Comparative Analysis of Reduced Commensurate Fractional-Order Interval System Based on Artificial Bee Colony Method ^†

Abstract

Large dimensional systems are complicated and very tough to control. The effective solution is to reduce the large dimension of the systems to a lower dimension. This paper aims to reduce the dimension of a higher-order fractional commensurate interval system to a low-order fractional commensurate interval system by using evolutionary techniques. Kharitonov’s theorem and artificial bee colony optimization technique are used to determine the interval numerator and denominator polynomials for the simplified models. The algorithm is very modest and generates a stable reduced dimensional commensurate fractional interval model preserving the properties of the original system. The efficiency of the suggested strategy is illustrated with a numerical example.

1. Introduction

Order reduction is a technique for reducing the difficulty levels in dynamical systems and control systems; present day researchers use model order reduction theory in numerical systems [1]. Numerical problems contain many equations and/or variables. It takes a lot of time to find a solution to these problems. To overcome this problem, researchers have considered model reduction techniques with an acceptable timeframe and limited storage capacity but with a reliable output [2]. Some of the conventional methods [3] for model order reduction include the Padè approximation technique [4], aggregation method [5], time moment matching method [6], Routh approximation method [7], etc. All of the aforementioned strategies are straightforward, but they do not guarantee stable reduced order models. Various metaheuristic methods are employed to solve these issues, including the genetic algorithm [8], particle swarm optimization [9], the firefly algorithm, the cuckoo search algorithm [10], and ant colony optimization. These metaheuristic methods are used to solve the optimization problems. Some researchers have implemented a combination of two metaheuristic techniques to address the optimization issue [11]. For the last few years, model order reduction has been used for both continuous systems and discrete systems [12]. There are many techniques to investigate the model order reduction used for interval systems [13]. There are not many techniques available in the literature for reducing large-scale fractional interval systems, however an application of a modified least squares method to commensurate fractional systems was recently added to the literature [14].

In this paper, the authors would like to propose a new technique for the reduction of commensurate fractional interval systems using an artificial bee colony and Kharitonov’s theorem.

One of the most effective ways to assess the order reduction of a fractional-order interval system is using Kharitonov’s theorem. It is relevant to both commensurate and incommensurate classes of FOIS systems [15]. The usage of interval arithmetic for model order reduction is avoided owing to Kharitonov’s theorem, which also produces fixed reduced order models [16]. The authors proposed applying this method to the order reduction of fractional commensurate interval systems because of the simplicity of Kharitonov’s theorem [17]. According to [18], time moment matching for continuous interval systems is used to obtain the numerator and a modified least squares approach is used to obtain the reduced denominator. Additional extensions of the approach are made to correspond with fractional interval systems [19]. Ref. [20] provides an explanation of the Comments on the Interval Routh Approximants calculations. According to [21], the Routh array is built using the generalized Routh table to derive the denominator of higher-order systems. Utilizing the cuckoo optimization algorithm, the numerator of the reduced order system is obtained. The primary contribution of [22] is an attempt to use the differential evolution (DE) method to lower the complexity of a high-dimensional non-integer commensurate system. The authors convey the order reduction of a fractional commensurate interval system by using artificial bee colony optimization. The numerator and denominator of the commensurate fractional-order interval system are obtained by using Kharitonov’s theorem.

The following is the present paper’s structure: Segment II contains an overview of artificial bee colony optimization, Segment III contains the problem statement, and Segment IV contains a flowchart of ABC optimization. A numerical example is presented in Segment V and segment VI contains the conclusion.

2. A Summary of ABC Optimization

Some methods of model order reduction are inspired from nature. In this category, ABC optimization is the latest addition. This method is inspired by the lifestyle of bees and was developed in 2005 by Dervis Kara Boga [23]. The ABC optimization algorithm is based on bees’ behavior. It is a metaheuristic algorithm and this algorithm combines both local and global search. This method is mainly used to solve different engineering problems. The ABC algorithm gives a better solution compared to PSO. The ABC algorithm is divided into three groups: food sources, employed foragers, and jobless foragers. Food source value depends on its richness and the case of extraction; it can be represented by a single quantity called profitability [24]. Employed foragers are currently related with food sources; they exploit a food source and they contain information on distance, profitability, and direction from the nest. These employed foragers transfer the information to other bees with a certain probability. The other bees gather nectar and return to the hive to empty their loads. It falls into one of three kinds. First, it can stop pursuing the food source and become an uncommitted follower. The second is to take part in a dance ceremony known as a waggle dance in the dancing area. So, the other bees which are unemployed as of now watch this dance and may choose to follow the particular working forager to go to that particular food source to gather the nectar and then return back to the food source and keep gathering the nectar. The third choice is not to dance, but rather to carry on foraging at the nectar source and return to the hive [25]. There are two types of unemployed bees: one is onlookers and the other one is scouts. So, these onlookers watch the waggle dance and they may choose to follow that particular bee and start searching for a food source. Scouts, on the other hand, do not always observe the waggle dance but instead begin a tour of the area around the nest on their own.

3. Problem Statement

Consider an asymptotically fixed high-order fractional commensurate interval system defined as follows:

$$\mathrm{G}(\mathrm{s})={\displaystyle \frac{\mathrm{N}(\mathrm{s},\mathrm{X})}{\mathrm{D}(\mathrm{s},\mathrm{Y})}}={\displaystyle \frac{\left[{\mathrm{X}}_0^{-},{\mathrm{X}}_0^{+}\right]{\mathrm{s}}^{{\mathsf{\beta }}_0}+[{\mathrm{X}}_1^{-},{\mathrm{X}}_1^{+}]{\mathrm{s}}^{{\mathsf{\beta }}_1}+\dots \dots +[{\mathrm{X}}_{\mathrm{m}}^{-},{\mathrm{X}}_{\mathrm{m}}^{+}]{\mathrm{s}}^{{\mathsf{\beta }}_{\mathrm{m}}}}{\left[{\mathrm{Y}}_0^{-},{\mathrm{Y}}_0^{+}\right]{\mathrm{s}}^{{\mathsf{\alpha }}_0}+\left[{\mathrm{Y}}_1^{-},{\mathrm{Y}}_1^{+}\right]{\mathrm{s}}^{{\mathsf{\alpha }}_1}+\dots \dots +[{\mathrm{Y}}_{\mathrm{n}}^{-},{\mathrm{Y}}_{\mathrm{n}}^{+}]{\mathrm{s}}^{{\mathsf{\alpha }}_{\mathrm{n}}}}}.$$

(1)

$\left[{\mathrm{X}}_{\mathrm{i}}^{-},{\mathrm{X}}_{\mathrm{i}}^{+}\right] \mathrm{a}\mathrm{n}\mathrm{d}\left[{\mathrm{Y}}_{\mathrm{j}}^{-},{\mathrm{Y}}_{\mathrm{j}}^{+}\right]$ are lower and upper boundary coefficients in ith and jth perturbation of the numerator and denominator, respectively. For i = 0, 1, 2, 3, 4, …, m and j = 0, 1, 2, 3, 4, …, n, the above equation can also be represented as

$$\mathrm{G}(\mathrm{s})={\displaystyle \frac{\sum _{\mathrm{i}=0}^{\mathrm{m}}[{\mathrm{X}}_{\mathrm{i}}^{-},{\mathrm{X}}_{\mathrm{i}}^{+}]{\mathrm{s}}^{{\mathsf{\beta }}_{\mathrm{i}}}}{\sum _{\mathrm{i}=0}^{\mathrm{n}}[{\mathrm{Y}}_{\mathrm{i}}^{-},{\mathrm{Y}}_{\mathrm{i}}^{+}]{\mathrm{s}}^{{\mathsf{\alpha }}_{\mathrm{i}}}}},$$

(2)

where ${\mathsf{\beta }}_{\mathrm{m}}$ > ${\mathsf{\beta }}_{\mathrm{m}-1}$ > ……${\mathsf{\beta }}_1$ ≥ 0 and ${\mathsf{\alpha }}_{\mathrm{n}}$ > ${\mathsf{\alpha }}_{\mathrm{n}-1}$ > ……${\mathsf{\alpha }}_1$ ≥ 0.

If all of the derivative powers of the aforementioned equations have multiple integers of $\mathsf{\gamma },$ that is ${\mathsf{\alpha }}_{\mathrm{i}}$, ${\mathsf{\beta }}_{\mathrm{i}}$ = ${\mathsf{\gamma }}^{\mathrm{i}},$ then it becomes known as an equivalent fractional-order interval system. The above equation can be modified as follows:

$$\mathrm{G}(\mathrm{s})={\displaystyle \frac{\sum _{\mathrm{i}=0}^{\mathrm{m}}[{\mathrm{X}}_{\mathrm{i}}^{-},{\mathrm{X}}_{\mathrm{i}}^{+}]{\mathrm{s}}^{\mathsf{\gamma }\mathrm{i}}}{\sum _{\mathrm{i}=0}^{\mathrm{n}}[{\mathrm{Y}}_{\mathrm{i}}^{-},{\mathrm{Y}}_{\mathrm{i}}^{+}]{\mathrm{s}}^{\mathsf{\gamma }\mathrm{i}}}}$$

(3)

$$\mathrm{G}(\mathrm{s})={\displaystyle \frac{\sum _{\mathrm{i}=0}^{\mathrm{m}}\left[{\mathrm{X}}_{\mathrm{i}}^{-},{\mathrm{X}}_{\mathrm{i}}^{+}\right]({\mathrm{s}}^{\mathsf{\gamma }}{)}^{\mathrm{i}}}{\sum _{\mathrm{i}=0}^{\mathrm{n}}[{\mathrm{Y}}_{\mathrm{i}}^{-},{\mathrm{Y}}_{\mathrm{i}}^{+}]({\mathrm{s}}^{\mathsf{\gamma }}{)}^{\mathrm{i}}}}$$

(4)

By substituting ${\mathrm{s}}^{\mathsf{\gamma }}$ = λ into Equation (4), it becomes an integer-order FOIS transfer function as follows:

$$\mathrm{G}(\mathsf{\lambda })={\displaystyle \frac{\sum _{\mathrm{i}=0}^{\mathrm{m}}[{\mathrm{X}}_{\mathrm{i}}^{-},{\mathrm{X}}_{\mathrm{i}}^{+}]{\mathsf{\lambda }}^{\mathrm{i}}}{\sum _{\mathrm{i}=0}^{\mathrm{n}}[{\mathrm{Y}}_{\mathrm{i}}^{-},{\mathrm{Y}}_{\mathrm{i}}^{+}]{\mathsf{\lambda }}^{\mathrm{i}}}}$$

(5)

$$\mathrm{G}(\mathsf{\lambda })={\displaystyle \frac{\left[{\mathrm{X}}_0^{-},{\mathrm{X}}_0^{+}\right]{\mathsf{\lambda }}^0+\left[{\mathrm{X}}_1^{-},{\mathrm{X}}_1^{+}\right]{\mathsf{\lambda }}^1+\cdots \cdots +[{\mathrm{X}}_{\mathrm{m}}^{-},{\mathrm{X}}_{\mathrm{m}}^{+}]{\mathsf{\lambda }}^{\mathrm{m}}}{\left[{\mathrm{Y}}_0^{-},{\mathrm{Y}}_0^{+}\right]{\mathsf{\lambda }}^0+\left[{\mathrm{Y}}_1^{-},{\mathrm{Y}}_1^{+}\right]{\mathsf{\lambda }}^1+\cdots \cdots +[{\mathrm{Y}}_{\mathrm{n}}^{-},{\mathrm{Y}}_{\mathrm{n}}^{+}]{\mathsf{\lambda }}^{\mathrm{n}}}}$$

(6)

The higher-order commensurate fractional-order interval system is represented as a high-order integer interval system shown in Equation (6). When the numerator and denominator interval polynomials of Equation (6) are subjected to Kharitonov’s theorem, four fixed transfer functions are produced as

$$\begin{gathered} {\mathrm{N}}_1(\mathsf{\lambda },\mathrm{X})={\mathrm{X}}_{\mathrm{m}}^{-}{\mathsf{\lambda }}^{\mathrm{m}}+\dots \dots +{\mathrm{X}}_2^{+}{\mathsf{\lambda }}^2+{\mathrm{X}}_1^{-}\mathsf{\lambda }+{\mathrm{X}}_0^{-},\hspace{1em}\hspace{1em}\hspace{1em}{\mathrm{N}}_2(\mathsf{\lambda }, \mathrm{X})={\mathrm{X}}_{\mathrm{m}}^{-}{\mathsf{\lambda }}^{\mathrm{m}}+\cdots \cdots +{\mathrm{X}}_2^{+}{\mathsf{\lambda }}^2+{\mathrm{X}}_1^{+}\mathsf{\lambda }+{\mathrm{X}}_0^{-}, \\ {\mathrm{N}}_3(\mathsf{\lambda }, \mathrm{X})={\mathrm{X}}_{\mathrm{m}}^{+}{\mathsf{\lambda }}^{\mathrm{m}}+\dots \dots +{\mathrm{X}}_2^{-}{\mathsf{\lambda }}^2+{\mathrm{X}}_1^{-}\mathsf{\lambda }+{\mathrm{X}}_0^{+},\hspace{1em}\hspace{1em}\hspace{1em}{\mathrm{N}}_4(\mathsf{\lambda }, \mathrm{X})={\mathrm{X}}_{\mathrm{m}}^{+}{\mathsf{\lambda }}^{\mathrm{m}}+\dots \dots +{\mathrm{X}}_2^{-}{\mathsf{\lambda }}^2+{\mathrm{X}}_1^{+}\mathsf{\lambda }+{\mathrm{X}}_0^{+}. \end{gathered}$$

(7)

Kharitonov’s theorem can be applied to the denominator of Equation (1):

$$\begin{gathered} {\mathrm{D}}_1(\mathsf{\lambda }, \mathrm{Y})={\mathrm{Y}}_{\mathrm{n}}^{-}{\mathsf{\lambda }}^{\mathrm{n}}+\dots \dots +{\mathrm{Y}}_2^{+}{\mathsf{\lambda }}^2+{\mathrm{Y}}_1^{-}\mathsf{\lambda }+{\mathrm{Y}}_0^{-},\hspace{1em}\hspace{1em}\hspace{1em}{\mathrm{D}}_2(\mathsf{\lambda }, \mathrm{Y})={\mathrm{Y}}_{\mathrm{n}}^{-}{\mathsf{\lambda }}^{\mathrm{n}}+\dots \dots +{\mathrm{y}}_2^{+}{\mathsf{\lambda }}^2+{\mathrm{y}}_1^{+}\mathsf{\lambda }+{\mathrm{Y}}_0^{-}, \\ {\mathrm{D}}_3(\mathsf{\lambda }, \mathrm{Y})={\mathrm{Y}}_{\mathrm{n}}^{+}{\mathsf{\lambda }}^{\mathrm{n}}+\dots \dots +{\mathrm{Y}}_2^{-}{\mathsf{\lambda }}^2+{\mathrm{Y}}_1^{-}\mathsf{\lambda }+{\mathrm{Y}}_0^{+},\hspace{1em}\hspace{1em}\hspace{1em}{\mathrm{D}}_4(\mathsf{\lambda }, \mathrm{Y})={\mathrm{Y}}_{\mathrm{n}}^{+}{\mathsf{\lambda }}^{\mathrm{n}}+\dots \dots +{\mathrm{Y}}_2^{-}{\mathsf{\lambda }}^2+{\mathrm{Y}}_1^{+}\mathsf{\lambda }+{\mathrm{Y}}_0^{+}. \end{gathered}$$

(8)

The four Kharitonov’s transfer functions below are obtained by using Equations (7) and (8):

$${\mathrm{G}}_1(\mathrm{s})={\displaystyle \frac{{\mathrm{N}}_1(\mathsf{\lambda },\mathrm{X})}{{\mathrm{D}}_1(\mathsf{\lambda },\mathrm{Y})}},\hspace{1em}\hspace{1em}{\mathrm{G}}_2(\mathrm{s})={\displaystyle \frac{{\mathrm{N}}_2(\mathsf{\lambda },\mathrm{X})}{{\mathrm{D}}_2(\mathsf{\lambda },\mathrm{Y})}},\hspace{1em}\hspace{1em}{\mathrm{G}}_3(\mathrm{s})={\displaystyle \frac{{\mathrm{N}}_3\left(\mathsf{\lambda },\mathrm{X}\right)}{{\mathrm{D}}_3(\mathsf{\lambda },\mathrm{Y})}},\hspace{1em}\hspace{1em}{\mathrm{G}}_4(\mathrm{s})={\displaystyle \frac{{\mathrm{N}}_4(\mathsf{\lambda },\mathrm{X})}{{\mathrm{D}}_4(\mathsf{\lambda },\mathrm{Y})}}$$

(9)

We can apply artificial bee colony optimization to the above four Kharitonov’s transfer functions. Figure 1 shows the flowchart of the ABC algorithm for the reduction of a commensurate fractional-order interval system.

Karaboga created the artificial bee colony (ABC) method in 2005 as a swarm-based metaheuristic technique for addressing numerical problems. It was influenced by the way honey bees employed intelligence when foraging. The algorithm is based on [25]’s notion for honey bee colony foraging behavior. The model consists of food sources and both functioning and non-functioning foraging bees. Foraging bees, which make up the first two elements, are employed and jobless. The third component is that they search for enough food sources close to their hive. The model also pinpoints two essential behaviors for the emergence of collective intelligence and self-organization: foragers need to be pushed away from low food sources, which produces negative feedback, and drawn toward rich food sources, which produces positive feedback.

In the ABC method, a colony of artificial bees, or agents, searches for an abundance of artificial food sources (ideal solutions to a particular issue). In order to use the ABC method, the optimization issue under consideration is initially converted into the task of finding the best parameter vector that reduces an objective function. Subsequently, by moving toward better solutions and removing worse ones, the artificial bees repeatedly seek neighbors using a neighbor search approach, improving the population of the first solution vectors that they have haphazardly found. The steps involved in obtaining the reduced transfer function are clearly given in Figure 1.

We obtained four $\mathrm{r}$-th-order reduced transfer functions:

$$\begin{array}{cc}{\mathrm{k}}_{1\mathrm{r}}(\mathsf{\lambda })={\displaystyle \frac{{\mathrm{U}}_{10}+{\mathrm{U}}_{11}\mathsf{\lambda }+\cdots \cdots +{\mathrm{U}}_{1(\mathrm{r}-1)}{\mathsf{\lambda }}^{\mathrm{r}-1}}{{\mathrm{V}}_{10}+{\mathrm{V}}_{11}\mathsf{\lambda }+\cdots \cdots +{\mathrm{V}}_{1(\mathrm{r}-1)}{\mathsf{\lambda }}^{\mathrm{r}-1}+{\mathsf{\lambda }}^{\mathrm{r}}}},& \hspace{1em}\hspace{1em}\hspace{1em}{\mathrm{k}}_{2\mathrm{r}}(\mathrm{s})={\displaystyle \frac{{\mathrm{U}}_{20}+{\mathrm{U}}_{21}\mathsf{\lambda }+\cdots \cdots +{\mathrm{U}}_{1(\mathrm{r}-1)}{\mathsf{\lambda }}^{\mathrm{r}-1}}{{\mathrm{V}}_{20}+{\mathrm{V}}_{21}+\cdots \cdots +{\mathrm{V}}_{2(\mathrm{r}-1)}{\mathsf{\lambda }}^{\mathrm{r}-1}+{\mathsf{\lambda }}^{\mathrm{r}}}},\\ {\mathrm{k}}_{3\mathrm{r}}(\mathsf{\lambda })={\displaystyle \frac{{\mathrm{U}}_{30}+{\mathrm{U}}_{31}\mathsf{\lambda }+\cdots \cdots +{\mathrm{U}}_{3(\mathrm{r}-1)}{\mathsf{\lambda }}^{\mathrm{r}-1}}{{\mathrm{V}}_{30}+{\mathrm{V}}_{31}\mathsf{\lambda }+\cdots \cdots +{\mathrm{V}}_{3(\mathrm{r}-1)}{\mathsf{\lambda }}^{\mathrm{r}-1}+{\mathsf{\lambda }}^{\mathrm{r}}},}& \hspace{1em}\hspace{1em}\hspace{1em}{\mathrm{k}}_{4\mathrm{r}}(\mathsf{\lambda })={\displaystyle \frac{{\mathrm{U}}_{40}+{\mathrm{U}}_{41}\mathsf{\lambda }+\cdots \cdots +{\mathrm{U}}_{3(\mathrm{r}-1)}{\mathsf{\lambda }}^{\mathrm{r}-1}}{{\mathrm{V}}_{40}+{\mathrm{V}}_{41}\mathsf{\lambda }+\cdots \cdots +{\mathrm{V}}_{3(\mathrm{r}-1)}{\mathsf{\lambda }}^{\mathrm{r}-1}+{\mathsf{\lambda }}^{\mathrm{r}}}}.\end{array}$$

(10)

The lower and higher values of the coefficients are considered from the above four transfer functions and then formulate the reduced fractional commensurate interval model represented below:

$${\mathrm{G}}_{\mathrm{r}}(\mathsf{\lambda })={\displaystyle \frac{\left[{\mathrm{U}}_{\mathrm{w}0\mathrm{m}\mathrm{i}\mathrm{n}},{\mathrm{U}}_{\mathrm{w}0\mathrm{m}\mathrm{a}\mathrm{x}}\right]+\left[{\mathrm{U}}_{\mathrm{w}1\mathrm{m}\mathrm{i}\mathrm{n}},{\mathrm{U}}_{\mathrm{w}1\mathrm{m}\mathrm{a}\mathrm{x}}\right]\mathsf{\lambda }+\cdots \cdots +[{\mathrm{U}}_{\mathrm{w}\mathrm{r}-1\mathrm{m}\mathrm{i}\mathrm{n}},{\mathrm{U}}_{\mathrm{w}\mathrm{r}-1\mathrm{m}\mathrm{a}\mathrm{x}}]{\mathsf{\lambda }}^{\mathrm{r}-1}}{\left[{\mathrm{V}}_{\mathrm{w}0\mathrm{m}\mathrm{i}\mathrm{n}},{\mathrm{V}}_{\mathrm{w}0\mathrm{m}\mathrm{a}\mathrm{x}}\right]+\left[{\mathrm{V}}_{\mathrm{w}1\mathrm{m}\mathrm{i}\mathrm{n}},{\mathrm{V}}_{\mathrm{w}1\mathrm{m}\mathrm{a}\mathrm{x}}\right]\mathsf{\lambda }+\cdots \cdots +[{\mathrm{V}}_{\mathrm{w}\mathrm{r}\mathrm{m}\mathrm{i}\mathrm{n}},{\mathrm{V}}_{\mathrm{w}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{x}}]{\mathsf{\lambda }}^{\mathrm{r}}}}$$

(11)

The integer-order transfer function mentioned above is converted into the fractional commensurate interval transfer function by substituting λ = ${\mathrm{s}}^{\mathsf{\gamma }}$ as follows:

$${\mathrm{G}}_{\mathrm{r}}(\mathrm{s})={\displaystyle \frac{\left[{\mathrm{U}}_0^{-},{\mathrm{U}}_0^{+}\right]+\left[{\mathrm{U}}_1^{-},{\mathrm{U}}_1^{+}\right]{\mathrm{s}}^{\mathsf{\gamma }}+\cdots \cdots +\left[{\mathrm{U}}_{\mathrm{r}-1}^{-},{\mathrm{U}}_{\mathrm{r}-1}^{+}\right]{\mathrm{s}}^{\mathsf{\gamma }\mathrm{r}-1}}{\left[{\mathrm{V}}_0^{-},{\mathrm{V}}_0^{+}\right]+\left[{\mathrm{V}}_1^{-},{\mathrm{V}}_1^{+}\right]{\mathrm{s}}^{\mathsf{\gamma }}+\cdots \cdots +\left[{\mathrm{V}}_{\mathrm{r}-1}^{-},{\mathrm{V}}_{\mathrm{r}-1}^{+}\right]{\mathrm{s}}^{\mathsf{\gamma }\mathrm{r}-1}+[{\mathrm{V}}_{\mathrm{r}}^{-},{\mathrm{V}}_{\mathrm{r}}^{+}]{\mathrm{s}}^{\mathsf{\gamma }\mathrm{r}}}}$$

(12)

${\mathrm{R}\mathrm{I}\mathrm{S}\mathrm{E}}_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}, \mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{I}\mathrm{S}\mathrm{E}}_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}, \mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{I}\mathrm{S}\mathrm{E}}_{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{s}\mathrm{e}}$ are defined as follows:

$${\mathrm{R}\mathrm{I}\mathrm{S}\mathrm{E}}_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}={\int }_0^{\mathrm{t}\mathrm{s}\mathrm{i}\mathrm{m}}[\mathrm{y}(\mathrm{t})-{\mathrm{y}}_{\mathrm{r}}(\mathrm{t}){]}^2\mathrm{d}\mathrm{t}/{\int }_0^{\mathrm{t}\mathrm{s}\mathrm{i}\mathrm{m}}[\mathrm{y}(\mathrm{t})-\mathrm{y}(\mathrm{\infty }){]}^2\mathrm{d}\mathrm{t}$$

(13)

$${\mathrm{I}\mathrm{S}\mathrm{E}}_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}={\int }_0^{\mathrm{t}\mathrm{s}\mathrm{i}\mathrm{m}}[\mathrm{y}\left(\mathrm{t}\right)-{\mathrm{y}}_{\mathrm{r}}(\mathrm{t}){]}^2\mathrm{d}\mathrm{t}$$

(14)

$${\mathrm{I}\mathrm{S}\mathrm{E}}_{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{s}\mathrm{e}}={\int }_0^{\mathrm{t}\mathrm{s}\mathrm{i}\mathrm{m}}[\mathrm{g}\left(\mathrm{t}\right)-{\mathrm{g}}_{\mathrm{r}}(\mathrm{t}){]}^2\mathrm{d}\mathrm{t}$$

(15)

4. Numerical Examples

4.1. Example 1

Consider the following commensurate fractional-order interval system (16) illustrated in [17].

$$\mathrm{G}(\mathrm{s})={\displaystyle \frac{\begin{array}{c}\left[\mathrm{1.9,2.1}\right]{\mathrm{s}}^{6.4}+\left[\mathrm{24.7,27.3}\right]{\mathrm{s}}^{3.2}+\left[\mathrm{157.7,174.3}\right]{\mathrm{s}}^{1.6}+\\ \left[\mathrm{542,599}\right]{\mathrm{s}}^{0.8}+\left[\mathrm{930,1028}\right]{\mathrm{s}}^{0.4}+\left[\mathrm{721.8,797.8}\right]{\mathrm{s}}^{0.2}+\left[\mathrm{187.1,206.7}\right]\end{array}}{\begin{array}{c}\left[\mathrm{0.95,1.05}\right]{\mathrm{s}}^{12.8}+\left[\mathrm{8.779,9.703}\right]{\mathrm{s}}^{6.4}+\left[\mathrm{52.23,57.73}\right]{\mathrm{s}}^{3.2}\\ \left[\mathrm{182.9,202.1}\right]{\mathrm{s}}^{1.6}+\left[\mathrm{429,474.2}\right]{\mathrm{s}}^{0.8}+\left[\mathrm{572.5,632.7}\right]{\mathrm{s}}^{0.4}+\left[\mathrm{325.3,359.5}\right]{\mathrm{s}}^{0.2}+\left[\mathrm{57.35,63.39}\right]\end{array}}}$$

(16)

By substituting λ = ${\mathrm{s}}^{0.2}$, a third-order FOIS transfer function is expressed in terms of λ.

$$\mathrm{G}(\mathrm{s})={\displaystyle \frac{\begin{array}{c}\left[\mathrm{1.9,2.1}\right]{\mathrm{\lambda }}^6+\left[\mathrm{24.7,27.3}\right]{\mathrm{\lambda }}^5+\left[\mathrm{157.7,174.3}\right]{\mathrm{\lambda }}^4\\ +\left[\mathrm{542,599}\right]{\mathrm{\lambda }}^3+\left[\mathrm{930,1028}\right]{\mathrm{\lambda }}^2+\left[\mathrm{721.8,797.8}\right]{\mathrm{\lambda }}^1+\left[\mathrm{187.1,206.7}\right]\end{array}}{\begin{array}{c}\left[\mathrm{0.95,1.05}\right]{\mathrm{\lambda }}^7+\left[\mathrm{8.779,9.703}\right]{\mathrm{\lambda }}^6\left[\mathrm{52.23,57.73}\right]{\mathrm{\lambda }}^5+\\ \left[\mathrm{182.9,202.1}\right]{\mathrm{\lambda }}^4+\left[\mathrm{429,474.2}\right]{\mathrm{\lambda }}^3+\left[\mathrm{572.5,632.7}\right]{\mathrm{\lambda }}^2+\left[32 \mathrm{5.3,359.5}\right]{\mathrm{\lambda }}^1+\left[\mathrm{57.35,63.39}\right]\end{array}}}$$

(17)

Equation (9) is used to derive the higher-order commensurate fractional-order interval system’s four fixed-parameter Kharitonov’s transfer functions:

$$\begin{gathered} {\mathrm{G}}_1(\mathsf{\lambda })={\displaystyle \frac{2.1{\mathsf{\lambda }}^6+24.7{\mathsf{\lambda }}^5+157.7{\mathsf{\lambda }}^4+599{\mathsf{\lambda }}^3+1028{\mathsf{\lambda }}^2+721.8\mathsf{\lambda }+187.1}{1.05{\mathsf{\lambda }}^7+9.703{\mathsf{\lambda }}^6+52.23{\mathsf{\lambda }}^5+182.9{\mathsf{\lambda }}^4+474.2{\mathsf{\lambda }}^3+632.7{\mathsf{\lambda }}^2+325.3\mathsf{\lambda }+57.35}}, {\mathrm{G}}_2(\mathsf{\lambda })={\displaystyle \frac{2.1{\mathsf{\lambda }}^6+27.3{\mathsf{\lambda }}^5+157.7{\mathsf{\lambda }}^4+542{\mathsf{\lambda }}^3+1028{\mathsf{\lambda }}^2+797.8\mathsf{\lambda }+187.1}{0.95{\mathsf{\lambda }}^7+9.703{\mathsf{\lambda }}^6+57.73{\mathsf{\lambda }}^5+182.9{\mathsf{\lambda }}^4+429{\mathsf{\lambda }}^3+632.7{\mathsf{\lambda }}^2+359.5\mathsf{\lambda }+57.35}}, \\ {\mathrm{G}}_3(\mathsf{\lambda })={\displaystyle \frac{1.9{\mathsf{\lambda }}^6+24.7{\mathsf{\lambda }}^5+174.3{\mathsf{\lambda }}^4+599{\mathsf{\lambda }}^3+930{\mathsf{\lambda }}^2+721.8\mathsf{\lambda }+206.7}{1.05{\mathsf{\lambda }}^7+8.779{\mathsf{\lambda }}^6+52.23{\mathsf{\lambda }}^5+202.1{\mathsf{\lambda }}^4+474.2{\mathsf{\lambda }}^3+572.5{\mathsf{\lambda }}^2+325.3\mathsf{\lambda }+63.39}}, {\mathrm{G}}_4\left(\mathsf{\lambda }\right)={\displaystyle \frac{1.9{\mathsf{\lambda }}^6+27.3{\mathsf{\lambda }}^5+174.3{\mathsf{\lambda }}^4+542{\mathsf{\lambda }}^3+930{\mathsf{\lambda }}^2+797.8\mathsf{\lambda }+206.7}{0.95{\mathsf{\lambda }}^7+8.779{\mathsf{\lambda }}^6+57.73{\mathsf{\lambda }}^5+202.1{\mathsf{\lambda }}^4+429{\mathsf{\lambda }}^3+572.5{\mathsf{\lambda }}^2+359.5\mathsf{\lambda }+63.39}}. \end{gathered}$$

(18)

Equation (12) is used to create the second-order reduced model from the initial higher-order system.

$${\mathrm{G}}_{\mathrm{r}}(\mathrm{s})={\displaystyle \frac{\left[\mathrm{13.186,18}\right]{\mathsf{\lambda }}^1+\left[\mathrm{16.30,16.31}\right]}{\left[\mathrm{1,1}\right]{\mathsf{\lambda }}^2+\left[\mathrm{11.93,14.88}\right]{\mathsf{\lambda }}^1+\left[\mathrm{5,5}\right]}}$$

(19)

The fractional commensurate interval transfer function is created from the aforementioned integer-order transfer function by substituting λ = ${\mathrm{s}}^{0.2}$:

$${\mathrm{G}}_{\mathrm{r}}(\mathrm{s})={\displaystyle \frac{\left[\mathrm{13.186,18}\right]{\mathrm{s}}^{0.2}+\left[\mathrm{16.30,16.31}\right]}{\left[\mathrm{1,1}\right]{\mathrm{s}}^{0.4}\left[\mathrm{11.93,14.88}\right]{\mathrm{s}}^{0.2}+\left[\mathrm{5,5}\right]}}$$

(20)

The step response of the proposed reduced fractional commensurate interval system (both lower and upper boundaries) compared with the original system and the gamma-delta method is portrayed in Figure 2 and Figure 3.

Figure 2 and Figure 3 make it abundantly evident that in contrast to the step response generated by the gamma–delta technique, the proposed technique yields a reduced model whose reaction closely resembles the original system’s step response.

Table 1. The performance indices of the proposed method and the gamma-delta method with respect to the original system.

Example 1		ISE for Impulse Response	ISE for Step Response	RISE for Step Response
Original with respect to proposal method	L.B	1.000000	0.100000	0.010000
	U.B	1.217891	0.289190	0.042670
Original with respect to gamma-delta method	L.B	3.357806	2.435341	0.358393
	U.B	2.417664	0.879348	0.129747

shows the ISE values (${\mathrm{I}\mathrm{S}\mathrm{E}}_{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{s}\mathrm{e}},{\mathrm{I}\mathrm{S}\mathrm{E}}_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}},{\mathrm{R}\mathrm{I}\mathrm{S}\mathrm{E}}_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}$) and

Table 2. The rise time, settling time, and delay times for the original system, proposed method, and gamma-delta method.

Example 1		Rise Time	Settling Time
Original response	L.B	0.02	1.96
	U.B	0.034	2.077
Proposed method response	L.B	0.05	1.753
	U.B	0.034	2.077
Gamma-delta method	L.B	0.21	3.01
	U.B	0.037	2.12

shows the time-domain specifications. The proposed model’s step response closely resembles that of the original system as determined by Kharitonov’s theorem and the ABC technique. To check the superiority of the suggested approach, it is compared to other known methods employing ISE values and time-domain parameters.

4.2. Example 2

An example of a commensurate fractional-order interval system is as follows [18]:

$$\mathrm{G}(\mathrm{s})={\displaystyle \frac{\left[\mathrm{1,1}\right]{\mathrm{s}}^{2.4}+\left[\mathrm{7,8}\right]{\mathrm{s}}^{1.6}+\left[\mathrm{24,25}\right]{\mathrm{s}}^{0.8}+\left[\mathrm{24,25}\right]}{\left[\mathrm{1,1}\right]{\mathrm{s}}^{3.2}+\left[\mathrm{10,11}\right]{\mathrm{s}}^{2.4}+\left[\mathrm{35,36}\right]{\mathrm{s}}^{1.6}+\left[\mathrm{50,51}\right]{\mathrm{s}}^{0.8}+\left[\mathrm{24,25}\right]}}.$$

(21)

By substituting λ = ${\mathrm{s}}^{0.8}$, a third-order FOIS transfer function articulated in terms of λ is expressed as follows:

$$\mathrm{G}(\mathrm{s})={\displaystyle \frac{\left[\mathrm{1,1}\right]{\mathsf{\lambda }}^3+\left[\mathrm{7,8}\right]{\mathsf{\lambda }}^2+\left[\mathrm{24,25}\right]{\mathsf{\lambda }}^1+\left[\mathrm{24,25}\right]}{\left[\mathrm{1,1}\right]{\mathsf{\lambda }}^4+\left[\mathrm{10,11}\right]{\mathsf{\lambda }}^3+\left[\mathrm{35,36}\right]{\mathsf{\lambda }}^2+\left[\mathrm{50,51}\right]{\mathsf{\lambda }}^1+\left[\mathrm{24,25}\right]}}.$$

(22)

Equation (9) is used to obtain Kharitonov’s transfer functions with four fixed parameters for the original higher-order commensurate fractional-order interval system:

$${\mathrm{G}}_1(\mathrm{s})={\displaystyle \frac{1{\mathsf{\lambda }}^3+8{\mathsf{\lambda }}^2+24\mathsf{\lambda }+24}{1{\mathsf{\lambda }}^4+11{\mathsf{\lambda }}^3+36{\mathsf{\lambda }}^2+50\mathsf{\lambda }+24}}, {\mathrm{G}}_2(\mathrm{s})={\displaystyle \frac{1{\mathsf{\lambda }}^3+8{\mathsf{\lambda }}^2+25\mathsf{\lambda }+24}{1{\mathsf{\lambda }}^4+10{\mathsf{\lambda }}^3+36{\mathsf{\lambda }}^2+51\mathsf{\lambda }+24}}, {\mathrm{G}}_3(\mathrm{s})={\displaystyle \frac{1{\mathsf{\lambda }}^3+7{\mathsf{\lambda }}^2+24\mathsf{\lambda }+25}{1{\mathsf{\lambda }}^4+11{\mathsf{\lambda }}^3+35{\mathsf{\lambda }}^2+50\mathsf{\lambda }+25}}, {\mathrm{G}}_4(\mathrm{s})={\displaystyle \frac{1{\mathsf{\lambda }}^3+7{\mathsf{\lambda }}^2+25\mathsf{\lambda }+25}{1{\mathsf{\lambda }}^4+10{\mathsf{\lambda }}^3+35{\mathsf{\lambda }}^2+51\mathsf{\lambda }+25}}.$$

(23)

The original higher-order system is reduced to a second-order model using Equation (12):

$${\mathrm{G}}_{\mathrm{r}}(\mathrm{s})={\displaystyle \frac{\left[\mathrm{5,5}\right]{\mathsf{\lambda }}^2+\left[\mathrm{12.02,12.57}\right]}{\left[\mathrm{1,1}\right]{\mathsf{\lambda }}^2+\left[\mathrm{18,18}\right]{\mathsf{\lambda }}^1+\left[\mathrm{12.02,12.57}\right]}}.$$

(24)

The integer-order transfer function is changed to the fractional commensurate interval transfer function by substituting λ = s^0.8

$${\mathrm{G}}_{\mathrm{r}}(\mathrm{s})={\displaystyle \frac{\left[\mathrm{5,5}\right]{\mathrm{s}}^{0.8}+\left[\mathrm{12.02,12.57}\right]}{\left[\mathrm{1,1}\right]{\mathrm{s}}^{1.6}+\left[\mathrm{18,18}\right]{\mathrm{s}}^{0.8}+\left[\mathrm{12.02,12.57}\right]}}$$

(25)

The step response of the recommended reduced fractional commensurate interval system (both lower and upper boundaries) compared with the original system and the least squares method is shown in Figure 4 and Figure 5, respectively. The rise time, settling time, and delay times for the original system, proposed method, and least squares method are calculated and compared, which are illustrated in

Table 3. The rise time, settling time, and delay times of the original system, proposed method, and least squares method.

Example 2		Rise Time	Settling Time	Delay Time
Original system	L.B	2.07	7.104	0.710
	U.B	2.013	7.473	0.688
Proposed method	L.B	2.037	7.006	0.7041
	U.B	1.949	7.245	0.5731
Least squares method	L.B	1.925	7.915	1.109
	U.B	1.775	7.702	0.699

. Table 3 shows the time-domain specifications. The step response of the proposed model obtained from Kharitonov’s theorem and the ABC method nearly replicate the original high-order fractional commensurate interval system. Using integral square error values and time-domain specifications, the recommended approach is compared with other current methods in order to determine its superiority. Finally, it is observed that the proposed method shows better performance compared to other methods illustrated in recent studies. Finally, Figure 2, Figure 3, Figure 4 and Figure 5 show that the replies from the suggested technique closely resemble the responses from the original system, and the time-domain specifications of the responses obtained from the proposed method are illustrated in Table 2 and Table 3, which are predominantly extracted from the Figure 2, Figure 3, Figure 4 and Figure 5.

5. Conclusions

In the proposed work, the application of Kharitonov’s theorem and the ABC technique to reduce a higher-order fractional commensurate interval system to a lower-order fractional commensurate interval system is carried out. If the original system is fixed, the recommended method ensures the flexibility of the proposed model. When compared to other reference systems, the suggested method is relatively easy to understand. Examples from the literature are used for verification of the proposed method and further compared with the models obtained from other reference strategies. This approach may potentially be expanded to continuous, discrete, fractional non-commensurate systems; however, this process is time-consuming.