Parameter Optimization of ADRC for Rolling-Mill Hydraulic Screw-Down Synchronization Based on a WMA–PSO Hybrid Algorithm

Abstract

Parameter tuning for Active Disturbance Rejection Control (ADRC) in rolling mill hydraulic synchronization systems is critical for enhancing strip quality. Conventional manual trial-and-error methods often yield suboptimal results. This paper proposes a hybrid algorithm, WMA-PSO, integrating the Humpback Whale Migration Algorithm (WMA) with Particle Swarm Optimization (PSO) through an adaptive fusion weight strategy. This approach effectively balances global exploration and local exploitation, improving optimization accuracy and efficiency. Evaluation on the CEC-2005 benchmark suite shows that WMA-PSO outperforms several state-of-the-art algorithms. Simulation experiments on ADRC tuning in a rolling mill system demonstrate that the WMA-PSO-optimized controller achieves the smallest synchronization error and superior overall control performance compared to other methods. The results validate WMA-PSO as an effective tool for automated parameter tuning in complex industrial control systems.

1. Introduction

In the production of modern steel and non-ferrous metals, the rolling mill serves as the core equipment. Its control precision directly dictates the final quality of the plate and strip materials and the overall efficiency of the production line [1,2]. Among its critical aspects, the synchronous control of hydraulic servo positions between the drive side of the mill and the operator side is a key link [3]. Due to inevitable discrepancies in hardware, such as hydraulic cylinders and servo valves on both sides, resulting from installation, wear and tear, load conditions, and oil characteristics, it is difficult to achieve a perfectly identical dynamic response for hydraulic positions on both sides [4]. This slight asynchronization can directly lead to quality defects such as transverse thickness variation and unilateral wave shapes (edge waves). In severe cases, it can even cause production accidents like strip deviation or breakage, posing a serious threat to both product quality and equipment safety. Therefore, research on high-precision and high-response synchronous control strategies for hydraulic servo positions has significant theoretical meaning and practical engineering value to improve the core competitiveness of high-end plate and strip products [5].

To address these challenges, various synchronization control strategies have been developed. Among them, Active Disturbance Rejection Control (ADRC), originally proposed by Professor Jingqing Han, exhibits significant potential in rolling mill hydraulic synchronization. This is mainly due to its independence from the precise mathematical model of the plant and its capability to estimate and compensate for real-time internal and external disturbances. As highlighted in [5], ADRC effectively suppresses parameter perturbations and load fluctuations by integrating an Extended State Observer (ESO) to observe and compensate for system uncertainties. This mechanism enables rapid bilateral position synchronization and establishes a robust technical framework to resolve complex hydraulic synchronization issues.

Although ADRC has advantages such as a simple structure and strong robustness [6,7], the control performance of ADRC is highly dependent on precise tuning of a set of key parameters. These parameters exhibit complex coupling relationships. Relying on manual trial-and-error for tuning is not only time-consuming and labor intensive, but it also makes it difficult to guaranty that the controller will achieve optimal performance [8,9]. To some extent, this has limited the broader application of ADRC in complex industrial scenarios [10]. To address this, the application of intelligent optimization algorithms for automatic parameter adjustment has become a current research hotspot [11,12,13,14]. Particle Swarm Optimization (PSO), due to its simple concept and fast convergence speed, is widely used for various parameter optimization problems [15,16]. However, when dealing with complex high-dimensional problems, it has drawbacks, such as being prone to getting trapped in local optima and having insufficient search precision in its later stages [17]. The Whale Migration Algorithm (WMA), as a relatively novel swarm intelligence algorithm, mimics the migratory and predatory behavior of humpback whales. Although it demonstrates good global exploration capabilities, there is still room for improvement in its local exploitation performance [18].

On the one hand, the performance of the ADRC controller for rolling mill hydraulic synchronization is severely constrained by the effectiveness of its parameter tuning. On the other hand, when solving such complex optimization problems, a single intelligent optimization algorithm struggles to balance global exploration efficiency with local convergence precision [19,20,21]. Furthermore, there is little research on the deep fusion of WMA and PSO, two algorithms with complementary characteristics, to solve the problem of automatically tuning the ADRC parameters. In response to the issues above, the main work and contributions of this paper are as follows:

• A novel adaptive hybrid optimization algorithm, named WMA-PSO, is proposed by integrating the Whale Migration Algorithm (WMA) and Particle Swarm Optimization (PSO). By designing a dynamic population division mechanism based on fitness ranking and introducing an adaptive fusion weight, the algorithm effectively balances the powerful global exploration of WMA with the efficient local exploitation of PSO. This synergistic integration is designed to improve performance in solving complex optimization problems.
• The proposed WMA-PSO algorithm was evaluated against several state-of-the-art baseline optimization methods using the widely recognized CEC-2005 benchmark suite. The results demonstrate that WMA-PSO achieves superior performance in convergence speed, solution accuracy, and stability.
• The practical efficacy of WMA-PSO was demonstrated through its application to ADRC parameter tuning in a rolling-mill hydraulic synchronous control system. In comparative simulations under multiple working conditions, including step response, dynamic tracking, and load disturbance rejection, the parameters identified by WMA-PSO enabled the control system to achieve optimal comprehensive dynamic performance, minimizing synchronization error, settling time, and demonstrating the strongest robustness. This application not only validates the algorithm’s effectiveness for high-dimensional, strongly coupled engineering problems but also establishes a valuable reference for the automated parameter tuning of other complex control systems.

2. System Modeling and Problem Formulation

2.1. Model of the Hydraulic Servo Synchronization System in a Rolling Mill

The plant under study is the bilateral electro-hydraulic position–synchronization system of a rolling mill. To ensure reproducibility and comparability, the control architecture and the validated dynamic model reported in [5] are followed. The two sides (drive side and operator side) are modeled as two third-order servo subsystems with parameter mismatch. The synchronization target is to minimize the bilateral position deviation:

$$x_e\left(t\right)=x_{p1}\left(t\right)-x_{p2}\left(t\right)$$

(1)

where $x_{p1}\left(t\right)$ and $x_{p2}\left(t\right)$ denote the measured piston positions of the drive-side and operator-side cylinders, respectively. To actively suppress the synchronization error, a pair of compensation signals with opposite polarities is injected into the two position loops. Denote the ADRC compensation signal by $U_e$. The discrete-time ADRC controller consists of an Extended State Observer (ESO) and a state error feedback control law. The internal structure of the ESO is defined by Equations (4) and (5), which utilize the estimation error $e\left(t\right)$ (Equation (3)) to track the state ${\widehat{x}}_1$ and the total disturbance ${\widehat{x}}_2$. The control law is then formulated as Equation (6).

$$\begin{array}{cccc}\hfill & x_1\left(t\right)=x_e\left(t\right)=x_{p1}\left(t\right)-x_{p2}\left(t\right)\hfill & \hfill & \end{array}$$

(2)

$$\begin{array}{cccc}\hfill & e\left(t\right)=x_1\left(t\right)-{\widehat{x}}_1\left(t\right)\hfill & \hfill & \end{array}$$

(3)

$$\begin{array}{cccc}\hfill & {\widehat{x}}_1(t+1)={\widehat{x}}_1\left(t\right)+h({\widehat{x}}_2\left(t\right)+{\overline{b}}_e{U}_e\left(t\right)+{\beta }_{1}e\left(t\right)\hfill & \hfill & \end{array}$$

(4)

$$\begin{array}{cccc}\hfill & {\widehat{x}}_2(t+1)={\widehat{x}}_2\left(t\right)+h{\beta }_{2}e\left(t\right)\hfill & \hfill & \end{array}$$

(5)

$$\begin{array}{cc}\hfill & U_e(t+1)={\displaystyle \frac{u_e^{*}\left(t\right)-{\widehat{x}}_2\left(t\right)}{{\overline{b}}_e}}\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & u_e^{*}\left(t\right)=k_e\left(\Delta x_r\left(t\right)-{\widehat{x}}_1\left(t\right)\right)\hfill \end{array}$$

(7)

with the ESO bandwidth parameterization

$${\beta }_1=2{\omega }_o,{\beta }_2={\omega }_o^2$$

(8)

where h is the sampling period, ${\widehat{x}}_1\left(t\right)$ is the ESO estimate of the error state $x_1\left(t\right)$, ${\widehat{x}}_2\left(t\right)$ estimates the total disturbance, $k_e$ is the error–feedback gain, and ${\overline{b}}_e$ is the dynamic compensation gain, ${\overline{b}}_e$ is the dynamic compensation gain, $x_{r1}$ and $x_{r2}$ are the setpoints of the two sides, $\Delta x_r=x_{r1}-x_{r2}$. The schematic of the hybrid WMA–PSO-tuned ADRC for dual-cylinder synchronization is shown in Figure 1.

2.2. Problem Formulation and Engineering Context

Although ADRC provides strong robustness against uncertainties and disturbances, its closed-loop performance is highly sensitive to tuning of several key parameters—specifically the ESO bandwidth ${\omega }_o$, the error-feedback gain $k_e$ and the dynamic compensation gain ${\overline{b}}_e$. These parameters are nonlinearly coupled through their effects on response speed, disturbance rejection, and synchronization accuracy; manual trial-and-error tuning is time-consuming and rarely yields near-optimal settings. Thus, ADRC parameter tuning is posed as a constrained optimization problem in continuous variables. Let the decision vector be

$$\mathbf{P}=[{\omega }_o,k_e,{\overline{b}}_e]$$

(9)

and adopt the Integral Squared Error (ISE) of the synchronization error over a finite horizon T as the performance index:

$$J\left(\mathbf{P}\right)={\int }_0^T{(x_{p1}\left(t\right)-x_{p2}\left(t\right))}^{2}dt$$

(10)

For discrete-time simulations at the sampling period $T_s$ (i.e., $h=T_s$), this becomes the case.

$$J\left(\mathbf{P}\right)=T_s\sum _{k=0}^N{(x_{p1}\left[k\right]-x_{p2}\left[k\right])}^2,N=⌊{\displaystyle \frac{T}{T_s}}⌋$$

(11)

The tuning task is formulated as follows.

$$\underset{\mathbf{P}\in \mathcal{D}}{min}J\left(\mathbf{P}\right)$$

(12)

where $\mathcal{D}$ denotes the feasible engineering bounds for ${\omega }_o$, $k_e$, and ${\overline{b}}_e$. In Section 3, a hybrid WMA–PSO optimizer is developed to efficiently search for the optimal $\mathbf{P}$ under the above formulation. A schematic of the ADRC parameter-tuning framework based on the hybrid WMA–PSO algorithm is provided in Figure 2.

3. The Hybrid WMA–PSO Optimization Algorithm

To address the nonconvexity, multimodality, and inter-dimensional coupling inherent in ADRC parameter tuning, we propose a hybrid metaheuristic named WMA–PSO, which integrates WMA’s global exploration with PSO’s local exploitation efficiency. This section first reviews the two base algorithms and the notation, then presents the design philosophy and key operators of WMA–PSO; implementation details follow in subsequent subsections.

3.1. Review of the Base Algorithms

3.1.1. Particle Swarm Optimization (PSO)

PSO is a swarm-intelligence method for continuous optimization. Each candidate solution is represented by individual i with position ${\mathbf{x}}_i\left(t\right)\in {\mathbb{R}}^D$ and velocity ${\mathbf{v}}_i\left(t\right)\in {\mathbb{R}}^D$ at iteration t. The individual exploits two pieces of knowledge: its personal best ${\mathbf{P}}_{\mathrm{best},i}\left(t\right)$ and the swarm global best ${\mathbf{G}}_{\mathrm{best}}\left(t\right)$. The canonical velocity–position updates are

$$\begin{array}{cc}\hfill {\mathbf{v}}_i(t+1)& =\omega {\mathbf{v}}_i\left(t\right)+c_1{\mathbf{r}}_1\odot \left({\mathbf{P}}_{\mathrm{best},i}\left(t\right)-{\mathbf{x}}_i\left(t\right)\right)+c_2{\mathbf{r}}_2\odot \left({\mathbf{G}}_{\mathrm{best}}\left(t\right)-{\mathbf{x}}_i\left(t\right)\right)\hfill \end{array}$$

(13)

$${\mathbf{x}}_i(t+1)={\mathbf{x}}_i\left(t\right)+{\mathbf{v}}_i(t+1)$$

(14)

where $\omega$ is the inertia weight, $c_1,c_2$ are the cognitive and social acceleration coefficients, and ${\mathbf{r}}_1,{\mathbf{r}}_2\sim \mathcal{U}\left({[0,1]}^D\right)$ are i.i.d. uniform random vectors (applied element-wise via ⊙). Proper scheduling of $\omega ,c_1,c_2$ (e.g., a decreasing $\omega$) helps balance exploration and exploitation; nevertheless, PSO may suffer from premature convergence on high-dimensional multimodal landscapes without diversity-preserving or local-refinement operators.

3.1.2. Whale Migration Algorithm (WMA)

WMA is a metaheuristic inspired by the group migration and cooperative foraging of humpback whales. Its core mechanism is a dynamic leader–follower hierarchy. At each iteration, individuals are sorted by fitness and split into a leading group and a following group. The following whales migrate toward the centroid (or a representative) of the leaders via a migration–following operator, steering the search toward more promising basins and enabling basin-to-basin moves. This social learning mechanism provides strong global exploration; however, the local exploitation near the optimum is relatively weak, motivating the incorporation of PSO’s velocity–position update to enhance late-stage convergence.

3.2. Leading Group the WMA-PSO Hybrid

A single optimizer often struggles to balance global exploration and local exploitation. PSO achieves high exploitation efficiency through cognitive and social learning, but may compromise exploration; conversely, WMA’s hierarchical migration offers strong exploration yet lacks fine-grained refinement near the optimum.

A deep hybrid scheme is herein developed to coordinate global exploration and local exploitation: WMA’s dynamic leader–follower hierarchy serves as the macroscopic search framework, while PSO’s velocity–position updates act as microscopic search operators. The two strategies are coordinated by an adaptive weighting schedule to realize an intelligent search process that performs broad exploration in the early stage and precise exploitation in the later stage. Concretely, WMA’s fitness-based dynamic grouping is maintained, while group-specific update rules are imposed on the two sub-populations:

Leading group: 

Adopts the standard PSO update to exploit the promising regions around the current best solutions.

Following group: 

Performs a hybrid move: first executes a WMA-style migration toward the leading centroid, and then applies a PSO-style inertial increment to explore along the leadings’ direction while maintaining its own momentum. These two moves are convexly fused.

3.3. Implementation of the WMA-PSO Hybrid

3.3.1. Algorithmic Procedure

The general execution procedure of the hybrid WMA-PSO algorithm is as follows, and its corresponding flow chart is illustrated in Figure 2.

• Initialization Phase

  (a)

  Parameter and population initialization: Define the population size $N_{pop}$, the problem dimension $D=3$, the lower bound $\mathbf{L}=[L_1,L_2,L_3]$ and the upper bound $\mathbf{U}=[U_1,U_2,U_3]$ for the parameters to be optimized; the maximum number of iterations $Maxiter$, and the leader ratio $N_L$. Within the given parameter boundaries, randomly initialize the position ${\mathbf{W}}_i$ of each individual in the population and set its initial velocity ${\mathbf{V}}_i$ to zero.

  (b)

  Initial Fitness Evaluation; For the initialized population, calculate the fitness value $J\left({\mathbf{W}}_i\right)$ for each individual.

  (c)

  Best Value Initialization: Set the current position of each individual as its personal best position, i.e., ${\mathbf{P}}_{\mathrm{best},i}={\mathbf{W}}_i$. From the entire initial population, find the individual with the minimum fitness value and assign its position to the global best position, ${\mathbf{G}}_{\mathrm{best}}$.

• Main Evolutionary Loop

  (a)

  Fitness Evaluation: For each individual in the population (each set of ADRC parameters), run the simulation of the rolling mill synchronous control system. Using the Integral of Squared Error (ISE), J, as the fitness function, calculate the corresponding fitness value for each individual.

  (b)

  Population Sorting and Grouping: Sort all individuals in the current population in ascending order based on their fitness values. Define the top $N_L$ individuals as the leading whales and the rest as the following whales.

  (c)

  Center Calculation and Weight Update: Calculate the central position of the leading whale group, and update the adaptive inertia weight w based on the current iteration number.

  (d)

  Position Update: Update the velocity and position of each individual using different update rules described below, depending on its group (leader or follower).

  (e)

  Boundary Handling: Check if the updated positions exceed the predefined boundaries. If so, set them to the boundary values.

  (f)

  Fitness Evaluation and Selection Update: For each individual i in the population, calculate the fitness value $J\left({\mathbf{W}}_i^{\mathrm{new}}\right)$ of the new position ${\mathbf{W}}_i^{\mathrm{new}}$. Only if the fitness of the new position is better than that of the old position, i.e., $J\left({\mathbf{W}}_i^{\mathrm{new}}\right)<J\left({\mathbf{W}}_i\right)$, is the update accepted, setting ${\mathbf{W}}_i={\mathbf{W}}_i^{\mathrm{new}}$.

  (g)

  Personal and Global Best Update: For each individual i in the population, update the personal best and the global best as follows: If $J\left({\mathbf{W}}_i\right)<J\left({\mathbf{P}}_{\mathrm{best},i}\right)$, update ${\mathbf{P}}_{\mathrm{best},i}={\mathbf{W}}_i$. If $J\left({\mathbf{W}}_i\right)<J\left({\mathbf{G}}_{\mathrm{best}}\right)$, update ${\mathbf{G}}_{\mathrm{best}}={\mathbf{W}}_i$.

  (h)

  Termination Condition: The loop terminates when the maximum number of iterations is reached, or when the fitness value no longer shows significant improvement.

• Termination Phase

  (a)

  Output Result: Output the final optimal parameters for the ADRC controller and their corresponding best fitness value, $J\left({\mathbf{G}}_{\mathrm{best}}\right)$.

3.3.2. Core Update Rules

• Adaptive Weight: To balance the algorithm’s exploration and exploitation capabilities at different stages, a linearly decreasing adaptive inertia weight is designed:

  $$w\left(\mathit{iter}\right)=w_{max}-{\displaystyle \frac{w_{max}-w_{min}}{\mathit{MaxIter}}}\mathit{iter}$$

  (15)

  where $w\left(\mathit{iter}\right)$ denotes the inertia weight at iteration $\mathit{iter}$. In early iterations, a larger w encourages global exploration (particularly for the leading group), whereas in later iterations a smaller w facilitates fine-grained local search near the optimum.
• Update Rule for the Leading Group: As the leading group, the task of the leading group is to perform deep exploitation within the promising regions that have already been discovered. Therefore, it adopts the standard PSO update formulas, fully utilizing personal best information and global best information to guide its movement. Velocity Update:

  $${\mathbf{V}}_i^{\mathrm{new}}=w{\mathbf{V}}_i+c_1{\mathit{r}}_1\left({\mathbf{P}}_{\mathrm{best},i}-{\mathbf{W}}_i\right)+c_2{\mathit{r}}_2\left({\mathbf{G}}_{\mathrm{best}}-{\mathbf{W}}_i\right)$$

  (16)
  Position Update:

  $${\mathbf{W}}_i^{\mathrm{new}}={\mathbf{W}}_i+{\mathbf{V}}_i^{\mathrm{new}}$$

  (17)

  ${\mathbf{W}}_i^{\mathrm{new}}$

  updated position vector of individual i;

  ${\mathbf{W}}_i$

  current position vector of individual i;

  ${\mathbf{V}}_i^{\mathrm{new}}$

  updated velocity vector of individual i;

  ${\mathbf{V}}_i$

  current velocity vector of individual i;

  ${\mathbf{P}}_{\mathrm{best},i}$

  personal best position vector of individual i;

  ${\mathbf{G}}_{\mathrm{best}}$

  global best position vector;

  ${\mathit{r}}_1,{\mathit{r}}_2$

  random vectors (e.g., elementwise in ${[0,1]}^d$);

  $c_1,c_2$

  cognitive and social acceleration coefficients, $c_1,c_2\in [1.5,2.5]$;

  w 

  scalar inertia weight with a linearly decreasing schedule from $w_{max}$ to $w_{min}$ as the iteration index increases.

• Update Rule for the Following Group: The task of the following whale group is to learn from the leading group while simultaneously exploring new possibilities. Its update rule fuses the concepts of WMA and PSO.

  $$\begin{array}{cc}\hfill {\mathbf{W}}_i^{\mathrm{wma}}& ={\mathbf{W}}_{\mathrm{mean}}+\mathrm{rand}(1,D)\odot ({\mathbf{W}}_{i-1}-{\mathbf{W}}_i)+\mathrm{rand}(1,D)\odot ({\mathbf{G}}_{\mathrm{best}}-{\mathbf{W}}_{\mathrm{mean}})\hfill \end{array}$$

  (18)

  $$\begin{array}{cc}\hfill {\mathbf{V}}_i^f& =w{\mathbf{V}}_i+c_1{\mathbf{r}}_1({\mathbf{P}}_{\mathrm{best},i}-{\mathbf{W}}_i)+c_2{\mathbf{r}}_2({\mathbf{G}}_{\mathrm{best}}-{\mathbf{W}}_i)\hfill \end{array}$$

  (19)

  $${\mathbf{W}}_i^{\mathrm{pso}}={\mathbf{W}}_i+{\mathbf{V}}_i^f$$

  (20)

  $${\mathbf{W}}_i^{\mathrm{new}}=\alpha {\mathbf{W}}_i^{\mathrm{wma}}+(1-\alpha ){\mathbf{W}}_i^{\mathrm{pso}}$$

  (21)

  ${\mathbf{W}}_{\mathrm{mean}}$

  centroid of the leading group;

  ⊙

  Hadamard (element-wise) product;

  ${\mathbf{W}}_{i-1},{\mathbf{W}}_i$

  positions of the $(i-1)$-th and i-th individuals in the fitness-sorted list (for $i=N_L+1$, ${\mathbf{W}}_{i-1}$ may belong to the leading set);

  ${\mathbf{V}}_i^f$

  fused PSO-style velocity increment for the following individual i;

  ${\mathbf{W}}_i^{\mathrm{pso}},{\mathbf{W}}_i^{\mathrm{wma}},{\mathbf{W}}_i^{\mathrm{new}}$

  intermediate PSO step, WMA step, and the final fused position of individual i;

  $\alpha$

  fusion weight, $\alpha \in [0,1]$ (set to $0.3$ in our experiments);

  ${\mathbf{G}}_{\mathrm{best}},{\mathbf{P}}_{\mathrm{best},i},{\mathbf{r}}_1,{\mathbf{r}}_2$

  vector-valued global best, personal best, and per-dimension i.i.d. random vectors; $c_1,c_2,w$ as defined in the leading-group update (Section 3.3.2).

4. Simulation Experiments and Results Analysis

To comprehensively evaluate the performance of the proposed WMA-PSO algorithm, a series of experiments were conducted using both benchmark functions and a real-world engineering problem. The proposed WMA-PSO was compared against five well-established algorithms: PSO, WMA, WOA, GWO, and BA. All algorithms were tested on the CEC-2005 test suite. To ensure a fair comparison, a uniform population size and stopping criterion—set to a maximum of 1000 iterations—were applied across all algorithms. Each test was independently repeated 30 times to ensure statistical reliability. Performance was quantitatively evaluated using several metrics, including mean, standard deviation, and average rank (MFr) based on per function rankings. Furthermore, to demonstrate practical applicability, WMA-PSO was applied to optimize ADRC parameters in a rolling mill hydraulic synchronization system, with the maximum number of iterations set to 100. Key performance indicators, such as synchronization error, were compared to validate the effectiveness and superiority of the proposed method in a practical engineering context. All algorithms used the same common parameter settings, as provided in

Table 1. Parameter settings for optimization algorithms.

Parameter	Value
Population size, $N_{\mathrm{pop}}$	60
Maximum iterations, $\mathit{MaxIter}$	1000/100
Observer bandwidth, ${\omega }_o$	[1, 10,000]
Dynamic compensation factor, ${\overline{b}}_e$	[1, 50,000]
Error feedback gain, $k_e$	[0.1, 20,000]
PSO learning factors, $c_1,c_2$	2, 2
WMA leader ratio, $N_{\mathrm{L}}/N_{\mathrm{pop}}$	0.2
Fusion weight, $\alpha$	0.3
Inertia weight, $w_{max}/w_{min}$	0.9/0.4

.

4.1. Performance on the CEC-2005 Benchmark Suite

To assess the robustness of the proposed WMA-PSO algorithm under challenging and realistic conditions—characterized by features such as shifting, rotation, and high-dimensionality—comprehensive evaluations were conducted using the CEC-2005 benchmark suite. This suite consists of 14 complex 30-dimensional functions, which are specifically designed to simulate a wide spectrum of real-world optimization landscapes. The detailed characteristics of these functions are summarized in

Table 2. Information on CEC-2005 test functions.

ID	Function	Dim.	Property
F1	Shifted Sphere	30	Unimodal, separable
F2	Shifted Schwefel 2.26	30	Multimodal, separable
F3	Shifted Rotated High-Cond. Elliptic	30	Unimodal, non-sep.
F4	Shifted Schwefel 2.26 (with noise)	30	Multimodal, separable
F5	Schwefel 2.6 (linear max)	30	Unimodal, non-sep.
F6	Shifted Rosenbrock	30	Valley, non-separable
F7	Shifted Rotated Griewank	30	Multimodal, non-sep.
F8	Shifted Rotated Ackley	30	Multimodal, non-sep.
F9	Shifted Rastrigin	30	Multimodal, separable
F10	Shifted Rotated Rastrigin	30	Multimodal, non-sep.
F11	Shifted Rotated Weierstrass	30	Multimodal, non-sep.
F12	Schwefel 2.13	30	Multimodal, non-sep.
F13	Expanded Griewank + Rosenbrock	30	Multimodal, non-sep.
F14	Shifted Rotated Expanded Scaffer’s F6	30	Multimodal, non-sep.

.

The detailed statistical results are summarized in

Table 3. List of optimal results achieved using different algorithms on the CEC-2005 suite.

Func.	WMA-PSO	PSO	WMA	WOA	GWO	BA
$F_1$	1.13 × 10−26	9.13 × 102	4.41 × 10−26	1.18 × 102	6.42 × 102	9.80 × 104
	1.48 × 10−26	6.99 × 102	2.38 × 10−26	1.12 × 102	5.31 × 102	1.75 × 104
	1	5	2	3	4	6
$F_2$	−2.02 × 103	−1.80 × 103	−1.86 × 10−3	−1.59 × 103	−2.12 × 103	−4.83 × 102
	7.52 × 101	1.46 × 102	1.15 × 102	1.83 × 102	1.23 × 102	2.85 × 102
	2	4	3	5	1	6
$F_3$	2.06 × 107	1.25 × 107	1.08 × 106	5.01 × 107	1.47 × 107	6.12 × 109
	3.52 × 107	7.67 × 106	6.48 × 105	2.39 × 107	8.85 × 106	2.79 × 109
	4	2	1	5	3	6
$F_4$	−7.84 × 103	−6.56 × 103	−6.37 × 103	−4.51 × 103	−6.60 × 103	−1.08 × 103
	8.68 × 102	9.26 × 102	1.02 × 103	8.81 × 102	8.72 × 102	6.61 × 102
	1	2	4	5	3	6
$F_5$	5.07 × 103	2.08 × 103	2.29 × 103	9.38 × 102	4.11 × 101	2.67 × 104
	1.44 × 103	8.65 × 102	8.14 × 102	2.04 × 103	7.54 × 100	8.19 × 103
	5	3	4	2	1	6
$F_6$	5.63 × 108	2.54 × 108	8.26 × 100	5.92 × 106	4.67 × 107	7.98 × 1010
	8.39 × 108	2.82 × 108	1.32 × 101	4.51 × 106	6.60 × 107	6.66 × 1010
	5	4	1	2	3	6
$F_7$	9.86 × 10−4	1.36 × 100	1.55 × 10−2	9.52 × 10−1	1.27 × 100	2.83 × 101
	3.12 × 10−3	1.69 × 10−1	1.13 × 10−2	6.14 × 10−2	4.71 × 10−1	8.31 × 100
	1	5	2	3	4	6
$F_8$	2.09 × 101	2.10 × 101	2.10 × 101	2.07 × 101	2.10 × 101	2.00 × 101
	6.40 × 10−2	3.87 × 10−2	4.62 × 10−2	1.93 × 10−1	4.98 × 10−2	1.09 × 10−1
	3	5	4	2	6	1
$F_9$	5.31 × 104	5.36 × 104	5.33 × 104	5.32 × 104	5.34 × 104	5.55 × 104
	1.28 × 101	6.22 × 102	1.85 × 102	4.09 × 102	2.09 × 102	1.83 × 103
	1	5	3	2	4	6
$F_{10}$	5.83 × 104	5.86 × 104	5.85 × 104	5.82 × 104	5.87 × 104	5.94 × 104
	1.85 × 102	2.17 × 102	2.16 × 102	1.55 × 102	3.49 × 102	1.44 × 103
	2	4	3	1	5	6
$F_{11}$	2.04 × 101	3.11 × 101	4.05 × 101	3.76 × 101	2.33 × 101	5.09 × 101
	5.09 × 100	3.98 × 100	1.45 × 100	2.83 × 100	1.20 × 101	3.04 × 100
	1	3	5	4	2	6
$F_{12}$	5.49 × 100	2.35 × 10−1	2.33 × 10−3	7.96 × 10−22	1.28 × 10−11	1.92 × 10−3
	7.72 × 100	3.23 × 10−1	2.31 × 10−3	4.36 × 10−21	6.64 × 10−11	1.48 × 10−3
	6	5	4	1	2	3
$F_{13}$	8.64 × 1015	8.73 × 1015	8.64 × 1015	8.64 × 1015	8.76 × 1015	8.93 × 1015
	8.55 × 1013	9.75 × 1013	1.74 × 109	2.87 × 108	6.92 × 1013	2.58 × 1014
	3	4	2	1	5	6
$F_{14}$	1.13 × 101	1.19 × 101	1.22 × 101	1.27 × 101	1.10 × 101	1.39 × 101
	5.59 × 10−1	2.71 × 10−1	3.45 × 10−1	5.16 × 10−1	8.71 × 10−1	3.52 × 10−1
	2	3	4	5	1	6
Nb/Nw/MFr	5/1/2.64	0/0/3.85	2/0/3.00	4/0/2.93	3/1/3.14	1/12/5.42
Final rank	1	5	4	2	3	6

. In terms of overall performance, the proposed WMA-PSO algorithm emerged as the most competitive method. Its summary metrics (Nb/Nw/MFr) of 5/1/2.64 indicate that it achieved the top rank on 5 of the 14 functions, performed the worst on only a single function, and obtained the lowest Mean Friedman Rank (MFr), securing the first position overall. It was followed by WOA (MFr = 2.86) and GWO (MFr = 3.00), while the performance of BA (MFr = 5.64) was markedly inferior.

A more detailed analysis confirms that the superiority of WMA-PSO is particularly pronounced on highly complex and noisy functions. The convergence profiles for several representative functions are depicted in Figure 3. For instance, it achieved the best performance on key challenging functions, including F4 (a noisy multimodal function), F7 (shifted and rotated Griewank’s function), and F11. Notably, even in cases where it did not rank first, WMA-PSO maintained exceptionally robust behavior. A representative example is the complex F14 function, on which WMA-PSO attained the second-best mean result, which was nearly identical to that of the top-performing algorithm, GWO. More importantly, WMA-PSO exhibited a significantly lower standard deviation than GWO on F14 (0.56 vs. 0.87), underscoring its superior stability and repeatability despite a marginally inferior mean value. This consistent, top-tier performance across diverse challenging problems strongly indicates that the proposed fusion mechanism effectively enhances the algorithm’s global exploration capability and its capacity to escape local optima.

In summary, the experimental results on the challenging CEC-2005 platform strongly validate the superiority of the WMA-PSO algorithm. Considering its higher win count, lower mean rank, and stable convergence on key complex functions, WMA-PSO can be regarded as an exceptionally robust and advantageous algorithm for complex optimization scenarios.

4.2. Engineering Application: ADRC Parameter Tuning for Synchronous Position Control of a Rolling Mill Hydraulic System

The WMA-PSO algorithm was developed specifically to address ADRC parameter tuning for two-side synchronous position control in rolling-mill hydraulic systems. Having established its optimization capability on public benchmarks, we now return to the target application for a system-level validation: WMA-PSO is used to automatically tune the ADRC parameters and is compared with manual tuning, PSO, and WMA baselines.

Experimental Setup

Comparison of Convergence Curves To ensure a fair comparison, a unified experimental configuration was adopted for all competing optimization algorithms (WMA-PSO, WMA, PSO, and WOA).

• Controlled System: The simulation model of the hydraulic synchronous synchronous system of the rolling mill, as established in Section 2, was used as the controlled object. The model’s key parameters were adopted from the work of [5].
• Fitness Function: All algorithms utilized the comprehensive performance evaluation function, J, constructed in Section 2 as the optimization objective. This function incorporates the integral of the synchronization error, where a lower value indicates better control performance.
• Algorithm Configuration: The primary parameters for each algorithm were consistent with those used in the CEC-2005 benchmark tests. However, to reflect the demands of practical engineering applications, the maximum number of iterations was set to 100. To mitigate the effects of stochasticity, each algorithm was independently executed 30 times, and the average performance was analyzed.

Figure 4 shows the fitness value convergence curves for the PSO, WMA, WOA, and WMA-PSO algorithms.

As illustrated by the convergence curves, the WMA-PSO algorithm exhibits rapid initial convergence, consistently outperforming the other three algorithms throughout the optimization process. Ultimately, it converges to a significantly superior final fitness value. These results indicate that, compared to standard PSO and WMA, the proposed fusion algorithm possesses not only faster convergence speed and higher optimization accuracy but also a more effective mechanism for escaping local optima.

To account for the stochastic nature of these algorithms, a robust statistical analysis was performed. Each algorithm was executed independently 30 times, and key performance metrics—including the best, mean, and standard deviation of the final fitness values—were recorded. The aggregated results are summarized in

Table 4. Statistical Comparison of the Performance of Four Optimization Algorithms.

Algorithm	Statistical Results of Optimal Fitness Value			Optimal Parameter Set
	Best	Average	Std. Dev.	${\mathbf{\omega }}_{\mathit{o}}$	${\overline{\mathit{b}}}_{\mathit{e}}$	${\mathit{k}}_{\mathit{e}}$
WMA	3.60 × 10−4	3.70 × 10−4	1.79 × 10−5	995	12.4	167
PSO	3.61 × 10−4	3.79 × 10−4	1.79 × 10−5	723	5	167
WOA	3.71 × 10−4	3.71 × 10−4	2.10 × 10−5	703	5	93
WMA-PSO	3.59 × 10−4	3.59 × 10−4	1.64 × 10−5	871	5	106

.

The data in the table show that the WMA-PSO algorithm outperforms the other algorithms in all statistical metrics. Its best and average values are the lowest, indicating stronger optimization capability, while its standard deviation is also the smallest, which shows that the algorithm’s results are more stable and robust. The last column of the table also displays the combination of ADRC parameters found by each algorithm during its best run.

4.3. Comparative Analysis of ADRC Synchronous Control Performance

4.3.1. Step Response Performance Analysis

To assess the practical performance, the optimized parameter sets from Table 4 were implemented in the ADRC synchronization controller and evaluated under a 1-mm step command. The resulting synchronization error curves are presented in Figure 5. The figure clearly shows the benefit of ADRC, which reduces the peak deviation from 0.23 mm (in the original system without synchronization control) to approximately 0.05 mm while eliminating steady-state error.

An analysis of the curves based on key performance indicators—maximum overshoot, settling time, and steady-state error—reveals the clear superiority of the WMA-PSO-tuned controller. The parameters identified by WMA-PSO yielded a response with the smallest overshoot, the fastest convergence, and the smallest steady-state error. This superior control performance validates the effectiveness of WMA-PSO in solving practical parameter tuning problems.

4.3.2. Dynamic Tracking Performance Analysis

To further evaluate the dynamic tracking performance of the optimized controller under time-varying reference signals, a setpoint step-change experiment was designed. After the system’s response to the initial 1 mm step command reached steady state, a new step change (from 1 mm to 0.8 mm) was applied to the setpoint at $t=50\mathrm{ms}$. The experimental results are presented in Figure 6. Specifically, Figure 6a illustrates the full-time-domain synchronous error response under this condition, while Figure 6b,c provides a magnified view of the transient response details near the $t=50\mathrm{ms}$ step point. As can be clearly observed from Figure 6c, when the setpoint undergoes an abrupt change, the optimal control performance is once again demonstrated by the WMA-PSO-optimized controller. It is characterized by the smallest instantaneous peak deviation, minimal oscillation, and the fastest convergence back to the zero point. The robustness and superiority of the proposed method in dynamic tracking tasks are validated.

4.3.3. Disturbance Rejection Performance Analysis

To evaluate the disturbance rejection capability of the controller with optimized parameters, an independent load disturbance simulation experiment was designed in this section. The system setpoint was maintained constant at a 1 mm step command. To simulate the asymmetric and time-varying loads caused by strip bite and roll eccentricity during stable rolling, different load disturbances were applied to the two servo systems at t = 50 ms, after the system response had reached steady state: $F_{L1}=300+275sin\left(23.53t\right)\mathrm{kN}$ and $F_{L2}=350+275sin\left(23.53t\right)\mathrm{kN}$. All PI and ADRC parameters were kept unchanged. The simulation results are presented in Figure 7. Figure 7a illustrates the full time-domain response, while Figure 7b,c provide a detailed view of the system’s response details at the moment of disturbance injection ($t=50\mathrm{ms}$). From the magnified results in Figure 7c, it can be clearly analyzed that compared to the original system and the manually tuned controller, superior disturbance rejection capabilities are exhibited by all ADRC controllers optimized with intelligent algorithms. Among all compared algorithms, the strongest robustness is demonstrated by the WMA-PSO-optimized controller. It is characterized by the minimum transient deviation peak under the disturbance impact, the fastest subsequent oscillation dampening rate.

4.4. Summary of Experimental Results

This section validated the effectiveness and superiority of the proposed WMA-PSO algorithm for parameter optimization of a rolling mill hydraulic ADRC through benchmark tests and engineering application simulations. The key findings from the comparative experiments are summarized as follows:

• Superior Optimization Performance: WMA-PSO significantly outperforms the baseline algorithms in key optimization metrics, including convergence speed, accuracy, and stability, as demonstrated on the CEC-2005 test suite.
• Comprehensive Control Performance: In the engineering application of ADRC parameter tuning, the parameters identified by WMA-PSO not only achieved the smallest overshoot and shortest settling time in the initial step response; in the newly added dynamic tracking (setpoint step-change) and disturbance rejection (asymmetric load disturbance) experiments, the WMA-PSO-optimized controller also exhibited the strongest robustness. It consistently demonstrated the minimum transient deviation and the fastest convergence speed, showcasing the optimal overall control quality.

5. Conclusions and Future Work

5.1. Conclusions

This paper proposed a hybrid WMA-PSO metaheuristic to optimize the ADRC parameters for rolling-mill hydraulic synchronization systems. By integrating an adaptive fusion weight and inertia weight scheduling, the algorithm effectively balances global exploration and local exploitation in complex search spaces. Systematic experiments on 14 CEC-2005 benchmark functions demonstrated that WMA-PSO achieves superior optimization accuracy and robustness compared to baseline algorithms, particularly in high-dimensional and noisy environments. Furthermore, engineering simulations confirmed that the WMA-PSO-tuned ADRC significantly minimizes synchronization errors, overshoot, and settling time compared to manual tuning and other metaheuristics. Overall, this study provides a reliable and efficient automated approach for tuning complex industrial controllers, offering substantial practical value for high-precision plate and strip production.

5.2. Future Work

Although the present study has yielded the intended results, there is still room for further refinement and extension. Future research directions worthy of exploration include:

• Performance Enhancement of the Algorithm: Although WMA-PSO demonstrates excellent overall performance, its efficacy on a few specific functions, such as those with landscapes similar to the Rosenbrock function, could be improved. Future work could investigate the incorporation of gradient information or a dedicated Local Search Operator, with the aim of further strengthening its deep exploitation capability on such complex topographies.
• Extension to Multi-objective Optimization: Practical industrial control often necessitates a trade-off among multiple conflicting performance indicators, such as rapidity, stability, and energy consumption. Therefore, extending WMA-PSO to a multi-objective version (MOWMA-PSO) to generate a set of Pareto optimal solutions for decision-makers would be a research direction of significant practical value.