Chaotic-Based Improved Henry Gas Solubility Optimization Algorithm: Application to Electric Motor Control

Abstract

This study presents a novel meta-heuristic optimization method that combines the Henry Gas Solubility Optimization (HGSO) technique with symmetric chaotic systems. By leveraging the randomness of chaotic systems, the parameters of the HGSO algorithm that require random generation are produced through chaotic processes, allowing the algorithm to exhibit chaotic behavior in its pursuit of optimal values. This innovative approach is termed Chaotic Henry Gas Solubility Optimization (CHGSO), with the primary objective of enhancing the performance of the HGSO method. The randomness of the data obtained from chaotic systems was validated using NIST-800-22 tests. The CHGSO method was applied to both 47 benchmark functions and the optimization of parameters for a PID controller utilized in the speed control of a DC motor. To evaluate the effectiveness of the proposed method, it was compared with several widely recognized algorithms in the literature, including PSO, WOA, GWO, EA, SA, and the original HGSO algorithm. The results demonstrate that the proposed method achieved the best performance in 43 of the benchmark functions, outperforming the other algorithms. In the context of controller design, the PID parameters were optimized using the error-based ITSE objective function. According to the controller responses, the proposed method has achieved the best results in the simulation studies, with a settling time of 0.035 and a rise time of 0.014 without overshooting, and in the experimental studies, with a settling time of 0.15 and a settling time of 1.4%. When the results are examined, it is observed that it has achieved successful results in the controller design problem.

1. Introduction

Optimization has an important place in every field in the world. Recently, meta-heuristic optimization methods have been frequently preferred. In the current literature, various optimization methods are being applied across different fields. For instance, the Bonobo Optimization Algorithm has been utilized to optimize Support Vector Machine for establishing a relationship between entropy-based features and fault types in bearing fault diagnosis [1]. Horse Herd Optimization has been employed for the optimal sizing of photovoltaic/hydropower/fuel cell systems to minimize energy costs [2]. Ant Colony Optimization has been applied to a distributed flexible job shop problem developed for maintenance and transportation operations [3]. Multi-Objective Hybrid Teaching–Learning-Based Optimization–Grey Wolf Optimizer (MOHTLBOGWO) has been used to solve the optimal placement problem of energy resources, aiming to reduce losses and improve reliability in radial distribution systems [4]. A sinusoidal-cosine algorithm has been developed for the optimal design and energy management of hybrid photovoltaic/wind/fuel cell systems [5]. Brain Storm Optimization and the Learning-Driven Brain Storm Optimization Algorithm have been proposed for open shop, vehicle routing, and energy efficiency problems [6,7]. Additionally, a multi-objective evolutionary algorithm has been implemented for scheduling in distributed flexible job shops with random processing times [8]. As these examples illustrate, a wide variety of current optimization methods are employed in various domains. One of the optimization methods that maintains its relevance in the literature, demonstrate advantages in computational cost, and has shown convergence to optimal values is the Henry Gas Solubility Optimization method, which is discussed in this study. The Henry Gas Solubility Optimization (HGSO) method is based on Henry’s Law, formulated by William Henry in the 19th century. Henry’s Law states that the solubility of gases in liquids at a constant temperature is directly proportional to the partial pressure of the gas. HGSO mimics this law and thus belongs to the class of physics-based metaheuristic optimization methods. Integration with various test functions and engineering problems has shown that HGSO yields more successful results than other algorithms, such as particle swarm optimization, the gravitational search algorithm, the cuckoo search algorithm, the grey wolf optimization algorithm, the whale optimization algorithm, the elephant herding algorithm, and the simulated annealing algorithm [9]. HGSO is also used in fields, such as optimizing maximum power point tracking (MPPT) [10], feature selection for data mining and the classification of big data [11], and optimizing power system stabilizer (PSS) [12] and PID [13] parameters.

Chaotic systems are a subgroup of nonlinear systems. They were first introduced to the scientific world through the work of Lorenz. This field has gained recognition over time and continues to be the focus of numerous academic studies. The most notable characteristic of chaotic systems is their high sensitivity to initial conditions. Different initial conditions lead to entirely distinct trajectories for chaotic systems. Hence, when dealing with chaotic systems, having knowledge about not only the mathematical model of the system but also the specific initial conditions used in the system is essential.

In the studies found in the literature, the chaos-based Henry Gas Solubility Optimization method has been proposed for optimizing various engineering problems [14], classifying soil images in smart agriculture systems [15], feature selection for an LSTM (Long Short-Term Memory) model [16], and for optimizing the same 47 benchmark functions and 3 engineering problems compared to the original HGSO method [17].

In this conducted research, the pseudorandomness of chaotic systems is integrated with the HGSO method, proposing a new optimization approach named Chaotic Henry Gas Solubility Optimization (CHGSO). This proposal has been rigorously evaluated through international standards, such as NIST tests, supporting the randomness aspect.

In this study, an application for the speed control of a dc motor has been realized by tuning PID parameters with CHGSO. There are many studies in the literature that focus on setting the parameters of PID controllers. Recently, metaheuristic optimization methods have become frequently used to tune these parameters. The HGSO [5], Particle Swarm Optimization (PSO) [18,19], Evolutionary Algorithm (EA) [20], Simulated Annealing (SA) [21], Artificial Bee Colony (ABC) [22,23,24], Jaya Optimization Algorithm (JOA) [25], Invasive Weed Optimization (IWO) [26], Gravitational Search Algorithm (GSA) [27], Harris Hawks Optimization (HHO) [28], Genetic Algorithm (GA) [29,30], and Whale Optimization Algorithm (WOA) [31] are some of the meta-heuristic optimization methods that are used for this purpose.

A literature review shows that the HGSO algorithm is a modern approach to optimization. Studies demonstrate that it consistently converges closer to the optimal value while incurring lower computational costs compared to other methods. Therefore, the HGSO algorithm was chosen for this study.

The effectiveness of the proposed algorithm is tested by optimizing the PID parameters for the speed control of a DC motor. The optimization process has been assessed through comprehensive tests, such as the unit step response, unit parabolic response, error-based performance criterion analysis, and disturbance load analyses. The obtained results detail the control performance of CHGSO.

The research is supported not only by simulations but also by experimental studies. Both simulation and experimental work demonstrate a significant improvement in CHGSO over other algorithms, such as WOA, GWO, EA, SA, PSO, and HGSO, in terms of optimization time and PID controller performance, as commonly used in the literature.

This study offers a new perspective on optimization methods utilizing chaotic systems, highlighting the potential of obtaining more effective and faster solutions in control systems through the CHGSO method. The findings obtained from this research may have a broad impact not only on industrial applications but also on scientific research, serving as inspiration for future studies.

The paper is structured into several chapters. The first chapter includes a brief introduction, along with evaluations of recent studies and the current study. Chapter 2 provides information about the HGSO method, the chaotic systems used, and NIST randomness test results. The third chapter presents the proposed method, CHGSO, along with its performance analysis based on 47 different benchmark functions. Chapter 4 describes the optimization of PID parameters for DC motor speed control using the CHGSO method. The designed controller’s speed responses and performance analyses are compared to commonly used optimization methods in the literature, both in simulation and real-time environments. Chapter 5 concludes the study with assessments of the findings.

2. Technical Background

2.1. Henry Gas Solubility Optimization

2.1.1. Henry’s Law

Henry’s law states that the solubility of gases in a liquid at a constant temperature is directly proportional to the partial pressure [9].

Henry’s Law is given below (1).

$$S_g=H\ast P_g$$

(1)

In Equation (1), $S_g$ is the gas solubility, $H$ is Henry’s constant, and $P_g$ is the partial pressure of the gas. Note that Henry’s constant is temperature-dependent and changes with the change in temperature.

$${\displaystyle \frac{d lnH}{d (\frac{1}{T})}}={\displaystyle \frac{-\nabla E}{R}}$$

(2)

In Equation (2), $\nabla E$ is the enthalpy of dissolution, $R$ is the gas constant, and $T$ is the temperature parameters [9].

$$H\left(T\right)=\exp \left({\displaystyle \frac{B}{T}}\right)\ast A$$

(3)

In Equation (3), $A$ and $B$ are two parameters dependent on $T$. The reference temperature is [9].

$$T=298.15 K$$

When the equations are solved together and the dissolution enthalpy is assumed to be constant, Equation (4) is obtained.

$$H\left(T\right)=\exp \left(-C\ast ({\displaystyle \frac{1}{T}}-{\displaystyle \frac{1}{T^0}})\right)\ast H^0$$

(4)

2.1.2. HGSO Algorithm

The HGSO algorithm takes place in 8 steps.

Step 1: Initialization process. In this step, the gas particle number (N) is determined. The initial positions of these gas particles are established. Initial values of Henry’s constant, partial gas pressure, and the $C$ constant (enthalpy) are assigned.

$$X_i\left(t+1\right)=X_{min}+r\ast (X_{max}-X_{min})$$

(5)

The initial positions of the gas particles are determined by Equation (5). $X_{min}$ and $X_{max}$ define the limits of the problem, $r$ is random number between $0$ and $1$, and $X_i$ represents the position of the $i\mathrm{t}\mathrm{h}$ gas particle in the gas particle population.

$$\begin{array}{l}{H}_j\left(t\right)=l_1\ast r\\ P_{ij}\left(t\right)=l_2\ast r\\ C_j\left(t\right)=l_3\ast r\end{array}$$

(6)

Based on Equation (6), values of Henry’s constant, the partial gas pressure, and $C$ constant are assigned.$H_j$ is the Henry constant of the cluster $j$, $P_{ij}$ represents the partial gas pressure of the $i\mathrm{t}\mathrm{h}$ gas particle of the cluster $j$, $C_j$ is the enthalpy constant of the cluster $j$, and $t$ is the number of iterations. Note that $l_1,l_2$, and $l_3$ are constant values. ($l_1=0.05 , l_2=100 , l_3=0.01)$ [9].

Step 2: Clustering. The gas particle population is divided into clusters consisting of an equal number of gas particles. Since similar gases are present in the clusters, the Henry and enthalpy constants of the cluster are the same.

Step 3: Evaluation. All gas particles in the clusters are evaluated according to the objective function. Based on this evaluation, the gas particles are sorted from good to bad results.

Step 4: Updating the Henry constant. The Henry constant is updated according to Equation (7).

$$\begin{array}{l}{H}_j\left(t+1\right)=H_j\left(t\right)\ast \exp \left(-C\ast ({\displaystyle \frac{1}{T\left(t\right)}}-{\displaystyle \frac{1}{T^0}})\right)\\ T\left(t\right)=\exp \left(-{\displaystyle \frac{t}{iter}}\right)\end{array}$$

(7)

In Equation (7), $H_j$ is Henry’s constant of cluster $j$, $T$ is temperature, $T^0=298.15$ is the reference temperature, $t$ is the iteration, and $iter$ is the total number of iterations.

Step 5: Updating the solubility. Gas solubility is updated according to Equation (8).

$$S_{ij}\left(t\right)=K\ast H_j\left(t+1\right)\ast P_{ij}\left(t\right)$$

(8)

In Equation (8), $S_{ij}$ is the solubility of the $i\mathrm{t}\mathrm{h}$ gas particle of cluster $j$, $P_{ij}$ is the partial gas pressure of $i\mathrm{t}\mathrm{h}$ gas particle of cluster $j$, and $K$ is a constant value.

Step 6: Updating the positions. The positions of gas particles are updated according to Equation (9).

$$\begin{array}{c}{X}_{ij}\left(t+1\right)=X_{ij}\left(t\right)+F\ast r\ast \gamma \ast \left(X_{ibest}\left(t\right)-X_{ij}\left(t\right)\right)\\ +F\ast r\ast a\ast \left(S_{ij}\left(t\right) \times X_{best}\left(t\right)-X_{ij}\left(t\right)\right)\\ \gamma =\beta \ast \exp \left(-{\displaystyle \frac{F_{best}\left(t\right)+\epsilon }{F_{ij}\left(t\right)+\epsilon }}\right)\\ \epsilon =0.05\end{array}$$

(9)

In Equation (9), $X_{ij}$ represents the position of the $i\mathrm{t}\mathrm{h}$ gas particle in cluster $j$, and $r$ is a random number between $0$ and $1$. $X_{ibest}$ refers to the position of optimal gas particle, while $X_{best}$ represents the position of optimal gas particle in the entire population. $\gamma$ represents the ability of the $i\mathrm{t}\mathrm{h}$ gas particle in cluster $j$ to interact with other gases in its cluster. Parameter $a$ determines the effect of gas particles on the $i\mathrm{t}\mathrm{h}$ gas particle in cluster $j$ ($a=1)$, while $\beta$ is a constant value. $F_{ij}$ denotes the value of the objective function for the $i\mathrm{t}\mathrm{h}$ gas particle in cluster $j$, while $F_{best}$ refers to the value of the objective function for the optimal gas particle in the entire population. $\epsilon$ is a constant value with a small value to avoid the error of dividing by zero. Additionally, $F$ is a flag ($F=\pm 1)$ that changes the direction of advancement at the gas particle’s position.

Step 7: Escape from local optima. To escape the local optimum, the gas particles are aligned and the worst gas particles are identified. The selection process is performed by Equation (10).

$$\begin{array}{l}{N}_w\left(t\right)=N\ast \left(r\ast \left(c_2-c_1\right)+c_1\right) \\ c_1=0.1\\ c_2=0.2\end{array}$$

(10)

Here, $N$ is the gas particle number, and $N_w$ is the worst gas particle number to be selected.

Step 8: Updating the positions of the worst agents. The positions of the gas particles selected in step 7 are randomly updated within the global boundaries of the problem.

$$G_{ij}\left(t+1\right)=G_{\mathrm{m}\mathrm{i}\mathrm{n}}+r\ast (G_{max}-G_{min})$$

(11)

In Equation (11), $G_{ij}$ is the position of the $i\mathrm{t}\mathrm{h}$ gas particle in cluster $j$, and $G_{\mathrm{m}\mathrm{i}\mathrm{n}}$ and $G_{max}$ are the global limits of the problem. To facilitate better comprehension of the equations given above, a flowchart depicting the optimization method for Henry gas solubility is presented as Figure 1.

2.2. Chaotic Systems

Chaotic systems exhibit high sensitivity to initial conditions and parameters. Small perturbations in the initial conditions of a chaotic system can cause significant changes in its long-term behavior. However, despite the seemingly random and unpredictable nature of chaotic systems, they exhibit some level of self-organization and coherence. This study aims to harness the random characteristics of chaotic systems to create the necessary stochastic behavior required in metaheuristic optimization methods. To achieve this, various widely-used chaotic systems from the existing literature are considered and analyzed.

2.2.1. Duffing-Van Der Pol Chaotic System

$$\begin{array}{l}\dot{x}=y\\ \dot{y}=\mathrm{a}\ast (1-x^2)\ast \mathrm{y}-{\mathrm{x}}^3+\mathrm{b}\ast \mathrm{c}\mathrm{o}\mathrm{s}(\mathrm{c}\ast \mathrm{z})\\ \dot{z}=1\end{array}$$

(12)

The parameters $\mathrm{a}$, $b$, and $c$ are taken as $0.2$, $5.8$, and $3$, respectively [32]. The initial conditions of the system are chosen as $x\left(0\right)=0, y\left(0\right)=0,$ and $z\left(0\right)=0$. It should be noted that this non-autonomous system has been made autonomous by transforming $z=t$.

2.2.2. Lorenz Chaotic System

$$\begin{array}{l}\dot{x}=\mathsf{\sigma }\ast (\mathrm{y}-\mathrm{x})\\ \dot{y}=-\mathrm{x}\ast \mathrm{z}+\mathsf{\rho }\ast \mathrm{x}-\mathrm{y}\\ \dot{z}=\mathrm{x}\ast \mathrm{y}-\mathsf{\beta }\ast \mathrm{z} \end{array}$$

(13)

The parameters are σ = 10, ρ = 28, and β = 8/3 [33].

The initial conditions are $x\left(0\right)=0,y\left(0\right)=-0.01, \mathrm{a}\mathrm{n}\mathrm{d} z\left(0\right)=9$.

2.2.3. Rucklidge Chaotic System

$$\begin{array}{l}\dot{x}=-\mathrm{k}\ast \mathrm{x}+\mathrm{l}\ast \mathrm{y}-\mathrm{y}\ast \mathrm{z}\\ \dot{y}=x\\ \dot{z}=-\mathrm{z}+y^2 \end{array}$$

(14)

The parameters are $\mathrm{k}=2$ and $\mathrm{l}=6.7$ [34].

The initial conditions are $x\left(0\right)=0,y\left(0\right)=0, \mathrm{a}\mathrm{n}\mathrm{d} z\left(0\right)=4.5$.

2.2.4. Rössler Chaotic System

$$\begin{array}{l}\dot{x}=-\mathrm{y}-\mathrm{z}\\ \dot{y}=\mathrm{x}+\mathrm{a}\ast \mathrm{y}\\ \dot{z }=\mathrm{b}+\mathrm{z}\ast (\mathrm{x}-\mathrm{c}) \end{array}$$

(15)

The parameters are a = 0.2, b = 0.2, and c = 5.7 [35].

The initial conditions are $x\left(0\right)=-9, y\left(0\right)=0, \mathrm{a}\mathrm{n}\mathrm{d} z\left(0\right)=0$.

2.2.5. Rikitake Chaotic System

$$\begin{array}{l}\dot{x}=-\mathrm{m}\ast \mathrm{x}+\mathrm{z}\ast \mathrm{y}\\ \dot{y}=\mathrm{m}\ast \mathrm{x}+(\mathrm{z}-\mathrm{a})\ast \mathrm{x}\\ \dot{z}=1-\mathrm{x}\ast \mathrm{y}\end{array}$$

(16)

The parameters are m = 2 and a = 5 [36].

The initial conditions are $x\left(0\right)=0,y\left(0\right)=0.1, \mathrm{a}\mathrm{n}\mathrm{d} z\left(0\right)=0$.

2.2.6. Duffing Chaotic System

$$\begin{array}{l}\dot{x}=y\\ \dot{y}=-\mathrm{k}\ast \mathrm{y}-{\mathrm{x}}^3+\mathrm{b}\ast \mathrm{c}\mathrm{o}\mathrm{s}\left(\mathrm{z}\right)\\ \dot{z}=1 \end{array}$$

(17)

The parameters are k = 0.1 and b = 11 [37].

The initial conditions are $x\left(0\right)=0, y\left(0\right)=0, \mathrm{a}\mathrm{n}\mathrm{d} z\left(0\right)=2$.

Note: This non-autonomous system has been made autonomous by transforming $z=t$.

2.2.7. Chen Chaotic System

$$\begin{array}{l}\dot{x}=\mathrm{a}\ast (\mathrm{y}-\mathrm{x})\\ \dot{y}=-\mathrm{x}\ast \mathrm{z}+(\mathrm{c}-\mathrm{a})\ast \mathrm{x}+\mathrm{c}\ast \mathrm{y}\\ \dot{z}=\mathrm{x}\ast \mathrm{y}-\mathrm{b}\ast \mathrm{z}\end{array}$$

(18)

The parameters are a = 35, b = 3, and c = 28 [38].

The initial conditions are $x\left(0\right)=-10,y\left(0\right)=0, \mathrm{a}\mathrm{n}\mathrm{d} z\left(0\right)=37$.

Chaotic systems often lack traditional symmetry due to their complex and nonlinear nature. However, their state spaces frequently exhibit fractal patterns that imply a form of pseudo-symmetry. For example, the renowned Lorenz attractor displays this pseudo-symmetry through its fractal-like geometry, which arises from its chaotic trajectories. Thus, although chaotic systems do not adhere to conventional symmetry, their fractal structures reveal a kind of pseudo-symmetry. The phase portraits of the chaotic systems used in this study are presented in Figure 2.

2.3. NIST Tests

Random numbers obtained from chaotic systems are converted into random bit sequences and subjected to NIST randomness tests. NIST tests are considered the highest international standard for randomness. NIST-800-22 consists of 15 statistical tests [39]. Float numbers obtained from chaotic systems are converted into 32-bit number arrays. A 1 million-bit random bit sequence is created using these arrays and precise digits. The randomness of the generated 1 million-bit sequence is evaluated by running NIST-800-22 tests.

Table 1. Test results of chaotic systems (NIST-800-22).

	Duffing-Van Der Pol	Lorenz	Rössler	Rikitake	Duffing	Chen	Rucklidge
Frequency (monobit) test	Passed	Passed	Passed	Passed	Passed	Passed	Passed
Block frequency test	Passed	Passed	Passed	Passed	Passed	Passed	Passed
Cumulative sum test	Passed	Passed	Passed	Passed	Passed	Passed	Passed
Runs test	Passed	Passed	Passed	Passed	Passed	Passed	Passed
Longest run test	Passed	Passed	Passed	Passed	Passed	Passed	Passed
Binary matrix rank test	Passed	Passed	Passed	Passed	Passed	Passed	Passed
Discrete Fourier transform test	Passed	Passed	Passed	Passed	Passed	Passed	Passed
Non overlapping templates test	Passed	Passed	Passed	Passed	Passed	Failed	Failed
Overlapping templates test	Passed	Passed	Passed	Passed	Passed	Passed	Passed
Maurer’s universal statistical test	Passed	Passed	Passed	Passed	Passed	Passed	Passed
Approximate entropy test	Passed	Passed	Passed	Passed	Passed	Passed	Passed
Random excursions test	Passed	Passed	Passed	Passed	Passed	Passed	Passed
Random excursions variant test	Passed	Passed	Passed	Passed	Passed	Passed	Passed
Serial test	Passed	Passed	Passed	Passed	Passed	Passed	Passed
Linear complexity test	Passed	Passed	Passed	Passed	Passed	Passed	Passed

shows the pass/fail results of chaotic systems in NIST-800-22 randomness tests.

The findings of the test results presented in Table 1 categorize the randomness of chaotic systems based on relevant tests. It is evident from these results that the use of chaotic equation systems does not necessarily guarantee the required level of randomness. This observation underscores the importance of employing randomness tests. The requirement for a chaotic system to pass all randomness tests has been a crucial criterion in the selection of the system to be hybridized.

3. A Novel Optimization Algorithm (CHGSO)

The aim of this study is to investigate the integration of chaotic systems to enhance the performance of the HGSO algorithm. HGSO is an algorithm that models the physical processes of gas solutions and has the potential to yield effective results in solving complex optimization problems. However, a fundamental challenge with such optimization algorithms is ensuring that the algorithm effectively explores a vast search space. In this context, while the use of stochastic processes is common, these processes can be prone to getting trapped in local minima under certain conditions. Chaotic systems can be employed to address this issue. Despite their deterministic structure, chaotic systems exhibit a high degree of complexity and randomness due to their dynamic characteristics. These features make chaotic systems effective tools for exploration in optimization algorithms. One key attribute of chaotic systems is their sensitivity to initial conditions, where small changes can lead to significant differences in outcomes. These systems can exhibit unpredictable behaviors over the long term, thereby reducing the risk of getting trapped in local minima and allowing the algorithm to explore a broader search space. The conducted study aims to integrate the random-number-generation capabilities of chaotic systems into the HGSO algorithm, enhancing both exploration and exploitation capabilities. Stochastic search methods are often limited by their inability to reach a global optimum or effectively explore the solution space. To overcome these issues, the complex, ergodic [40], and random characteristics offered by chaotic systems will be utilized to improve the performance of the HGSO algorithm. The ergodic property of chaotic systems ensures that all possible states in the solution space emerge over time. This allows chaotic systems to create a comprehensive search strategy. In ergodic systems, the likelihood of visiting each point over time is higher, thus increasing the chances of reaching a global optimum. With this characteristic of chaotic systems, it is anticipated that the HGSO algorithm may develop a more balanced and effective exploration strategy. Consequently, the performance of the algorithm can be enhanced during both exploration and exploitation phases. The proposed Chaotic HGSO (CHGSO) algorithm integrates these chaotic characteristics into the search processes of HGSO. This integration occurs through the influence of chaotic systems, from the initial positions of gas particles to the position updates during the search process. The CHGSO algorithm involves the use of random numbers obtained from chaotic systems in both the initial and subsequent stages. This process is visualized as a flowchart in Figure 3.

The algorithm of the HGSO method proposed by Hashim et al. (2019) [9] is provided as pseudocode in Algorithm 1, with updates clearly defined to facilitate an understanding of the chaotic hybridization.

Algorithm 1 Pseudo-code of CHGSO algorithm [9]

1: Chaotically initialization $X_i$ (i = 1, 2,… N), number of gas types i, $H_j$, $P_{ij}$, $C_j$, $l_1$, $l_2$ and $l_3$. Equations (19) and (20) 2: Divide the population agents into number of gas types (cluster) with the same Henry’s constant value ($H_j$). 3: Evaluate each cluster $j$. 4: Get the best gas $X_{ibest}$ in each cluster, and the best search agent $X_{best}$. 5: while $t$ < maximum number of iterations do 6:     for each search agent do 7:       Chaotically update the positions of all search agents using Equation (21) 8:     end for 9:     Update Henry’s constant of each gas type using Equation (7) 10:   Update solubility of each gas using Equation (8) 11:   Rank and chaotically select number of worst agents using Equation (22). 12:   Chaotically update the position of the worst agents using Equation (23). 13:   Update the best gas $X_{ibest}$, and the best search agent $X_{best}$. 14: end while 15: t = t + 1 16: return $X_{best}$

Chaotic initialization process: One of the most fundamental characteristics of chaotic systems is their extreme sensitivity to initial conditions. When this property is applied to the determination of initial states in optimization algorithms, it may significantly enhance their performance. In the traditional HGSO algorithm, the initial positions of gas particles are determined by random numbers. However, in the proposed CHGSO algorithm, these initial positions are determined using random numbers generated by chaotic systems. This makes the initialization phase both dynamic and unpredictable. Through chaotic systems, the positions of the gas particles are not merely random but are set according to a complex, chaotic structure. This approach reconfigures the Equations (5) and (6) of the HGSO algorithm to fit a chaotic framework. The equations have been adapted to the chaotic initialization process, and the formulation for determining the initial positions of the gas particles is as follows:

$$X_i\left(t+1\right)=X_{min}+r_c\ast (X_{max}-X_{min})$$

(19)

In this context, $r_c$ represents a random number derived from a chaotic system, while $X_{min}$ and $X_{max}$ denote the lower and upper bounds of the solution space, respectively. In the traditional HGSO algorithm, this value is randomly chosen between 0 and 1, whereas in the proposed CHGSO algorithm, it is derived from a chaotic system within the same range. This modification ensures that the initial positions are distributed across a more diverse and expansive solution space. Consequently, it reduces the risk of getting trapped in local minima early in the process, allowing for a more comprehensive exploration of the solution space.

Additionally, other critical parameters in the HGSO algorithm, such as the Henry constant, partial gas pressure, and constant C, are also determined by random numbers generated from chaotic systems. These parameters play a crucial role in enhancing the algorithm’s performance. The use of chaotic systems to determine these values increases randomness in the initialization phase, enabling the implementation of a more complex and effective exploration strategy. In this regard, Equation (20) has been reformulated as follows:

$$\begin{array}{l}{H}_j\left(t\right)=l_1\ast r_c\\ P_{ij}\left(t\right)=l_2\ast r_c\\ C_j\left(t\right)=l_3\ast r_c\end{array}$$

(20)

These equations illustrate how key parameters, such as the Henry constant ($H_j$), partial gas pressure ($P_{ij}$), and constant ($C_j$), are determined using random numbers derived from chaotic systems. By integrating chaotic dynamics into the initialization phase, the algorithm’s inherent randomness is significantly enhanced, allowing for a broader and more thorough exploration of the solution space. This chaotic structure reduces the likelihood of early convergence to local minima and promotes a more effective search for global optima.

Updating the positions chaotically: The update of gas particle positions is a critical step in optimization processes. These position updates form the foundation of the algorithm’s search strategy and play a vital role in reaching local or global optima. In the traditional HGSO algorithm, these updates are determined by stochastic processes. However, in the CHGSO algorithm, this process is transformed into a chaotic structure. Random numbers generated by chaotic systems are used for updating the positions of gas particles, enabling them to follow a more complex and dynamic search strategy.

The chaotic position update process is performed through the modification of Equation (9) as follows:

$$\begin{array}{c}{X}_{ij}\left(t+1\right)=X_{ij}\left(t\right)+F\ast r_c\ast \gamma \ast \left(X_{ibest}\left(t\right)-X_{ij}\left(t\right)\right)\\ +F\ast r_c\ast a\ast \left(S_{ij}\left(t\right) \times X_{best}\left(t\right)-X_{ij}\left(t\right)\right)\\ \gamma =\beta \ast \exp \left(-{\displaystyle \frac{F_{best}\left(t\right)+\epsilon }{F_{ij}\left(t\right)+\epsilon }}\right)\epsilon =0.05\end{array}$$

(21)

In this equation, the new positions of the gas particles are updated based on a chaotic progression derived from their previous positions. Here, $r_c$ represents a random number generated from a chaotic system. This randomness prevents the gas particles from focusing solely on local optima during position updates, thereby facilitating a faster convergence toward the global optimum. Additionally, the parameter $\gamma$ is adjusted according to the best and worst fitness values of the gas particles, and it is further influenced by chaotic dynamics.$\epsilon$ is a constant value with a small value to avoid the error of dividing by zero.

Chaotically select number of worst agents: In the HGSO algorithm, selecting the agents with the worst performance and updating their positions is a critical step in enhancing the algorithm’s efficiency. During this phase, rather than selecting the worst-performing agents randomly, the chaotic systems guide the selection. In this context, the original Equation (10) of the algorithm has been modified as follows:

$$\begin{array}{l}{N}_w\left(t\right)=N\ast \left(r_c x \left(c_2-c_1\right)+c_1\right) \\ c_1=0.1\\ c_2=0.2\end{array}$$

(22)

This equation ensures that the agents with the worst performance are selected in a chaotic manner.

Chaotically updating the position of the worst agents: The positions of these selected agents are updated within the global boundaries of the problem space using chaotic dynamics. This process involves adjusting the positions of the agents based on values generated by chaotic systems. The update operation is expressed by the following equation:

$$G_{ij}\left(t+1\right)=G_{\mathrm{m}\mathrm{i}\mathrm{n}}+r_c\ast (G_{max}-G_{min})$$

(23)

This equation illustrates how the positions of the worst-performing agents are updated chaotically. By utilizing chaotic behavior in the position updates, the exploration capability of the algorithm is significantly enhanced, enabling a more efficient search process.

The proposed method was evaluated with 47 benchmark functions, listed in Appendix A [1]. The benchmark function results are compared with two different HGSO studies based on Mohammadi et al.’s (2022) proposed QHGSO algorithm with chaos [17], the original HGSO study with the same parameters as CHGSO, and the optimization methods commonly used in literature, PSO, WOA, GWO, EA, and SA. The optimization parameters used in these studies are given in

Table 2. Parameter value of compered algorithms.

CHGSO (proposed method)	Number of iterations: 200 Number of gas particles: 30 Number of clusters: 5 M1 = 0.1, M2 = 0.2, L1 = 0.005, l2 = 100, l3 = 0.01, a, b, k = 1, e = 0.05
HGSO	Number of iterations: 200 Number of gas particles: 30 Number of clusters: 5 M1 = 0.1, M2 = 0.2, L1 = 0.005, l2 = 100, l3 = 0.01, a, b, k = 1, e = 0.05
QHGSO	Number of iterations: 200 Number of gas particles: 30 Number of clusters: N/A M1 = 0.1, M2 = 0.2, L1 = 0.005, l2 = 100, l3 = 0.01, a, b, k = 1, e = 0.05
PSO	Number of iterations: 200 Number of swarm: 30 C1 = 2.1, C2 = 2.1
EA	Number of iterations: 200 Number of parents: 20 Number of children: 4
GWO	Number of iterations: 200 Number of wolves: 30
SA	Number of iterations: 200 Number of materials: 30 Cooling rate: 0.98
WOA	Number of iterations: 200 Number of whales: 30

.

Benchmark functions are commonly employed to evaluate the performance of optimization methods. The effectiveness of optimization algorithms is assessed based on these benchmark functions. Therefore, the benchmark functions presented in the Appendix A are utilized to compare the performance of the proposed algorithm.

Utilizing the parameters specified in Table 2, CHGSO, HGSO, PSO, GWO, WOA, EA, and SA algorithms are used to optimize the benchmark functions provided in Appendix A. The results are compared with QHGSO [17], obtained using the same benchmark functions, with other optimization algorithms. The comparisons are presented in

Table 3. Optimization results of the benchmark functions.

	Fmin		CHGSO	HGSO	QHGSO	PSO	EA	GWO	SA	WOA
F1	0.00 × 100	MEAN	0.00 × 100	4.13 × 10−158	0.00 × 100	7.62 × 101	4.35 × 109	1.26 × 10−16	2.00×109	9.28 × 10−43
		STD	0.00 × 100	8.13 × 10−159	0.00 × 100	1.31 × 102	9.81 × 108	3.87 × 10−16	3.79×108	3.39 × 10−42
F2	0.00 × 100	MEAN	0.00 × 100	1.60 × 10−81	0.00 × 100	8.18 × 100	1.51 × 10−3	2.64 × 10−11	1.22×102	1.51 × 10−24
		STD	0.00 × 100	1.12 × 10−80	0.00 × 100	5.71 × 100	9.07 × 10−4	3.62 × 10−11	1.07×101	7.03 × 10−24
F3	0.00 × 100	MEAN	0.00 × 100	7.72 × 10−80	0.00 × 100	5.74 × 104	1.10 × 102	2.06 × 10−2	5.96×104	1.92 × 10−24
		STD	0.00 × 100	5.37 × 10−79	0.00 × 100	6.40 × 104	6.17 × 101	3.4 × 10−2	3.72×104	1.21 × 10−23
F4	0.00 × 100	MEAN	0.00 × 100	1.03 × 10−79	0.00 × 100	7.61 × 104	8.21 × 10−1	2.01 × 10−7	1.50×10−1	4.39 × 10−24
		STD	0.00 × 100	4.35 × 10−79	0.00 × 100	1.04 × 105	5.96 × 10−1	2.76 × 10−7	6.10×10−2	2.93 × 10−23
F5	0.00 × 100	MEAN	0.00 × 100	1.87 × 10−125	0.00 × 100	5.00 × 1015	3.45 × 10−6	4.02 × 10−37	9.05×103	2.16 × 10−36
		STD	0.00 × 100	1.09 × 10−124	0.00 × 100	3.11 × 1016	1.48 × 10−6	1.69 × 10−37	1.34×103	1.03 × 10−35
F6	0.00 × 100	MEAN	0.00 × 100	6.82 × 10−41	0.00 × 100	5.38 × 100	9.28 × 100	5.52 × 10−5	9.25×102	1.94 × 10−16
		STD	0.00 × 100	3.40 × 10−40	0.00 × 100	2.14 × 100	5.21 × 100	2.98 × 10−5	5.36×101	4.61 × 10−16
F7	0.00 × 100	MEAN	0.00 × 100	5.43 × 10−40	0.00 × 100	8.47 × 100	2.17 × 101	3.08 × 10−2	7.48×101	6.07 × 101
		STD	0.00 × 100	2.75 × 10−39	0.00 × 100	1.49 × 100	8.44 × 100	2.01 × 10−2	2.48×100	2.78 × 101
F8	0.00 × 100	MEAN	0.00 × 100	8.82 × 10−41	0.00 × 100	1.29 × 102	6.09 × 1028	8.71 × 10−5	1.50×1037	1.35 × 10−16
		STD	0.00 × 100	3.91 × 10−40	0.00 × 100	1.87 × 102	3.37 × 1029	4.77 × 10−5	3.47×1037	4.05 × 10−16
F9	0.00 × 100	MEAN	0.00 × 100	0.00 × 100	0.00 × 100	6.95 × 107	1.19 × 109	4.61 × 10−30	1.66×109	3.70 × 10−43
		STD	0.00 × 100	0.00 × 100	0.00 × 100	1.28 × 108	8.54 × 108	1.92 × 10−29	6.23×108	2.60 × 10−42
F10	0.00 × 100	MEAN	0.00 × 100	−1.79 × 103	0.00 × 100	−3.00 × 103	−2.96 × 103	−2.77 × 103	−2.09×103	−3.00 × 103
		STD	0.00 × 100	2.21 × 102	0.00 × 100	4.80 × 10−1	3.55 × 101	1.48 × 102	5.02×101	0.00 × 100
F11	0.00 × 100	MEAN	0.00 × 100	1.89 × 10−77	0.00 × 100	8.48 × 101	2.94 × 100	9.47 × 10−10	6.04×103	2.65 × 10−23
		STD	0.00 × 100	1.28 × 10−76	0.00 × 100	7.17 × 101	2.56 × 100	1.24 × 10−9	6.11×102	8.28 × 10−23
F12	0.00 × 100	MEAN	8.88 × 10−16	8.88 × 10−16	8.70 × 100	2.03 × 101	1.41 × 101	1.98 × 10−5	2.03×101	8.61 × 10−13
		STD	2.06 × 10−31	8.96 × 10−31	6.12 × 100	3.31 × 100	8.19 × 100	7.83 × 10−6	1.73×10−1	1.93 × 10−12
F13	0.00 × 100	MEAN	0.00 × 100	9.43 × 10−43	8.65 × 10−1	4.80 × 101	4.11 × 100	4.17 × 10−3	4.61×101	5.10 × 10−1
		STD	4.13 × 10−139	4.44 × 10−42	2.31 × 100	3.10 × 101	2.02 × 100	1.76 × 10−3	2.97×100	3.61 × 100
F14	0.00 × 100	MEAN	0.00 × 100	1.51 × 10−81	7.17 × 102	1.56 × 10−9	1.94 × 10−3	1.97 × 10−11	6.81×101	6.56 × 10−26
		STD	0.00 × 100	9.48 × 10−81	1.90 × 103	1.07 × 10−9	1.90 × 10−3	2.97 × 10−11	1.44×101	2.06 × 10−25
F15	0.00 × 100	MEAN	0.00 × 100	2.62 × 10−73	9.98 × 10−1	4.49 × 106	6.58 × 1010	5.26 × 10−3	4.29×1010	5.28 × 10−15
		STD	0.00 × 100	1.56 × 10−72	−9.98 × 10−1	2.61 × 106	6.74 × 109	4.84 × 10−3	3.76×109	3.53 × 10−14
F16	−1.00 × 100	MEAN	−1.00 × 100	−1.00 × 100	1.48 × 100	−1.00 × 100	−1.00 × 100	−1.00 × 100	−1.03×10−1	−1.00 × 100
		STD	0.00 × 100	0.00 × 100	1.49 × 100	0.00 × 100	2.25 × 10−5	0.00 × 100	1.26×10−2	0.00 × 100
F17	0.00 × 100	MEAN	0.00 × 100	0.00 × 100	1.61 × 101	3.66 × 10−1	9.15 × 10−1	1.18 × 10−2	3.99×102	7.20 × 10−3
		STD	0.00 × 100	0.00 × 100	1.36 × 101	2.29 × 10−1	1.28 × 10−1	1.50 × 10−2	3.75×101	5.09 × 10−2
F18	2.00 × 100	MEAN	2.00 × 100	1.89 × 101	4.35 × 100	−1.54 × 106	1.11 × 107	8.10 × 100	3.80×1010	2.00 × 100
		STD	0.00 × 100	8.87 × 101	1.78 × 101	8.96 × 106	6.72 × 107	1.39 × 101	5.58×1010	0.00 × 100
F19	2.00 × 100	MEAN	2.00 × 100	4.84 × 101	7.35 × 10−19	−6.49 × 106	3.89 × 106	2.30 × 101	7.79×1010	2.00 × 100
		STD	0.00 × 100	3.03 × 102	4.08 × 10−17	4.58 × 107	2.56 × 107	3.85 × 101	1.54×1011	0.00 × 100
F20	0.00 × 100	MEAN	0.00 × 100	0.00 × 100	4.16 × 10−1	2.89 × 10−5	9.01 × 10−12	5.38 × 10−8	1.37×10−1	0.00 × 100
		STD	0.00 × 100	0.00 × 100	2.56 × 10−1	9.79 × 10−5	9.31 × 10−12	4.00 × 10−8	2.80×10−2	0.00 × 100
F21	0.00 × 100	MEAN	6.57 × 10−6	3.01 × 10−4	6.56 × 101	1.71 × 103	3.03 × 10−1	7.10 × 10−3	5.53×101	1.18 × 10−2
		STD	1.63 × 10−5	2.79 × 10−4	4.64 × 101	1.42 × 103	1.16 × 10−1	3.61 × 10−3	7.56×100	1.28 × 10−2
F22	0.00 × 100	MEAN	0.00 × 100	0.00 × 100	2.69 × 102	2.20 × 102	1.08 × 102	1.36 × 101	3.58×102	2.55 × 100
		STD	0.00 × 100	0.00 × 100	1.95 × 102	4.79 × 101	2.35 × 101	1.08 × 101	1.65×101	1.78 × 101
F23	0.00 × 100	MEAN	5.14 × 10−1	1.61 × 101	3.27 × 10−9	2.20 × 102	8.74 × 10−2	5.15 × 100	1.06×104	1.61 × 101
		STD	4.17 × 100	2.32 × 100	1.03 × 10−9	4.79 × 101	5.83 × 10−2	1.93 × 100	1.96×103	6.44 × 100
F24	0.00 × 100	MEAN	4.17 × 10−12	3.15 × 10−11	1.94 × 10−8	2.91 × 10−4	1.91 × 10−11	1.61 × 10−7	1.48×10−5	5.72 × 10−12
		STD	1.07 × 10−11	6.19 × 10−13	1.94 × 10−8	3.78 × 10−4	1.22 × 10−11	5.58 × 10−7	1.34×10−5	3.25 × 10−12
F25	0.00 × 100	MEAN	4.34 × 10−236	4.34 × 10−232	2.43 × 102	0.00 × 100	1.07 × 10−97	1.27 × 10−103	1.72×10−53	−1.40 × 10−1
		STD	3.89 × 10−1	0.00 × 100	9.38 × 101	0.00 × 100	7.60 × 10−97	8.97 × 10−103	8.26×10−53	3.51 × 10−1
F26	0.00 × 100	MEAN	0.00 × 100	1.49 × 10−75	9.38 × 101	1.15 × 109	2.94 × 1010	2.90 × 10−9	3.86×108	1.02 × 10−21
		STD	0.00 × 100	1.06 × 10−74	−2.00 × 102	2.10 × 109	3.49 × 1010	4.12 × 10−9	1.39×108	4.66 × 10−21
F27	−2.00 × 102	MEAN	−2.00 × 102	−2.00 × 102	−2.00 × 102	−2.00 × 102	−2.00 × 102	−2.00 × 102	−1.98×102	−2.00 × 102
		STD	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	9.13×10−1	0.00 × 100
F28	1.00 × 100	MEAN	1.00 × 100	1.00 × 100	1.00 × 100	1.00 × 100	1.00 × 100	1.00 × 100	8.40×101	1.00 × 100
		STD	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	7.99 × 10−6	0.00 × 100	6.57×101	0.00 × 100
F29	0.00 × 100	MEAN	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	8.88 × 10−18	0.00 × 100	4.25×100	4.44 × 10−18
		STD	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	6.28 × 10−17	0.00 × 100	3.43×100	3.14 × 10−17
F30	0.00 × 100	MEAN	1.80 × 10−1	1.80 × 10−1	1.80 × 10−1	1.80 × 10−1	1.80 × 10−1	1.80 × 10−1	4.93×100	1.92 × 10−1
		STD	2.90 × 10−17	1.12 × 10−16	0.00 × 100	1.12 × 10−16	1.12 × 10−16	1.12 × 10−16	3.72×100	4.61 × 10−2
F31	0.00 × 100	MEAN	0.00 × 100	0.00 × 100	0.00 × 100	7.77 × 10−18	0.00 × 100	0.00 × 100	2.93×100	2.05 × 10−4
		STD	0.00 × 100	0.00 × 100	0.00 × 100	2.25 × 10−17	0.00 × 100	0.00 × 100	2.67×100	5.25 × 10−4
F32	0.00 × 100	MEAN	0.00 × 100	3.79 × 10−110	0.00 × 100	9.42 × 10−32	4.01 × 10−21	2.53 × 10−74	1.25×10−2	3.09 × 10−28
		STD	0.00 × 100	2.68 × 10−109	0.00 × 100	5.05 × 10−31	6.57 × 10−21	1.66 × 10−73	1.13×10−2	1.93 × 10−27
F33	−4.33 × 101	MEAN	−4.35 × 101	−4.29 × 101	−4.26 × 101	−4.27 × 101	−4.27 × 101	−4.27 × 101	−4.26×101	−4.25 × 101
		STD	5.02 × 10−2	3.38 × 10−2	1.54 × 10−1	2.26 × 10−1	2.17 × 10−1	2.10 × 10−1	1.17×10−1	1.18 × 10−1
F34	−2.06 × 100	MEAN	−2.10 × 100	−2.11 × 100	−2.06 × 100	−2.11 × 100	−2.11 × 100	−2.11 × 100	−2.10×100	−2.11 × 100
		STD	4.00 × 10−4	3.14 × 10−7	1.17 × 10−7	8.97 × 10−16	8.97 × 10−16	8.97 × 10−16	1.06×10−3	2.81 × 10−6
F35	−1.00 × 100	MEAN	−1.00 × 100	−7.40 × 10−2	6.55 × 104	−3.29 × 10−1	−1.92 × 10−2	−2.17 × 10−4	−1.75×10−4	−6.84 × 10−3
		STD	4.52 × 10−1	2.38 × 10−1	0.00 × 100	3.14 × 10−1	3.63 × 10−3	5.12 × 10−5	2.16×10−5	4.46 × 10−2
F36	0.00 × 100	MEAN	0.00 × 100	4.45 × 10−111	0.00 × 100	1.24 × 10−30	1.90 × 10−1	6.00 × 10−78	1.86×10−1	5.50 × 10−37
		STD	0.00 × 100	2.40 × 10−110	0.00 × 100	8.44 × 10−30	1.34 × 100	4.24 × 10−77	1.88×10−1	3.59 × 10−36
F37	−3.32 × 100	MEAN	−2.81 × 100	−3.01 × 100	−2.33 × 100	−1.33 × 10−2	−3.26 × 100	−3.25 × 100	−2.97×100	−3.11 × 100
		STD	2.67 × 10−1	8.62 × 10−2	3.79 × 10−1	6.56 × 10−2	6.03 × 10−2	8.72 × 10−2	1.16×10−1	2.45 × 10−1
F38	0.00 × 100	MEAN	0.00 × 100	3.88 × 10−111	0.00 × 100	5.39 × 10−33	2.44 × 10−21	6.65 × 10−82	5.40×10−3	2.67 × 10−42
		STD	0.00 × 100	2.33 × 10−110	0.00 × 100	2.54 × 10−32	3.98 × 10−21	4.51 × 10−81	4.56×10−3	1.70 × 10−41
F39	9.00 × 10−1	MEAN	9.00 × 10−1	9.00 × 10−1	9.00 × 10−1	9.90 × 10−1	9.80 × 10−1	9.32 × 10−1	9.29×10−1	9.28 × 10−1
		STD	2.32 × 10−16	8.97 × 10−16	4.52 × 10−16	3.03 × 10−2	4.04 × 10−2	4.72 × 10−2	2.75×10−2	4.54 × 10−2
F40	0.00 × 100	MEAN	0.00 × 100	−1.90 × 1034	0.00 × 100	−1.93 × 1021	−1.85 × 1017	−1.25 × 1069	1.93×106	−8.10 × 1069
		STD	0.00 × 100	1.08 × 1035	0.00 × 100	7.16 × 1021	5.46 × 1017	5.45 × 1069	9.58×106	5.43 × 1070
F41	0.00 × 100	MEAN	0.00 × 100	1.84 × 10−107	0.00 × 100	1.31 × 10−30	6.73 × 10−17	2.99 × 10−65	4.29×101	6.05 × 10−52
		STD	0.00 × 100	1.18 × 10−106	0.00 × 100	6.25 × 10−30	2.03 × 10−16	2.05 × 10−64	4.98×101	4.28 × 10−51
F42	0.00 × 100	MEAN	0.00 × 100	7.33 × 10−110	0.00 × 100	4.59 × 10−31	5.33 × 10−20	7.34 × 10−74	1.59×10−1	4.98 × 10−25
		STD	0.00 × 100	3.62 × 10−109	0.00 × 100	3.21 × 10−30	1.10 × 10−19	5.04 × 10−73	1.48×10−1	3.45 × 10−24
F43	0.00 × 100	MEAN	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	3.39 × 10−1	0.00 × 100	7.61×10−2	1.57 × 10−4
		STD	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	2.17 × 10−1	0.00 × 100	7.75×10−2	4.75 × 10−4
F44	0.00 × 100	MEAN	0.00 × 100	0.00 × 100	0.00 × 100	5.24 × 10−3	4.42 × 10−1	1.93 × 10−2	9.03×10−2	2.18 × 10−2
		STD	0.00 × 100	0.00 × 100	0.00 × 100	1.43 × 10−2	1.21 × 10−1	2.18 × 10−2	7.74×10−2	2.21 × 10−2
F45	0.00 × 100	MEAN	0.00 × 100	2.49 × 10−13	0.00 × 100	2.80 × 10−7	1.10 × 10−8	6.44 × 10−9	3.11×10−4	4.95 × 10−12
		STD	0.00 × 100	1.44 × 10−12	0.00 × 100	6.68 × 10−7	4.21 × 10−8	9.10 × 10−9	4.16×10−4	1.65 × 10−11
F46	0.00 × 100	MEAN	0.00 × 100	3.60 × 10−109	0.00 × 100	−1.88 × 10−15	−1.78 × 10−15	2.00 × 10−6	3.73×10−3	1.32 × 10−5
		STD	0.00 × 100	1.80 × 10−108	0.00 × 100	1.77 × 10−15	1.79 × 10−15	5.65 × 10−6	3.27×10−3	4.04 × 10−5
F47	−4.00 × 102	MEAN	−4.00 × 102	−4.00 × 102	−4.00 × 102	−4.00 × 102	−3.78 × 102	−4.00 × 102	−3.86×102	−4.00 × 102
		STD	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	6.05 × 101	0.00 × 100	1.45×101	0.00 × 100

.

In Table 3, the green background indicates the results that converge most closely with the minimum value of the function, while the blue background indicates the results that converge second most effectively. The proposed method achieved the best result for a total of 43 functions and the second-best result for 2 functions. The chaotic version of the HGSO (QHGSO) attained the best results in 31 instances and the second-best results in 2 instances. In contrast, the original HGSO algorithm produced the best results in 15 cases and the second-best results in 23 cases. The PSO algorithm achieved the best results in 9 cases and the second-best results in 3 cases. The EA algorithm yielded the best results in 6 cases and the second-best results in 2 cases. The GWO algorithm produced the best results in 8 cases, while the SA algorithm achieved the second-best results in 6 cases. The WOA algorithm attained the best results in 7 cases and the second-best results in 9 cases. Notably, when the CHGSO algorithm was run with the same parameters, a significant improvement in the results was observed.

A key characteristic of metaheuristic algorithms is their sensitivity to parameters that significantly influence the algorithm’s performance. The efficiency and effectiveness of these algorithms are directly tied to the parameters that guide the search processes. These parameters determine how the solution space is explored, which solutions are emphasized, and which strategies are followed in the search for an optimal solution. Even small changes in these parameters can lead to notable differences in the algorithm’s performance, greatly affecting the quality of the results obtained. Therefore, to observe the sensitivity of the model to parameter variations, the proposed method was executed with different sets of parameters. The selected parameters are provided in

Table 4. Different parameter values of the proposed method.

CHGSO_1	Number of iterations: 100 Number of gas particles: 50 Number of clusters: 5 M1 = 0.1, M2 = 0.2, L1 = 0.005, l2 = 100, l3 = 0.01, a, b, k = 1, e = 0.05
CHGSO_2	Number of iterations: 50 Number of gas particles: 50 Number of clusters: 5 M1 = 0.1, M2 = 0.2, L1 = 0.005, l2 = 100, l3 = 0.01, a, b, k = 1, e = 0.05
CHGSO_3	Number of iterations: 100 Number of gas particles: 30 Number of clusters: 5 M1 = 0.1, M2 = 0.2, L1 = 0.005, l2 = 100, l3 = 0.01, a, b, k = 1, e = 0.05
CHGSO_4	Number of iterations: 50 Number of gas particles: 30 Number of clusters: 5 M1 = 0.1, M2 = 0.2, L1 = 0.005, l2 = 100, l3 = 0.01, a, b, k = 1, e = 0.05
CHGSO_5	Number of iterations: 100 Number of gas particles: 50 Number of clusters: 10 M1 = 0.1, M2 = 0.2, L1 = 0.005, l2 = 100, l3 = 0.01, a, b, k = 1, e = 0.05
CHGSO_6	Number of iterations: 50 Number of gas particles: 50 Number of clusters: 10 M1 = 0.1, M2 = 0.2, L1 = 0.005, l2 = 100, l3 = 0.01, a, b, k = 1, e = 0.05
CHGSO_7	Number of iterations: 100 Number of gas particles: 30 Number of clusters: 10 M1 = 0.1, M2 = 0.2, L1 = 0.005, l2 = 100, l3 = 0.01, a, b, k = 1, e = 0.05
CHGSO_8	Number of iterations: 50 Number of gas particles: 30 Number of clusters: 10 M1 = 0.1, M2 = 0.2, L1 = 0.005, l2 = 100, l3 = 0.01, a, b, k = 1, e = 0.05

, and the results produced by the algorithm corresponding to these parameters are presented in

Table 5. Benchmark function results for different parameter values of the proposed method.

	Fmin		CHGSO_1	CHGSO_2	CHGSO_3	CHGSO_4	CHGSO_5	CHGSO_6	CHGSO_7	CHGSO_8
F1	0.00 × 100	MEAN	1.68 × 10−118	3.50 × 10−58	6.31 × 10−108	2.47 × 10−53	1.18 × 10−114	1.27 × 10−53	2.18 × 10−103	9.94 × 10−51
		STD	8.18 × 10−31	8.29 × 10−17	2.33 × 10−27	8.85 × 10−14	1.11 × 10−27	6.83 × 10−14	4.74 × 10−24	9.85 × 10−12
F2	0.00 × 100	MEAN	1.14 × 10−60	8.66 × 10−31	6.82 × 10−60	6.51 × 10−29	4.51 × 10−59	5.34 × 10−30	1.83 × 10−56	3.33 × 10−29
		STD	5.76 × 10−17	1.58 × 10−9	6.62 × 10−15	3.70 × 10−9	3.81 × 10−14	1.41 × 10−8	1.15 × 10−13	3.91 × 10−8
F3	0.00 × 100	MEAN	2.94 × 10−62	5.40 × 10−30	1.99 × 10−59	2.29 × 10−30	1.51 × 10−59	2.95 × 10−29	1.69 × 10−56	3.25 × 10−28
		STD	5.87 × 10−17	3.21 × 10−11	9.86 × 10−18	4.65 × 10−10	4.65 × 10−15	1.84 × 10−10	4.54 × 10−14	1.26 × 10−8
F4	0.00 × 100	MEAN	5.22 × 10−60	2.92 × 10−29	2.08 × 10−56	1.54 × 10−47	1.80 × 10−58	1.14 × 10−28	5.95 × 10−54	1.23 × 10−26
		STD	4.25 × 10−15	3.63 × 10−9	5.72 × 10−15	6.28 × 10−9	1.57 × 10−14	3.48 × 10−10	1.62 × 10−13	1.07 × 10−7
F5	0.00 × 100	MEAN	5.50 × 10−136	1.51 × 10−71	5.42 × 10−141	4.77 × 10−65	3.28 × 10−126	1.29 × 10−65	1.43 × 10−142	2.01 × 10−64
		STD	7.85 × 10−31	4.38 × 10−22	4.38 × 10−30	1.72 × 10−8	6.86 × 10−28	2.27 × 10−20	4.46 × 10−29	7.54 × 10−16
F6	0.00 × 100	MEAN	5.78 × 10−30	2.11 × 10−14	1.40 × 10−27	9.41 × 10−14	8.35 × 10−28	2.30 × 10−13	2.76 × 10−26	4.92 × 10−13
		STD	9.44 × 10−9	1.49 × 10−4	1.66 × 10−08	1.49 × 10−4	1.86 × 10−8	1.64 × 10−4	1.10 × 10−6	1.48 × 10−3
F7	0.00 × 100	MEAN	1.60 × 10−29	1.03 × 10−14	3.17 × 10−27	2.62 × 10−14	2.69 × 10−29	4.72 × 10−14	1.12 × 10−26	2.22 × 10−13
		STD	2.92 × 10−9	7.53 × 10−6	7.68 × 10−8	2.75 × 10−4	2.91 × 10−8	1.05 × 10−4	3.23 × 10−7	9.12 × 10−4
F8	0.00 × 100	MEAN	5.12 × 10−30	1.46 × 10−14	3.63 × 10−28	4.98 × 10−85	1.82 × 10−27	1.09 × 10−13	2.92 × 10−26	7.06 × 10−13
		STD	6.77 × 10−8	5.79 × 10−5	8.36 × 10−8	6.73 × 10−4	1.79 × 10−6	1.36 × 10−3	2.04 × 10−6	2.15 × 10−3
F9	0.00 × 100	MEAN	1.21 × 10−298	2.10 × 10−153	4.02 × 10−295	−3.00 × 103	1.67 × 10−297	1.48 × 10−142	1.46 × 10−262	3.65 × 10−138
		STD	1.55 × 10−75	5.67 × 10−43	1.76 × 10−70	8.66 × 102	2.32 × 10−64	2.64 × 10−38	6.98 × 10−63	8.81 × 10−35
F10	0.00 × 100	MEAN	−3.00 × 103	−3.00 × 103	−3.00 × 103	−3.00 × 103	−3.00 × 10−3	−3.00 × 103	−3.00 × 103	−3.00 × 10−3
		STD	1.58 × 102	2.85 × 102	2.96 × 102	8.74 × 102	6.22 × 101	3.28 × 102	3.83 × 102	3.86 × 102
F11	0.00 × 100	MEAN	2.83 × 10−60	6.91 × 10−30	3.36 × 10−57	1.10 × 10−26	5.92 × 10−59	2.15 × 10−27	8.95 × 10−56	4.27 × 10−26
		STD	4.32 × 10−17	2.89 × 10−9	7.13 × 10−14	5.16 × 10−8	4.93 × 10−14	9.64 × 10−8	4.55 × 10−14	5.53 × 10−7
F12	0.00 × 100	MEAN	8.88 × 10−16	4.44 × 10−15	8.88 × 10−16	7.99 × 10−15	8.88 × 10−16	7.99 × 10−15	8.88 × 10−16	4.44 × 10−15
		STD	1.08 × 10−8	9.70 × 10−6	2.65 × 10−9	9.58 × 10−6	8.07 × 10−8	2.19 × 10−5	4.10 × 10−8	2.99 × 10−5
F13	0.00 × 100	MEAN	3.83 × 10−31	6.79 × 10−16	6.69 × 10−33	2.17 × 10−15	7.72 × 10−31	2.85 × 10−15	1.27 × 10−28	2.10 × 10−15
		STD	1.25 × 10−09	4.64 × 10−06	4.80 × 10−8	5.11 × 10−5	6.25 × 10−9	1.36 × 10−5	1.28 × 10−7	3.36 × 10−5
F14	0.00 × 100	MEAN	1.71 × 10−61	5.47 × 10−32	8.58 × 10−63	8.16 × 10−29	1.56 × 10−61	1.04 × 10−30	2.66 × 10−56	2.67 × 10−27
		STD	1.07 × 10−15	9.19 × 10−10	9.57 × 10−17	9.97 × 10−10	2.29 × 10−15	2.49 × 10−8	6.64 × 10−14	6.49 × 10−9
F15	0.00 × 100	MEAN	7.52 × 10−55	6.71 × 10−25	1.47 × 10−49	7.43 × 10−21	5.18 × 10−51	6.56 × 10−22	4.41 × 10−47	5.63 × 10−21
		STD	3.54 × 10−13	4.54 × 10−5	1.70 × 10−10	2.90 × 10−18	6.45 × 10−9	1.43 × 10−2	1.82 × 10−9	7.95 × 10−2
F16	−1.00 × 100	MEAN	−1.00 × 100	−1.00 × 100	−1.00 × 100	−1.00 × 100	−1.00 × 100	−1.00 × 100	−1.00 × 100	−1.00 × 100
		STD	0.00 × 100	0.00 × 100	0.00 × 100	2.89 × 10−1	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100
F17	0.00 × 100	MEAN	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100
		STD	1.02 × 10−14	2.21 × 10−8	4.96 × 10−13	5.77 × 10−1	2.64 × 10−13	2.79 × 10−7	5.47 × 10−14	1.72 × 10−6
F18	2.00 × 100	MEAN	2.00 × 100	2.00 × 100	2.00 × 100	2.00 × 100	2.00 × 100	2.00 × 100	2.00 × 100	2.00 × 100
		STD	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100
F19	2.00 × 100	MEAN	2.00 × 100	2.00 × 100	2.00 × 100	0.00 × 100	2.00 × 100	2.00 × 100	2.00 × 100	2.00 × 100
		STD	0.00 × 100	0.00 × 100	0.00 × 100	5.77 × 10−1	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100
F20	0.00 × 100	MEAN	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100
		STD	0.00 × 100	0.00 × 100	0.00 × 100	1.10 × 10−4	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100
F21	0.00 × 100	MEAN	3.81 × 10−5	4.87 × 10−5	5.12 × 10−5	0.00 × 100	7.12 × 10−5	7.11 × 10−5	5.20 × 10−5	2.19 × 10−4
		STD	2.08 × 10−4	1.48 × 10−3	6.81 × 10−4	6.57 × 10−4	3.28 × 10−4	7.67 × 10−4	4.95 × 10−4	1.41 × 10−3
F22	0.00 × 100	MEAN	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100
		STD	0.00 × 100	6.12 × 10−9	1.15 × 10−13	3.63 × 100	2.13 × 10−13	4.61 × 10−7	1.90 × 10−11	3.05 × 10−6
F23	0.00 × 100	MEAN	6.62 × 10−1	8.53 × 10−1	3.88 × 10−1	2.45 × 10−11	6.04 × 10−1	6.05 × 10−1	1.34 × 100	1.34 × 100
		STD	4.90 × 100	5.03 × 100	5.10 × 100	5.64 × 100	4.14 × 100	4.26 × 100	4.42 × 100	4.45 × 100
F24	0.00 × 100	MEAN	4.17 × 10−12	4.17 × 10−12	5.92 × 10−12	4.34 × 10−232	4.22 × 10−12	4.22 × 10−12	4.35 × 10−12	4.35 × 10−12
		STD	1.32 × 10−11	5.53 × 10−11	1.16 × 10−11	1.21 × 10−11	1.53 × 10−11	1.22 × 10−8	3.86 × 10−8	3.93 × 10−8
F25	0.00 × 100	MEAN	4.34 × 10−232	4.34 × 10−232	4.34 × 10−232	4.34 × 10−232	4.34 × 10−232	4.34 × 10−232	4.34 × 10−232	4.34 × 10−232
		STD	0.00 × 100	0.00 × 100	0.00 × 100	2.60 × 10−15	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100
F26	0.00 × 100	MEAN	1.89 × 10−62	2.70 × 10−30	9.20 × 10−60	−2.00 × 102	6.15 × 10−60	6.83 × 10−29	5.17 × 10−54	8.69 × 10−27
		STD	5.30 × 10−16	1.85 × 10−10	7.45 × 10−15	5.77 × 101	4.03 × 10−16	4.14 × 10−10	5.04 × 10−14	6.72 × 10−9
F27	−2.00 × 102	MEAN	−2.00 × 102	−2.00 × 10−2	−2.00 × 102	−2.0 × 102	−2.00 × 102	−2.00 × 102	−2.00 × 102	−2.00 × 102
		STD	0.00 × 100	0.00 × 100	0.00 × 100	5.80 × 101	0.00 × 100	2.89 × 10−5	0.00 × 100	2.89 × 10−5
F28	1.00 × 100	MEAN	1.00 × 100	1.00 × 100	1.00 × 100	0.00 × 100	1.00 × 100	1.00 × 100	1.00 × 100	1.00 × 100
		STD	0.00 × 100	0.00 × 100	0.00 × 100	2.89 × 10−1	0.00 × 100	8.66 × 10−7	0.00 × 100	2.89 × 10−7
F29	0.00 × 100	MEAN	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100
		STD	0.00 × 100	2.93 × 10−12	3.20 × 10−17	5.20 × 10−2	0.00 × 100	2.30 × 10−11	1.67 × 10−15	1.80 × 10−10
F30	0.00 × 100	MEAN	1.80 × 10−1	1.80 × 10−1	1.80 × 10−1	0.00 × 100	1.80 × 10−1	1.80 × 10−1	1.80 × 10−1	1.80 × 10−1
		STD	2.90 × 10−17	2.90 × 10−17	2.90 × 10−17	5.20 × 10−2	2.90 × 10−17	2.90 × 10−17	2.90 × 10−17	2.90 × 10−17
F31	0.00 × 100	MEAN	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100
		STD	0.00 × 100	3.30 × 10−12	1.60 × 10−17	3.07 × 10−10	0.00 × 100	5.89 × 10−10	2.68 × 10−15	6.02 × 10−10
F32	0.00 × 100	MEAN	9.57 × 10−154	1.09 × 10−78	7.22 × 10−131	−4.20 × 101	8.85 × 10−110	8.87 × 10−57	1.51 × 10−105	4.71 × 10−52
		STD	7.36 × 10−27	1.54 × 10−14	1.18 × 10−21	1.21 × 101	3.61 × 10−22	5.25 × 10−12	3.98 × 10−18	2.64 × 10−12
F33	−4.33 × 101	MEAN	−4.25 × 101	−4.25 × 101	−4.25 × 101	−4.25 × 101	−4.25 × 101	−4.25 × 101	−4.25 × 101	−4.25 × 101
		STD	2.02 × 10−1	1.65 × 10−1	8.95 × 10−1	1.15 × 101	2.98 × 10−1	5.53 × 10−1	1.83 × 100	1.82 × 100
F34	−2.06 × 100	MEAN	−2.11 × 100	−2.11 × 100	−2.11 × 100	−2.11 × 100	−2.11 × 100	−2.11 × 100	−2.11 × 100	−2.11 × 100
		STD	6.91 × 10−4	1.03 × 10−3	1.38 × 10−2	6.04 × 10−1	6.99 × 10−4	1.55 × 10−3	3.48 × 10−3	4.20 × 10−3
F35	−1.00 × 100	MEAN	−1.15 × 10−2	−9.01 × 10−3	−9.94 × 10−1	−1.30 × 10−2	−2.54 × 10−4	−1.65 × 10−4	−1.00 × 100	−2.48 × 10−2
		STD	3.90 × 10−3	2.55 × 10−3	3.87 × 10−1	5.00 × 10−3	3.62 × 10−5	1.63 × 10−5	3.86 × 10−1	9.54 × 10−3
F36	0.00 × 100	MEAN	1.60 × 10−138	3.77 × 10−68	1.05 × 10−104	−2.35 × 100	9.41 × 10−107	6.89 × 10−54	9.50 × 10−109	3.73 × 10−51
		STD	5.77 × 10−27	3.09 × 10−14	6.43 × 10−19	6.79 × 10−1	1.38 × 10−21	7.88 × 10−12	2.64 × 10−18	7.33 × 10−13
F37	−3.32 × 100	MEAN	−2.66 × 100	−2.52 × 100	−2.63 × 100	−2.58 × 100	−2.59 × 100	−2.59 × 100	−2.49 × 100	−2.33 × 100
		STD	3.00 × 10−1	2.73 × 10−1	2.90 × 10−1	6.71 × 10−1	2.88 × 10−1	3.05 × 10−1	2.37 × 10−1	2.06 × 10−1
F38	0.00 × 100	MEAN	2.24 × 10−139	3.06 × 10−70	9.83 × 10−112	1.84 × 10−56	3.23 × 10−110	9.51 × 10−57	6.56 × 10−111	3.14 × 10−52
		STD	5.50 × 10−26	2.54 × 10−13	6.57 × 10−21	2.60 × 10−1	2.61 × 10−24	1.28 × 10−12	1.46 × 10−18	2.65 × 10−13
F39	9.00 × 10−1	MEAN	9.00 × 10−1	9.00 × 10−1	9.00 × 10−1	−3.37 × 103	9.00 × 10−1	9.00 × 10−1	9.00 × 10−1	9.00 × 10−1
		STD	2.32 × 10−16	2.32 × 10−16	2.32 × 10−16	9.74 × 102	2.32 × 10−16	2.32 × 10−16	2.32 × 10−16	2.32 × 10−16
F40	0.00 × 100	MEAN	−1.39 × 107	−1.94 × 105	−4.41 × 106	−1.87 × 105	−1.92 × 105	−2.42 × 104	−2.23 × 105	−6.54 × 104
		STD	4.00 × 106	5.52 × 104	1.27 × 106	6.08 × 104	7.12 × 104	8.24 × 103	7.91 × 104	1.87 × 104
F41	0.00 × 100	MEAN	1.94 × 10−131	5.56 × 10−68	3.53 × 10−104	3.82 × 10−47	1.13 × 10−106	2.52 × 10−50	3.95 × 10−107	3.32 × 10−45
		STD	5.02 × 10−24	1.24 × 10−12	1.07 × 10−18	3.12 × 10−11	7.49 × 10−20	1.78 × 10−11	1.08 × 10−15	5.46 × 10−10
F42	0.00 × 100	MEAN	2.60 × 10−139	5.67 × 10−68	1.93 × 10−108	0.00 × 100	9.50 × 10110	1.87 × 10−53	8.12 × 10−113	5.64 × 10−52
		STD	5.65 × 10−25	2.86 × 10−13	4.54 × 10−21	7.86 × 10−14	9.55 × 10−22	7.16 × 10−13	1.15 × 10−17	4.75 × 10−12
F43	0.00 × 100	MEAN	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100
		STD	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100
F44	0.00 × 100	MEAN	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100
		STD	0.00 × 100	8.23 × 10−8	1.26 × 10−2	6.98 × 10−5	6.41 × 10−17	1.39 × 10−8	3.77 × 10−11	1.30 × 10−7
F45	0.00 × 100	MEAN	1.13 × 10−102	1.75 × 10−47	3.39 × 10−43	4.85 × 10−41	8.97 × 10−38	9.94 × 10−31	8.84 × 10−34	1.70 × 10−26
		STD	2.72 × 10−16	1.43 × 10−13	3.49 × 10−14	2.12 × 10−14	1.45 × 10−18	9.71 × 10−15	1.98 × 10−14	9.87 × 10−13
F46	0.00 × 100	MEAN	2.01 × 10−147	1.17 × 10−79	3.29 × 10−109	5.83 × 10−54	2.67 × 10−110	1.42 × 10−56	3.44 × 10−115	1.12 × 10−54
		STD	1.21 × 10−25	2.74 × 10−13	6.83 × 10−22	3.12 × 10−14	1.36 × 10−21	3.52 × 10−13	5.24 × 10−18	3.19 × 10−12
F47	−4.00 × 102	MEAN	−4.00 × 102	−4.00 × 102	−4.00 × 102	−4.00 × 102	−4.00 × 102	−4.00 × 102	−4.00 × 102	−4.00 × 102
		STD	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100	0.00 × 100

.

4. Optimization of PID Parameters for Speed Control of DC Motor with Proposed CHGSO Method

The performance of the proposed algorithm was tested on a real engineering problem: the DC motor speed control application. For the purpose of comparison, other recent and commonly used optimization methods, such as HGSO, PSO, WOA, EA, and SA, were also applied to the same problem.

The mathematical model of the DC motor is given in Equation (24). In Equation (24), $V_a$ is the applied voltage to the motor, $R_a$ is the resistance of the motor windings, $L_a$ is the inductance of the motor windings, $e_b$ is the back electromotive force, $K_e$ is the electrical constant, $T$ is the generated torque, $J$ is the moment of inertia, $B$ is the damping constant, $T_l$ is the load torque, $K_m$ is the mechanical constant, $i$ is the current, and $w$ is the angular velocity of the motor.

$$\begin{array}{l}{V}_a\left(t\right)=i_a{R}_a+L_a{\displaystyle \frac{{di}_a}{dt}}+e_b\\ e_b\left(t\right)=K_{e}w\left(t\right)\\ T\left(t\right)=J{\displaystyle \frac{dw}{dt}}+Bw+T_l\\ T\left(t\right)=K_{m}i\left(t\right)\end{array}$$

(24)

By applying the Laplace transform to the equations given in Equation (24), the block diagram of the motor is shown in Figure 4.

Based on the block diagram given above, it can be seen that the input of the motor system is voltage and the output is angular velocity. No load is applied to the DC motor. The transfer function of the DC motor is given in Equation (25).

$$G_m(s)={\displaystyle \frac{K_m}{\left(L_{a}s+R_a\right)\left(Js+B\right)+K_e{K}_m}}$$

(25)

The speed of the DC motor is controlled by the applied voltage. The PID controller determines the applied voltage depending on the desired speed by adjusting the coefficients of $K_p$, $K_i,$ and $K_d$. The mathematical model of the PID controller is given in Equation (26).

$$G_{PID}(s)=K_p+{\displaystyle \frac{K_i}{s}}+K_{d}s$$

(26)

The closed−loop block diagram of the system designed with the PID controller for which the parameters are set by CHGSO for the speed control of the DC motor is given in Figure 5.

The parameters of the motor are Ra = 0.517 ohm, La = 0.057 mH, J = 14.4 gcm², B = 0.00024 Nms/rad, Km = 0.0115, and Ke = 0.0115. The motor reduction gear ratio is 1/53.

The ITSE (integrated of time−weighted−squared−error) performance criterion is used as the objective function for tuning PID parameters. The objective function is given in Equation (27).

$$ITSE=\int t{e}^2\left(t\right)dt$$

(27)

4.1. Results

The main objective of this study is to minimize the error by optimizing the Kp, Ki, and Kd coefficients of the PID controller used to control the speed of the DC motor with optimization algorithms. The HGSO algorithm is combined with the chaotic systems described in Section 2.2. The HGSO algorithm was run with each system.

Table 6. Results of the CHGSO for different chaotic systems.

	Kp	Ki	Kd	Fitness	Optimization Time(s)
Duffing−VanDerPol	11.88994	196.9933	0.078988	0.000040	90.17311
Rucklidge	12.35782	196.81	0.106938	0.000056	88.16251
Chen	11.74649	192.8399	0.101465	0.000060	85.78418
Duffing	10.81011	173.7897	0.094326	0.000069	86.83223
Rössler	12.52376	191.5878	0.117096	0.000075	86.21741
Rikitake	12.52376	191.5878	0.117096	0.000075	86.9942
Lorenz	10.96456	179.5598	0.103868	0.000079	89.74064

shows the optimization times, results of the objective function, and Kp, Ki, and Kd values.

Optimization time refers to the operations an algorithm performs to determine the most appropriate Kp, Ki, and Kd values for the control system. This time is the completion time of 50 iterations required for the algorithm to reach the best performance in these parameters. That is, it is a measure of how effective and fast the algorithm is in the process of optimizing these critical parameters. As shown in Table 6, the best result in terms of the objective function was obtained with the Duffing–Van Der Pol chaotic system.

For the practical implementation of the addressed problem, a Maxon DC motor was utilized. The motor parameters were modeled in a computer environment, and simulation studies were conducted. Experimental investigations were carried out by transferring data between the motor and the computer using a data acquisition card. All simulation and experimental studies were implemented using MATLAB R2022a installed on a Windows 10 64−bit system with a 2.60 GHz processor and 12 GB RAM.

4.1.1. Simulation Results

The CHGSO algorithm is obtained by hybridizing HGSO with the chaotic systems described in the study. The HGSO algorithms hybridized with each chaotic system and were utilized to optimize the PID controller parameters. The simulation results for the unit step response of the DC motor controlled by these controllers are presented in Figure 6.

According to Figure 6, all seven controllers reach the reference value without exceeding the system responses. Among them, the CHGSO−PID system achieved the 1% settling band in the shortest time of 0.035 s. The performance of the other controllers is as follows: the Chen CHGSO−PID reached this settling time in 0.045 s, the Rucklidge CHGSO−PID in 0.047 s, the Duffing CHGSO−PID in 0.049 s, the Lorenz CHGSO−PID in 0.050 s, the Rössler CHGSO−PID in 0.052 s, and the Rikitake CHGSO−PID in 0.053 s.

The Duffing–Van Der Pol chaotic system provides the fastest response and converges to a value close to the minimum in the objective function, successfully passing the NIST tests. Therefore, the Duffing–Van Der Pol chaotic system has been hybridized with the HGSO algorithm.

The proposed CHGSO algorithm is compared with that widely used in the literature, PSO [41], WOA [42], SA [43], EA [44], GWO [45], and HGSO [9] optimization methods in the literature. The parameters used in the comparison of optimization algorithms were selected through experimental methods, ensuring alignment with commonly used approaches in the literature. In this process, the effects of the parameters on optimization results were considered, and parameters, such as the number of search agents and the number of iterations, were kept constant, aiming to compare the algorithms solely based on solution quality. To ensure a fair and objective comparison of the optimization methods, all computational parameters are presented in detail in

Table 7. Parameters of the algorithms.

CHGSO	Number of iterations: 50 Number of gas particles: 50 Number of clusters: 5 M1 = 0.1, M2 = 0.2 L1 = 0.005, l2 = 100, l3 = 0.01 a, b, k = 1, e = 0.05 Global boundaries = [0–200]
HGSO	Number of iterations: 50 Number of gas particles: 50 Number of clusters: 5 M1 = 0.1, M2 = 0.2 L1 = 0.005, l2 = 100, l3 = 0.01 a, b, k = 1, e = 0.05 Global boundaries = [0–200]
PSO	Number of iterations: 50 Number of swarm: 50 C1 = 2.1 C2 = 2.1 Global boundaries = [0–200]
WOA	Number of iterations: 50 Number of whales: 50 Global boundaries = [0–200]
EA	Number of iterations: 50 Number of parents: 20 Number of children: 4 Global boundaries = [0–200]
SA	Number of iterations: 50 Number of materials: 50 Cooling rate: 0.98 Global boundaries = [0–200]
GWO	Number of iterations: 50 Number of wolves: 50 Global boundaries = [0–200]

.

The HGSO, PSO, GWO, WOA, EA, and SA algorithms were executed 30 times each. The minimum, maximum, and mean values of the objective function obtained from these 30 runs and the corresponding optimization times are presented in

Table 8. Results of the optimization algorithms.

	MIN_Fitness	MAX_Fitness	Mean_Fitness	Optimization Time (s)
CHGSO	0.000040	0.000043	0.000041	90.17311
HGSO	0.000039	0.000059	0.000049	103.72647
GWO	0.000052	0.000097	0.000075	944.2524
PSO	0.000028	0.000273	0.000093	286.7835
WOA	0.000026	0.000109	0.000101	357.4827
EA	0.000105	0.000589	0.000198	1369.41
SA	0.000116	0.000545	0.000310	911.7728

. These results are compared with those obtained from the CHGSO (Duffing–Van Der Pol chaotic HGSO) system.

Table 6 indicates that the CHGSO algorithm is more effective in converging with the minimum value of the objective function. In terms of the optimization time, it is observed that CHGSO is the most efficient method, followed by the original HGSO. HGSO reduces the optimization time by approximately 64% compared to the PSO method, while CHGSO reduces the optimization time by approximately 13% compared to HGSO. The simulation responses of the PID controllers optimized with these methods are listed in Table 7 and illustrated in Figure 7 for a unit step input.

According to Figure 7, the PSO−PID controller has an overshoot of 15% and the WOA−PID controller has an overshoot of 10.2% and the GWO_PID controller has a overshoot of 3.3% relative to the reference value. It is observed that the CHGSO−PID controller is the fastest system, settled within the 1% settling band in 0.035 s. The second fastest is the HGSO−PID controller, which settled within the 1% band in 0.040 s. The settling time, rise time, and overshoot values for all systems are given below in

Table 9. Comparison of controller responses.

	Overshoot (%)	Settling Time (s)	Rise Time (s)
CHGSO−PID	0	0.035	0.014
HGSO−PID	0	0.040	0.016
EA−PID	0.3	0.047	0.028
PSO−PID	15	0.053	0.012
SA−PID	0	0.099	0.033
WOA−PID	10.2	0.046	0.014
GWO−PID	3.3	0.17	0.020

.

As can be seen from Table 7, the controller with the best results is CHGSO−PID, followed by the original HGSO−PID. In the simulation environment, a disturbance effect of 0.01 Nm in 0.05 s was applied to the DC motor controlled by all PID controllers. The results are shown in Figure 8.

The performance of the controllers is demonstrated in Figure 8. The CHGSO−PID controller exceeds the reference value by 2.8%, and the HGSO−PID controller exceeds it by 4.4%. The settling times within the 1% band for the CHGSO−PID and HGSO−PID controllers are 0.021 s and 0.022 s, respectively. After the disturbance, both controllers’ system responses are reduced by 25% compared to the reference value. During the disturbance, the HGSO−PID controller overshoots 1.4% and settles within the 1% band at 0.67 s, whereas the CHGSO−PID controller overshoots 0.6% and settles within the 1% band at 0.63 s.

A quarter−period sinusoidal reference signal was applied to all PID controllers. The responses of controllers to the reference signal are presented in Figure 9.

Figure 9 shows that the CHGSO−PID controller tracks the reference signal with less error than the other controller. The type 1 norms of the errors produced by all controllers are shown in Figure 10. The type 1 norm of the error produced by the CHGSO−PID is 2.3685, and the type 1 norm of the error produced by the HGSO−PID controller is 3.6477. The norm values of the errors produced by the other controllers, in order, are as follows: EA−PID = 5.142, PSO−PID = 5.249, SA−PID = 6.812, WOA−PID = 5.288, and GWO−PID = 5.428.

4.1.2. Experimental Results

An experimental setup has been established for the studies, as shown in Figure 11. The experimental setup includes the following: (1) a computer in which the software is installed and control is executed, (2) a DAQ card that facilitates data transfer between the controlled system and the computer, (3) the driver circuit board for the controlled system (DC motor), (4) a line driver circuit board, (5) a power supply, (6) an oscilloscope, and (7) the DC motor.

Unit Step Response of Unloaded DC Motor

In the studies carried out in the simulation environment, HGSO−PID and CHGSO−PID controllers provided very accurate results. In the experimental study, the controllers were tested by applying it to a DC motor for real−time speed control. The unit step reference input responses of the controllers are given in Figure 12.

In Figure 12, performances of the all controllers are observed. The overshoots of the controllers are as follows: PSO−PID exhibited 14.5%, WOA−PID 13.2%, HGSO−PID 8.9%, GWO−PID 5.4%, and CHGSO−PID 1.4%. The controllers reached the 1% settling band in the following times: CHGSO−PID in 0.15 s, HGSO−PID in 0.18 s, WOA−PID in 0.21 s, PSO−PID in 0.22 s, EA−PID in 0.36 s, GWO−PID in 0.55 s, and SA−PID in 0.65 s.

Unit Step Response of Under Load DC Motor

The motor under a load of 10 kg, CHGSO−PID, and other controllers were applied. The responses of the controllers for a unit step input are given in Figure 13.

As seen in Figure 13, CHGSO−PID produces a better response. The overshoot percentages of the controllers are as follows: PSO−PID exhibited 33.5%, WOA−PID 28.9%, HGSO−PID 24.1%, and CHGSO−PID 21%. The controllers reached the 1% settling band in the following times: CHGSO−PID and HGSO−PID both in 1.95 s, WOA−PID in 1.98 s, and GWO−PID in 1.99 s. The other controllers were unable to reach the settling band within 2 s.

Parabolic Input Response of Unloaded DC Motor

Real−time control of the DC motor using controllers was carried out by applying a quarter−period sinusoidal speed reference signal. The speed responses of the motor running without load for the all controllers are given in Figure 14.

As seen from Figure 14, the unloaded DC motor controlled by CHGSO−PID follows the speed reference signal with a lower error compared to the other controller. The type 1 norms of the errors generated with these controllers are given in Figure 15.

The Type 1 norm values of the errors generated by the controllers for the motor running without load are as follows: CHGSO−PID = 2.6981, HGSO−PID = 5.7012, EA−PID = 6.89, PSO−PID = 5.25, SA−PID = 7.382, WOA−PID = 5.953, and GWO−PID = 6.133.

Parabolic Input Response of Under Load DC Motor

A load of 10 kg was applied to the DC motor, which was controlled without load in Section 4.1.2. A quarter period sinusoidal signal is applied as input, and the speed response of the motor is presented in Figure 16.

As seen in Figure 16, the DC motor controlled by the CHGSO−PID tracks the speed reference signal with a lower error compared to those controlled by the other controllers. The Type 1 norms of the errors produced by these controllers are presented in Figure 17.

The Type 1 norm values of the errors generated by the controllers for the motor operating under load are as follows: CHGSO−PID = 3.3278, HGSO−PID = 7.1539, EA−PID = 8.012, PSO−PID = 5.849, SA−PID = 9.5, WOA−PID = 6.73, and GWO−PID = 7.118.

5. Conclusions

In this study, the CHGSO method is proposed and applied to both benchmark functions and tuning PID parameters. Benchmark functions are utilized to evaluate the performance of optimization algorithms. The results from these benchmark functions demonstrate that the proposed method converges most closely with the global minimum values compared to the other algorithms analyzed. The main objective of this research is to enhance the HGSO algorithm. When both the HGSO and CHGSO algorithms are tested with the same parameters, the HGSO algorithm achieves the best results in 15 functions, while the CHGSO algorithm excels in 43 functions. The experimental findings reveal that the proposed optimization algorithm demonstrates superior convergence performance compared to other optimization methods evaluated under the same conditions. Notably, the algorithm successfully reaches the global minimum within the search boundaries of the function in 39 different test functions. This demonstrates that the CHGSO algorithm effectively converges with global minimum values, outperforming both commonly used algorithms in the literature and the original HGSO algorithm.

The PID control with tuned parameters is applied to DC motor application in both simulation and experimental environments. To demonstrate the capabilities of the proposed CHGSO−PID, comparative performance analyses of PID controllers optimized with the original HGSO, PSO, GWO, WOA, EA, and SA algorithms commonly used in the literature are carried out. It has been shown that the proposed method provides superiority compared to other methods in the simulation time, performance analysis, operation under load, and sudden changes due to disturbing the load moment. In practice, when the motor starts with zero initial velocity, it must overcome mechanical limitations and the motor’s inertia to reach the reference signal. This situation leads to an initial error, but the optimization process aims to minimize this error. Theoretically, the global minimum value of the objective function is zero, but experimental results have shown that the proposed method reaches an average value very close to this, specifically 0.000041. During the experiments, the objective function values ranged between 0.000040 and 0.000043, indicating results that are quite close to the global minimum. When compared to other optimization algorithms, the results demonstrate that the original HGSO algorithm reached a value of 0.000049, the GWO algorithm reached 0.000075, the PSO algorithm 0.00093, the WOA algorithm 0.00101, the EA algorithm 0.000198, and the SA algorithm 0.000310. These results indicate that the proposed method converges with the global minimum more effectively than both the HGSO and other algorithms. This demonstrates that the proposed optimization algorithm is more capable of approaching the global minimum, even in complex problems.

Computational complexity expresses the extent to which an optimization method utilizes resources. In this study, the computational complexity of various optimization methods was compared while evaluating their effectiveness. This comparison focused on the time taken by the optimization methods to reach the optimal solution. When examining the computational complexity, the CHGSO algorithm was found to require the least amount of time to achieve the optimal result. Compared to other widely used algorithms in the literature, the speed of the HGSO algorithm is a key reason for its selection in this study. Additionally, the proposed method for improving the HGSO algorithm resulted in a 13% reduction in running time.

In future studies, the utilization of various algorithms for optimizing controller parameters holds significant potential across a broad spectrum of engineering applications. For instance, methods, such as the Artificial Bee Colony Algorithm, the Harris Hawks Optimization Algorithm, the Grey Wolf Optimization Algorithm, the cuckoo search algorithm, and the grasshopper optimization algorithm can be concurrently tested in both computer simulations and real−time applications. This approach will allow for the identification of optimal values for controller parameters, and the performances of these algorithms can be compared to determine the most effective approach. Furthermore, successful results obtained from these studies can be utilized to create a more robust and flexible optimization strategy by hybridizing different algorithms with chaotic systems. This hybridization has the potential to enhance the process of optimizing control parameters. In future studies, this integration can be extended to address specific topics, such as structural, thermal, and hydraulic–pneumatic engineering problems. For instance, structural design problems, like the material selection, shape, and topology optimization of a beam, can be used to assess the effectiveness of these algorithms. Such investigations could represent a crucial step in the development of effective optimization strategies for a variety of engineering applications.