Economical Design of Drip Irrigation Control System Management Based on the Chaos Beetle Search Algorithm

Abstract

In the realm of existing intelligent drip irrigation control systems, traditional PID control encounters challenges in delivering satisfactory control outcomes, primarily owing to issues related to non-linearity, time-varying behavior, and hysteresis. In order to solve the problem of the unstable operation of the drip irrigation system in an intelligent irrigation system, this paper proposes chaotic beetle swarm optimization (CBSO) based on the BAS (beetle antennae search) longicorn search algorithm, with inertial weights, variable learning factors, and logistic chaos initialization improving global search capabilities. This was accomplished by formulating the optimization objective, which involved integrating the control input’s time integral term, the square term, and the absolute value of the error. Subsequently, PID parameter tuning was performed. In order to verify the actual effect of the CBSO algorithm on the PID drip irrigation control system, MATLAB was used to simulate and compare PID control optimized by the GA algorithm, PSO algorithm, and BSO (beetle search optimization) algorithm. The results show that PID control based on CBSO optimization has a short response time, small overshoot, and no oscillation in the steady state process. The performance of the controller is improved, which provides a basis for PID parameter setting for a drip irrigation control system.

1. Introduction

In recent years, China has recognized the need to modernize its agricultural irrigation practices to improve efficiency and conserve water resources. Traditional irrigation methods, such as border irrigation and furrow irrigation, have been used widely in the country for centuries [1]. However, these methods often suffer from inefficiencies, leading to water wastage and inadequate water supply for different crop requirements [2]. To address these challenges, there has been a growing emphasis on the development and adoption of advanced irrigation technologies in China [3]. One such approach is the automation and intellectualization of agricultural irrigation systems. This involves the integration of sensors, data analysis, and control systems to optimize water usage and tailor irrigation schedules to the specific needs of the crops [2]. By deploying modern technologies, farmers can monitor soil moisture levels, weather conditions, and crop water requirements in real-time. This allows for precise and targeted irrigation, ensuring that crops receive the right amount of water at the right time. For example, drip irrigation systems, which deliver water directly to the plant’s root zone, have gained popularity due to their efficiency and water-saving potential [4]. Furthermore, the use of automation and intelligent control systems enables farmers to remotely manage and control irrigation operations. This reduces the reliance on manual labor and ensures that irrigation activities are carried out consistently and accurately [5]. Additionally, these systems can be integrated with weather forecasting models to adjust irrigation schedules based on predicted rainfall or evapotranspiration rates. The adoption of automated and intelligent irrigation practices in China offers several benefits. Firstly, it helps increase crop yields and improve the quality of agricultural products by providing optimal irrigation conditions. Secondly, it contributes to the conservation of freshwater resources by reducing water wastage and improving water use efficiency. Lastly, it can enhance the economic viability of farming operations by reducing labor costs and improving overall productivity [6].

In the field of agricultural water-saving irrigation, the PID (proportional-integral-derivative) controller has garnered significant attention and research due to its simplicity, reliability, and robustness. However, researchers have explored various modifications and enhancements to the traditional PID controller to overcome certain limitations [7]. The PSO algorithm is also a popular optimization technique; however, it is prone to becoming trapped in local optimal solutions during the search process. This limitation may affect the overall performance and efficiency of the irrigation control system [8]. To address some of these challenges, some researchers designed a water-saving drip irrigation control system in a greenhouse based on the fuzzy-PID algorithm [9]. This approach incorporates fuzzy logic into the PID control scheme to enhance its adaptability and performance. The system was verified using real-world objects, demonstrating its potential effectiveness in water-saving irrigation applications. However, it is important to note that traditional PID control methods may encounter difficulties when dealing with controlled objects that have significant time delays or high order dynamics. In such cases, optimizing and tuning PID parameters can be challenging, leading to issues such as large overshoot and harmonic oscillations in the system [10]. Researchers continue to explore advanced control techniques and algorithms to overcome these limitations. For instance, advanced model-based control strategies, such as model predictive control (MPC), have shown promise in addressing the challenges posed by complex and time-delayed systems. These approaches utilize mathematical models of the controlled object to predict its behavior and optimize control actions in real-time. While the PID controller remains widely used in agricultural water-saving irrigation due to its simplicity and reliability, researchers have explored various modifications and alternative control strategies to enhance its performance. The integration of fractional order control, optimization algorithms like PSO, and fuzzy logic can provide improvements in specific contexts [11]. However, for systems with significant time delays and high-order dynamics, alternative control strategies like model predictive control may be more suitable for achieving optimization and tuning of control parameters.

Swarm intelligence and metaheuristics represent essential paradigms in solving complex optimization problems. Swarm intelligence draws inspiration from the collective behavior of social animals to devise algorithms capable of solving intricate problems. It involves the collaboration of multiple agents or entities to achieve a common goal, mimicking the behavior of natural swarms [12]. Metaheuristic algorithms, within the realm of optimization techniques, are heuristic strategies designed to efficiently explore large solution spaces by balancing exploration and exploitation. They are iterative and stochastic, inspired by natural phenomena or human-made systems, and have demonstrated success in solving diverse optimization problems. Recent advancements in the field of metaheuristics have led to the development of novel algorithms such as the giant trevally optimizer (GTO) [13] and elephant clan optimization (ECO). The GTO algorithm, inspired by the hunting behavior of giant trevally fish, incorporates aggressive and prey-searching mechanisms to achieve global optimization. On the other hand, ECO draws inspiration from the social structure and coordination of elephant clans to optimize complex problems through hierarchical communication and cooperation among individuals [14]. These cutting-edge metaheuristic algorithms aim to overcome various challenges encountered in traditional optimization techniques by introducing innovative strategies inspired by natural processes. They exhibit promising potential in addressing complex optimization problems and have attracted significant attention in recent research endeavors.

The BAS (beetle antennae search) algorithm is an intelligent swarm optimization algorithm inspired by the feeding behavior of the longicorn beetle. It has found applications in various fields, including photovoltaic maximum power point tracking, low-illumination texture image enhancement, and 3D path planning for mobile robots. In photovoltaic maximum power point tracking, the BAS algorithm has been applied to optimize the operation of solar panels and extract the maximum power output under varying environmental conditions. By mimicking the foraging behavior of beetles, the algorithm aims to efficiently locate the global maximum power point, improving the overall energy conversion efficiency of photovoltaic systems [15]. In the domain of low-illumination texture image enhancement, the BAS algorithm has been utilized to enhance the details and visual quality of images captured in low-light conditions [16]. By leveraging the search capabilities of the beetle-inspired algorithm, it can effectively enhance the texture information and improve the overall visibility of images, making them more visually appealing and informative. By simulating the exploration behavior of beetles, the algorithm can efficiently navigate complex environments and find optimal paths, considering various constraints and objectives [17]. This enables mobile robots to autonomously plan their paths, avoiding obstacles and reaching their destinations in an efficient and reliable manner. This means that the algorithm may struggle to find the global optimal solution in complex optimization problems with multiple peaks or valleys [18]. Examples include incorporating local search mechanisms, adaptive strategies, and hybridizing with other optimization techniques to overcome the limitations associated with local optima [19]. While it has demonstrated effectiveness, challenges has faced related to search accuracy and local optima still exist, necessitating further research and development to enhance its performance and applicability in complex optimization problems.

To address the challenges mentioned earlier, the CBSO-PID (chaos beetle search optimization-PID) drip irrigation control system was proposed as a solution for water-saving irrigation [2]. This system integrates the chaos algorithm, BSO algorithm, and PID controller to achieve improved performance and efficiency. The CBSO-PID system takes advantage of the chaos algorithm to enhance the search capabilities of the BSO algorithm. By introducing chaotic behavior into the search process, the algorithm can explore the search space more effectively and overcome the limitations of local optima. This allows for better optimization of the PID controller parameters, which are essential for achieving stable and efficient drip irrigation control [20,21]. The tuning of the proportional, integral, and differential (PID) parameters of the controller is transformed into an optimized problem. By formulating the stability of the control system as an optimization objective, the system can search for the optimal combination of PID parameters [22]. This optimization process aims to improve the stability, accuracy, and responsiveness of the drip irrigation control system. Simulation studies were conducted to evaluate the performance of the CBSO-PID optimized PID drip irrigation controller. A comparison was made with other methods such as BAS-PID (Beetle Antennae Search-PID) and PSO-PID (particle swarm optimization-PID). The results demonstrate that the CBSO-PID method exhibits better robustness and practical value compared to these alternative approaches. The integration of the chaos algorithm, BSO algorithm, and PID controller in the CBSO-PID drip irrigation control system offers several advantages. By integrating the chaos algorithm, BSO algorithm, and PID controller, it offers improved robustness, stability, and practical value compared to alternative methods like BAS-PID and PSO-PID. This research contributes to the development of intelligent and optimized control systems for sustainable and efficient agricultural practices.

CBSO dynamically adjusts the PID parameters during optimization to enhance control system performance in response to changing system dynamics or environmental conditions. By integrating the PID controller with the CBSO algorithm, the goal is to optimize the PID parameters effectively, enhancing control system performance and robustness while overcoming the limitations of traditional manual tuning methods. In this paper, we delve into the domain of metaheuristic optimization by proposing the chaotic beetle swarm optimization (CBSO) algorithm, drawing on the principles of swarm intelligence, and incorporating novel techniques inspired by nature. The effectiveness of CBSO is evaluated in the context of optimizing PID parameters for intelligent drip irrigation control systems, aiming to enhance their performance in addressing non-linearity, time-varying behavior, and hysteresis. Additionally, we benchmark the CBSO approach against established metaheuristic algorithms, including PSO, and BSO, to assess its efficacy and provide insights into its comparative performance. In the subsequent sections, this manuscript unfolds as follows. Section 2 provides an in-depth review and elucidation of swarm intelligence, metaheuristic optimization, and the principles underlying the proposed CBSO-PID algorithm. Following this, Section 3 delineates the empirical evaluation, featuring comprehensive simulations and comparisons between CBSO-PID and other prominent metaheuristic algorithms such as PSO and BSO. Section 4 presents the results and discussions are expounded, delineating the efficacy and comparative performance of the CBSO approach in enhancing intelligent drip irrigation control systems. This structured organization aims to provide a comprehensive insight into the efficacy and applicability of the CBSO algorithm in optimizing PID parameters for intelligent drip irrigation control systems.

2. Description of CBSO-PID

2.1. PID Control

In the field of drip irrigation control systems, the PID controller is commonly utilized. The PID controller combines the control deviation, represented as e(t) = r(t) − y(t), where r(t) is the desired input value and y(t) is the actual output value. This control deviation is linearly combined using proportional, integral, and differential terms to generate the control signal that regulates the drip irrigation process. The proportional coefficient (Kp), integral time constant (Ki), and differential time constant (Kd) are crucial parameters in determining the controller’s performance. To optimize the performance of the drip irrigation system, it is important to select the best values for these PID parameters [23]. In this paper, the authors propose using a combination of the chaos algorithm and the BAS algorithm to optimize the three parameters of the PID controller. The chaos algorithm introduces chaotic behavior into the optimization process, which helps in exploring the search space more effectively. Chaos-based optimization algorithms have shown promise in handling complex and nonlinear optimization problems, as they can escape local optima and search for global optima. The BAS algorithm, inspired by the feeding behavior of the longicorn beetle, is an intelligent swarm optimization algorithm. It leverages the collective intelligence of a population of individuals to search for optimal solutions. By integrating the BAS algorithm with the chaos algorithm, the authors aim to enhance the search capabilities and overcome the limitations of local optima during the optimization process [24].

Through the combined chaos–BAS optimization approach, the authors intend to find the optimal values for Kp, Ki, and Kd of the PID controller. By optimizing these parameters, the performance of the drip irrigation system can be improved in terms of stability, accuracy, and responsiveness. The use of intelligent optimization algorithms like chaos–BAS in conjunction with the PID controller offers a promising approach to enhance the performance of drip irrigation control systems. By effectively tuning the PID parameters, the system can achieve better regulation of the irrigation process, leading to improved water efficiency and crop productivity.

2.2. BAS Algorithm

The BAS algorithm is an optimization technology designed to achieve multi-objective function based on the principle of longicorn foraging. It can realize automatic optimization without knowing the specific form of function and gradient information, and the optimization process is efficient [25].

(1)

Creating spatial coordinates of a single longicorn of the left and right whiskers

$$\{\begin{array}{c}{x}_{ir}=x^t+d^{it}\cdot b\\ x_{il}=x^t-d^{it}\cdot b\end{array}$$

(1)

where, x_ir represents the spatial position of the right whisker of the i-th longicorn after the t-th iteration, while x_il represents the spatial position of the left whisker of the i-th longicorn after the t-th iteration. d^it represents the distance between the left and right whisker of the beetle, and x^t represents the centroid coordinates of the location of the longicorn.

(2)

Determining the smell intensity of a single longicorn of the left and right whiskers

According to the fitness function, the odor intensity of the left and right whisker can be determined, so as to update the location of the left and right whisker iteratively.

$$\{\begin{array}{c}{x}^{it+1}=x^{it}-{\delta }^{it}*b*sing[f(x_{ir})-f(x_{il})]\\ fitness=\frac{1}{N}{\displaystyle \sum _{j=1}^N{(t_{sim(j)}-y_j)}^2}\end{array}$$

(2)

In the formula, δ^it represents the step size factor corresponding to the i-th longicorn at the t-th iteration; sing() is a sign function, in the fitness function; t_sin(j) represents the output value of the j-th sample; and y_j represents the actual value of the j-th sample.

(3)

Determining the step size factor

The step size factor was used to control the search range of the longicorn whiskers. In order to avoid too small a search area, a large initial step size can be set. At the same time, in order to ensure the fineness of the search, a step size of a linear decreasing weight was adopted in this paper [22].

$${\delta }^t=\eta *{\delta }^{t-1}$$

(3)

η is the attenuation coefficient of the step size, and a number close to 1 between [0, 1] is taken. However, up to now, the setting of the step size factor has not formed complete theoretical system guidance. In this paper, through several experiments, 0.95 was selected after repeated tests. At the same time, the initial step size δ = 3 was determined, and the number of iterations was set as n = 100. In the drip irrigation system, in order to make the algorithm continue to be applicable in the high-dimensional parameter space and avoid the output of wrong control parameters due to the randomness of the initial value of a single longicorn, the BAS can be established by using the notion of particle swarm optimization. The location of each longicorn in the group is initialized, where the initial location of each longicorn should take the random number between [−0.5, 0.5] and saved it in the bestA set. At the same time, according to the fitness function, the global optimal fitness values of all longicorns at this time are recorded in the bestfinessA set. After that, according to Formula (2), the locations of each longicorn are updated iteratively. Then, the corresponding fitness function values are obtained according to the positions of the left and right whiskers iterated the Formula (3), and the bestA set and bestfitnessA set are updated in time. Finally, the best initial location bestB and the optimal fitness value bestfitnessB of the whole longicorn group are obtained by the comparing values in the two sets.

The Chaos Beetle Search Algorithm, being a novel method in the realm of control systems, presents both advantages and disadvantages, which are crucial to understand for a comprehensive evaluation.

Advantages: The Chaos Beetle Search Algorithm demonstrates strong global optimization capabilities. It can efficiently explore the search space, aiding in finding optimal or near-optimal solutions for complex problems. This algorithm exhibits robustness against local optima, allowing it to avoid becoming stuck in suboptimal solutions and promoting better convergence to the global optimum. Its ability to adapt and respond to dynamic environments or changing conditions is advantageous, making it suitable for systems with varying parameters or uncertainties. The Chaos Beetle Search Algorithm can be applied to problems of different complexities and dimensions, showcasing scalability across various control system scenarios.

Disadvantages: The performance of the Chaos Beetle Search Algorithm might be sensitive to its parameters, requiring careful tuning for optimal results. Inappropriate parameter settings can affect convergence and search efficiency. In certain cases, the algorithm may demand higher computational resources, especially with larger problem sizes, which could impact its efficiency and practical application in real-time systems. While proficient in global exploration, the algorithm might not always guarantee the absolute global optimum due to the stochastic nature of its search process. Implementing the Chaos Beetle Search Algorithm might pose challenges in its understanding and practical application due to its complex mathematical and computational nature. Understanding these advantages and disadvantages is crucial for the proper utilization and assessment of the Chaos Beetle Search Algorithm’s suitability for specific control system applications. Further research and analysis could help mitigate its limitations and enhance its strengths for practical deployment in various domains.

2.3. CBSO Algorithm

In order to prevent the BSO algorithm from falling into a local optimal solution, this paper introduces the chaos idea into the BSO algorithm and establishes the CBSO algorithm. Through the ergodicity of the chaotic motion and the optimal position of the whole longicorn group, a chaotic sequence is generated and is randomly assigned to every longicorn. At this point, the update formula of the longicorn position is:

$$V_i^{k+1}=\omega V_i^k+c_1\times rands(1)\times (P{b}_i^k-X_i^k)+c_2\times rands(1)\times (P{g}_{}^k-X_i^k)+c_2\times rands(1)\times {\xi }_i^k$$

(4)

where, V_i^k+1 represents the speed of the ith longicorn in the group after the kth iteration, X_i^k+1 represents the position of the ith longicorn in the group after the kth iteration, and Pb_i^k represents the historical optimal position of the ith longicorn in the group at the kth iteration. Pg^k is the optimal population position of the longicorn group [26].

c₁, c₂ and c₃ are learning factors of the speed of updating, and ω is the inertia weight, which is updated by Formula (5); k is the iteration time; and M is the time-varying constant.

$$\{\begin{array}{l}{\mathrm{c}}_i{=\mathrm{c}}_i+\frac{2*k}{M}\\ \omega ={\omega }_0-\frac{0.5*k}{M}\end{array}$$

(5)

Logistic chaos initialization is used to improve the global search ability, whose formula is:

$$x_{n+1}=\mu x_n(1-x_n)$$

(6)

In the optimization process described in the paper, the iterative search for the best control parameters of the drip irrigation control system using the chaos–BAS algorithm continues until certain termination conditions are met. There are two termination conditions specified in the paper:

• Fitness function value: The fitness function evaluates the performance of the drip irrigation control system based on certain criteria, such as control deviation, stability, or water-saving efficiency. The optimization process continues until the fitness function value reaches a predefined set value, which is 0.001 in this paper. Reaching this set value indicates that the system has achieved a satisfactory level of performance [27].
• Maximum iteration limit: To ensure that the optimization does not continue indefinitely, a maximum number of iterations is set. In this paper, the maximum iteration limit is defined as 100. Once the maximum number of iterations is reached, the optimization process is terminated, regardless of the fitness function value achieved [28].

When either of these termination conditions is met, the PID drip irrigation control parameters at that point are considered the best solutions obtained through the training process. These control parameters represent the optimal values of Kp, Ki, and Kd that yield the desired performance of the drip irrigation control system. By iteratively refining the control parameters using the chaos–BAS algorithm, the optimization process aims to converge towards the best possible solution, where the system achieves the desired performance criteria. The termination conditions ensure that the optimization process stops when either the performance reaches a satisfactory level (fitness function value) or when the search process has reached its maximum limit (iteration count). Overall, the chaos-BAS algorithm continually repeats the optimization process until the termination conditions are met, allowing for the identification of the best control parameters for the drip irrigation control system. This iterative approach enables the system to adapt and improve its performance over time, leading to enhanced water-saving irrigation and better control of the irrigation process.

When integrating the PID (proportional-integral-derivative) control with the Chaos Beetle Search Algorithm (CBSO), several challenges or difficulties may arise, impacting the combined system’s performance. The PID controller relies on tuning parameters (such as proportional gain, integral time, and derivative time) to achieve optimal control. Integrating it with the CBSO introduces additional parameters or settings, leading to increased complexity in parameter tuning. Balancing the settings for both algorithms simultaneously to ensure synergy without conflicts can be challenging. Combining the PID control and CBSO involves interaction between two different algorithms. Coordinating their actions to achieve effective control without causing instability, overshoot, or oscillations can be complex. Ensuring that the actions of one algorithm do not contradict or interfere with the other is crucial. Ensuring convergence and stability of the combined system is a critical challenge. The CBSO’s global exploration tendencies and the PID controller’s stability requirements need to be harmonized. Maintaining stability while achieving optimization goals is a delicate balance that may pose difficulties. The CBSO might demand significant computational resources due to its iterative search nature. Integrating this with the real-time requirements of PID control might lead to increased computational overhead, potentially impacting the system’s responsiveness and efficiency. Combining the algorithms may affect the robustness and adaptability of the control system. Changes in the system dynamics or disturbances might challenge the adaptability of the combined system, requiring a careful balance between robustness and adaptability. Ensuring that the optimization objectives of the CBSO align with the control objectives of the PID controller can be challenging. Optimizing one aspect of the system might inadvertently affect the other, requiring trade-offs and compromises. Overcoming these difficulties involves careful integration, parameter tuning, and optimization to harness the strengths of both algorithms while mitigating potential conflicts or challenges in their combined operation. Robust testing, simulation, and analysis are crucial to understanding and addressing these difficulties for successful implementation in practical control systems. Here are the highlighted contributions of the paper:

• Proposing an Innovative Approach: The paper introduces a novel method named Chaotic Beetle Swarm Optimization (CBSO) based on the BAS (beetle antennae search) longicorn search algorithm. This approach aims to address the challenges posed by non-linearity, time-varying behavior, and hysteresis in intelligent drip irrigation control systems.
• Enhanced Control Methodology: By incorporating inertial weights, variable learning factors, and logistic chaos initialization, the CBSO algorithm is designed to improve global search capabilities, which is crucial for stabilizing the operation of drip irrigation systems.
• Optimization Goal and Parameter Tuning: The paper focuses on using the integral term of the control input square and the absolute value of the error as the optimization goal for PID parameter tuning. This methodology aims to enhance the control performance of the drip irrigation system.
• Empirical Validation: MATLAB simulations were conducted to evaluate and compare the performance of the proposed CBSO-optimized PID control against other optimization algorithms (GA, PSO, and BSO). The results demonstrated that the CBSO-based PID control exhibited advantages such as a shorter response time, minimal overshoot, and stable state behavior without oscillation. This validation provides practical evidence of the effectiveness of the proposed algorithm in improving controller performance.

These contributions collectively establish the paper’s significant contributions towards addressing the challenges in intelligent drip irrigation control systems by introducing a novel optimization approach and validating its effectiveness through simulations and comparisons.

3. Controller Optimization Based on the CBSO-PID

3.1. CBSO-PID Controller

According to the above algorithm principle, a drip irrigation control system based on the PID control optimized by the CBSO algorithm was designed, as shown in Figure 1. In the three-dimensional search space constructed by the three parameters Kp, Ki and Kd, the optimal parameters for the controller effect are found.

The essence of parameter optimization lies in solving the minimum problem of the optimization objective function. In order to meet the control requirements of the PID controller, the objective function selected the square integral of the time square and error product. At this point, the fitness function can be expressed as Equation (7), where t is the time and e(t) represents the deviation between the actual output and the expected value.

$$F(IBSTE)={\displaystyle \underset{0}{\overset{\infty }{\int }}t\left|e(t)\right|}dt$$

(7)

3.2. CBSO-PID Control Process

The parameter optimization of the PID controller is a multi-objective optimization problem, and the stability and rapidity of system control should be considered comprehensively [29], as shown in Figure 2. The concrete steps are as follows:

• Step 1: Initialize the algoricorn parameters, determine the group size i, the maximum number of iterations itermax, the basic parameters of each longicorn, and the value range of the controller number Kp, Ki, and Kd [30].
• Step 2: Calculate the fitness of each position, by comparing the fitness values of longicorn individuals in each group Kp, Ki, and Kd, obtain individual optimal solution p_best in the current position, and finally obtain the current global optimal solution G_best by comparison [31].
• Step 3: Update the position and speed of each longicorn in the group through Formulas (3) and (6).
• Step 4: Compare the individual position of each longicorn with the historical position. If the current position is better, the individual optimal position is replaced. Then, compare the position of each longicorn individual with the historical position of the group, if the current position is better, it is replaced by the group optimal [32].
• Step 5: Generate the chaotic sequence based on the optimal position of the whole longicorn group, and randomly assign positions in the chaos sequence to every longicorn.
• Step 6: Judge whether the termination condition is met. If not, return to the step (3); if yes, end the iteration [33].

3.3. Simulation Process

In order to verify the effectiveness of the CBSO-PID controller, Simulink was used to conduct modeling and simulation of a greenhouse drip irrigation system, as shown in Figure 3.

The transfer function represents the mathematical relationship between the input and output of a dynamic system. It characterizes the system’s behavior by describing how the system responds to inputs, enabling analysis and prediction of system dynamics. The transfer function is pivotal in control system analysis, providing insights into system stability, frequency response, and overall performance. Understanding the system’s transfer function is vital for the effective optimization of the PID controller using the CBSO algorithm. It helps in selecting appropriate control parameters for optimal systems. In drip irrigation systems, the transfer function represents the relationship between input control signals (e.g., water flow rates) and output variables (e.g., soil moisture levels). Understanding this function is crucial for efficient water delivery and crop management.

Given that the optimal soil moisture level falls within the range of 50% to 60%, we have established the desired soil moisture value at 55%, with an initial value of 0%. Soil moisture naturally decreases due to factors such as substrate infiltration and plant transpiration. Following the methodology outlined for the attenuation factor of soil moisture [26], we define the system’s loss function using Formula (7). The nutrient delivery system of CBSO-PID and the drip irrigation system is shown in Figure 4. The drip irrigation system is a complexly controlled object, which is approximated by a second-order pure lag model:

$$G(s)=\frac{1}{2s^2+3s+1}{e}^{-0.2s}$$

(8)

3.4. Optimization Parameters

In the research conducted by Zhang Yinyan in 2019, it was proposed that the BAS (beetle antennae search) algorithm can achieve asymptotic convergence with a probability of 1 by adjusting the step size [13]. This finding suggests that by appropriately tuning the step size parameter, the optimization effectiveness of the CBSO (chaos beetle search optimization) algorithm can be improved. In the CBSO-PID algorithm, the optimization effect is indeed influenced by the parameter settings of δ and η in Equation (4). To further enhance the robustness of the CBSO-PID algorithm, the authors of this paper adjusted these parameters. Specifically, they set η as an arithmetic series ranging from 0.55 to 0.99, and δ as an arithmetic series ranging from 2 to 8. By varying the values of η and δ within these ranges, the authors aimed to explore different combinations and determine the optimal parameter values for the CBSO-PID algorithm. In this paper, they conducted verification experiments to evaluate the performance of the algorithm for different parameter settings [34].

The results of these experiments indicate that the best performance is achieved when η is set to 0.95 and δ is set to 3. This combination of parameter values yields the most desirable optimization outcomes and robustness for the CBSO-PID algorithm in the context of the drip irrigation control system. By adjusting the parameters η and δ, the authors were able to finetune the behavior of the CBSO-PID algorithm and improve its effectiveness in finding optimal solutions. This approach contributes to enhancing the overall performance and robustness of the algorithm in optimizing the PID controller parameters for drip irrigation control. The research suggests that adjusting the step size parameters in the CBSO-PID algorithm, as inspired by findings for the BAS algorithm, can optimize the algorithm’s performance. Through experiments and evaluations, the authors determined that setting η to 0.95 and δ to 3 yield the best results for the CBSO-PID algorithm in the context of the drip irrigation control system.

4. Simulation Results

The results show that CBSO-PID algorithm can eliminate the comprehensive influence factors under natural conditions and improve the parameter optimization of the drip irrigation control system through a large amount of data training. By adjusting the step length δ and attenuation coefficient η of single longicorns, the prediction accuracy of soil moisture content was correlated with that of soil moisture content. The steady-state performance and dynamic performance of the system can be greatly improved after this adjustment, which reflects the robustness of the algorithm. In the research mentioned, the optimization effect of the CBSO-PID algorithm was evaluated by comparing it with three other algorithms: PSO-PID (particle swarm optimization-PID), GA-PID (genetic algorithm-PID), and BSO-PID (beetle swarm optimization-PID). The performance of these algorithms was depicted in Figure 5, Figure 6 and Figure 7, with each figure presenting the PID parameter optimization curves in a specific order.

4.1. Optimization Curve

Starting from the top and moving to the bottom of the figures, the curves represent the optimization progress of the respective algorithms. Here is a brief explanation of each algorithm’s curve:

• PSO-PID Algorithm: The topmost curve in Figure 4 represents the PID parameter optimization curve of the PSO-PID algorithm. This curve showcases the progress and convergence behavior of the PSO-PID algorithm in optimizing the PID controller parameters for the drip irrigation control system.
• GA-PID Algorithm: The next curve in Figure 5 corresponds to the PID parameter optimization curve of the GA-PID algorithm. It illustrates the optimization process and convergence characteristics of the GA-PID algorithm in finding the optimal PID controller parameters.
• BSO-PID Algorithm: The third curve in Figure 6 represents the PID parameter optimization curve of the BSO-PID algorithm. This curve demonstrates the optimization behavior and convergence performance of the BSO-PID algorithm in the context of the drip irrigation control system.
• CBSO-PID Algorithm: Finally, the bottom curve in Figure 6 exhibits the PID parameter optimization curve of the CBSO-PID algorithm. This curve showcases the optimization progress and convergence properties of the CBSO-PID algorithm, which is the algorithm proposed in this research.

By comparing these curves, the researchers were able to analyze and evaluate the performance of each algorithm in terms of convergence speed, stability, and optimization effectiveness. The purpose of this comparison is to demonstrate the superiority of the CBSO-PID algorithm over the other algorithms in optimizing the PID controller parameters for the drip irrigation control system. The figures presented in the research show the PID parameter optimization curves for the PSO-PID, GA-PID, BSO-PID, and CBSO-PID algorithms. These curves provide a visual representation of the optimization progress and convergence behavior of each algorithm, facilitating a comparative analysis of their performance.

4.2. Control Performance and System Response

Based on the information provided, Figure 5, Figure 6 and Figure 7 demonstrate that among the four algorithms (CBSO-PID, PSO-PID, GA-PID, and BSO-PID), the CBSO-PID algorithm has the most efficient performance in terms of PID parameter optimization. It achieves the best effect in significantly fewer iterations, specifically within 20 iterations.

On the other hand, the GA-PID algorithm performs poorly compared to the other algorithms, including PSO-PID and BSO-PID. This suggests that the genetic algorithm (GA) is less effective in optimizing the PID controller parameters for the drip irrigation control system in this particular study. The PSO-PID algorithm also exhibits suboptimal performance, although it is relatively better than the GA-PID algorithm. It implies that the particle swarm optimization (PSO) algorithm has limitations in achieving satisfactory results within the given optimization constraints. The comparison presented in Figure 4, Figure 5 and Figure 6 highlights the superior performance of the CBSO-PID algorithm in terms of efficiency and effectiveness. It converges quickly within a small number of iterations, leading to optimal PID parameter values for the drip irrigation control system. Conversely, the GA-PID algorithm performs poorly, and the PSO-PID algorithm shows relatively better but still suboptimal results compared to the CBSO-PID algorithm.

To further verify the optimization effect of CBSO-PID, the control curve of the fitness function, the output curve of the system step response, and the output error curve of the system step response of the above four algorithms were plotted, respectively, as shown in Figure 8, Figure 9 and Figure 10.

Based on the information provided, Figure 8 illustrates the performance of the four optimization algorithms (CBSO, BSO, PSO, and GA) in finding the optimal fitness value of the fitness function. The results indicate the following observations:

• CBSO Algorithm: The CBSO algorithm is the fastest among the four algorithms in finding the optimal fitness value. It achieves this within only 18 iterations, indicating its efficiency in converging towards the optimal solution.
• BSO Algorithm: The BSO algorithm is the second fastest, taking 26 iterations to find the optimal fitness value. Although slightly slower than the CBSO algorithm, it still demonstrates a relatively efficient performance.
• PSO Algorithm: The PSO algorithm requires 47 iterations to find the optimal fitness value. It is slower compared to the CBSO and BSO algorithms but still manages to converge towards the optimal solution.
• GA Algorithm: The GA algorithm does not find the optimal fitness value within a finite number of iterations. It falls into a local optimum and is unable to achieve the desired optimal solution. This indicates that the GA algorithm performs poorly in this particular optimization task.

In terms of stability, the CBSO and BSO algorithms are described as more stable in determining the optimal value of the fitness function. This implies that they consistently converge towards the optimal solution across different runs or iterations [35]. On the other hand, the GA and PSO algorithms are characterized as slower and less stable. This suggests that they may exhibit more variability in their convergence behavior and require a larger number of iterations to reach the optimal fitness value reliably. Based on the findings from Figure 8, the CBSO and BSO optimization algorithms demonstrate faster convergence and greater stability in determining the optimal value of the fitness function. Meanwhile, the GA and PSO algorithms are slower and exhibit lower stability in achieving the desired optimization outcome [36].

Based on the information provided, Figure 9 and Figure 10 provide additional insights into the performance and characteristics of the CBSO-PID algorithm compared to the other algorithms (PSO, BSO, and GA) in terms of stability, overshoot, control accuracy, and robustness. Figure 9:

• CBSO Algorithm: The CBSO algorithm reaches stability in 2.72 s and exhibits a maximum overshoot of less than 0.3%. It demonstrates fast convergence and high control accuracy.
• PSO and BSO Algorithms: Both the PSO and BSO algorithms take more than 3.50 s to reach stability. They also have a maximum overshoot greater than 0.5%. The PSO algorithm shows an additional increase in rise time (0.14 s) compared to the CBSO algorithm.
• GA Algorithm: The GA algorithm requires a long time for both rise and convergence. It exhibits a large overshoot and significant fluctuations in the step signal within a finite number of iterations.

The CBSO algorithm stands out among the four algorithms due to its fast convergence, high control accuracy, and relatively low overshoot. In contrast, the PSO and BSO algorithms show slower convergence, higher overshoot, and the PSO algorithm introduces additional rise time compared to the CBSO algorithm. The GA algorithm performs poorly in terms of both convergence time and control accuracy. The CBSO-PID intelligent irrigation control system, as depicted in Figure 10, exhibits better control effectiveness and robustness compared to the other algorithms. The maximum errors of the GA and PSO algorithms are 0.18 Pa and 0.12 Pa, respectively. In contrast, the maximum errors of the CBSO and BSO algorithms are less than 0.02 Pa, indicating higher control accuracy and smaller deviations from the desired values. Both Figure 9 and Figure 10 further support the superiority of the CBSO-PID algorithm over the PSO, BSO, and GA algorithms in terms of fast convergence, high control accuracy, and robustness. The CBSO algorithm demonstrates faster convergence, lower overshoot, and smaller errors, highlighting its effectiveness in intelligent irrigation control systems.

4.3. Discussion about Economical Management

Based on the results obtained from the comparison of the CBSO-PID algorithm with the PSO-PID, BSO-PID, and GA-PID algorithms, there are implications for the economical management of systems or processes.

• Efficiency: The CBSO-PID algorithm consistently demonstrates fast convergence and efficient optimization compared to the other algorithms. This indicates that implementing the CBSO-PID algorithm can potentially lead to improved efficiency in various systems and processes. Faster convergence means quicker attainment of optimal parameter values, which can translate to reduced operation time and enhanced resource utilization.
• Control Accuracy: The CBSO-PID algorithm exhibits high control accuracy with minimal overshoot and low error rates compared to the other algorithms. This implies that by adopting the CBSO-PID algorithm, one can achieve precise control and maintain stability in various applications. Improved control accuracy is crucial for minimizing waste, optimizing resource usage, and reducing production costs.
• Robustness: The CBSO-PID algorithm demonstrates robust performance, as evidenced by its consistent and reliable results across different iterations. Robustness is essential for ensuring consistent and stable operation, which can contribute to minimizing operational disruptions and associated economic losses.
• Cost Savings: The superior performance of the CBSO-PID algorithm, characterized by faster convergence, high control accuracy, and robustness, can potentially lead to cost savings in terms of energy consumption, material utilization, and maintenance. Optimized control parameters obtained through the CBSO-PID algorithm can result in improved system efficiency, reduced resource waste, and enhanced overall productivity, all of which can contribute to cost reduction.
• System Optimization: The CBSO-PID algorithm’s ability to optimize PID controller parameters efficiently can have broader implications for system optimization in various industries. By fine-tuning control parameters, system performance can be enhanced, leading to increased productivity, improved product quality, and better overall operational efficiency.

The results obtained from the comparison of the CBSO-PID algorithm with other optimization algorithms highlight its potential for economical management. Implementing the CBSO-PID algorithm can contribute to improved efficiency, control accuracy, robustness, cost savings, and system optimization, thereby positively impacting economic factors in different domains.

5. Conclusions

The implementation of CBSO-PID control within the water supply management system of intelligent drip irrigation demonstrates remarkable efficacy, characterized by swift iteration, heightened control precision, minimized control duration, and enhanced robustness. In summary, the integration of CBSO within the PID control strategy presents a compelling solution for water supply management in intelligent drip irrigation systems. The demonstrated improvements in control precision, operational speed, and system stability underscore its potential for enhancing agricultural irrigation practices, signifying a substantial step towards intelligent and efficient water resource management in agriculture.

• Within the drip irrigation framework, PID control employing the CBSO algorithm exhibits notable advantages, showcasing reduced regulation time and minimal overshoot. These attributes signify the realization of an optimal control strategy, amplifying the efficiency of the irrigation system compared with GA/PSO.
• Throughout the CBSO algorithm’s optimization process, the adjustments in step size and attenuation factor significantly enhance the practical numerical optimization of PID. This optimization mitigates traditional PID control challenges, including nonlinearity, time-varying behavior, and hysteresis, thereby improving its efficacy in real-world applications.
• Empirical results from simulations affirm the stability and dynamic performance of CBSO-PID control. Its demonstrated stability aligns with the requisites of automatic control in agricultural irrigation systems, thereby contributing to the system’s overall intelligence and efficiency.