FOMCON Toolbox-Based Direct Approximation of Fractional Order Systems Using Gaze Cues Learning-Based Grey Wolf Optimizer

Abstract

This study discusses a new method for the fractional-order system reduction. It offers an adaptable framework for approximating various fractional-order systems (FOSs), including commensurate and non-commensurate. The fractional-order modeling and control (FOMCON) toolbox in MATLAB and the gaze cues learning-based grey wolf optimizer (GGWO) technique form the basis of the recommended method. The fundamental advantage of the offered method is that it does not need intermediate steps, a mathematical substitution, or an operator-based approximation for the order reduction of a commensurate and non-commensurate FOS. The cost function is set up so that the sum of the integral squared differences in step responses and the root mean squared differences in Bode magnitude plots between the original FOS and the reduced models is as tiny as possible. Two case studies support the suggested method. The simulation results show that the reduced approximations constructed using the methodology under consideration have step and Bode responses more in line with the actual FOS. The effectiveness of the advocated strategy is further shown by contrasting several performance metrics with some of the contemporary approaches disseminated in academic journals.

1. Introduction

Fractional-order (FO) system models make it possible to describe real-world systems more accurately, and FO modeling has been used to solve several scientific and technical issues more reliably [1,2]. For example, FO differential equations are used for thermal dynamics of buildings [3], uncertain systems [4,5], batteries, fuel cells, supercapacitors [6], life sciences problems [7], robotics [8], and control design [9,10,11,12]. For about two decades, control engineering has significantly profited from the benefits of FO modeling, improving control capability and producing highly accurate plant modeling [13]. While fractional calculus promises to enhance the efficiency of modeling and control systems significantly, the optimum enactment of FO transfer function (FOTF) models involves far greater computing costs than integer-order (IO) models. The fundamental cause is that the FO derivative operator is nonlocal and needs all previous values of a function [13,14]. Consequently, the memory required to retain all past values of signals or functions increases. It drastically increases the computational load on calculating FO components [13,14,15]. Reduced IO models were developed to address these issues and enable using FO components with a tolerably low computational burden [16]. These models can simulate the reactions of FO elements within specific operating ranges. Numerous other numerical methods that offer an IO transfer function are usually used.

Both continuous and discrete models are commonly used to describe these techniques in the time and frequency domains [14,15,16]. The IO approximation approaches include continued fraction expansion (CFE), and Chareff, Matsuda, Carlson, and Oustaloup approximations [17,18,19,20,21]. In addition to these techniques, stability boundary locus (SBL) fitting, modified SBL, and time reaction-based curve fitting have recently been utilized [22]. IO approximation transfer functions of higher order are generated using these techniques for greater accuracy [2,16,23,24]. The study of control systems does not favor high-order transfer function calculations because they might result in computational errors. Model order reduction (MOR) is a technique used in system theory that simplifies a vast, complicated structure into one with fewer variable parameters [25]. This operational challenge is crucial to understanding the system dynamics better, minimizing the computational load of simulation, and enhancing the design of controllers. While MOR has been used for decades in IO systems, it has only recently been applied to FO models. Most relevant published works describe two distinct FOS types: commensurate and non-commensurate FO. The fractional powers of integrators and differentiators, which characterize the class of systems known as commensurate-type FOSs, are multiples of real numbers. As a result, it is possible to convert this type of system into an IO system by using straightforward assumptions. Fractional terms in non-commensurate FOSs are not always multiples of real numbers. FOSs with non-integer terms and nonrational numbers may be discovered in nature.

In the literature, only some authors covered reduced-order approximation techniques for FOSs. Conventional methods, usually established for IO systems approximation, were used by Tavakoli-Kakhki et al. [26] for MOR of comparable FO linear systems. This category includes direct truncation, Padé approximation, rational interpolation, shifted Padé approximation, and singular perturbation. Khanra et al. [27] proposed a unique two-phase mitigation strategy established on time moment and Markov parameters approximation to reduce the complexity of multi-input multi-output FOSs. Jagodziński and Lampasiak [28] used the Padé approximation method to approximate FOS. Gao [29] introduced a reliable method of reduction that uses the asymmetric Lanczos technique. Jiang et al. [30] reduced FOSs using Arnoldi’s method. Rydel et al. [31] demonstrated how balanced truncation and singular perturbation may be used to construct reduced-order discrete-time commensurate FOSs. Sarkar et al. [32] adopted the approach of approximate frequency fitting to get a close approximation of FOSs in the delta realm. Stanisawski et al. [33] created a balanced truncation reduction approach based on SVD for discrete-time FO state-space-characterized SISO systems. Garappa et al. [34] explored an order reduction strategy for linear time-invariant (LTI) FOSs established on Krylov subspaces. Verma et al. [35] used balanced truncation, Hankel norm approximation, and singular perturbation to approximate non-commensurate FOSs. Badri and Sojoodi [36] examined the robust stability of interval FOSs with time-varying delays and performed a stabilization analysis. Al-Saggaf et al. [37] designed an FO active disturbance rejection controller using Bode’s ideal transfer function and an IO filter. Various optimization strategies and methodologies have emerged recently to get a close approximation of the FOTF. Gao and Liao [38] used particle swarm optimization to obtain a rational approximation of FOSs. Colliding bodies optimization was used by Mahata et al. [39] to construct FO low-pass Butterworth filters for IO rational approximations that met an exact magnitude response.

Table 1. Overview of FOSs reduction techniques in the recent literature.

Author, Year	Reduction Method	Remarks
Ali Yüce [13]	Equilibrium optimization algorithm	(i). The non-commensurate FOSs are directly reduced to integer models without employing operator-based approximation strategies. (ii). The lowering of orders in FOSs with commensurate orders was not covered. (iii). The dynamics of FOSs are approximated with 5th-order IO models.
Soloklo and Bigdeli [16]	Genetic algorithm	(i). Directly obtains the reduced integer/fractional-order models for commensurate and non-commensurate FOSs. (ii). A weighted cost function is minimized to get reduced-order models for SISO and MIMO FOSs. As an additional constraint, the Routh–Hurwitz stability criteria were considered by this technique.
Krajewski and Viaro [40]	Rational L2 approximation	(i). The commensurate and non-commensurate FO state-space systems are approximated using integer-order state-space models. (ii). Requires the Oustaloup approximation to obtain an equal IO system. (iii). The iterative interpolation algorithm obtains the optimal reduced model by minimizing the L2 norm.
Duddeti [41]	Modified balanced truncation method	(i). It reduced the order of commensurate and non-commensurate FOSs. (ii). Obtaining the IO equivalent models requires simple mathematical substitution and Oustaloup approximation. (iii). It requires performing an inverse substitution to get the reduced commensurate FO model. (iv). It may fail to create optimum reduced models.
Saxena et al. [42]	Time moment matching and Big Bang Big Crunch optimization algorithm (BBBCOA)	(i). A mathematical substitution is required to convert the commensurate FOS into IO. The time moment matching approach generates the denominator, and the BBBCOA calculates the numerator. (ii). It requires an inverse substitution to convert the reduced IO model into the FO model’s commensurate form. (iii). It does not discuss non-commensurate order FOSs order reduction.
Bourouba et al. [43]	Differential evaluation algorithm (DEA)	(i). Equivalent 5th-order IO models directly approximate the non-commensurate FOSs. (ii). It does not talk about commensurate order FOSs order reduction.
Jain and Hote [44]	BBBCOA and particle swarm optimization (PSO) algorithm	(i). Non-commensurate FOSs are directly approximated with reduced IO models using the hybrid BBBCOA and PSO algorithms. (ii). It does not talk about commensurate order FOSs order reduction.
Jain et al. [45]	BBBCOA	(i). A mathematical substitution is required to convert the commensurate FOS into IO. The BBBCOA obtains the reduced IO model. (ii). It requires an inverse substitution to convert the reduced IO model into the FO model’s commensurate form. (iii). It does not discuss non-commensurate order FOSs order reduction.
Ganguli et al. [46]	Hybrid firefly metaheuristic algorithms	(i). The reduced IO equivalent models for the non-commensurate FOSs were obtained. (ii). Requires the transformation of the FOS to IO using the Oustaloup approximation. (iii). It is necessary to transform the integer order configuration to the delta domain related to a predetermined sampling period. (iv). It does not talk about commensurate order FOSs order reduction.
Mouhou and Badri [47]	Grey wolf optimizer-based cuckoo search algorithm (GWO-CSA)	(i). Equivalent IO models approximate the non-commensurate and irrational-order FOSs. (ii). First, the Oustaloup approximation obtains a higher-order IO system, and then GWO-CSA receives a reduced-order model. (iii). It does not talk about the commensurate order FOSs order reduction.
Kumar et al. [48]	DEA	(i). A mathematical substitution is required to convert the commensurate FOS into IO. The DEA obtains the reduced IO model. (ii). It requires an inverse substitution to convert the reduced IO model into the FO model’s commensurate form. (iii). It does not discuss non-commensurate order FOSs order reduction.
Singh et al. [49]	Stability equation method and colliding bodies optimization (CBO)	(i). A mathematical substitution is needed to convert the commensurate FOS into IO. The stability equation approach generates the denominator, and the CBO calculates the numerator. (ii). It requires an inverse substitution to convert the reduced IO model into the FO model’s commensurate form. (iii). It does not talk about non-commensurate order FOSs order reduction.

briefly describes FOS reduction techniques in the recent literature.

Based on our literature assessment, most reduction techniques are intended for commensurate or non-commensurate fractional orders of magnitude. The MOR techniques for commensurate and non-commensurate modeling are uncommon except [16,40,41]. The MOR of fractionally ordered commensurate and non-commensurate systems is the main topic of this work. The suggested approach is based on the gaze cues learning-based grey wolf optimizer (GGWO) method [50] using the FOMCON toolkit [51,52,53]. The primary advantage of the offered method is that it may reduce the order of commensurate and non-commensurate FOSs without requiring operator-based approximation, mathematical replacement, or intermediate phases. The RMSE discrepancies in Bode magnitude graphs and the ISE disparities between the reduced models and the original FOS are the metrics for which the cost function is set to optimize. Two case studies support the proposed technique. Empirical data demonstrate that reduced systems built with the considered method have step and Bode responses closer to the original FOS.

The remaining article is structured here: The problem setup for order reduction of FOSs is attempted in Section 2. The GGWO algorithm is drafted briefly in Section 3. The proposed reduction methodology is detailed in Section 4. The simulation outcomes and performance assessment were detailed in Section 5. In Section 6, the paper concludes with several suggestions for more study.

The recommended method’s main contributions are as follows:

(i)

Provide a novel, efficient, fast optimization approach to compute direct reduced-order models of complex FOSs without employing mathematical substitutions or operator-based approximation techniques.

(ii)

Verify the MOR approach with the work performed on the GGWO by examining the specified test system’s frequency and step response data.

(iii)

Compare time domain properties and performance measures like integral square error (ISE), integral absolute error (IAE), integral time absolute error (ITAE), and root-mean-square error (RMSE).

(iv)

Demonstrate that the suggested approach is more successful than the algorithms found in recent literature.

2. Outline of Fractional Calculus and FOMCON Toolbox for MATLAB and Problem Statement

Fractional calculus (FC): Numerous well-known mathematicians, including Euler, Laplace, Fourier, Liouville, Riemann, Abel, and Laurent, have expressed interest in FC. Nevertheless, the theory of generalized operators did not evolve to a level appropriate for serving as the foundation for contemporary mathematics until 1884 [54].

Definition 1.

FC is a generalization of integration and differentiation to a fractional-order operator ${}_{p}D_t^{\delta }$. Here, $p$ and $t$ represent the lower and upper limit of the differ-integral operator and $\delta$ denotes the fractional order such that [55]

$${}_{p}D_t^{\delta }=\left\{\begin{array}{ll}{\displaystyle \frac{d^{\delta }}{d{t}^{\delta }}}& \Re (\delta )>0,\hfill \\ 1\hfill & \Re (\delta )=0,\hfill \\ {\displaystyle \underset{p}{\overset{t}{\int }}{(d\tau )}^{-\delta }}\hfill & \Re (\delta )<0,\hfill \end{array}\right.$$

(1)

It is generally assumed that $\delta \in \Re$, but it may also be a complex number [55]. We solely take into account the former scenario in our work. Riemann–Liouville, Caputo, and Grünwald–Letnikov are among the authors who have defined the fractional operator, among others, in the literature [55].

Several tools have been created for modeling, synthesizing, and analyzing fractional-order systems. MATLAB toolboxes NINTEGER [56] and CRONE [57] are examples of these tools. One well-known FOS modeling and control design tool is the FOMCON toolbox for MATLAB. It is an FC-based toolkit for control design and system modeling. It is an expansion of the mini toolbox FOTF (“Fractional-order Transfer Functions”), made available by [55,58,59]. An overview of the FOMCON toolbox explains the rationale for its creation and discusses how it relates to existing toolboxes focused on FC, which is discussed in [53]. It offers graphical user interfaces (GUIs), convenience functions, ways to identify models in the time and frequency domains, fractional PID controller design and optimization, and Simulink blocks.

Problem statement:

Let the input-output behavior of a non-commensurate system with fractional derivative be described as follows.

$$G_{NCFO}(s)={\displaystyle \frac{{\displaystyle \sum _{j=0}^m{f}_j{s}^{{\alpha }_j}}}{{\displaystyle \sum _{i=0}^n{h}_i{s}^{{\beta }_i}}}}={\displaystyle \frac{f_0{s}^{{\alpha }_0}+f_1{s}^{{\alpha }_1}+\dots +f_{m-1}{s}^{{\alpha }_{m-1}}+f_m{s}^{{\alpha }_m}}{h_0{s}^{{\beta }_0}+h_1{s}^{{\beta }_1}+\dots +h_{n-1}{s}^{{\beta }_{n-1}}+h_n{s}^{{\beta }_n}}}$$

(2)

where $f_j,h_i\in \Re$, ${\alpha }_j,j=0,1,2,\dots ,m$; ${\beta }_i,i=0,1,2,\dots ,n$; ${\beta }_n>{\beta }_{n-1}>\dots {\beta }_1>{\beta }_0\ge 0;$ ${\alpha }_m>{\alpha }_{m-1}>\dots {\alpha }_1>{\alpha }_0\ge 0$.

Let any FOS input-output behavior with commensurate order $q$ because of the Caputo derivative, which can be described as follows:

$$G_{CFO}(s)={\displaystyle \frac{{\displaystyle \sum _{j=0}^m{f}_{1j}{s}^{{\alpha }_{1j}}}}{{\displaystyle \sum _{i=0}^n{h}_{1i}{s}^{{\beta }_{1i}}}}}={\displaystyle \frac{f_{10}{s}^{{\alpha }_{10}}+f_{11}{s}^{{\alpha }_{11}}+\dots +f_{1m-1}{s}^{{\alpha }_{1m-1}}+f_{1m}{s}^{{\alpha }_{1m}}}{h_{10}{s}^{{\beta }_{10}}+h_{11}{s}^{{\beta }_{11}}+\dots +h_{1n-1}{s}^{{\beta }_{1n-1}}+h_{1n}{s}^{{\beta }_{1n}}}}$$

(3)

where $f_{1j},h_{1i}\in \Re$, ${\alpha }_{1j}=jq;j=0,1,2,\dots ,m$, ${\beta }_{1i}=iq;i=0,1,2,\dots ,n$; ${\beta }_{1n}>{\beta }_{1n-1}>\dots {\beta }_{11}>{\beta }_{10}\ge 0;$ ${\alpha }_{1m}>{\alpha }_{1m-1}>\dots {\alpha }_{11}>{\alpha }_{10}\ge 0$, $q\in {\Re }^{+}$.

The primary objective of this article is to discover lower-order models for non-commensurate (2) and commensurate (3) FOSs, with integer-order (4), commensurate order (5), and non-commensurate order (6), respectively. This ensures that the ISE and RMSE for the step response and the Bode magnitude response for both the original system and the diminished models are as small as feasible using the GGWO algorithm.

$$R(s)={\displaystyle \frac{{\displaystyle \sum _{j=0}^{k-1}{a}_j{s}^j}}{{\displaystyle \sum _{i=0}^k{b}_i{s}^i}}}={\displaystyle \frac{a_0{s}^0+a_1{s}^1+\dots +a_{k-1}{s}^{k-1}}{b_0{s}^0+b_1{s}^1+\dots +b_{k-1}{s}^{k-1}+b_k{s}^k}}$$

(4)

where $a_j,b_i\in \Re$

$$R_{CFO}(s)={\displaystyle \frac{{\displaystyle \sum _{j=0}^{k-1}{a}_{1j}{s}^{{\alpha }_{1j}}}}{{\displaystyle \sum _{i=0}^k{b}_{1i}{s}^{{\beta }_{1i}}}}}={\displaystyle \frac{a_{10}{s}^{{\alpha }_{10}}+a_{11}{s}^{{\alpha }_{11}}+\dots +a_{1k-1}{s}^{{\alpha }_{1k-1}}}{b_{10}{s}^{{\beta }_{10}}+b_{11}{s}^{{\beta }_{11}}+\dots +b_{1k-1}{s}^{{\beta }_{1k-1}}+b_{1k}{s}^{{\beta }_{1k}}}}$$

(5)

where $a_{1j},b_{1i}\in \Re$, ${\alpha }_{1j}=jq;j=0,1,2,\dots ,k-1$, and ${\beta }_{1i}=iq;i=0,1,2,\dots ,k$; ${\beta }_{1n}>{\beta }_{1n-1}>\dots {\beta }_{11}>{\beta }_{10}\ge 0;$ ${\alpha }_{1m}>{\alpha }_{1m-1}>\dots {\alpha }_{11}>{\alpha }_{10}\ge 0$ and $q\in {\Re }^{+}$.

$$R_{NCFO}(s)={\displaystyle \frac{{\displaystyle \sum _{j=0}^{k-1}{a}_{2j}{s}^{{\alpha }_j}}}{{\displaystyle \sum _{i=0}^k{b}_{2i}{s}^{{\beta }_i}}}}={\displaystyle \frac{a_{20}{s}^{{\alpha }_0}+a_{21}{s}^{{\alpha }_1}+\dots +a_{2k-1}{s}^{{\alpha }_{k-1}}}{b_{20}{s}^{{\beta }_0}+b_{21}{s}^{{\beta }_1}+\dots +b_{2k-1}{s}^{{\beta }_{k-1}}+b_{2k}{s}^{{\beta }_k}}}$$

(6)

where $a_{2j},b_{2i}\in \Re$, ${\alpha }_j,j=0,1,2,\dots ,k-1$; ${\beta }_i,i=0,1,2,\dots ,k$; ${\beta }_n>{\beta }_{n-1}>\dots {\beta }_1>{\beta }_0\ge 0;$ ${\alpha }_m>{\alpha }_{m-1}>\dots {\alpha }_1>{\alpha }_0\ge 0$.

The reduced models are obtained using the GGWO algorithm by minimizing a weighted error function, which is the combination of the ISE between the step responses and RMSE between the Bode magnitude responses of the original and reduced models.

$$J=W_1{J}_{\mathrm{ISE}}+W_2{J}_{\mathrm{RMSE}}$$

(7)

with $J_{\mathrm{ISE}}={\displaystyle \underset{0}{\overset{t_{sim}}{\int }}{(y(t)-r(t))}^2(t)dt};J_{\mathrm{RMSE}}=\sqrt{{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^{N_{\omega }}{\left({\left|G(j{w}_i)\right|}_{dB}-{\left|R(j{w}_i)\right|}_{dB}\right)}^2}}{N_{\omega }}}}$ subject to $J<\epsilon$.

In the equation, $y(t)$ and $r(t)$ characterize the unit step reaction of the original system and reduced model. $t_{sim}$ refers to simulation time. ${\left|G(j{w}_i)\right|}_{dB}$, ${\left|R(j{w}_i)\right|}_{dB}$ are the magnitude of the original system and reduced model, respectively. $\epsilon$ is the tolerance error, $N_{\omega }$ is the number of frequency samples in the frequency range ${\omega }_{low}\le \omega \le {\omega }_{high}$ and $W_1$, and $W_2$ the weighting factors. Along with ISE and RMSE, IAE and ITAE are also measured to assess performance.

3. Gaze Cues Learning-Based Grey Wolf Optimizer (GGWO)

GWO algorithm [60,61,62,63]:

One of the most often referenced swarm intelligence algorithms is the grey wolf optimizer (GWO) [60,61,62,63]. The GWO algorithm draws influence from the hunting behavior and hierarchical structure of grey wolves in the wild. Alpha, beta, delta, and omega are the four categories of grey wolves, who are grouped into structured packs and adhere to a social hierarchy. In GWO, the top three most effective solutions were chosen as leading wolves. Following the lead wolf’ lead, the other wolves comply. Wolves pursue and encircle their prey, seek for and approach the prey, and then attack the victim. This is the three-step hunting process. The GWO model process of encircling the prey by grey wolves and approaching the prey to hunt can be depicted in the form of mathematical equations [50,60,61,62] as

$$G=\left|B\times Y_p(t)-Y_i(t)\right|$$

(8)

$$Y(t+1)=Y_p(t)-A\times G$$

(9)

In the above equations, t represents the present generation, $(t+1)$ denotes an updated generation, $Y_i$ is the moment vector of grey-wolf, $Y_p$ is the location vector of prey, $G$ represents the grey wolf encirclement pattern for hunting, and $A,B$ are coefficient vectors [50,60,61,62].

$$A=2\times a(t)\times r_1-a(t)$$

(10)

$$B=2\times r_2$$

(11)

$$a(t)=2-(2\times t)/MaxIter$$

(12)

where $a$ is a constant decrease linearly from 2 to 0 as generation progresses, and $r_1,r_2$ are random vectors varies $[0,1]$.

By employing (8) and (9) to randomly move its position around the prey, a grey wolf can get closer to its target. According to the GWO algorithm, since $\alpha , \beta , \mathrm{and} \delta$ are assumed to possess superior locational knowledge, the other wolves must follow them by considering their location. This behavior is explicated by (13)–(15) [50,60,61,62]:

$$G_{\alpha }=\left|B_1\times Y_{\alpha }-Y(t)\right|,G_{\beta }=\left|B_2\times Y_{\beta }-Y(t)\right|,G_{\delta }=\left|B_3\times Y_{\delta }-Y(t)\right|$$

(13)

$$\begin{array}{l}{Y}_{i1}(t)=Y_{\alpha }(t)-A_{i1}\times G_{\alpha }(t)\\ Y_{i2}(t)=Y_{\beta }(t)-A_{i2}\times G_{\beta }(t)\\ Y_{i3}(t)=Y_{\delta }(t)-A_{i3}\times G_{\delta }(t)\end{array}$$

(14)

In the iteration t, $Y_{\alpha },Y_{\beta }, \mathrm{and} Y_{\delta }$ represent the top three best wolves.

$$Y_{i-GWO}(t+1)={\displaystyle \frac{Y_{i1}(t)+Y_{i2}(t)+Y_{i3}(t)}{3}}$$

(15)

The variables in (13)–(15) are comparable to the variables specified in (8) and (9).

GGWO algorithm [50]

The GWO’s search approach applies intense selection pressure to populate the total population by repeatedly finding the solutions. The early convergence results in a lack of population variety and an unbalanced relationship between extraction and exploration. Therefore, specific improvements have been made to address these issues. Unfortunately, most of these enhancements are intrinsically flawed by the high selective pressure issue since they rely on the GWO’s search approach. Mohammad et al. [50] recently presented an improved GWO method known as gaze cues learning-based grey wolf optimizer (GGWO). In this, neighbor gaze cues learning (NGCL) and random gaze cues learning (RGCL) are two novel search algorithms that are introduced [50]. Gaze cue learning is the source of inspiration for these search tactics. The main goals are to lessen the GWO algorithm’s current intense selection pressure and limited diversity, which lead to issues with premature convergence, local optima entrapment, and stagnation. Exploiting and avoiding local optima are made more accessible with the help of the NGCL method. The RGCL increases the population variety by balancing exploration and exploitation. Diversification, exploration, and exploitation are enhanced when the three search techniques, GWO, NGCL, and RGCL, work together. The picking and updating process involves competition among candidates to revamp the present position of the wolf.

Initialization [50]: This stage involves the distribution of $N$ random wolves using a predetermined range $[v,l]$.

$$y_{ij}=v_j+(q_j-v_j)\times rand(0,1)$$

(16)

where $y_{ij}$ is the position of the ith wolf is in the jth dimension, $q_j$ and $v_j$ are the maximum and minimum bound values of the jth dimension. A matrix Pop with $N$ rows and Dim columns store the locations of $N$ wolves. For every wolf $y_i$, in each generation t, the fitness value is calculated by $f(y_i(t))$.

NGCL search strategy [50]: In this, through gaze cueing $Y_i$’s close neighbors and the alpha wolf, a candidate position $Y_i$-NGCL is formed for wolf $Y_i$ so that each dimension d is understood, and a collection of four elements, represented by the letter $N_i$, which are chosen from the population and arranged in an ascending fitness order, comprise the near neighbors of $Y_i$. These elements include two wolves that come before and after $Y_i$. The NGCL is defined by Equation (17).

$$Y_{i-NGCL,d}(t+1)=Y_{\alpha ,d}(t)+F_i\times (N_{i,d}(t)-Y_{i,d}(t))$$

(17)

where $Y_{\alpha ,d}, N_{i,d}, \mathrm{and} Y_{i,d}$ are the $D$-th dimension of the alpha wolf, a random element of $N_i$ and $Y_i$, respectively [50]. Also, Fi is a random number yielded by the Cauchy distribution; its value must be between (0 and 1) [50].

RGCL search strategy [50]: In RGCL, each dimension of the candidate’s position $Y_i$-RGCL is created by Equation (18), and the wolf $Y_i$ is learned by gaze cueing two wolves, $Y_{r1}$ and $Y_{r2}$ randomly [50].

$$Y_{i-RGCL,d}(t+1)=Y_{i,d}(t)+F_i\times (Y_{r1,d}(t)-Y_{r2,d}(t))$$

(18)

where $Y_{i,d}$ are the $D$-th dimension of the ith wolf and $Y_{r1,d} \mathrm{and} Y_{r2,d}$ are two random wolves selected from the population [50].

Selection and updating step: The fitness scores for the candidate locations $Y_{i-GWO}, Y_{i-NGCL}, \mathrm{and} Y_{i-RGCL}$ are contrasted to determine which candidate location has the lowest fitness value, $Y_{i-can}$. The location of the wolf $Y_i$ is then adjusted for the following iteration $(t+1)$ using Equation (19).

$$Y_i(t+1)=\left\{\begin{array}{cc}{Y}_{i-can}(t+1), & \mathrm{if} f(Y_{i-can})\le f(Y_i)\hfill \\ Y_i(t)& \mathrm{otherwise}\hfill \end{array}\right.$$

(19)

4. The Proposed Method for Direct Approximation of Reduced-Order Models

The functional block schematic of the offered direct approximate technique and the cost function to be minimized to obtain the reduced transfer function coefficients’ using the GGWO algorithm are shown in Figure 1 and Figure 2. Consider a stable LTI system with fractional orders and a model with reduced orders with a predefined structure transfer function with unknown coefficients and orders. The goal is to determine the values of the reduced-order model’s unknown parameters so that the FOSs’ critical properties and the reduced-order models are virtually comparable. The simplified model order may be either a positive integer or a positive real number, depending on whether the integer or fractional deceased-order model is used. In the block diagram, the inputs of the FOTF and the recommended model transfer function are each given a signal consisting of a unit step and a sinusoidal signal, respectively. One can find the reduced model optimal coefficients by reducing the weighted error signal between the actual and simplified models’ step and Bode magnitude reactions. The GGWO algorithm arrives at the best possible diminished model by minimizing the cost function during optimization. The maximum number of iterations allowed for this research’s purposes was 150, and the number of particles chosen for this investigation was 50. The lower and upper solution space borders are [−100, 10,000] to calculate reduced model coefficient values. The cost function progressively declines as the iteration progresses. As a result, the cycle continues this way until the prerequisite conditions are fulfilled. When the required criteria have been satisfied, the optimal parameters for the model’s transfer function may be derived. The flow chart of the proposed algorithm is detailed in Figure 3.

The suggested technique is outlined below.

4.1. Non-Commensurate FOS Approximation

The following steps generate integer, commensurate, and non-commensurate order-reduced models for non-commensurate FOS.

Step 1: With the help of the MATLAB FOMCON toolbox, compute the step response and the Bode magnitude reaction of the non-commensurate FOS as separate vectors.

Step 2: Assume a recommended framework for a simplified integer, commensurate, and non-commensurate order-approximated model with undetermined numerator and denominator coefficients.

Step 3: Use the GGWO and find the adequate values for the parameters of the reduced order system as close as possible to an incommensurable FOS by minimizing the cost function shown in Equations (20)–(22).

The cost function is formulated as follows:

$$J_{Int}=W_1{\displaystyle \underset{0}{\overset{t_{sim}}{\int }}{(y_{NCFO}(t)-r_{Int}(t))}^2(t)dt}+W_2\sqrt{{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^{N_{\omega }}{\left({\left|G_{NCFO}(j{w}_i)\right|}_{dB}-{\left|R_{Int}(j{w}_i)\right|}_{dB}\right)}^2}}{N_{\omega }}}}$$

(20)

$$J_{CFO}=W_1{\displaystyle \underset{0}{\overset{t_{sim}}{\int }}{(y_{NCFO}(t)-r_{CFO}(t))}^2(t)dt}+W_2\sqrt{{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^{N_{\omega }}{\left({\left|G_{NCFO}(j{w}_i)\right|}_{dB}-{\left|R_{CFO}(j{w}_i)\right|}_{dB}\right)}^2}}{N_{\omega }}}}$$

(21)

$$J_{NCFO}=W_1{\displaystyle \underset{0}{\overset{t_{sim}}{\int }}{(y_{NCFO}(t)-r_{NCFO}(t))}^2(t)dt}+W_2\sqrt{{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^{N_{\omega }}{\left({\left|G_{NCFO}(j{w}_i)\right|}_{dB}-{\left|R_{NCFO}(j{w}_i)\right|}_{dB}\right)}^2}}{N_{\omega }}}}$$

(22)

4.2. Commensurate FOS Approximation

The following steps generate integer, commensurate, and non-commensurate order-reduced models for commensurate FOSs.

Step 1: With the help of the MATLAB FOMCON toolbox, compute the step response and the Bode magnitude reaction of the commensurate FOS as separate vectors.

Step 2: Assume a recommended framework for a simplified integer, commensurate, and non-commensurate order-approximated model with undetermined numerator and denominator coefficients.

Step 3: Use the GGWO and find the adequate values for the parameters of the reduced order system as close as possible to an incommensurable FOS by minimizing the cost function shown in Equations (23)–(25).

The cost function is formulated as follows:

$$J_{Int}=W_1{\displaystyle \underset{0}{\overset{t_{sim}}{\int }}{(y_{CFO}(t)-r_{Int}(t))}^2(t)dt}+W_2\sqrt{{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^{N_{\omega }}{\left({\left|G_{CFO}(j{w}_i)\right|}_{dB}-{\left|R_{Int}(j{w}_i)\right|}_{dB}\right)}^2}}{N_{\omega }}}}$$

(23)

$$J_{CFO}=W_1{\displaystyle \underset{0}{\overset{t_{sim}}{\int }}{(y_{CFO}(t)-r_{CFO}(t))}^2(t)dt}+W_2\sqrt{{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^{N_{\omega }}{\left({\left|G_{CFO}(j{w}_i)\right|}_{dB}-{\left|R_{CFO}(j{w}_i)\right|}_{dB}\right)}^2}}{N_{\omega }}}}$$

(24)

$$J_{NCFO}=W_1{\displaystyle \underset{0}{\overset{t_{sim}}{\int }}{(y_{CFO}(t)-r_{NCFO}(t))}^2(t)dt}+W_2\sqrt{{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^{N_{\omega }}{\left({\left|G_{CFO}(j{w}_i)\right|}_{dB}-{\left|R_{NCFO}(j{w}_i)\right|}_{dB}\right)}^2}}{N_{\omega }}}}$$

(25)

where $y_{NCFO}(t)$, $y_{CFO}(t)$ characterize the unit step reaction of the ono-commensurate and commensurate FOS; $r_{Int}(t)$, $r_{CFO}(t)$, $r_{NCFO}(t)$ refer to the unit step response of the reduced IO, commensurate order, and non-commensurate order model, respectively; ${\left|G_{NCFO}(j{w}_i)\right|}_{dB}$, ${\left|G_{CFO}(j{w}_i)\right|}_{dB}$ are the magnitude of the ono-commensurate and commensurate FOS, respectively; ${\left|R_{Int}(j{w}_i)\right|}_{dB}$, ${\left|R_{CFO}(j{w}_i)\right|}_{dB}$, ${\left|R_{NCFO}(j{w}_i)\right|}_{dB}$ are the magnitude of the reduced integer, commensurate, and non-commensurate order models, respectively.

The superiority of the proposed reduction algorithm (PRA) over some existing techniques in the literature can be demonstrated by calculating the percentage enhancement (PE) in the values of performance measures (PM) like ISE, IAE, ITAE, and RMSE. The PE in the performance measures for the PRA regarding other existing approaches (OEA) in the literature is evaluated using the following formula.

$${\mathrm{PE}}_{\mathrm{PM}}={\displaystyle \frac{\mathrm{PM} \mathrm{with} \mathrm{OEA}-\mathrm{PM} \mathrm{with} \mathrm{PRA}}{\mathrm{PM} \mathrm{with} \mathrm{OEA}}}$$

(26)

5. Case Studies and Result Analysis

The presented strategy may be used effectively in several different typical situations. The findings are contrasted with the well-established models already used for pollution control to evaluate how effectively the proposed approach performs. The simulation outcomes are shown to exhibit how successful the process is. To model fractional-order systems, MATLAB 2020a users have resorted to using the FOMCON toolbox [51,52,53].

Case study 1: Consider the input-output behavior of a commensurate FOS with the structure stated in (2), as examined by [41,45].

$$G_{CFO}(s)={\displaystyle \frac{\begin{array}{l}40320+185760s^{1.2}+222088s^{2.4}+122664s^{3.6}+36380s^{4.8}+\\ 5982s^6+514s^{7.2}+18s^{8.4}\end{array}}{\begin{array}{l}40320+109584s^{1.2}+118124s^{2.4}+67284s^{3.6}+22449s^{4.8}+4536s^6+\\ 546s^{7.2}+36s^{8.4}+s^{9.6}\end{array}}}$$

(27)

Step 1: Using the FOMCON toolbox, compute the commensurate FOSs unit step and Bode magnitude response.

Step 2: Let the desired arrangement for the simplified model with integer, commensurate order $q=1.2$, and non-commensurate order as follows:

$$R_{Int}(s)={\displaystyle \frac{a_{10}s+a_{00}}{b_{20}{s}^2+b_{10}s+b_{00}}}$$

(28)

$$R_{CFO}(s)={\displaystyle \frac{a_{11}{s}^{1.2}+a_{01}}{b_{21}{s}^{2.4}+b_{11}{s}^{1.2}+b_{01}}}$$

(29)

$$R_{NCFO}(s)={\displaystyle \frac{a_{12}{s}^{{\rho }_3}+a_{02}}{b_{22}{s}^{{\rho }_1}+b_{12}{s}^{{\rho }_2}+b_{02}}}$$

(30)

Step 3: The cost function represented by (23)–(25) is reduced by utilizing GGWO to calculate the coefficients of the simplified model (28)–(30). Section 4 describes the GGWO settings that were employed in this testing system. Finally, the commensurate FOS (26) is approximated with the following reduced models:

$$R_{Int}(s)={\displaystyle \frac{8.2052s+3.4048}{s^2+2.8335s+3.4972}}$$

(31)

$$R_{CFO}(s)={\displaystyle \frac{17.8783s^{1.2}+5.4449}{s^{2.4}+7.3083s^{1.2}+5.4505}}$$

(32)

$$R_{NCFO}(s)={\displaystyle \frac{9979.1234s^{1.2025}+3368.5147}{542.6223s^{2.4067}+3977.7466s^{1.2404}+3369.1563}}$$

(33)

Figure 4 and Figure 5 represent the original system and reduced-order models’ step response and Bode plots, respectively. To simplify things, these models were compared with reduced systems developed using a combination of several conventional and mixed heuristic approaches. Figure 4 and Figure 5 show that the time and frequency responses of the proposed reduced approximations closely follow that of the original CFO system than the reduced models [64,65,66]. This proves that the recommended strategy offers a better response approximation. Apart from Figure 4 and Figure 5,

Table 2. Evaluation of the proposed strategy’s effectiveness compared with other MOR strategies.

Reduction Technique	Simplified System	ISE	IAE	ITAE	Rise Time	MP (%)	Settling Time	RMSE
Commensurate FOS					0.1076	130.1406	10.7291
Proposed reduced CFO model	${\displaystyle \frac{17.8783s^{1.2}+5.4449}{s^{2.4}+7.3083s^{1.2}+5.4505}}$	3.0492 × 10−4	0.1879	1.9629	0.1063	132.3894	11.1208	0.1272
Proposed reduced NCFO model	${\displaystyle \frac{9979.1234s^{1.2025}+3368.5147}{\begin{array}{l}542.6223s^{2.4067}+3977.7466s^{1.2404}\\ +3369.1563\end{array}}}$	3.1086 × 10−5	0.0614	0.4793	0.1075	131.0211	9.8924	0.0358
Proposed reduced IO model	${\displaystyle \frac{8.2052s+3.4048}{s^2+2.8335s+3.4972}}$	0.0050	1.2213	26.3641	0.1137	136.3334	4.0058	1.7692
Duddeti, (2023) [41]	${\displaystyle \frac{17.7688s^{1.2}+4.8356}{s^{2.4}+7.3649s^{1.2}+4.8356}}$	0.0018	0.3645	2.4063	0.1076	131.2609	12.0827	0.2049
Jain et al., 2020 [45]	${\displaystyle \frac{150s^{1.2}+48.94}{8.995s^{2.4}+61.49s^{1.2}+48.28}}$	0.0013	0.7565	17.6012	0.1140	127.3962	10.5345	0.4097
Kumar et al., 2023 [48]	${\displaystyle \frac{13.117209s^{1.2}+3.845182}{s^{2.4}+5.288137s^{1.2}+3.845182}}$	0.0033	0.3638	1.3518	0.1326	133.8951	11.4179	1.6778
Duddeti and Naskar 2024 [67]	${\displaystyle \frac{16.5007s^{1.2}+4.8356}{s^{2.4}+7.3649s^{1.2}+4.8356}}$	0.0032	0.4086	1.2501	0.1174	116.2430	11.3969	0.5372
Duddeti et al., 2023 [68]	${\displaystyle \frac{17.3435s^{1.2}+4.8313}{s^{2.4}+7.3649s^{1.2}+4.8356}}$	0.0017	0.3932	3.0981	0.1106	126.4081	11.8604	0.2661
Adamou-Mitiche and Mitiche 2017 [64]	${\displaystyle \frac{3.657s^{1.2}+1}{0.5s^{2.4}+1.5s^{1.2}+1}}$	0.0425	1.2482	3.5776	0.2041	129.3815	11.9291	4.9196
Prajapati et al., 2022 [65]	${\displaystyle \frac{6.7786s^{1.2}+2}{s^{2.4}+3s^{1.2}+2}}$	0.0403	1.0750	1.9546	0.2212	115.4959	11.0853	5.3360
Desai and Prasad 2013 [66]	${\displaystyle \frac{2.06774s^{1.2}+0.43184}{s^{2.4}+1.17368s^{1.2}+0.43184}}$	0.2879	3.9168	12.9649	0.5856	81.5678	12.5800	12.2987

presents the transfer functions of the proposed reduced models and simplified order models obtained using conventional and modern MOR methodologies [41,45,48,64,65,66,67,68]. In addition to that, it compares the time domain performance analysis specifications like rise time, peak overshoot, and settling time and various system performance analysis metrics like ISE, IAE, ITAE, and RMSE. The rise time of the proposed CFO model is nearly the same as the original system of the models proposed by [45,48,64,65,66,67,68]. The settling time is nearly the same as the original system of the models proposed by [41,48,64,66,67,68]. Similarly, the peak overshoot is enhanced compared with [48,65,66,67] models. Hence, Table 2 shows that the proposed models’ time domain data approximately follow that of the original CFO system compared with other existing models.

Table 3. The percentage enhancement in the values of performance measures compared to other MOR strategies.

Reduction Technique	Percentage Enhancement in Performance Measures
	%ISE	%IAE	%ITAE	%RMSE
Concerning the proposed reduced CFO model
Duddeti, (2023). [41]	83.0602	48.4499	18.4266	37.9209
Jain et al., 2020 [45]	76.5446	75.1619	88.8479	68.9529
Kumar et al., 2023 [48]	90.7601	48.3507	No improvement	92.4186
Duddeti and Naskar 2024 [67]	90.4712	54.0137	No improvement	76.3217
Duddeti et al., 2023 [68]	82.0635	52.2126	36.6418	52.1984
Adamou-Mitiche and Mitiche 2017 [64]	99.2825	84.9463	45.1336	97.4144
Prajapati et al., 2022 [65]	99.2434	82.5209	No improvement	97.6162
Desai and Prasad 2013 [66]	99.8941	95.5027	84.8599	98.9657

demonstrates the percentage enhancement in the proposed CFO model’s performance measures compared with existing techniques. Table 3 confirms that the proposed CFO model’s ISE, IAE, and RMSE are more enhanced than the models presented in [41,45,48,64,65,66,67,68]. The ITAE is more enhanced than the [41,45,64,66,68] models. Hence, from Figure 4 and Figure 5 and Table 2 and Table 3, it can be observed that the proposed algorithm can accurately generate reduced-order approximations compared with the state-of-the-art techniques.

Case study 2. Let us take into consideration the non-commensurate FOTF, which is utilized extensively in the literature for IO approximation presented in [13]

$$G_{NCFO}(s)={\displaystyle \frac{0.6s^{1.24}+0.7}{0.5s^{4.16}+3.2s^{3.28}+4.4s^{2.12}+5.5s^{1.57}+1.2s^{0.94}+1}}$$

(34)

The reduced commensurate, non-commensurate, and integer order models are obtained using the same process as test system 1.

$$R_{Int}(s)={\displaystyle \frac{-0.0004744s^3+0.5989s^2+1021.6169s+443.7429}{659.4237s^4+5179.7s^3+9189.0363s^2+2578.2993s+630.8166}}$$

(35)

$$R_{CFO}(s)={\displaystyle \frac{0.0895s^{1.3}+1085.9}{3686.2769s^{2.6}+8325.6158s^{1.3}+1559.0396}}$$

(36)

$$R_{NCFO}(s)={\displaystyle \frac{716.2751s^{0.0136}+251.262}{1669.538s^{2.7968}+8518.568s^{1.4255}+1355.6488}}$$

(37)

A comparison is made between the proposed reduced models and operator-based models of order 22 made with techniques [14,18,19,20] and a fourth-order model made with an equilibrium optimizer [13]. Figure 6 and Figure 7 represent the original system and reduced-order models’ step response and Bode plots, respectively. The proposed reduced models were compared with reduced systems developed using a combination of several conventional and mixed heuristic approaches. Figure 6 shows the time response of the proposed reduced approximations closely following that of the original NCFO system compared with the reduced models [14,20]. Figure 7 shows the frequency response of the proposed reduced approximations closely following that of the original NCFO system compared with the reduced models [13]. Apart from Figure 6 and Figure 7,

Table 4. Comparative performance study of the suggested approach and several MOR methodologies found in the relevant research literature, case study 2.

Method	Reduced Order	Rise Time	Overshoot	Settling Time	ISE	IAE	ITAE	RMSE
Original NCFO system		4.4850	23.5790	26.8889
Proposed IO reduced model	4	4.3709	22.1384	27.4269	1.5245 × 10−4	0.3529	17.4160	1.0013
Proposed CFOS	---	4.4099	14.1633	24.0917	0.0037	0.9739	18.1996	2.0405
Proposed NCFOS	---	4.1215	26.5213	22.7568	0.0023	1.4216	80.7507	0.8536
Yüce 2023 [13]	4	4.4293	23.7923	28.0934	1.0934 × 10−4	0.1923	4.8988	19.3946
Deniz et al., 2020 [14]	22	4.5646	17.1837	21.0800	0.0022	0.7976	17.3631	0.3980
Vinagre et al., 2000 [18]	22	4.3655	25.5066	28.4451	3.9870 × 10−4	0.5623	25.5314	0.5843
Matsuda 1993 [19]	22	4.3033	22.6922	17.8920	6.6320 × 10−4	0.4685	11.4574	0.2969
Oustaloup et al., 2000 [20]	22	4.2968	22.5894	18.0120	0.0015	1.0909	54.4230	0.4245

shows the time domain performance analysis specifications like rise time, peak overshoot, and settling time and various system performance analysis metrics, such as ISE, IAE, ITAE, and RMSE. The rise time of the proposed IO model is nearly the same as the original system of the models proposed by [14,18,19,20]. The settling time is nearly the same as the original system compared with the models described in [13,14,18,19,20]. Similarly, the peak overshoot is more enhanced than in [14,18] models. Hence, Table 2 shows that the proposed models’ time domain data approximately follow that of the original NCFO system compared with other models [14,18].

Table 5. The percentage enhancement in the values of performance measures compared with other MOR strategies.

Method	Reduced Order	Percentage Enhancement in Performance Measures
		%ISE	%IAE	%ITAE	%RMSE
Proposed IO reduced model	4
Yüce 2023 [13]	4	No improvement	No improvement	No improvement	94.8372
Deniz et al., 2020 [14]	22	93.0705	55.7548	1.2756	No improvement
Vinagre et al., 2000 [18]	22	61.7632	37.2399	32.8607	No improvement
Matsuda 1993 [19]	22	77.0129	24.6745	No improvement	No improvement
Oustaloup et al., 2000 [20]	22	89.8367	67.6506	68.5030	No improvement

demonstrates the percentage enhancement in the proposed IO model’s performance measures compared with existing techniques. Table 5 confirms that the proposed IO model’s ISE and IAE are more enhanced than the models described in [14,18,19,20]. The ITAE is more enhanced than the [14,18,20] models. The RMSE is more enhanced than [13]. Hence, from Figure 6 and Figure 7 and Table 4 and Table 5, it can be observed that the performance of the reduced system derived using the recommended direct approximation technique from the fourth order reveals that it is more satisfactory than operator-based approximation approaches from the 22nd order.

5.1. Potential Advantages of the Proposed Reduction Strategy

➢

The proposed method directly obtains the reduced model for general FOSs.

➢

It avoids the intermediate step of transforming the FOS into an equivalent IO system, i.e., the use of mathematical and inverse mathematical substitutions for commensurate FOSs and operator-based techniques like Chareff, Oustaloup, Mastuda, and Carlson approximations for non-commensurate FOSs.

➢

The advised technique lessens the need to use the intricate and higher-order FO model while conducting simulations and computations.

➢

The suggested strategy produces a model that is ideally simplified and offers a better approximation of Bode diagrams and higher-order system step responses.

5.2. Stability Analysis

Here, the stability of commensurate and non-commensurate fractional systems is analyzed using Matignon’s stability theorem [69]. According to it, a fractional system with commensurate order $G_{CFO}(s)={\displaystyle \frac{P(s)}{Q(s)}}$ can be stable if and only if $\left|arg(s)\right|>q{\displaystyle \frac{\pi }{2}},\forall \sigma C,Q(\sigma )=0$, in the $\sigma$ plane where $\sigma =sq$.

The unstable regions of the $G_{CFOS}(s)$, $R_{CFO}(s)$, $R_{NCFO}(s)$, and $R_{Int}(s)$ are plotted in Figure 8. It can be observed that the original system does not have any poles in the unstable region with $\left|arg(\sigma )\right|=3.1416$. Similarly, the proposed commensurate and non-commensurate order models do not have any poles in the instability region with $\left|arg(\sigma )\right|=3.1416$ and $\left|arg(\sigma )\right|=0.1002$, respectively. The information above concludes that the proposed algorithm generates a stable reduced system.

The unstable regions of the $G_{NCFO}(s)$, $R_{CFO}(s)$, $R_{NCFO}(s)$, and $R_{Int}(s)$ are plotted in Figure 9. It can be observed that the original system does not have any poles in the unstable region with $\left|arg(\sigma )\right|=0.0198$. Similarly, the proposed commensurate and non-commensurate order models do not have any poles in the instability region with $\left|arg(\sigma )\right|=3.1416$ and $\left|arg(\sigma )\right|=0.0221$ respectively. The information above concludes that the proposed algorithm generates a stable reduced system.

6. Conclusions and Future Prospects

This study proposes a new algorithm for reduced-order approximation of continuous-time FOS. The proposed technique is established using the FOMCON toolbox and GGWO algorithm. The proposed reduction algorithm directly approximates the FOS as a simplified-order model. The FOMCON toolbox obtains the original FOS’s step and frequency response data. Later, the GGWO algorithm is used to determine the reduced model optimal coefficients by minimizing a weighted error function. This process does not rely on mathematical substitution or operator-based techniques like Oustaloap, Mastuda, Carason, and Chareff, which result in higher IO systems for FOSs. Two case studies are used to substantiate the proposed algorithm’s efficacy. From Figure 4, Figure 5, Figure 6 and Figure 7, it can be observed that the proposed reduced models match the original system closer than some state-of-the-art techniques in the literature. To further demonstrate the effectiveness of the reduction algorithm, the often-used error indices in systems and control theory are also assessed and contrasted with the current methods in Table 2 and Table 4. Table 3 and Table 5 demonstrate the percentage enhancement in the proposed CFO and IO model’s performance measures compared with existing techniques. It can be observed from Table 2 and Table 4 that the suggested technique produces a reduced-order system with nearly the same time response characteristics as the actual system. From Table 3 and Table 5, there is a considerable improvement in the performance measures compared with some existing methods; the proposed method generates reduced models with enhanced ISE, IAE, ITAE, and RMSE. The suggested method is thus computationally effective as it does not involve any intermediate steps and offers an enhanced estimate of the original FOS.

In the future, the method may be used to reduce fractional-order MIMO systems. There is a connection between weights and fitness functions. Weight ascertainment techniques are used to calculate these weights to produce a better reduced model. The methods for determining weights that are often used include the equal-weight (EW), reciprocal weight (RW), rank-sum weight (RSW), rank-order-centroid (ROC), rank-exponent (RE), and so on.