© Y. Li et al., Published by EDP Sciences, 2025

1 Introduction

Electro-hydraulic position servo systems exhibit advantages such as rapid response speed, high control accuracy, and strong load adaptability [1,2]. They have been widely utilized in fields requiring high-precision dynamic control, including aerospace applications, robotics technology, and construction machinery. However, electro-hydraulic servo systems are subject to parametric changes, external disturbances, and unmodeled uncertainties. These factors making the tracking accuracy of conventional PID controllers inadequate to meet the requirements [3].

Active Disturbance Rejection Control (ADRC) employs an observer to estimate model inaccuracies and exogenous perturbations, actively compensates for these factors, and integrates them with a control law to achieve desired control performance, thereby overcoming the limitations of conventional PID controllers [4]. However, the effectiveness of ADRC in practical applications is constrained by its complex and experience-dependent parameter tuning process. To further enhance control performance, numerous scholars have integrated existing control methodologies with fuzzy control strategies, exemplified by Fuzzy-PID control [5] and Fuzzy-ADRC control [6,7]. Tao et al. [8] utilized a fuzzy logic-based adaptive mechanism to dynamically optimize the Extended State Observer parameters, thereby enhancing the system's robustness under dynamic parametric and external interference conditions. Nevertheless, the method exhibited limitations in addressing suboptimal parameter calibration within the fuzzy logic framework, leading to insufficient dynamic tracking performance.

To address the parameter tuning challenges in ADRC controllers, scholars have introduced optimization algorithms to achieve online self-tuning of parameters. Cai et al. [9] proposed an approach based on an improved PSO algorithm. Compared to traditional PID control and conventional ADRC, the PSO-optimized ADRC offers advantages such as precise position tracking and strong immunity to interference. Wang et al. [10] employed the Grey Wolf Optimizer (GWO) to optimize controller parameters, achieving enhanced control accuracy. However, this algorithm remains susceptible to convergence factor constraints and tends to get stuck in local optima. Current research has predominantly adopted conventional optimization algorithms (e.g., PSO and GWO) for parameter tuning; however, these algorithms suffer from limitations such as slow convergence rates and insufficient global search capabilities. To address the aforementioned challenges, the NRBO [11] effectively circumvents the tendency of conventional algorithms to stagnate in local optima while exhibiting accelerated convergence rates and robust global search capabilities, rendering it particularly applicable to ADRC parameter optimization.

This study employs NRBO to perform online self-tuning of Fuzzy-ADRC parameters in an electro-hydraulic position servo system, thereby further enhancing system controllability. Finally, the simulations demonstrate the functionality of the proposed method.

2 NRBO-Fuzzy-ADRC control algorithm

2.1 Design of Fuzzy-ADRC

The ADRC consists mainly of a tracking differentiator (TD), a nonlinear error feedback controller (NLSEF) and an extended state observer (ESO) [12]. The ADRC strategy in this study can effectively handle system nonlinearities and unmodeled dynamics. It does not rely on a precise mathematical model. This control approach uses the ESO to lump together all complex internal and external disturbances into a unified “total disturbance,” enabling real-time observation and compensation. The ESO estimates these disturbances as an integrated entity and achieves real-time compensation through the control law, thereby effectively overcoming model uncertainties and unmodeled dynamic characteristics. Furthermore, this study integrates a fuzzy logic module to enhance the controller's adaptive capability under varying operating conditions. This enhancement enables the controller to respond rapidly and accurately to dynamic changes in the electro-hydraulic servo system, effectively addressing the challenges posed by high system nonlinearity.

By incorporating fuzzy logic [13], the disturbance values and state estimates obtained from the ESO are combined with the fuzzy compensation term to synthesize the final control input. The steps involved in the design of adaptive fuzzy compensation are outlined below, in four distinct phases.

Step 1: Fuzzy Logic System Design

The fuzzy system employs dual input signals (e₁: tracking error and e₂: error variation rate), each partitioned into 5 functions, thereby generating a rule base of 25 fuzzy inference rules. By selecting the membership functions and determining the fuzzy rules, the fuzzy compensation quantity u_f(x) is obtained through the defuzzification process.

In the selection of membership functions, e₁ and e₂ use Gaussian-type (Gaussmf) functions, while the membership functions for other domains are triangular-type (trimf) functions. The universe of discourse of the fuzzy variables is [−3, 3], and the fuzzy inference uses the Mamdani-type method, with the defuzzification process employing the centroid method [14]. The fuzzy inference rules are comprehensively outlined in Table 1.

Step 2: Construction of a Fuzzy Compensation Control Mechanism

The ESO estimates the unknown disturbance. After compensation, the actual control input u is formed as follows [12]:

$$u\left(k\right)=u_0\left(k\right)-\frac{z_3\left(k\right)}{b_0}$$(1)

where u₀ is the error feedback control input; b₀ is the compensation factor; z₃ is the output signal of ESO; $\frac{z_3\left(k\right)}{b_0}$is the disturbance compensation quantity estimated by the ESO.

The fuzzy compensation quantity u_f (x) is added to equation (1) to compensate for the residual disturbances not covered by the ESO, and the actual control input u₁ is modified as follows:

$$u_1\left(k\right)=u_0\left(k\right)-\frac{z_3\left(k\right)}{b_0}+u_{f\left(x\right)}$$(2)

Step 3: Fuzzy Output and Parameter Adaptation Law

The weighted average defuzzification method is adopted to calculate the fuzzy compensation quantity u_f (x), which is expressed as:

$$u_{f\left(x\right)}=\frac{{\displaystyle \sum _{\text{i}=1}^5}{\theta }_i{\mu }_i\left(e\left(x\right),\mathrm{\Delta }e\left(x\right)\right)}{{\displaystyle \sum _{\text{i}=1}^5}{\mu }_i\left(e\left(x\right),\mathrm{\Delta }e\left(x\right)\right)}$$(3)

where θ_i (k) is the adjustable parameter; μ_i serves as the critical weighting factor; e(k) represents the tracking error; Δe(k) represents the rate of change in tracking error.

Based on Lyapunov stability theory, the adjustment of θ_i (k) minimizes e(k). The parameter adaptation law is designed as:

$${\theta }_i\left(k\text{+1}\right)={\theta }_i\left(k\right)+\gamma e\left(k\right)\frac{{\mu }_i\left(e\left(x\right),\mathrm{\Delta }e\left(x\right)\right)}{{\displaystyle \sum _{\text{i}=1}^5}{\mu }_i\left(e\left(x\right),\mathrm{\Delta }e\left(x\right)\right)}$$(4)

where γ > 0 is the adaptive gain; e (k) represents the tracking error.

The control law must incorporate fuzzy compensation, and the updated control quantity u₁ is given by:

$$u_1\left(k\right)=u_0\left(k\right)-\frac{z_3\left(k\right)}{b_0}+\frac{{\displaystyle \sum _{\text{i}=1}^5}{\theta }_i{\mu }_i\left(e\left(x\right),\mathrm{\Delta }e\left(x\right)\right)}{{\displaystyle \sum _{\text{i}=1}^5}{\mu }_i\left(e\left(x\right),\mathrm{\Delta }e\left(x\right)\right)}$$(5)

Step 4: Stability Analysis via Lyapunov Function

Establish the Lyapunov function candidate as:

$$V\left(k\right)=\frac{1}{\text{2}}{e}^2\left(k\right)\text{+}\frac{1}{2\gamma }{\displaystyle \sum _{\text{i}=1}^5}{{\displaystyle \tilde{\theta }}}_{\text{i}}^2\left(k\right)$$(6)

where ${{\displaystyle \tilde{\theta }}}_{\text{i}}^2\left(k\right)$is the parameter error between the i th fuzzy rule adjustable parameter θ_i (k) and its ideal optimal value ${\theta }_i^{*}$, ${{\displaystyle \tilde{\theta }}}_{\text{i}}^2\left(k\right)={\theta }_i^{*}-{\theta }_i\left(k\right)$. γ is the adaptive gain coefficient, which adjusts the rate of parameter update.

Dynamic update of parameter error:

Based on equation (4), the dynamic update equation of the parameter error ${\displaystyle \tilde{\theta }}\left(k\right)$ at the next moment can be derived:

$${{\displaystyle \tilde{\theta }}}_{\text{i}}^2\left(k+1\right)=\theta \left(k+1\right)-\gamma e\left(k\right)\frac{{\displaystyle \sum _{\text{i}=1}^5}{\mu }_i}{{\displaystyle \sum _{\text{i}=1}^5}{\mu }_i}$$(7)

Based on equation (5), the ESO disturbance estimate [12] and fuzzy compensation. The dynamic equation for the tracking error is given by:

$$e\left(k+1\right)=e\left(k\right)-{\beta }_{01}fal\left(e_1,a_{11},\delta \right)-{\beta }_{02}fal\left(e_2,a_{12},\delta \right)+{\displaystyle \sum _{\text{i}=1}^5}{{\displaystyle \tilde{\theta }}}_i\left(k\right)\frac{{\mu }_i}{{\displaystyle \sum _{\text{i}=1}^5}{\mu }_i}+\delta \left(k\right)$$(8)

where δ (k) denotes bounded disturbance caused by the ESO estimation error; β₀₁ and β₀₂ represent gains of ESO; e₁ and e₂ represent error values; a₁₁ and a₁₂ are parameters of the nonlinear function; fal() is a nonlinear function. The fal function is utilized to deal with the state errors e₁, e₂, aiming to drive the error reduction.

Lyapunov function difference and stability proofs: The difference of the Lyapunov function between consecutive time steps is: ΔV (k) = V (k + 1) − V (k).

The tracking error dynamic equation (8) and the parameter error update equation (7) are substituted into the calculation of the differential ΔV(k). Based on the parameter adaptation law, detailed algebraic operations lead to the following expression: equation (9).

$$\mathrm{\Delta }V\left(k\right)=\frac{1}{\text{2}}{e}^2\left(k+1\right)-\frac{1}{\text{2}}{e}^2\left(k\right)+\frac{1}{2\gamma }{\displaystyle \sum _{\text{i}=1}^5}\left({{\displaystyle \tilde{\theta }}}_{\text{i}}^2\left(k+1\right)-{{\displaystyle \tilde{\theta }}}_{\text{i}}^2\left(k\right)\right)$$(9)

Assuming the controller parameters (e.g., β₀₁, β₀₂ > 0) are well-designed and the fuzzy logic system drives e₁, e₂ to zero efficiently, and further assuming the ESO estimation error Δ(k) is bounded (based on detailed algebraic operations and inequality relaxation),the following inequality can be obtained after simplification:

$$\mathrm{\Delta }V\left(k\right)\le 0$$

When the disturbance is bounded, e (k)and ${\displaystyle \tilde{\theta }}\left(k\right)$ will ultimately converge to a neighbourhood of zero, satisfying the stability conditions. This demonstrates that the Lyapunov stability condition ΔV (k) ≤ 0 is satisfied, which inherently accommodates unmodeled nonlinear effects. Furthermore, the fuzzy adaptive law equation (6) enhances the robustness of the electro-hydraulic position servo system by compensating for nonlinear drift through adaptation of θ_i (k).

2.2 Parameter optimization of the NRBO

To optimize the efficacy of the ADRC control, a fuzzy logic-based compensation [15] module is integrated into the controller. Concurrently, the NRBO algorithm is utilized to enable real-time adaptive parameter adjustment for the Fuzzy-ADRC hybrid controller, improving system dynamic performance. The NRBO-Fuzzy-ADRC controller structure diagram is presented in Figure 1.

In the ADRC controller, the parameters β₁ and β₂ of the NLSEF, as well as β₀₁, β₀₂, and β₀₃ of the ESO, are typically set as constants. However, in the tracking process of the electro-hydraulic positioning system, fixed parameters cannot adapt to the real-time changes in state errors, resulting in the system's inability to achieve precise trajectory tracking.

Figure 2 shows the optimization curves of β₁ and β₂ as well as β₀₁, β₀₂, and β₀₃ after 20 iterations

The optimization process was completed within 20 iterations, fully demonstrating the computational efficiency of the NRBO algorithm. For comparative analysis, an NRBO-optimized ADRC controller (NRBO-ADRC) without the fuzzy compensation module was also tuned using the same optimization process. The optimized parameters remain unchanged during online operation, thereby simplifying real-time calculations to simple arithmetic operations involving fuzzy compensation equation (5) and disturbance estimation equation (1). Since parameter tuning is performed only once before controller deployment, no additional computational burden is added during runtime.

Fig. 1 NRBO-Fuzzy-ADRC controller structure diagram.

Fig. 2 Optimization curves of β1, β2, β01, β02 and β03.

3 Newton-Raphson-based optimizer

3.1 Method principle

The NRBO is a numerical method for solving function zeros. It iteratively approximates equation roots using Taylor series expansions. The solution location is updated with first and second-order derivative information. The method integrates three key components: the Newton-Raphson Search Rule (NRSR) to enhance global exploration, the Trap Avoidance Operator (TAO) to reduce premature convergence, and multiple matrix configurations to refine the search trajectory. The algorithmic workflow includes the following key phases:

(1) Initialization phase

Initialize a population where each individual represents a combination of parameters and define upper and lower bounds for each parameter to ensure that the search space is reasonable.

$$x_j^n=l{b}_{\text{j}}+\text{r and}\times \left(u{b}_j-l{b}_j\right)n=\mathrm{1,2...}j=\mathrm{1,2,...},N_p,\dim$$(10)

where $x_j^n$ is the value of the nth individual on the nth parameter; rand is a random number (0,1); ub_j and lb_j are the upper and lower bounds of the j th parameter respectively; N_p denotes the population size; dim is the dimension.

The population matrix is:

$${\left[\begin{array}{cccc}\hfill x_1^{\text{1}}\hfill & \hfill x_2^{\text{1}}\hfill & \hfill \mathrm{...}\hfill & \hfill x_{\dim }^1\hfill \\ \hfill x_1^{\text{2}}\hfill & \hfill x_2^{\text{2}}\hfill & \hfill \dots \hfill & \hfill x_{\dim }^2\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill x_1^{N_{\text{p}}}\hfill & \hfill x_2^{N_{\text{p}}}\hfill & \hfill \mathrm{...}\hfill & \hfill x_{\dim }^{N_{\text{p}}}\hfill \end{array}\right]}_{N_{\text{p}}\times \mathrm{dim.}}$$(11)

(2) Calculation of fitness values

The fitness value of each individual is calculated using the Integral of Time-weighted Absolute Error (ITAE) as the primary optimization criterion. The fitness function is formulated as:

$$J={\int }_0^{t}t\left|e\left(t\right)\right|dt.$$(12)

(3) Search and update rule of NRSR

NRSR, as the core search mechanism of the NRBO algorithm, dynamically adjusts the search trajectory of individuals within the population. Furthermore, it modulates the exploration path of candidate solutions through Hessian matrix analysis of the objective function. This mechanism enhances both convergence speed and solution precision by leveraging second-order derivatives. The definition of the second derivative is as follows:

$$f\left(x+\mathrm{\Delta }x\right)=f\left(x\right)+f^{\prime }\left(x_0\right)+\frac{1}{2!}{f}^{″}\left(x_0\right)\mathrm{\Delta }{x}^2+\frac{1}{3!}{f}^{‴}\left(x_0\right)\mathrm{\Delta }{x}^3+\dots$$(13)

$$f\left(x-\mathrm{\Delta }x\right)=f\left(x\right)-f^{\prime }\left(x_0\right)+\frac{1}{2!}{f}^{″}\left(x_0\right)\mathrm{\Delta }{x}^2-\frac{1}{3!}{f}^{‴}\left(x_0\right)\mathrm{\Delta }{x}^3+\dots$$(14)

By subtracting or adding equations (13) and (14), the formulas for f^′ (x) 和 f^″ (x) are as detailed below:

$$f^{\prime }\left(x\right)=\frac{f\left(x+\mathrm{\Delta }x\right)-f\left(x-\mathrm{\Delta }x\right)}{2\mathrm{\Delta }x}$$(15)

$$f^{″}\left(x\right)=\frac{f\left(x+\mathrm{\Delta }x\right)+f\left(x-\mathrm{\Delta }x\right)-2\times f\left(x\right)}{\mathrm{\Delta }{x}^2}.$$(16)

The NRSR rules are used to update the positions of population individuals x_n+1, improving the current parameter combination. The update equation is:

$$x_{\text{n+1}}=x_{\text{n}}-\frac{f\left(x_{\text{n}}+\mathrm{\Delta }x\right)-f\left(x_{\text{n}}-\mathrm{\Delta }x\right)\times \mathrm{\Delta }x}{2\left(f\left(x_{\text{n}}+\mathrm{\Delta }x\right)+f\left(x_{\text{n}}-\mathrm{\Delta }x\right)-2f\left(x_{\text{n}}\right)\right)}$$(17)

where x_n is the current parameter combination; rand (1,dim) is a random number in the interval (1,dim); x_b is the best solution in the current population; $x_{\text{n}}^{\text{1T}}$ is the position information of the i th individual in the ongoing iteration step; Δx is the update step, $\mathrm{\Delta }x=\text{r and}\left(1,\dim \right)\times |x_{\text{b}}-x_{\text{n}}^{\text{1T}}|$.

The population search method of NRSR is defined as follows:

$$\text{NRSR}=\text{rand}\times \frac{x_w-x_{\text{b}}\times \mathrm{\Delta }x}{2\left(x_w+x_{\text{b}}-2x_{\text{n}}\right)}$$(18)

where x_w is the worst position.

(4) Trap Avoidance Operation

The Trap Avoidance Operator (TAO) dynamically adjusts the current position of a particle with a specific probability DF to avoid premature convergence. The update formula for the trap-avoidance operation is:

$$x_{\text{n}}^{\text{1T+1}}=x_{\text{TAO}}^{\text{1T}}$$(19)

$$x_{\text{TAO}}^{\text{1T}}=x_{\text{n}}^{\text{1T}}+{\theta }_1\left({\mu }_1{x}_b-{\mu }_2{x}_{\text{n}}^{\text{1T}}\right)+{\theta }_2\cdot \delta \left({\mu }_1\cdot Mean\left(x_{\text{1T}}-{\mu }_2{x}_{\text{n}}^{\text{1T}}\right)\right)$$(20)

where θ₁ and θ₂ are randomly generated values in the bounds [−1,1] and [−0.5,0.5] for each; Mean() is the mean function; δis the adaptive parameter, within [−1,1]; μ₁ and μ₂ are random individuals, defined by the following equation:

$$\begin{array}{l}{\mu }_1=\{\begin{array}{cc}\hfill 3\times \text{rand}\hfill & \hfill \text{if}\mathrm{\Delta }<0.5\hfill \\ \hfill 1\hfill & \hfill \text{otherwise}\hfill \end{array}\\ {\mu }_2=\{\begin{array}{cc}\hfill \text{rand}\hfill & \hfill \text{if}\mathrm{\Delta }<0.5\hfill \\ \hfill 1\hfill & \hfill \text{otherwise}\hfill \end{array}\end{array}$$(21)

where Δ is a randomly generated number within the range (0, 1).

(5) Updating the population position

The values of the current parameter combinations are updated based on the new positions after the NRSR and TAO operations with the goal of enhancing the quality of the solution. The updating formula is:

$$x_{\text{n}}^{\text{1T+1}}=x_{\text{n}}^{\text{1T}}-\text{NRSR}.$$(22)

(6) Termination conditions

Once the maximum allowed number of iterations is reached, terminate the optimization process and output the optimal solution; otherwise, proceed to step (2). The flowchart of NRBO-ADRC parameter optimization is shown in Figure 3.

Fig. 3 Parameter optimization flowchart of NRBO-ADRC.

3.2 Performance testing of the NRBO

To evaluate the solution quality enhancement of the NRBO algorithm, four typical test functions (F1–F4) are used to compare the performance of PSO, GWO, and NRBO. The average (Ave), standard deviation (Std), and best values (Bes) are recorded for all experimental results. The experiments are conducted with 500 iterations and a population size of 100. The test functions are listed in Table 2, while the results can be found in Table 3. Additionally, the corresponding three-dimensional plots of the test functions and their convergence behaviour are presented in Figures 4–7.

The test results show that the NRBO algorithm surpasses the other two algorithms in global convergence accuracy, stability, and optimization performance. In experiments with the unimodal test functions F1 and F2, as well as the multimodal test functions F3 and F4, the mean values, standard deviations, and optimal solution values of the NRBO algorithm were all the smallest.

Comprehensive experimental results indicate that when handling complex nonlinear mathematical models, the NRBO algorithm achieves a balance between high-precision solutions and stable output, demonstrating strong potential for engineering applications. This rapid convergence enables efficient offline optimization, eliminating runtime computational burdens.

Fig. 4 Test function.

Fig. 5 Test function.

Fig. 6 Test function.

Fig. 7 Test function.

4 Simulation analysis

4.1 Model establishment

To realistically simulate the hydraulic control system, a physical model was constructed in the MATLAB/Simulink environment. The model components include: a displacement pump providing a constant pressure source; an electro-hydraulic servo valve regulating the switching and flow rate of the hydraulic circuit; a single-rod hydraulic cylinder producing force and displacement; a spring-damper-mass system simulating intrinsic physical dynamics; applied loads and disturbances simulating operational loads and external perturbations; and the NRBO-Fuzzy-ADRC adjusting the control signals to achieve closed-loop operation.

The final physical model of the electro-hydraulic position servo system is shown in Figure 8, and the physical model of the electro-hydraulic servo valve is separately detailed in Figure 9. The overall simulation model of the NRBO-Fuzzy-ADRC control system is presented in Figure 10.

The NRBO algorithm is utilized to adapt the parameters of the Fuzzy-ADRC controller. A total of 14 parameters were optimized. The algorithm configuration used a population size of 50, 20 iterations, and a decision factor (DF) of 0.6.

Figure 11 presents the convergence curve of the fitness function after 20 iterations for a population size of 50. From the figure, it is evident that the curve initially decreases and then stabilizes. After the 16th iteration, the fitness value of the NRBO algorithm reached a steady state and no longer changed significantly. The better fitness value indicates that the solution found by the NRBO algorithm is of higher quality, highlighting the algorithm's superior optimization performance, faster convergence speed, and greater accuracy.

The parameter values for the NRBO-Fuzzy-ADRC controller are as follows: b₀ = 0.5; TD: r = 5, h₀ = 0.0006; NLSEF: β₁ = 200, β₂ = −1, a₁ = 0.19, a₂ = 0.87, δ₀ = 0.11; ESO: β₀₁ = 50, β₀₂ = 2000, β₀₃ = 70, a₁₁ = 0.18, a₁₂ = 0.10, δ = 0.1. The parameter configuration for the hydraulic cylinder and servo valve is shown in Table 4.

Based on actual physical constraints, plant parameters were determined using rated specifications of the following components: a Rexroth 4WE 6 J62/EG24N9K4 servo valve, a BAFANG MOB30 × 100 hydraulic cylinder, and Skydrol 500B-4 hydraulic fluid. The resulting parameter configuration for the electro-hydraulic servo physical model is provided in Table 4.

Fig. 8 Physical model of electro-hydraulic servo system.

Fig. 9 Physical model of electro-hydraulic servo valve.

Fig. 10 Overall simulation model of NRBO-Fuzzy-ADRC control system.

Fig. 11 Convergence curve of fitness function.

4.2 Step response

The system is loaded with 0.5 kN, a step signal of 0.04 m is input, the simulation time is set to 5 s. A load perturbation of 1 kN is applied to the hydraulic cylinder from 2 to 4 s. The simulation results are presented in Figure 12.

As illustrated in Figure 12, both the NRBO-ADRC and NRBO-Fuzzy-ADRC controllers exhibit rapid response characteristics upon disturbance occurrence, enabling the system to regain steady-state operation much more rapidly.

The values of rise time (t_r), adjustment time (t_s), steady state error (e_ss), ISE and ITSE calculated from Figure 12 are shown in Table 5.

Table 5 further demonstrates that, in comparison to the ADRC control algorithm, the NRBO-Fuzzy-ADRC algorithm demonstrates the fastest rise time and settling time while maintaining the minimal steady-state error.

From the simulation data, it is evident that the NRBO-Fuzzy-ADRC algorithm offers improved control performance and enhanced robustness in the electro-hydraulic position servo system.

Fig. 12 Step response characteristic curve.

4.3 Comparison of control input u

To rigorously assess the comparative performance of the three control methods during step response, the control input u of ADRC, NRBO-ADRC, and NRBO-Fuzzy-ADRC were analysed, with the results presented in Figure 13.

From the comparison of the control input u in Figure 13, it can be observed that after the disturbance occurs, the control input u of the ADRC shows a small variation, but its variation lasts for a relatively long time. In contrast, the control inputs of NRBO-ADRC and NRBO-Fuzzy-ADRC exhibit larger variations, but the variation processes are much shorter.

Fig. 13 Comparison of control input u.

4.4 Sine response

A sinusoidal input signal y = 0.04 sin(0.5πt) was applied to evaluate the tracking performance of the system under ADRC and NRBO-Fuzzy-ADRC controllers. The results from the simulation of the sine response (a) and the tracking error are presented (b) in Figure 14.

As illustrated in Figure 14, the tracking error under NRBO-Fuzzy-ADRC control is significantly smaller than that of the ADRC controller.

Based on the simulation results, the system's phase lag (PL), average error (e_a), relative amplitude attenuation (RAA), and ISE values calculated are listed in Table 6.

As Table 6 quantitatively demonstrates, the proposed NRBO-Fuzzy-ADRC controller significantly outperforms the conventional ADRC, reducing phase lag by 53.9%, improving tracking accuracy by 70.1%, and lowering the ISE by 90%.

These results conclusively confirm the remarkable superiority of the NRBO-Fuzzy-ADRC strategy in minimizing phase lag, suppressing tracking deviations, and enhancing precision, thereby ensuring high-fidelity trajectory tracking in the electro-hydraulic position servo system.

Fig. 14 Sine response characteristic curve and tracking error curves.

4.5 Robustness and sensitivity verification

The robustness of the controller under system parameter perturbations (±15%) and external disturbances is evaluated using Monte Carlo testing and standardized regression sensitivity analysis. The adaptability of the NRBO-Fuzzy-ADRC under varying operating conditions is then verified.

4.5.1 The robustness validation experimental design

The robustness of the controller was evaluated using Monte Carlo analysis [16] This simulation aimed to replicate real-world parameter fluctuations by applying ±15% variations to the system's key parameters. This tolerance band was chosen to conservatively cover the combined effects of component manufacturing tolerances and potential parameter drift during operation (Tab. 4). Based on the dynamic characteristics of the electro-hydraulic servo system and engineering experience, ten core parameters sensitive to disturbances are selected: m, K, B, p_m, q_m, degC, D_a, D_b, n_sv, ω_sv. This study sets the failure threshold to twice the optimal ITSE value (0.000014 from Tab. 5) based on common engineering practice.

The ITSE performance distribution of 500 samples is obtained through Monte Carlo simulation within MATLAB/Simulink. All sample ITSE values are below the failure threshold (maximum observed value: 1.1×10^‒5), confirming the reliability of the control strategy under ±15% parameter variations. The probability density distribution of ITSE, as shown in Figure 15, further illustrates the results.

The verified robustness under ±15% parameter perturbations, combined with the controller's deterministic structure, indicates that this method is suitable for deployment in embedded systems.

Fig. 15 Probability distribution density of ITSE.

4.5.2 Sensitivity analysis

To further investigate the influence of different parameter disturbances on control performance, the standardized regression coefficient method [17] is used to calculate the sensitivity index and identify the key sensitive parameters. The parameter sensitivity index chart is shown in Figure 16.

As illustrated in Figure 16, variations in damping coefficient B, rated flow q_m, and temperature degC exert the most substantial influence on control performance ITSE. Consequently, particular attention should be given to the variations of these parameters in practical applications.

Fig. 16 The parameter sensitivity index chart.

5 Conclusion

This study conducted numerical simulations on the MATLAB/Simulink platform to analyse the step response and sinusoidal response of an electro-hydraulic position servo system under three control algorithms: ADRC, NRBO-ADRC, and NRBO-Fuzzy-ADRC. The principal findings are summarized as follows:

• The simulation data demonstrate that, in the step response, the NRBO-Fuzzy-ADRC control algorithm outperforms the ADRC control algorithm, achieving the shortest tuning time of 0.20 s, and the phase lag and error of the sine response are also the smallest, which results in superior control performance.

• In the step response, the NRBO-Fuzzy-ADRC and NRBO-ADRC control algorithms demonstrate comparable performance in perturbation resistance; however, the NRBO-Fuzzy-ADRC exhibits superior steady-state accuracy with a significantly smaller steady-state error.

• After the parameters of ADRC are optimized by NRBO, it effectively prevents the parameters from getting stuck in local optima and resolves the precision issue of the electro-hydraulic position servo system. In addition, the controller parameters are optimized offline, which reduces the time and expert effort required for manual tuning.

• This paper discusses the robustness of the proposed method under ±15% parameter perturbations, supporting its potential deployment in embedded systems. Combined with its extremely low online computing requirements, this indicates that the NRBO-Fuzzy-ADRC strategy is suitable for real-time platforms.

Furthermore, this study has the following limitations that should be addressed in future research:

• Integrating high-fidelity actuator dynamics and sensor noise models to enhance validation robustness.

• Evaluating the computational cost and real-time performance of the algorithm when deployed on embedded hardware platforms, along with a quantitative assessment of its robustness under practical constraints.

• Experimental validation through Hardware-in-the-Loop (HIL) and physical bench tests to rigorously evaluate the controller's performance against real-world actuator limitations.

Funding

This research was funded by National Natural Science Foundation grant number China (51605145); Henan Provincial Natural Science Foundation (232300420085).

Conflicts of interest

The authors declare no conflict of interest.

Data availability statement

Data are contained within the article.

Author contribution statement

Conceptualization and investigation, Y.L.; resources and data curation, J.X.; software, N.S.; validation, L.M.; writing-original draft preparation, J.X.; writing-review and editing, Y.L. All authors have read and agreed to the published version of the manuscript.

References

All Tables

All Figures

	Fig. 1 NRBO-Fuzzy-ADRC controller structure diagram.
In the text

	Fig. 2 Optimization curves of β1, β2, β01, β02 and β03.
In the text

	Fig. 3 Parameter optimization flowchart of NRBO-ADRC.
In the text

	Fig. 4 Test function.
In the text

	Fig. 5 Test function.
In the text

	Fig. 6 Test function.
In the text

	Fig. 7 Test function.
In the text

	Fig. 8 Physical model of electro-hydraulic servo system.
In the text

	Fig. 9 Physical model of electro-hydraulic servo valve.
In the text

	Fig. 10 Overall simulation model of NRBO-Fuzzy-ADRC control system.
In the text

	Fig. 11 Convergence curve of fitness function.
In the text

	Fig. 12 Step response characteristic curve.
In the text

	Fig. 13 Comparison of control input u.
In the text

	Fig. 14 Sine response characteristic curve and tracking error curves.
In the text

	Fig. 15 Probability distribution density of ITSE.
In the text

	Fig. 16 The parameter sensitivity index chart.
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