Performance Enhancement of PID Controllers via the Golden Eagle Optimization (GEO) Algorithm
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Abstract
The precise tuning of Proportional-Integral-Derivative (PID) controller parameters remains a foundational challenge in control engineering, as industrial plant dynamics often exhibit non-linearities that traditional heuristic methods, such as Ziegler–Nichols, fail to address adequately. This paper investigates the application of the Golden Eagle Optimization (GEO) algorithm—a recently developed nature-inspired metaheuristic—to the simultaneous optimization of , , and gains.
Utilizing a MATLAB/Simulink implementation, the GEO algorithm was integrated with a benchmark plant model to minimize a defined objective function based on the Mean Absolute Error (MAE). The optimization process involved a population of 50 agents over 100 iterations. Simulation results indicate a robust convergence characteristic, with the objective function value reducing from an initial 2.81 to a global minimum of approximately 2.6. The resulting controller performance demonstrated significant improvements in transient response, specifically achieving a settling time of 2 seconds with negligible overshoot. This study confirms that the GEO algorithm provides a computationally efficient and reliable framework for enhancing the stability and precision of closed-loop control systems.
Keywords:
Golden Eagle Optimizer (GEO), PID Controller Tuning, Metaheuristic Algorithms, MATLAB/Simulink, Control System Optimization.
I. Introduction
The strategic importance of Proportional-Integral-Derivative (PID) controllers in industrial automation cannot be overstated; they constitute the vast majority of control loops in modern manufacturing and process industries due to their structural simplicity and functional reliability. However, the operational efficacy of these controllers is heavily contingent upon the precise determination of their three primary gains. Traditional tuning methodologies, while computationally inexpensive, often result in sub-optimal transient responses when applied to plants with complex or shifting dynamics. Consequently, the control systems community has increasingly shifted toward metaheuristic optimization algorithms to navigate these non-linear search spaces more effectively.
The Golden Eagle Optimizer (GEO) is a sophisticated nature-inspired algorithm, first introduced by Mohammadi-Balani et al. in the journal Computers and Industrial Engineering. The algorithm is mathematically modeled after the hunting behavior of golden eagles, which balance the need for exploration (cruising) and exploitation (attacking). By synthesizing these distinct phases of avian hunting—prey selection, spiral motion, and the transition between cruise and attack vectors—GEO provides a high-performance framework for solving multi-variable optimization problems.
The objective of this research is to utilize the GEO algorithm to perform the simultaneous tuning of the , , and parameters within a feedback control architecture. By iteratively minimizing the system's error, this work seeks to identify the global optima for these coefficients to enhance plant stability and response speed. The subsequent sections delineate the plant modeling, the mathematical mechanics of the GEO algorithm, and the empirical results obtained via the MATLAB/Simulink simulation environment.
II. System Configuration and Plant Modeling
A rigorous system configuration is essential for establishing a baseline for optimization efficacy. In this study, a benchmark feedback control loop is employed, utilizing a simple plant model to evaluate the controller’s ability to track a reference step signal. The feedback mechanism ensures that the error signal—the discrepancy between the desired setpoint and the actual plant output—is processed by the PID controller to generate the appropriate control effort.
The PID controller architecture is defined by three tuned variables: the Proportional gain ( ), the Integral gain ( ), and the Derivative gain ( ). In the time domain, the output of the controller is expressed by the governing equation:
For frequency-domain analysis and stability assessment, the controller is represented by its transfer function in the Laplace domain:
The performance of the closed-loop system is entirely dependent on the optimal selection of these variables. In this research, these gains are not static but are treated as coordinates within a three-dimensional search space, iteratively refined by the GEO algorithm to minimize the system’s cumulative error.
III. The Golden Eagle Optimization (GEO) Algorithm
The GEO algorithm offers a distinct advantage in control system optimization due to its unique handling of the exploration–exploitation trade-off. Unlike algorithms that rely on simple stochastic walks, GEO utilizes the geometry of hunting trajectories to maintain population diversity while converging on the global minimum.
Mathematical Modeling of Hunting Behavior
The algorithm simulates search agents (eagles) tracking a solution (prey). Each eagle’s movement is a composite of a cruise vector, facilitating global search, and an attack vector, facilitating local refinement. The position update for an eagle at iteration is defined symbolically as:
where:
· is the current position (current , , values).
· represents the Attack Vector, directing the agent toward the best prey found so far.
· represents the Cruise Vector, calculated as a tangent to the circle of the eagle’s spiral motion to encourage exploration.
· and are the attack and cruise factors, respectively, which adjust the eagle's trajectory to transition from high-exploration cruising to high-exploitation attacking as the iterations progress.
Algorithmic Flow and Implementation
The optimization begins with the initialization of 50 eagles within user-defined lower and upper bounds. Within the MATLAB environment, the implementation utilizes a case one function selection to define the PID tuning problem. In each iteration, the algorithm evaluates the fitness of each set of parameters. The current best values for the gains are stored in vector . Through the spiral motion and iterative updates of the attack and cruise factors, the population converges toward the set of gains that minimizes the objective function.
IV. Simulation Setup and Implementation
The simulation environment serves as the critical interface between metaheuristic theory and practical control application. To quantify the performance of each candidate solution, the Mean of the Absolute value of Error (MAE) was selected as the objective function :
This metric ensures the optimizer prioritizes the reduction of the absolute magnitude of the error throughout the simulation duration, leading to a balanced transient and steady-state response.
Implementation Parameters
Parameter | Value / Detail |
Population Size | 50 |
Maximum Iterations | 100 |
Optimization Variables | 3 ( ) |
Objective Function | Mean Absolute Error (MAE) |
Storage Variable | Optimal gains stored in vector |
Implementation Case | Function Case 1 (PID Tuning) |
The MATLAB/Simulink workflow involves the GEO script calling the Simulink model as a function. During each iteration, the script updates the PID block parameters in the workspace, executes the simulation, retrieves the error signal, and calculates the MAE. This error value is then fed back to the GEO algorithm to inform the next position update for the population.
V. Results and Discussion
The evaluation of the GEO-tuned PID controller focuses on the convergence of the optimization and the subsequent time-domain characteristics of the plant.
Optimization Performance
The convergence characteristic demonstrated that the GEO algorithm successfully navigated the search space. Early in the process, the objective function yielded a value of approximately 2.81, corresponding to a preliminary set of tuned parameters.
However, as the Attack and Cruise factors were dynamically adjusted over the 100 iterations, the MAE consistently decreased. The algorithm reached a stable global minimum of approximately 2.6, indicating that the GEO mechanics effectively refined the PID coefficients to minimize the cumulative system error.
Transient Response Analysis
The plant’s response using the optimal , , values (stored in vector ) shows superior performance compared to traditional tuning. The plant reached the reference setpoint with high precision and demonstrated the following characteristics:
· Settling Time: The system achieved a stable output in approximately 2 seconds.
· Damping: The optimized derivative gain provided sufficient damping to virtually eliminate overshoot, which is critical for maintaining mechanical integrity in industrial plants.
By minimizing the MAE to 2.6, the GEO algorithm ensures a level of stability and responsiveness that exceeds manual or trial-and-error tuning, which frequently results in sluggish responses or sustained oscillations.
VI. Conclusion and Future Scope
This study has successfully demonstrated the application of the Golden Eagle Optimization (GEO) algorithm for the automated tuning of PID controllers. By simulating the hunting trajectories of the golden eagle, the algorithm provided a robust mechanism for identifying the optimal Proportional, Integral, and Derivative gains.
The primary contributions of this work include the development of a functional MATLAB/Simulink interface for GEO-based tuning and the empirical verification of the algorithm’s effectiveness, achieving a significant MAE reduction to 2.6 and a 2-second settling time. These results highlight GEO's potential as a reliable tool for control engineers seeking to optimize complex industrial processes.
Future research will explore the application of GEO to Multi-Input Multi-Output (MIMO) systems and non-linear plant models with significant dead time. Furthermore, a comparative performance analysis against other metaheuristics, such as Particle Swarm Optimization (PSO) and Grey Wolf Optimizer (GWO), is recommended to further benchmark the computational efficiency and convergence reliability of the GEO algorithm.
VII. YouTube Video
VIII. Purchase link of the Model
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