, β₂, and β₃ are the gain parameters of the FO-LESO and the parameters are unknown and variable.

4.2. Design of Fractional-Order Active Disturbance Rejection Control for Buck Converter

The system control structure of the Buck converter based on FO-LADRC is shown in Figure 6.

In practical applications, circuit components such as inductors and capacitors in a Buck converter, as well as the input voltage, may be influenced by external environmental factors. The nominal values of the input voltage and load are represented by V_in0 and R₀. If the total disturbance of the system is denoted as f, the following state equation can be derived as follows:

$$\left\{\begin{array}{l}{\dot{x}}_1=x_2\\ {\dot{x}}_2={\ddot{x}}_1={\displaystyle \frac{D{V}_{in0}-x_1}{LC}}-{\displaystyle \frac{x_2}{RC}}+f\\ f={\displaystyle \frac{D}{LC}}\left(V_{in}-V_{in0}\right)+x_2\left({\displaystyle \frac{1}{R_{0}C}}-{\displaystyle \frac{1}{RC}}\right)\end{array}\right.$$

(7)

where f is the new state x₃, which denotes the overall disturbance of the system. Assuming that f is a bounded interference, the equation of the state can be reformulated as follows:

$$\left\{\begin{array}{l}{\dot{x}}_1=x_2\\ {\dot{x}}_2=x_3+bu\\ {\dot{x}}_3=\dot{f}\end{array}\right.$$

(8)

where control quantity u = ${\displaystyle \frac{1}{b}}\left({\displaystyle \frac{D{V}_{in0}-x_1}{LC}}-{\displaystyle \frac{x_2}{R_{0}C}}\right)$. b is the system gain. The actual output is y = x₁ = V_o.

According to Equation (8), the characteristic equation of the FO-LESO can be obtained as follows:

$$s^{3\alpha }+{\beta }_1{s}^{2\alpha }+{\beta }_2{s}^{\alpha }+{\beta }_3=0$$

(9)

To enhance the stability performance of the control system and reduce the adjustment time, it is assumed that the ideal characteristic equation can be expressed as follows:

$${\left(s^{\alpha }+{\omega }_0\right)}^3=0$$

(10)

Therefore, the mathematical expression for the relationship between parameters β₁, β₂, β₃, and ω₀ is as follows:

$$\left\{\begin{array}{l}{\beta }_1=3{\omega }_0\\ {\beta }_2=3{\omega }_0^2\\ {\beta }_3={\omega }_0^3\end{array}\right.$$

(11)

where ω₀ is the bandwidth of the observer. Its value will affect the observation accuracy of the Extended State Observer. When the ω₀ value is high, the accuracy of the observation will also be improved, but the noise signal will also be amplified. Therefore, the choice of the ω₀ value should fully consider the impact of noise on the system.

If the estimation error of z₃ is disregarded, the system can be simplified to a structure of double integral series.

$$D^{2\alpha }y=u_0$$

(12)

The control quantity u₀ generated by the PD controller can be expressed as follows:

$$u_0=K_p\left(V_{ref}-z_1\right)-K_d{z}_2$$

(13)

where K_p and K_d are the control parameters of the integer order PD controller. According to Equations (11) and (12), the transfer function of the PD control system can be obtained as follows:

$$G_{PD}(s)=\frac{K_p}{s^2+K_{d}s+K_p}$$

(14)

The control variable u, which was obtained through the compensation of disturbance u₀, can be expressed as follows:

$$u={\displaystyle \frac{-z_3+u_0}{b_0}}$$

(15)

4.3. Stability Analysis of FO-ADRC

4.3.1. Stability Analysis Based on System Transfer Function

Based on the structure of the FO-LESO shown in Figure 5, it can be inferred that the transfer function of the FO-LESO can be expressed as follows:

$$\left\{\begin{array}{l}{z}_1\left(s\right)={\displaystyle \frac{G\left(s\right)\left({\beta }_1{s}^{2\alpha }+{\beta }_2{s}^{\alpha }+{\beta }_3\right)+b_0{s}^{\alpha }}{G\left(s\right)\left(s^{3\alpha }+{\beta }_1{s}^{2\alpha }+{\beta }_2{s}^{\alpha }+{\beta }_3\right)}}Y\left(s\right)\\ z_2\left(s\right)={\displaystyle \frac{G\left(s\right)\left({\beta }_2{s}^{2\alpha }+{\beta }_3{s}^{\alpha }\right)+b_0{s}^{2\alpha }+b_0{\beta }_1{s}^{\alpha }}{G\left(s\right)\left(s^{3\alpha }+{\beta }_1{s}^{2\alpha }+{\beta }_2{s}^{\alpha }+{\beta }_3\right)}}Y\left(s\right)\\ z_3\left(s\right)={\displaystyle \frac{{\beta }_{3}G\left(s\right)s^{2\alpha }-b_0{\beta }_3}{G\left(s\right)\left(s^{3\alpha }+{\beta }_1{s}^{2\alpha }+{\beta }_2{s}^{\alpha }+{\beta }_3\right)}}Y\left(s\right)\end{array}\right.$$

(16)

According to Equation (16), the closed-loop transfer function of the whole system can be expressed as follows:

$$H\left(s\right)={\displaystyle \frac{K_p\left(s^{3\alpha }+{\beta }_1{s}^{2\alpha }+{\beta }_2{s}^{\alpha }+{\beta }_3\right)G\left(s\right)}{b_0{s}^{\alpha }\left(s^{2\alpha }+A_1{s}^{\alpha }+A_2\right)+G\left(s\right)\left(A_3{s}^{2\alpha }+A_4{s}^{\alpha }+A_5\right)}}$$

(17)

where

$$\left\{\begin{array}{l}{A}_1=K_d+{\beta }_1\\ A_2=K_p+K_d{\beta }_1+{\beta }_2\\ A_3=K_p{\beta }_1+K_d{\beta }_2+{\beta }_3\\ A_4=K_p{\beta }_2+K_d{\beta }_3\\ A_5=K_p{\beta }_3\end{array}\right.$$

(18)

According to Equation (2), the controlled object can be simplified as follows:

$$G\left(s\right)={\displaystyle \frac{b}{s^2+a_{1}s+a_2}}$$

(19)

where a₁ = 125, a₂ = 2.5 × 10⁷, and b = 2.5 × 10⁹. In this paper, the single-parameter tuning method is used to adjust the parameters, which are as follows:

$$\left\{\begin{array}{c}{\omega }_c=2500\\ {\omega }_o=\mathrm{10,000}\\ b_0=5\times {10}^8\end{array}\right.$$

(20)

According to Equations (17), (19), and (20), the closed-loop transfer function of the system can be obtained as follows:

$$H\left(s\right)={\displaystyle \frac{C_1{s}^{4.4}+C_2{s}^{3.6}+C_3{s}^{3.4}+C_4{s}^{2.8}+C_5{s}^{2.4}+C_6{s}^2+C_7{s}^{1.8}+C_8{s}^{1.6}+C_{9}s+C_{10}{s}^{0.8}+C}{D_1{s}^{6.4}+D_2{s}^{5.6}+D_3{s}^{5.4}+D_4{s}^{4.8}+D_5{s}^{4.6}+D_6{s}^{4.4}+D_7{s}^{3.8}+D_8{s}^{3.6}+D_9{s}^{3.4}+D_{10}{s}^{2.8}+D_{11}{s}^{2.4}+D_{12}{s}^2+D_{13}{s}^{1.8}+D_{14}{s}^{1.6}+D_{15}s+D_{16}{s}^{0.8}+D}}$$

(21)

where C_i and D_i (i = 0, 1, 2….) are shown in

Table 1. The parameter value of the closed-loop transfer function.

Parameter	Value	Parameter	Value	Parameter	Value
C	3.90625 × 1035	D	3.90625 × 1035	D11	9.4922 × 1023
C1	1.5625 × 1016	D1	5 × 108	D12	1.5625 × 1028
C2	4.6875 × 1020	D2	1.75 × 1013	D13	3.57422 × 1027
C3	1.953125 × 1018	D3	1.25 × 1011	D14	1.7890625 × 1029
C4	4.6875 × 1024	D4	2.28125 × 1017	D15	1.953125 × 1030
C5	5.859375 × 1022	D5	4.375 × 1015	D16	5.72265 × 1032
C6	1.5625 × 1028	D6	2.501 × 1016
C7	5.859375 × 1026	D7	5.703125 × 1019
C8	1.171875 × 1028	D8	7.59402 × 1021
C9	1.953125 × 1030	D9	3.125 × 1018
C10	1.171875 × 1032	D10	2.8597 × 1025

.

The Bode plot of the system according to Equation (17) is shown in Figure 7.

As illustrated in Figure 7, the phase margin of the FO-ADRC system is measured at 141.7959°, with a corresponding frequency of 8.3914 × 10³ rad/s. The system exhibits minimal steady-state error and high control accuracy in the low-frequency band. In the mid-frequency band, the curve slope is relatively flat, leading to reduced system overshoot and a shorter settling time. Moreover, in the high-frequency band, the curve demonstrates the system’s robust anti-disturbance capability through its low-amplitude response. Consequently, it can be concluded from this analysis that the FO-LADRC system possesses a superior response speed and anti-interference performance.

4.3.2. Stability Analysis Based on Errors

According to Equation (8), the transfer function of z₁, z₂, and z₃ can be obtained as follows:

$$z_1\left(s\right)={\displaystyle \frac{3{\omega }_0{s}^{2\alpha }+3{\omega }_0^2{s}^{\alpha }+{\omega }_0^3}{{\left(s^{\alpha }+{\omega }_0\right)}^3}}Y\left(s\right)+{\displaystyle \frac{b_{0}s}{{\left(s^{\alpha }+{\omega }_0\right)}^3}}U\left(s\right)$$

(22)

$$z_2\left(s\right)={\displaystyle \frac{3{\omega }_0^2{s}^{2\alpha }+{\omega }_0^3{s}^{\alpha }}{{\left(s^{\alpha }+{\omega }_0\right)}^3}}Y\left(s\right)+{\displaystyle \frac{b_0{s}^{\alpha }\left(s^{\alpha }+3{\omega }_0\right)}{{\left(s^{\alpha }+{\omega }_0\right)}^3}}U\left(s\right)$$

(23)

$$z_3\left(s\right)={\displaystyle \frac{{\omega }_0^3{s}^{2\alpha }}{{\left(s^{\alpha }+{\omega }_0\right)}^3}}Y\left(s\right)-{\displaystyle \frac{b_0{\omega }_0^3}{{\left(s^{\alpha }+{\omega }_0\right)}^3}}U\left(s\right)$$

(24)

Let $e_1=z_1-y_1;e_2=z_2-y_2;e_3=z_3-y_3$, the error equation of the state can be derived from Equation (8) as follows:

$$\left\{\begin{array}{l}{\dot{e}}_1=e_2-{\beta }_1{e}_1\\ {\dot{e}}_2=e_3-{\beta }_2{e}_1\\ {\dot{e}}_3=\dot{f}-{\beta }_3{e}_1\end{array}\right.$$

(25)

According to Equation (25), the transfer function of errors e₁ and e₂ is as follows:

$$E_1\left(s\right)=-{\displaystyle \frac{s^{3\alpha }}{{\left(s^{\alpha }+{\omega }_0\right)}^3}}Y\left(s\right)+{\displaystyle \frac{s^{\alpha }}{{\left(s^{\alpha }+{\omega }_0\right)}^3}}{b}_{0}U\left(s\right)$$

(26)

$$E_2\left(s\right)=-{\displaystyle \frac{\left(s^{\alpha }+3{\omega }_0\right)s^{3\alpha }}{{\left(s^{\alpha }+{\omega }_0\right)}^3}}Y\left(s\right)+{\displaystyle \frac{s^{\alpha }}{{\left(s^{\alpha }+{\omega }_0\right)}^3}}{b}_{0}U\left(s\right)$$

(27)

According to Equation (8), it can be obtained as follows:

$$f=x_3={\dot{x}}_2-b_{0}u=D^{2\alpha }y-b_{0}u$$

(28)

If both y and u are considered to be step signals with y(s) = K/s, u(s) = K/s, and amplitude K, the steady-state error of the system can be determined as follows:

$$\left\{\begin{array}{l}{e}_{1s}=\underset{s\to 0}{\lim }s{e}_1=0\\ e_{2s}=\underset{s\to 0}{\lim }s{e}_2=0\\ e_{3s}=\underset{s\to 0}{\lim }s{e}_3=0\end{array}\right.$$

(29)

As shown in Formula (29), the FO-LESO demonstrates excellent convergence and estimation capabilities, enabling errorless estimation of system state variables and generalized perturbations. This highlights its strong potential for accurate estimation in academic research and practical applications.

5. Simulation and Verification

To verify the effectiveness of the proposed FO-LADRC strategy, the circuit simulation model shown in Figure 6 was built on the MATLAB 2022/Simulink platform. The specific data are shown in

Table 2. Parameters of Buck converter system components.

Parameter	Symbol	Value	Parameter	Symbol	Value
Input voltage	Vin	100 V	Switching frequency	f	100 kHz
Output voltage	Vo	30 V	Order	α	0.83
Resistance	R	20 Ω	Closed-loop bandwidth	ωc	2500
Inductance	L	400 μH	Observer bandwidth	ωo	10,000
Capacitance	C	100 μF	Control gain	b0	5 × 108

.

First of all, to validate the convergence of the error estimation of FO-LADRC, the output waveforms of the system errors z₁, z₂, and z₃ are depicted as shown in Figure 8. As illustrated in Figure 8, the systematic errors converge to zero and achieve a steady state within a relatively brief period. Additionally, the duty cycle stabilizes at approximately 0.3.

Secondly, to further verify the superiority and effectiveness of FO-LADRC, simulation verification is carried out in two aspects: the dynamic response and anti-interference performance of the Buck converter. Figure 9 presents the dynamic response of the system. It can be observed from Figure 9 that the response speed of FO-LADRC during the ascending process of the system is conspicuously superior to that of PI and LADRC. At the same time, for the Buck converter controlled by FO-LADRC, there is essentially no voltage overshoot and it can enter the steady-state time rapidly. Thus, it can be inferred that FO-LADRC surpasses PI and LADRC in terms of response speed, voltage overshoot, and steady-state time.

Figure 10 depicts the anti-interference performance of the system. Considering the sudden change in the load resistance R during operation—20 Ω → 15 Ω—the system responses obtained by employing three control algorithms are presented in Figure 10. It can be observed from Figure 10 that the regulation time of the system controlled by the FO-LADRC algorithm proposed in this paper when the load changes from 20 Ω to 15 Ω is 0.032 s, and that he voltage fluctuation range is 29.12 to 30 V. For the PI-controlled system, the regulation time when the load changes from 20 Ω to 15 Ω is 0.121 s, and the voltage fluctuation range is 27.82 to 30 V. For the LADRC-controlled system, the regulation time when the load changes from 20 Ω to 15 Ω is 0.082 s, and the voltage fluctuation range is 28.42 to 30 V. Hence, when the load varies, the FO-LADRC algorithm proposed in this paper enables the output voltage to track the reference value more rapidly and stably, and possesses stronger anti-interference performance.

Figure 11 depicts the noise resistance performance of the system. Considering the noise, the system output voltage change obtained by employing three control algorithms is presented in Figure 11. It can be observed from Figure 11 that the average voltage fluctuation range of the system controlled by the PI algorithm proposed in this paper is 25.91 to 37.82 V. For the LADRC-controlled system, the voltage fluctuation range is 28.33 to 31.64 V. For the FO-LADRC-controlled system, the voltage fluctuation range is 29.42 to 30.51 V. Hence, when considering the noise, the FO-LADRC algorithm proposed in this paper has the smallest voltage fluctuation range and possesses stronger anti-noise robust performance.

As indicated in

Table 3. Dynamic performance indicators of the system.

Control Strategy	Response Time	Voltage Overshoot	Anti-Interference Performance	Fluctuation Range of Disturbance Voltage
PI	0.121 s	3.31 V	Poor	27.82 V~30 V
LADRC	0.082 s	2.12 V	Generally average	28.42 V~30 V
FOADRC	0.032 s	0.07 V	Strong	29.12 V~30 V

, the FO-LADRC strategy proposed in this study demonstrates a substantial enhancement in system performance and robustness against disturbances, particularly exhibiting superior anti-interference capabilities in response to abrupt load variations.

6. Experimental Verification

To verify the effectiveness of the control strategy, an experimental platform for the Buck converter system based on STM32 was developed, as illustrated in Figure 12.

In this paper, dynamic response experiments for the Buck converter system and anti-interference capability tests for the system were carried out separately, as illustrated in Figure 13 and Figure 14.

It can be observed from Figure 13 and Figure 14 that the control strategy proposed in this paper demonstrates significant superiority over PI and LADRC in terms of response speed, voltage overshoot, and anti-interference capability. The detailed performance metrics are presented in

Table 4. Dynamic performance indicators of experimental platform for Buck converter.

Control Strategy	Response Time	Voltage Overshoot	Voltage Recovery Time	Fluctuation Range of Disturbance Voltage
PI	0.121 s	3.31 V	0.152 s	27.82 V~30 V
LADRC	0.082 s	2.12 V	0.133 s	28.42 V~30 V
FOADRC	0.032 s	0.07 V	0.103 s	29.12 V~30 V

.

In the validation section of this paper, the primary focus has been on comparing the proposed fractional-order Extended State Observer with the conventional linear ESO and PI controller. To provide a more comprehensive evaluation, this study further incorporates a comparative analysis against the traditional nonlinear ESO and other control methods through simulation and experimentation. The comparison results are shown in

Table 5. Compared with the existing literature.

Literature	Control Strategy	Model Dependency Degree	Controller Complexity	Control Accuracy
This paper	FO-LADRC	Low	Medium	High
[8]	SMC	Medium	High	Medium
[14]	Cascaded Filter ADRC	Medium	High	Medium
[19]	DRL controller	High	High	Medium
[21]	ADRC with nonlinear ESO	Low	Medium	Medium

. Compared with the existing literature, the control strategy proposed int this paper has the best performance.

7. Conclusions

For the DC/DC Buck converter system, a fractional-order linear Active Disturbance Rejection Control (FO-LADRC) strategy is proposed. This strategy extends the traditional linear Active Disturbance Rejection Control (LADRC) into the fractional-order domain by incorporating the fractional order α, thereby enhancing the control accuracy and system performance. The research commences with the mathematical modeling of the DC/DC Buck converter, followed by the systematic design and implementation of the FO-LADRC controller to address the internal disturbances within the system. Subsequently, the system parameters and control rates are determined through rigorous analysis. On this foundation, the stability of the system is rigorously evaluated using the system transfer function and error analysis methods, while the Bode plot of the system is constructed to illustrate the robust stability achieved using the proposed FO-LADRC strategy. Simulation and experimental results demonstrate that, in comparison with conventional PI control and traditional LADRC, the FO-LADRC strategy presented in this study exhibits superior dynamic performance and disturbance rejection capabilities.