1 Introduction

The ability to exert control over systems stands as a paramount accomplishment that has propelled humanity to its current technological pinnacle. A prerequisite for effective control is the availability of accurate mathematical models that faithfully capture the essence of the systems. However, it is crucial to acknowledge that modelling these systems can present a formidable challenge, demanding a profound understanding of their intricacies and interplay. System identification methods, initially employed in the defence sector, particularly during the early twentieth century and notably during World War II, have been utilized for obtaining mathematical models from data collected through an experimental setup. Researchers have consistently highlighted the importance of minimizing complexity and maximizing efficiency in determining the transfer function through system identification methods [1].

Several methods have been suggested to obtain the transfer functions using information from both the time and frequency domains. System identification employs various methods, including white box, grey box, and black box, depending on the mathematical model's structure [2, 3]. In the white-box method, a complete understanding of the model is essential. The grey-box method involves indirectly acquiring parameters to construct the model. In the black-box method, the system's information is limited to the input/output data, without any further details available. Although the models derived from the white-box method provide very good results, it is often challenging to have all the information about the system. Therefore, grey-box and black-box methods are much more widely used [4]. Linear grey-box, nonlinear grey-box, and linear polynomial (auto-regressive) models (AR, ARX, ARMA, ARMAX) are employed for situations where the structure of systems is either unknown or partially known [5].

In the context of real-time applications, it is common to employ equipment with constrained resources, such as programmable logic controllers or microcontroller circuits. MATLAB, along with its system identification toolbox, is not suitable for devices with limited resources due to its substantial memory requirements and high cost. Recent advancements in the Python programming language have made it feasible to create real-time algorithms with comparable power to those in MATLAB. Especially, specific distributions like Micro Python are suitable for smaller, more flexible, and embedded systems compared to MATLAB, due to their low resource consumption and microcontroller compatibility. In addition, the Python programming language offers advantages such as a widespread developer community and being available for free licence. Therefore, the Python language has been chosen in the creation of this article. Meta-heuristic algorithms find application in diverse problem-solving scenarios, spanning tasks like optimizing company production processes to designing PI controllers. Consequently, these algorithms exhibit a wide range of practical applications. There have been great advances since these algorithms were first proposed, but many new algorithms are still being proposed. Meta-heuristic algorithms have been employed in addressing numerous single/multi-objective, continuous/discrete, and constrained/unconstrained problems.

One prospective domain for the application of meta-heuristic algorithms involves the parameter estimation of control systems, encompassing both continuous and discrete structures. The Ziegler-Nichols (ZN) open-loop or closed-loop transfer function determination methods partially address the problem in a straightforward manner. However, estimation of the transfer function of many systems is a challenging and time-consuming process. On the other hand, quickly determining the transfer function of a system after collecting the required data from an experimental setup simplifies the controller design process. In this study, research has been conducted to broaden the application scope of meta-heuristic algorithms developed to solve a wide range of problems. In one of the recent studies, meta-heuristic algorithms were used for system identification purposes, and it was stated that they are suitable for obtaining transfer functions depending on input/output data [6]. System identification methods find application in various areas, including boost converter applications. Despite possessing a comprehensive mathematical model, the authors faced challenges in parameterizing the PI (proportional, integral) controller when modelled in Simulink. Nevertheless, they successfully employed system identification methods, yielding satisfactory results in the modelling process. [7, 8].

Zhao et al. [9] proposed a nature-inspired meta-heuristic optimization algorithm called artificial ecosystem optimizer (O-AEO). Hassan et al. [10] suggested a chaotic AEO method for economic emission forecasting. Omotoso et al. [11] proposed an AEO-based optimization algorithm to perform demand-side management in a hybrid energy system. They compared the performance of the AEO algorithm with the performance of Harris hawk optimization (HHO) and the future search algorithm. Accordingly, they reported that the AEO algorithm was more efficient. Mahdad [12] applied the adaptive split AEO algorithm to solve non-smooth economic dispatching by considering some limits. Wireless sensor networks exemplify the practical use of optimization algorithms, particularly in the realm of efficient data transmission, given the constraints imposed by limited resources. A hybrid whale optimization based AEO method has been suggested to improve energy efficiency [13, 14]. Nguyen [15], who stated that reducing power losses in the distribution network is very important, proposed an AEO-based algorithm to reorganize a distribution network. He compared the proposed AEO algorithm with the cuckoo search algorithm and demonstrated that the AEO algorithm is highly effective. Izci et al. [16] developed a hybrid approach by combining the Nelder–Mead (NM) simplex method with the AEO algorithm, which was applied to estimate the proportional, integral, and derivative (PID) parameters of the controller in a DC-DC buck converter. They compared the hybrid method with simulated annealing, whale optimization, and genetic algorithm methods and reported that the performance was quite satisfactory. In another study on PID controller design, the AEO algorithm was used to determine the optimal design parameters of the controller of an automatic voltage regulator [17].

Kumar and Janamala [18] carried out a study based on AEO for the location and size selection of distributed solar photovoltaic systems. They pointed out that the AEO algorithm has very good performance for solving complex, nonlinear, multivariate optimization problems. Güvenç et al. [19] used the AEO method for the coordination of directional overcurrent relays. The authors compared their results against those achieved using differential evolution and opposition-based chaotic differential evolution algorithms, conclusively establishing the proposed algorithm as a potent and effective method for addressing the coordination problem. Similarly, Abdelhamid et al. [20] preferred the AEO method for the coordination of directional overcurrent relays. Shaheen et al. [21] proposed an AEO algorithm for solving the optimal capacitor placement problem in the electricity distribution network. Moussa et al. [22] employed the AEO algorithm to coordinate efficient power scheduling in smart homes and reported satisfactory results in terms of pricing. Niu et al. [23] calculated the adaptive infinite impulse response (IIR) filter parameters using the AEO algorithm in which dynamic opposite learning and nonlinear adaptive weighting coefficient are included. They also applied their algorithm to IIR-based system identification problems.

Menesy et al. [24] adapted their modified AEO (M-AEO) algorithm to the polymer electrolyte membrane fuel cell model. The proposed AEO-based optimization algorithm consists of an operator to ensure the balance between the exploitation and exploration phases. Sultan et al. [25] focus on the optimal design of a grid-connected renewable energy system consisting of photovoltaic, wind turbine, and fuel cells. The proposed system utilizes a fuel cell with a hydrogen gas tank to store energy in chemical form. The optimal component sizes of this hybrid generation system are derived using the improved AEO optimization technique. Eid et al. [26] utilized their AEO-based algorithm to solve optimal placement problems of distributed generation systems. They tried to achieve a balance between the exploitation and exploration phases in their proposed algorithm. In their enhanced AEO (E-AEO) algorithm, they attempted to strike a balance between the exploitation and exploration phases. They integrated the sine–cosine algorithm with the AEO algorithm and presented its performance; the sine–cosine algorithm is a technique that utilizes a mathematical model based on sine and cosine functions [27]. Similarly, Moussa et al. [28] proposed the use of AEO algorithms for reactive power distribution, which is an important field of study for the stable and safe operation of electric power systems. Elkholy et al. [29] carried out a study in which they aimed to develop an efficient tool based on AEO to calculate the unknown model parameters of the system in photovoltaic power generation. They compared the derived parameters with the experimental results and concluded that the performance of AEO is satisfactory. Some other studies on AEO algorithms calculated the parameters of photovoltaic systems and compared with other optimization algorithms [29,30,31,32,33,34]. Barshandeh et al. [35] proposed a hybrid structure where AEO and HHO algorithms are employed together. They applied the proposed algorithm to various engineering problems and common functions and presented its performance. The AEO algorithm has also been utilized in image processing applications as a hybrid with the deep neural network. Sahlol et al. [36] have presented a new hybrid AEO method for the effective classification of X-ray images using chest radiography and deep learning-based image segmentation techniques for tuberculosis diagnosis. Ewees et al. [37] proposed an AEO-based algorithm on the multi-level thresholding method, which is one of the effective image segmentation techniques. Their approach basically combines the differential evolution and AEO methods.

AEO-based algorithms have also been used in internet of things applications. Duhayyim et al. [38] proposed an AEO-based deep-learning model for sustainable waste management. Hosseini et al. [39] used the sine–cosine algorithm and advanced multi-objective hybrid AEO-based optimization algorithm for botnet detection in IoT. Essa et al. [40] proposed the integration of the random vector functional link (RVFL) network with the AEO algorithm to predict the power consumption and water efficiency of a seawater greenhouse. In another study, AEO algorithms have been used to solve the path-planning problem of unmanned combat aircraft [41]. Rizk-Allah and El-Fergany [42], added a nonlinear weight coefficient to the AEO algorithm. They employed this improved AEO (I-AEO) algorithm to define the model parameters of proton exchange membrane fuel cells. One of the derivatives of AEO algorithms was also applied to solve the transient stability constrained optimal power flow problem [43]. Thieu et al. [44] proposed a hybrid artificial intelligence model with a meta-heuristic algorithm to build a monthly groundwater level prediction model. The augmented AEO-based (A-AEO) multilayer perceptron model is constructed on a traditional MLP network, utilizing levy-flight and Gaussian distribution to enhance its optimization capability.

There are more than 80 meta-heuristic algorithms. However, the pursuit of meta-heuristic algorithms with superior performance is rapidly increasing the number of new and modified version of meta-heuristic algorithms. Meta-heuristic algorithms can be categorized as evolutionary, swarm, physics, human, bio, system, and math based (Fig. 1).

Fig. 1
figure 1

Systematic classification of meta-heuristic algorithms

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Meta-heuristic algorithms, a branch of computer science [48, 49], are designed to tackle large and complex optimization problems, relying on trial-and-error methods. However, the objective of these methods is to approach the solution space with greater effectiveness. This is achieved through the utilization of nature-inspired heuristic or intuitive methods. Likewise, certain machine learning algorithms may encompass various strategies operated with trial-and-error methods [50, 51]. For instance, the reinforcement learning method exhibits a systematic trial-and-error process when searching for solutions [52, 53]. Meta-heuristic algorithms are particularly useful in situations where the equation is known but the optimal values of the parameters are unknown. An illustrative example can be found in the context of designing PID controllers, where numerous studies are devoted to the endeavour of determining the optimal values for the three distinct parameters, namely the proportional gain, integral time, and derivative time, in a commonly known mathematical equation [54,55,56,57].

In order to compare performances of meta-heuristic algorithms in the article, various algorithms, including the AEO algorithm and its variants (O-AEO, I-AEO, E-AEO, M-AEO, A-AEO), as well as the water cycle algorithm (WCA), germinal centre optimization (GCO), grey wolf optimizer (GWO), Harris hawk optimization algorithm (HHOA), and ant lion optimizer (ALO) algorithms have also been considered. In Fig. 1, the selected system-based (green) and swarm-based (orange) algorithms are depicted in different colours. The WCA works by mimicking the natural process of the water cycle, incorporating precipitation, evaporation, and other hydrological phenomena to optimize solution search processes [58]. GCO imitates the competitive and adaptive features observed in the germinal centre reaction within the vertebrate immune system to create an optimization algorithm [59]. The GWO simulates the leadership hierarchy and hunting behaviour of grey wolves, employing hunting, stalking, and encircling strategies to optimize solution search processes [60]. The HHO algorithm is inspired by the hunting behaviour of Harris hawks, utilizing communication, cooperation, and vigilant strategies to optimize solution search processes [61]. The ALO algorithm emulates the trapping behaviour of antlions, employing a combination of random walk and spiral movement to optimize solution search processes [62]. Examining the conducted studies reveals numerous research efforts dedicated to the design of controller parameters using metaheuristic algorithms in the field of control systems. However, limited research has been conducted on deriving transfer functions based on input–output data using meta-heuristic methods. This study contributes to the field of control systems by employing metaheuristic algorithms for system identification through the utilization of the artificial ecosystem algorithm. Our contribution can be summarized as;

  • The AEO algorithms have been employed for the first time in system identification, a subfield of control systems, with the objective of predicting parameters of continuous transfer functions.

  • The efficiency of the AEO algorithms for the system identification problem has been validated through various analyses such as descriptive statistic, nonparametric test, transient response, frequency response, error performance/plot analysis, histogram, and pole zero map. Among the proposed algorithms, the E-AEO algorithm is found to offer better performance, and AEO, in general, can operate with fewer parameters compared to some other meta-heuristic algorithms. Early stopping (ES), function evaluation (FE), time bound (TB), and maximum generations (MG) benchmarks have been presented. The comparison with the differential evolution and opposition-based chaotic differential evolution algorithms has provided clear evidence of the effectiveness and power of the proposed algorithm in addressing the coordination problem.

2 Preliminary Analysis

A PC equipped with an 8-core Intel Core i7-9700 CPU running at 3 GHz, 8 GB of RAM, and a 256 GB SSD was used in acquiring the system model and developing the proposed algorithms. The codes used to predict transfer function parameters were written using Python (3.8) and libraries such as Mealpy 2.1.2 [45], and Control 0.9.2. Jupyter Notebook 3.0.14 were used as integrated development environments (IDEs). The experimental set's data was acquired by utilizing a dataset within MATLAB. These data were derived from the laboratory-scale air stream heater experimental set (Feedback's Process Trainer PT326) shown in Fig. 2. In this experiment set, air passing through a tube is heated at the nozzle, and the air temperature is measured by a thermocouple connected to the tube's outlet. The voltage applied to the inlet heater is denoted as the input reference signal (Volts), while the air temperature, measured by the thermocouple (Volts), is considered the output. The output data contains 1000 discrete measurements from the thermocouple transducer, which converts the temperature value to voltage. The input voltage is sampled randomly from one level to another. The measurement setup has a sampling time of 0.08 s. The model was trained using 80% of the collected data, while the remaining 20% was reserved for the testing phase. Figure 3 illustrates the thermocouple voltage measured at the output in response to the reference voltage applied to the heater input.

Fig. 2
figure 2

Feedback's process experiment set

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Fig. 3
figure 3

Heater input voltage vs. thermocouple output voltage

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3 Methodology

3.1 Artificial Ecosystem Algorithm (O-AEO)

The AEO algorithm was initially proposed by Zhao et al. [9], drawing inspiration from the energy flow within a natural ecosystem. As a concept, an ecosystem describes the environmental relationships between a group of living organisms. This concept is divided into two categories: abiotic and biotic ecosystems. Abiotic ecosystems involve non-living objects such as light, water, and air. However, biotic ecosystems include all living things. AEO is a population-based algorithm that mimics the production, consumption, and decomposition behaviour of organisms. Energy flow and nutrient cycling serve as a crucial balance in maintaining equilibrium in an ecosystem. Living things in an ecosystem are divided into three groups based on their behaviour: producers, consumers, and decomposers. Producers, categorized as green plants, derive their energy from the process of photosynthesis. Consumers are organisms that derive their energy by relying on producers or other consumers (animals). Depending on the type of food, consumers can be categorized into three groups: herbivores which consume only plants, carnivores which feed exclusively on animals, and omnivores which consume both plants and animals. Decomposers, such as bacteria, and fungi play a significant role in converting the remains of an organism's body into molecules. After the decomposition process, the remains are absorbed back into the soil by the producers (plants), and the cycle repeats.

Figure 4 illustrates the energy flow in an example ecosystem. Producers, fed by the sun, are situated at the beginning of the food chain. Consumers encompass a variety of living organisms, making them the most intricate part of the ecosystem. Production is responsible for improving the balance between exploration and exploitation. Consumption is tasked to improve the exploration process, while decomposition is tasked to improve the exploitation process. It is proposed that during the operation of the AEO algorithm, only one producer and one decomposer should be present in each population, while the other individuals should be considered consumers of the three predefined types. However, this proposition may vary depending on the specific requirements and design of the AEO algorithm. The energy level of each individual in the population is determined by the fitness function. The energy flow of the AEO algorithm is depicted in Fig. 5, where producers are highlighted in yellow, whereas decomposers are represented in red. In the algorithm, the worst individual X1 has the highest energy level (producer), while X9 is the best individual (decomposer) with the lowest energy level (fitness value). The other individuals are consumers; X2 and X5 are taken as herbivores, X3 and X7 as omnivores, and X4 and X6 as carnivores.

Fig. 4
figure 4

Representative view of an ecosystem

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Fig. 5
figure 5

Energy flow of AEO algorithm

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3.1.1 Production Process

In AEO, the producer mimics the role of the producer in the ecosystem, which needs the decomposer, sunlight, water, and the food supplied by the decomposer to produce food. In AEO, the producer with the lowest fitness value is updated according to the search limits and the best individual (the decomposer). As a consequence of this process, other individuals in the population will update their positions. Using the production operator, a new individual (producer) is generated between the best one (\(x_n\)) and a randomly generated individual (\(x_{{\text{rand}}}\)), expressed mathematically in Eqs. 1–3.

$$x_1 \left( {t + 1} \right) = \left( {1 - a} \right) \cdot x_n \left( t \right) + a \cdot x_{{\text{rand}}} \left( t \right)$$
(1)
$$a = \left( {1 - t/{\text{max}}_{{\text{iter}}} } \right) \cdot r_1$$
(2)
$$x_{{\text{rand}}} = r \cdot \left( {U - L} \right) + L$$
(3)

The variables in the equations such as \(n,{\text{max}}_{{\text{iter}}} , U, L,\) and \(r_1\). represent the population size, maximum iteration number, upper limits, lower limits, and a random number ranging between [0, 1]. The coefficients \(a\) and \(r\) represent a linear weight coefficient and a random vector in the range [0, 1].

3.1.2 Consumption Process

After the production process is realized by producers, the consumption process begins with consumers. In this process, each consumer can acquire energy from another consumer or producer linked to a lower energy level. Many animals' foraging mechanisms can be mimicked through a mathematical operator called levy-flight. This operator is also employed to improve the exploration phase in many algorithms. The consumption phase of an algorithm with levy-flight can be denoted by Eqs. 4–5.

$$C = \frac{1}{2}\frac{v_1 }{{\left| {v_2 } \right|}}$$
(4)
$$v_1 \ N\left( {0,1} \right), v_2 \ N\left( {0,1} \right)$$
(5)

where \(N (0,1)\) represents a normal distribution with a mean of 0 and a standard deviation of 1. This consumption factor will help each consumer to obtain food using different hunting strategies. If the consumer is chosen as herbivores at random, then only producers will be consumed. In this case, the consumption behaviour of the consumer (herbivore) will be presented mathematically as follows:

$$x_i \left( {t + 1} \right) = x_i \left( t \right) + C \cdot \left( {x_i \left( t \right) - x_1 \left( t \right)} \right), i \epsilon \left[ {2,..,n} \right]$$
(6)

If a consumer is randomly selected as a carnivore, it will eat another consumer with a higher-energy level. In this case, carnivore behaviour can be modelled by Eq. 7.

$$\begin{aligned} x_i \left( {t + 1} \right) & = x_i \left( t \right) + C \cdot \left( {x_i \left( t \right) - x_j \left( t \right)} \right), i \epsilon \left[ {3,..,n} \right] \\ j & = {\text{randi}}\left( {\left[ {2i - 1} \right]} \right) \\ \end{aligned}$$
(7)

If an omnivore is selected within the ecosystem, it can eat a consumer or producer with a high energy level. This behaviour is different from the behaviour of herbivores and carnivores. The mathematical model in this case can be represented in Eq. 8.

$$\begin{aligned} x_i \left( {t + 1} \right) & = x_i \left( t \right) + C \cdot \\ &\left( {r_2 \cdot \left( {x_i \left( t \right) - x_1 \left( t \right)} \right)} \right) + \left( {1 - r_2 } \right)\left( {x_i \left( t \right) - x_j \left( t \right)} \right), i \epsilon \left[ {3,..,n} \right] \\ j & = {\text{randi}}\left( {\left[ {2i - 1} \right]} \right) \\ \end{aligned}$$
(8)

In Eq. 8, r2 is a random number in the interval [0,1]. The updating of the individual positions is based on random selection or the consumption operator of the most unsuccessful individuals, respectively. In this way, it is possible to search for the global point.

3.1.3 Decomposition Process

This process is essential for AEO algorithms. Decomposers chemically break down deceased producers or consumers, releasing nutrients that are vital for the growth of other producers. In this case, the equation can be stated as Eqs. 9-12

$$x_i \left( {t + 1} \right) = x_n \left( t \right) + D \cdot \left( {e \cdot x_n \left( t \right) - h \cdot x_i \left( t \right)} \right), i \epsilon \left[ {1,..,n} \right]$$
(9)
$$D = 3 \cdot u, u\ N\left( {0,1} \right)$$
(10)
$$e = r_3 \cdot {\text{randi}}\left( {\left[ {1\,2} \right]} \right) - 1$$
(11)
$$h = 2 \cdot r_3 - 1$$
(12)

In Eq. 9, the decomposition factor \(D\) and the weight coefficients \(e\) and \(h\) are designed for mathematical modelling of decomposition process. The position represented by \(x_i\) of the ith individual can be determined based on the parameters \(D, e,\) and \(h\), using the position of \(x_n\).

AEO initiates the optimization process by randomly generating a population. At each iteration, the initial search individual updates its position in accordance with Eq. (1), whereas the other individuals have an equal probability of updating their positions using Eqs. (6), (7), or (8). If the calculated function value for the individual is better than the previous one, it is accepted. After this process, the position of each individual is updated according to Eq. 9. The update process in the AEO algorithm continues until the specified maximum number of epochs is reached or until termination criteria are met [9]. The pseudocode for the AEO algorithm is presented in Fig. 6.

Fig. 6
figure 6

Pseudocode of AEO algorithm

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3.2 Modified AEO Algorithm

For population-based meta-heuristic algorithms, exploration and exploitation are two important and conflicting phases. A well-defined balance between these two phases is crucial for narrowing the search space and solving the global optimum. The modified version of AEO algorithm by Menesy et al. [24] was used to the Polymer electrolyte membrane fuel cell model. In this study, the \(H\) operator, which is proposed to improve the performance, is represented in Eq. 13 and decreases linearly from 2 to 0 during the iteration time.

$$H=2\times \left(1-t/{{\text{max}}}_{{\text{iter}}}\right)$$
(13)

In Eq. (13),\({{\text{max}}}_{{\text{iter}}}\) indicates the maximum iteration number. After the addition of operator H to Eqs. 6–8, the new equations are represented as shown in Eqs. 14–16.

$$\begin{aligned} & x_i \left( {t + 1} \right) = x_i \left( t \right) + H \cdot C \cdot \\ & \quad \left( {x_i \left( t \right) - x_1 \left( t \right)} \right), i \epsilon \left[ {2,..,n} \right] \end{aligned}$$
(14)
$$\begin{aligned} x_i \left( {t + 1} \right) = & x_i \left( t \right) + H \cdot C \cdot \left( {x_i \left( t \right) - x_j \left( t \right)} \right), i \epsilon \left[ {3,..,n} \right] \\ j = & {\text{randi}}\left( {\left[ {2i - 1} \right]} \right) \\ \end{aligned}$$
(15)
$$\begin{aligned} x_i \left( {t + 1} \right) = & x_i \left( t \right) + H \cdot C \cdot\\ & \left( {r_2 \cdot \left( {x_i \left( t \right) - x_1 \left( t \right)} \right)} \right) + \left( {1 - r_2 } \right)\left( {x_i \left( t \right) - x_j \left( t \right)} \right), i \epsilon \left[ {3,..,n} \right] \\ j = & {\text{randi}}\left( {\left[ {2i - 1} \right]} \right) \\ \end{aligned}$$
(16)

3.3 Enhanced AEO Algorithm

Similar to the modified AEO, the enhanced AEO (E-AEO) algorithm proposed by Eid et al. [26], the E-AEO algorithm uses the G operator to balance between exploration and exploitation.

$$G=2\times \left(1-t/{{\text{max}}}_{{\text{iter}}}\right)$$
(17)

In Eq. 17, \({{\text{max}}}_{{\text{iter}}}\) indicates max. iteration number and \(t\) is the current iteration. In the production phase, this parameter was added to Eq. (6) and new Eq. 18 was formed as follows.

$${x}_{2}\left(t+1\right)={x}_{2}\left(t\right)+G\bullet C\bullet \left({x}_{2}\left(t\right)-{x}_{1}\left(t\right)\right), i \epsilon \left[2,..,n\right]$$
(18)

A sine–cosine based algorithm is also proposed in E-AEO algorithm. Equations 6–8 were rearranged as Eqs. 19–21 after the addition of these terms.

$$\left\{ {\begin{array}{l} {{\text{for}}\,r \le \frac{1}{3}\,{\text{and}}\,r_4 \le 0.5} \hfill \\ \qquad x_i \left( {t + 1} \right) = x_i \left( t \right) + \sin \left( {r_3 } \right) \times C \times \left[ {x_i \left( {t_3 } \right) - x_1 \left( t \right)} \right] \\ {{\text{for}}\, r_4 >0,5} \\ \qquad x_i \left( {t + 1} \right) = x_i \left( t \right) + \cos \left( {r_3 } \right) \times C \times \left[ {x_i \left( {t_3 } \right) - x_1 \left( t \right)} \right] \\ \end{array} } \right.$$
(19)
$$\left\{ \begin{array}{*{20}l} {{\text{for}}\,r < 1/3 < r \le 2/3 \,{\text{and}}\, r_4 \le 0.5} \\ \qquad x_i \left( {t + 1} \right) = x_i \left( t \right) + \sin \left( {r_3 } \right) \times C \times X_{ij} (t)\\{{\text{for}}\, r_4 > 0,5} \\ \qquad x_i \left( {t + 1} \right) = x_i \left( t \right) + \cos \left( {r_3 } \right) \times C \times X_{ij} \left( t \right), \\ \qquad j = {\text{randi}}\left( {\left[ {2i - 1} \right], X_{ij} \left( t \right) = x_i \left( t \right) + x_j \left( t \right)} \right) \\ \end{array} \right.$$
(20)
$$\left\{ \begin{array}{*{20}l} {\text{for}}\,r < 2/3 < r \le 1 \,{\text{and}}\, r_4 \le 0.5 \\ \qquad x_i \left( {t + 1} \right) = x_i \left( t \right) + \sin \left( {r_5 } \right) \times C[r_5 X_{i1} \left( t \right) + \left( {1 - r_5 } \right)X_{ij} (t)]\\ {\text{for}}\, r_4 > 0,5 \hfill \\ \qquad x_i \left( {t + 1} \right) = x_i \left( t \right) + \cos \left( {r_5 } \right) \times C [r_5 X_{i1} \left( t \right) + \left( {1 - r_5 } \right)X_{ij} ]\\ { j = {\text{randi}}\left( {\left[ {2i - 1} \right]} \right), X_{i1} \left( t \right) = x_i \left( t \right) + x_1 \left( t \right)} \\ \end{array} \right.$$
(21)

In Eqs. 19–21, \({r}_{3}, {r}_{4}, {r}_{5}\) are expressed as \({r}_{3}=2 \pi \times {\text{rand}}\), \({r}_{4}\)(random value between 0 and 1) and \({r}_{5}={\text{rand}}\).

3.4 Improved AEO Algorithm

In most of the meta-heuristic algorithms, there is a weighting coefficient to obtain the appropriate result in the search space during the exploration and exploitation phase. In most algorithms, this weighting algorithm is in linear form. In Rizk-Allah and El-Fergany [42] study, added a nonlinear adaptive weighting coefficient. Accordingly, the adaptive weight (\(a\)) coefficient is as follows:

$$a={r}_{1}(1-\frac{1}{{{\text{cos}}}^{-1}\left(0\right)}{({\text{cos}}}^{-1}(1-\frac{t}{{{\text{max}}}_{{\text{iter}}}})))$$
(22)

Equation 22 is actually a derived version of Eq. 2. Additionally, a dynamic crossover strategy has been added to the improved AEO (I-AEO) algorithm. According to this strategy, the improved equations are presented as follows:

$$CL=\left\{\begin{array}{c}\beta \bullet {x}_{R}+\left(1-\beta \right)\bullet {x}_{1} {\text{rand}}<0.5\\ \beta \bullet {x}_{1}+\left(1-\beta \right)\bullet {x}_{R} {\text{rand}}\ge 0.5\end{array}\right.$$
(23)
$$\beta ={a}^{{\text{max}}}-{(a}^{{\text{max}}}-{a}^{{\text{min}}})\frac{t}{T}$$
(24)

Rizk-Allah and El-Fergany [42], who developed the improved version of AEO algorithm, also added the Gaussian strategy to the AEO algorithm to enhance its performance. Gaussian mutation strategy is proposed in some of the meta-heuristic algorithms to improve exploitation capability. In this context, the density function of the Gaussian strategy is given Eq. 25.

$${f}_{{\text{gauss}}\left(0,{\sigma }^{2}\right)}(a)=\frac{1}{\sqrt{2\pi {\sigma }^{2}}}{e}^{\frac{-{a}^{2}}{{2a}^{2}}}$$
(25)

where \({\sigma }^{2}\) is defined variance between each individual.

3.5 Augmented AEO Algorithm (A-AEO)

Zhao et al. [9] initially suggest that the AEO algorithm has a strong global search capability; however, its convergence rate diminishes as it approaches the optimal solution due to a higher rate employed in the decomposition phase compared to the rate used in the exploitation phase. Therefore, Nguyen et al. [44] proposed the augmented AEO (A-AEO) algorithm to enhance the exploration (global search) and exploitation (local search) capabilities of the AEO algorithm by using levy-flight and Gaussian (normal) randomization methods together. In the updated consumption phase, the selected agent has a 50% chance of updating its location based on herbivore, omnivore, or carnivore organisms, and a 50% chance of using the proposed levy-flight method. The idea is to use more than one method for a better search in the exploration phase. The consumption phase is presented in Eq. 26.

$$x_i^{g + 1} = x_i^g + {\text{step}}^{g + 1} \cdot \left( {x_i^g - x_{{\text{best}}} } \right)$$
(26)

\(ste{p}^{g+1}\) is generated by the levy distribution with a randomly generated β value in the interval (1,2). In the proposed method, the update process has been revised to enhance the stability of the exploitation phase. In this stage, E factor was replaced by a D-dimensional Gaussian distribution vector. The update used in the algorithm is proposed in Eq. 27. In case the randomly generated value is greater than 0.5, Eq. 28 is taken into account.

$$x_i^{g + 1} = x_{{\text{best}}} + {\text{normal}}(0,1,D) \otimes \left( {x_{{\text{best}}} - x_i^g } \right)$$
(27)
$$x_{g + 1}^i = x_{{\text{best}}} + {\text{step}}^{g + 1} \cdot \left( {x_{{\text{best}}} - x_g^i } \right)$$
(28)

where ⊗ is defined as element-wise multiplication.

3.6 System Identification

Black-box modelling is characterized by the iterative process of estimating the coefficients of a proposed mathematical model, with the ultimate aim of identifying the optimal solution through the evaluation of performance results. Initially, a simple linear model is proposed, and if the performance is not sufficient, improvement is achieved by suggesting more complex models [5,6,7]. Equation 29 is a general expression of the black-box model.

$$y\left(k\right)= G\left(q\right)u\left(k\right)+ H\left(q\right)e\left(k\right)$$
(29)

In Eq. 29, y(k), u(k), and e(k) are defined as model output, input, and white noise, respectively. G(q) is a transfer function representing the system and H(q) is the disturbance effect. First, the input/output data of the system are collected, some filtering operations are applied, and an appropriate model is selected.

3.6.1 Transfer function model

Transfer functions are defined as a polynomial model that gives the relationship between input reference and output signal. In transfer functions, the degree of the model is equal to the degree of the denominator. The roots of the denominator are called model poles, and the roots of the numerator are called zeros [8]. A transfer function in continuous time is denoted in Eq. 30.

$${Y}_{\left(s\right)}= \frac{{\text{num}}\left(s\right)}{{\text{den}}\left(s\right)} {U}_{\left(s\right)}+ {e}_{\left(s\right)}$$
(30)

Here Y(s), U(s), and e(s) are defined as output, input, and disturbance, respectively. The \({\text{num}}(s)\) and \({\text{den}}(s)\) given in Eq. 30 are expressed as numerators and denominators describing the relationship between input and output.

3.7 Error Functions

Methods such as integral of square error (ISE), integral of the absolute value of error (IAE), integral of time-weighted squared error (ITSE), and integral of time multiply absolute error (ITAE) are available to determine system parameters and examine their performance [46]. ISE expresses the sum of the squares of the differences between the desired signal generated by a particular control system and the actual system output over time. A lower ISE value indicates better performance of the control system. IAE represents the sum of the absolute values of the differences between the reference signal and the actual system output. Unlike the squared error-based metrics, IAE directly considers the magnitude of the error without squaring it. ITSE involves summing the square of the error at each time step, where the error is typically the difference between the desired reference signal and the actual system output. ITAE is an integral-based metric that evaluates control system performance by considering the product of the absolute error and time at each step. Unlike metrics that solely focus on the absolute error (such as IAE) or squared error (such as ISE), ITAE combines the magnitude of the error with the time duration of the error [8]. The Integral of Time multiplied by Absolute Error (ITAE) serves as a metric in control systems, measuring the accumulated error over time. For this reason, the ITAE metric is proposed to measure the error. Table 1 presents the names and formulas of commonly known error objective functions.

Table 1 Common error objective function
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4 Results and Discussions

The performance of the selected AEO algorithms was comprehensively evaluated through comparative analyses with five different algorithms: water cycle algorithm (WCA), germinal centre optimization (GCO), grey wolf optimizer (GWO), Harris hawk optimization algorithm (HHOA), and ant lion optimizer (ALO).The selected algorithms have been used to predict the four parameters (z1, z2, p1, and p2) of the transfer function, with the lower and upper boundary set as [-100, 0, 0, 0] and [100, 100, 100, 20], respectively. Predefined parameters and the classification regarding the algorithms are presented in Table 2.

Table 2 Parameters of meta-heuristic algorithms
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4.1 Reliability Analysis and Constraints

Although many studies focusing on performance have not paid much attention to stopping criteria, the appropriate choice of stopping criteria is critical for system model parameter estimation. Especially for embedded hardware with constrained resources, it is very important to reach a solution in minimum time [47]. In this study, besides analysing the performance of meta-heuristic algorithms, the results of different stopping criteria are also compared.

4.1.1 Time Bound Constraint Performance

The meta-heuristic algorithms were tested by setting a maximum time bound of 20 s (TB = 20). In order to test the reliability of the algorithms, the independent run number for each algorithm was set to 100. The R2 performance of meta-heuristic algorithm under TB constraint is presented in Fig. 7. This concept implies finding the optimal parameters 100 cycle under the specified criteria. In this case, the success of the algorithms in making accurate predictions can be observed in Fig. 7. However, in cases where the performances are closely matched, it is necessary to conduct descriptive statistical analysis. Descriptive statistics results under TB constraint are presented in Table 3. The mean, minimum, maximum, and standard deviation values are presented. Furthermore, the number of runs with a performance (R2) less than 0.9 after 100 runs is also indicated. The designated value of 0.9 provides insights into the dependability of the algorithms, and its significance lies in the case of it being 0. The E-AEO algorithm achieved the highest R2 value of 0.9698, with an average R2 value of 0.9502. Additionally, the O-AEO (0.9509) and M-AEO (0.9510) algorithms also exhibited satisfied performance.

Fig. 7
figure 7

R2 performance of meta-heuristic algorithms under TB constraints

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Table 3 Descriptive statistics under TB constraints
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4.1.2 Maximum Generation Constraint Performance

The R2 values calculated considering the maximum number of generations (MG = 20) are plotted in Fig. 8. Table 4 presents descriptive statistical results under the MG constraint. The GWO algorithm, which has exhibited the highest performance and has never fallen below the 0.9 threshold, has shown successful performance (0.95). However, the I-AEO algorithm has reached a highest R2 value (0.9582) but has a value that falls below 0.9. M-AEO and E-AEO have high average performance values, but have performances fall below 0.9.

Fig. 8
figure 8

R2 performance of meta-heuristic algorithms under MG constraints

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Table 4 Descriptive statistics under MG constraints
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4.1.3 Function Evaluation Constraint Performance

The performance results were analysed based on 4000 iterations of the algorithms, taking into account the constraint of function evaluations (FE). Figure 9 demonstrates the performance variations of various algorithms under FE constraints.

Fig. 9
figure 9

R2 performance of meta-heuristic algorithms under FE Constraints

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Table 5 presents descriptive statistical results under the FE constraint. Considering the criteria stated in the previous section, the O-AEO algorithm has achieved the highest performance (0.9504). Moreover, the results obtained from the M-AEO and GWO algorithm independent run demonstrate notable proficiency. The ALO algorithm has the lowest mean R2 values in this comparison.

Table 5 Descriptive statistics under FE constraints
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4.1.4 Early Stopping Constraint

The early stopping (ES = 3) criterion can be defined as the end of the search process after the minimum calculated error value of the meta-heuristic algorithms remains unchanged for 3 cycles. This criterion gives the algorithms a long time to search for a solution as demonstrated in Fig. 10. Therefore, the meta-heuristic that try to calculate the optimal parameter without getting stuck. Figure 10. shows that the E-AEO algorithm was able to determine the global solution point every time after 100 iterations. Table 6 presents the average, minimum, maximum, count number (< 0.9), and standard deviation value of the algorithms considered in the paper when the ES criterion is taken into account. E-

Fig. 10
figure 10

R2 performance of meta-heuristic algorithms under ES constraint

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Table 6 Descriptive statistics under ES constraints
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AEO algorithm achieves the highest performance value (0.954) in the ES constraint when considered together with Table 6 and the Wilcoxon test presented in Table 7. When Table 6 and Table 7 are evaluated together, the E-AEO algorithm attains the highest performance value (0.954) in terms of the ES criterion. The O-AEO (0.951), A-AEO (0.9426), and WCA (0.9460) algorithms also demonstrate high performance values. However, the E-AEO algorithm has surpassed other methods in the Wilcoxon test.

Table 7 Wilcoxon signed-rank test E-AEO versus other meta-heuristic algorithm results under different type constraints
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4.2 Nonparametric test analysis

Analyses conducted using the nonparametric statistical method, the Wilcoxon test, are presented in Table 7. The Wilcoxon signed-rank test is a nonparametric test used to determine whether there is a significant difference between the means of two paired samples [63]. In this study, the Wilcoxon test was employed to compare algorithms based on their p-value values and determine the winner. Comparison between the E-AEO algorithm and other algorithms has been considered at a significance level of 5%. The calculated p-values for the considered TB, MG, FE, and ES constraints are provided in Table 7. Additionally, at the end of the Wilcoxon test, the winning algorithms are indicated by a positive sign ( +). It is evident that the E-AEO algorithm has demonstrated superior performance compared to other algorithms under the ES criterion. Moreover, under the TB and MG constraints, it outperformed the other 7 algorithms, and under the FE constraint, it outperformed the other 8 algorithms.

4.3 Variation Analysis of Model Parameters

While developing the model of the system, a transfer function is proposed with 2 coefficients in the numerator and denominator. For simplicity of representation, the numerator of the transfer function was chosen as a first-order term, while the denominator was selected as a second-order term for the system. The proposed second-order transfer function is presented in Eq. 31.

$${T}_{f}\left(s\right)=\frac{{{\text{z}}}_{2}s+{{\text{z}}}_{1}}{{{\text{s}}}^{2}+{p}_{2}{\text{s}}+{{\text{p}}}_{1}}$$
(31)

Considering the ES criterion of the E-AEO algorithm, the z1, z2, p1, and p2 values calculated over 100 iterations are given in Fig. 11. If Fig. 11 is analysed, the parameters z1, z2, p1, p2 vary between 12.58–12.605, −1.497–1.492, 12.92–12.89, 6.1–6.122, respectively.

Fig. 11
figure 11

Parameter variation of z1, z2, p1, and p2 obtained with E-AEO under ES constraints

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Figure 11 shows that the parameter variations remain nearly constrained throughout the iterations. The negative value of z2 indicates that there is a zero in the positive region on the root locus. If a pole root were in the positive region on the root locus, this would obviously result in instability, but having the zero in the positive region does not cause instability and provides a better representation of the model. Figure 12 shows the pole-zero placement on the root locus plot. Green colour indicates zero and red colour indicate pole placement on root locus. Most of the poles coloured in red are overlapped. Similarly, the green circles shown on the right positive side of the root-locus represent zeros. These overlapping parameters show the similarity of the estimates at a glance. Thus, the pz-map can be presented as evidence that the variation of the parameters is constrained. In conclusion, it can be stated that the E-AEO algorithm is both successful and stable in solving system identification problems. This success can be interpreted that a sine–cosine-based structure is more suitable for solving system identification problems.

Fig. 12
figure 12

Pole-zero placement of E-AEO algorithm on root-locus under ES constraints

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4.4 Comparison of Performances Indices

In Sect. 4.2, the number of independent runs was set to 100 to showcase that the algorithms yielded non-random outcomes and to evaluate their performance. In this section, the number of independent runs was set to 1. This concept denotes the execution of algorithms just once cycle to determine the optimal parameters. This allows for conducting the analyses presented in the subsections for the obtained optimal parameters. Table 8 shows the parameters calculated after the MG limit is set to 20. Under the MG constraint, the O-AEO (0.95092), E-AEO (0.95092), I-AEO (0.95099), A-AEO (0.95045), and GWO (0.94927) algorithms have exhibited high performances. In this case, the AEO algorithms except M-AEO have performed more successful optimization compared to the other presented algorithms. The parameter of transfer functions calculated with the AEO algorithms, excluding M-AEO, are very close to each other and the performance indicators MAPE, MAE, MSE, R2 and time are also very close to each other. The ALO (50.08 s) algorithm has exerted more effort for the solution compared to other algorithms.

Table 8 Performance indices of meta-heuristic algorithm under MG constraints
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The ES constraint ensures that the algorithm is stopped for 3 generations once there is no further decrease in the global best solution. Table 9 shows the optimized transfer functions MAPE, MAE, MSE, R2 and time values for meta-heuristic algorithms. When the ES criterion value is set to 3, the performance of all AEO algorithms is quite good, but the solution times increase considerably. Especially, E-AEO has reached the highest performance value (0.95105) while having a solution time of 59.07 s. The performance of M-AEO (0.95083) and A-AEO (095047) algorithms is also high; however, their respective solution times of 218.79 s. and 11.725 s. are much longer compared to E-AEO.

Table 9 Performance indices of meta-heuristic algorithm under ES constraints
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In the FE constraint criterion, the running algorithm is halted when it reaches the given maximum function number. The termination criterion used in this case closely resembles the MG stopping criterion. Throughout the study, a maximum constraint value of 4000 was set for fitness evaluations (FE), taking into account the dataset's distinct characteristics. Table 10 presents the performance indicators calculated under these specific conditions, illustrating the optimized transfer functions and performance metrics achieved through the utilization of meta-heuristic algorithms. Under the FE criterion, the E-AEO algorithm has achieved the highest performance (0.95117) with a solution time of 23.17 s. The M-AEO (0.80411), GCO (0.76997), and ALO (0.72665) algorithms have the lowest performance. The other algorithms perform adequately. The FE constraint enables the algorithms to solve within similar time bounds. The high performance of four different AEO algorithms under the FE criterion is important in demonstrating the effectiveness of the AEO.

Table 10 Performance indices of meta-heuristic algorithm under FE constraints
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The TB constraint is essentially based on stopping the algorithm at the end of the given time, regardless of the performance result. In this study, considering the characteristics of the data and the structure of the algorithms, the time limit was set to 20 s. In this case, the transfer functions and performance values are presented in Table 11. In accordance with Table 11, the I-AEO algorithm has demonstrated the best performance (0.95123). The algorithm with the lowest performance (0.67424) has been identified as ALO. Among the AEO algorithms, M-AEO has the lowest performance (0.72221).

Table 11 Performance indices of meta-heuristic algorithm under TB constraints
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The algorithms considered in problem-solving have been examined in terms of performance, solution times, and variations in transfer function parameters. Upon examining the presented statistical results, graphs, Wilcoxon test, and performance criteria, it is evident that AEO algorithms are quite sufficient in solving system identification problems. Among the AEO algorithms, particularly the E-AEO algorithm has stood out compared to other algorithms. Furthermore, for illustrative purposes, the changes in exploration and exploitation are also addressed in Fig. 13. The exploration and exploitation change of the E-AEO algorithm under the TB constraint is presented in Fig. 13a. Also, Fig. 13b shows the global best change, which indicates that the solution is almost found after approximately the 13th iteration.

Fig. 13
figure 13

E-AEO under TB constraints a) exploration/exploitation b) global best variations

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4.5 Time and Frequency Response Analysis

The time and frequency response of the optimized transfer functions is important for making an interpretation on behalf of the system. At this point, based on the TB constraint, the system response of the relevant transfer functions to the step input is presented in Fig. 14. In Fig. 14, O-AEO, I-AEO, E-AEO, and M-AEO algorithms have almost the same time response. It is difficult to comprehend the transient response performance through Fig. 14; therefore, Table 12 has been provided. In this table, considering the transient response of the optimized transfer functions, it can be observed that although there are slight differences in the rise times, the settling times of the AEO algorithms are rapid. Among the AEO algorithms, the E-AEO algorithm has the lowest settling time (1.30 s.) and peak time (1.71 s.).

Fig. 14
figure 14

Step response of optimized transfer functions with meta-heuristic algorithms under TB constraints

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Table 12 Transient responses of optimized transfer functions with meta-heuristic algorithms under TB constraints
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Table 12 presents the transient values such as rise time, settling time, overshoot, undershoot, peak value, and peak time related to the transfer functions derived using meta-heuristic algorithms. Figure 15 shows the frequency response (bode graph) of the transfer functions estimated by meta-heuristic algorithms. The bode diagram helps determine the system's gain and phase margins, which are key indicators of stability. If the gain or phase margin is too low, it suggests that the system may become unstable or exhibit undesired oscillations.

Fig. 15
figure 15

Frequency response of transfer functions generated by meta-heuristic algorithms under TB constraints

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If the frequency response is carefully observed, all algorithms, except the GCO algorithm, exhibit a similar frequency amplitude and phase, and the transfer functions obtained using meta-heuristic algorithms demonstrate stable behaviour. In Fig. 16, the error between the predicted transfer function response for each meta-heuristic algorithm and the original output data is presented. This figure indicates that the errors are within a constrained range. The main reason for the variability in error values in Fig. 16 is the variations in the original dataset. The error values being close to 0 indicate the prediction of a strong model.

Fig. 16
figure 16

Error plot of optimized transfer functions generated by meta-heuristic algorithms under TB constraints

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Figure 17 presents a histogram of the errors. In the histogram figure, the narrow range of error bars indicates that the performances are satisfactory. Drawing conclusions based on error plots is a relatively challenging task. Therefore, the histogram of errors is essential for observing the errors of the obtained transfer functions. The histogram of O-AEO, E-AEO, M-AEO, I-AEO, and GWO algorithms is found within a narrow range. Algorithms observed within a narrow range in the histogram have less error compared to other algorithms.

Fig. 17
figure 17

Error histogram of transfer functions generated by AEO algorithms under TB constraints

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Figure 18 presents the voltage amplitude versus time of the air stream dryer and the predicted transfer functions. The transfer functions produced by the WCA, GWO, GCO and M-AEO algorithms track the original data with a larger error while the O-AEO, I-AEO, E-AEO, and A-AEO successfully track the data.

Fig. 18
figure 18

Tracking ability of optimized transfer functions generated by meta-heuristic algorithms under TB constraints

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The presented graphs depict the comparison of meta-heuristic algorithms and demonstrate their performance under different types of constraints. To test the problem-solving abilities of the AEO algorithms, the reliability of the meta-heuristic algorithms was analysed by checking their R2 values after independent run number for each algorithm was set to 100. Additionally, after setting the independent run number to 1, optimized transfer functions were presented, and performance indicators such as transient response and frequency response were examined. Several analyses have demonstrated that the E-AEO algorithm has achieved higher success compared to other AEO and meta-heuristic algorithms. In the Wilcoxon test, the E-AEO algorithm has outperformed other algorithms under the ES criterion. This success demonstrates that meta-heuristic algorithms especially AEO algorithm can be easily applied to system identification problems.

5 Conclusion

The primary focus of this article is a research investigation that utilizes artificial ecosystem, water cycle, germinal centre, grey wolf, and Harris hawk optimizer algorithms for black-box system identification. The study involves processing reference signal and outlet temperature data obtained from the air stream heater. Algorithms have been compared under four different constraints (time bound, maximum generation, function evaluation, and early stopping constraints). Furthermore, reliability tests were conducted by examining descriptive statistics, the Wilcoxon nonparametric test, transient/frequency responses, and error performances.

It was shown that artificial ecosystem, and 5 state-of-the art optimization algorithms can be easily applied in system identification problems. Artificial ecosystem algorithms have exhibited more successful performance compared to other meta-heuristic algorithms. Under the early stopping criterion, the enhanced artificial eco system algorithm outperformed other algorithms. The mean R2 performances of the time bound, maximum generation, function evaluation, and early stopping constraints are 0.9502, 0.9425, 0.9466 and 0.954, respectively. When the parameter variation and pole-zero map of the transfer functions optimized with the artificial ecosystem algorithm under the early stopping criterion were examined, it was observed that stable and similar solutions were generated in each iteration. The algorithms were compared through the Wilcoxon test, and the E-AEO algorithm demonstrated superior performance based on the obtained p-values.