A hybrid incomplete decision system using fuzzy sets and rough set theory with varying object sets and values

Abstract

Researchers in science and engineering face various obstacles due to a lack of specific and full data. Many different approaches have been devised to deal with these restrictive requirements, but two notable schools of thought are the fuzzy set (FS) theory and the rough set (RS) theory, both of which have spawned many extensions and hybridizations. Although RS theory originated from an indiscernibility relation (also known as an equivalence relation in mathematics), emphasis rapidly shifted to similarity or coverings (and their fuzzy analogues). Many other hybrid schemes were suggested with this goal in mind. The gap between those concepts shrank because to this thorough analysis. Fuzzy set theory is a legitimate way to convey the ambiguity of assessment data, yet it is still inadequate for dealing with certain intricate problems in the actual world. In reality, decision makers will undoubtedly provide different kinds of ambiguous and nuanced assessments. Atanassov’s intuitionistic fuzzy set theory broadened the application of fuzzy set theory by imbuing it with an element of uncertainty. Sometimes in real life, you have to deal with a neutral element on top of the indeterminate one. Picture fuzzy sets were developed specifically for this purpose. Membership roles may be positive, neutral, or negative/refusal. In contrast, hesitant fuzzy sets and its hybrid models are useful when decision makers are on the fence about which option to choose. As a binary relation on a set, a graph is symmetric. It is a staple in mathematical modelling and is used in almost every scientific and technological discipline. Graph theory has been essential in the mathematical modelling and subsequent resolution of several real-world situations. Information about connections between things is often best represented using graph theory, which uses vertices to stand in for the items and edges for the relationships between them. The suggested dynamic algorithm is better to the static approach in dealing with the multidimensional dynamic changes of the hybrid incomplete decision system, according to a series of experiments carried out on nine UCI datasets.

Keywords

Intuitionistic fuzzy set theory graph theory rough set theory varying object sets and values

1 Introduction

The majority of the world’s technical, social and medical research, economic, environmental, and other problems are solved using mathematical approaches based on uncertainty and imprecision. Many theories have been proposed to address this issue. The terms “rough set “, “interval valued intuitionistic fuzzy set,” “fuzzy set,” “intuitional fuzzy set,” etc., all refer to different kinds of sets.

1.1 Fuzzy sets and their generalizations

If there is a degree of doubt about certain information. In 1965, [1] presented the concept of a fuzzy set as a logical extension of the idea of a regular (crisp) set. Classical set theory uses a binary criteria to determine whether or not an object belongs in a set. However, in fuzzy set theory, the membership function is evaluated in the real unit interval [0, 1], which allows for a more nuanced assessment of whether or not an element belongs to a set described by the function. From the time it was first imagined, this idea has been thoroughly examined by mathematicians and computer scientists. Fuzzy logic, neural networks, fuzzy automata, and even control systems are all offshoots of this branch of mathematics.

Topology is the branch of mathematics concerned with the study of how qualities of objects are preserved via continuous deformations like stretching without ripping or glueing. The Seven Bridges of Königsberg problem, in which Euler sought a solution to a geometrical challenge in which distance was irrelevant, is one of the first academic publications in contemporary topology [2]. Ideas from set theory provide the foundation of contemporary topology. The proposal will make certain topological assumptions about the reader. Many mathematical concepts have precise definitions thanks to set theoretic notions, which place them as “members” or “non-members” of a certain sphere of discussion. In groups and classes like “very hot days,” “beautiful places,” “real numbers much greater than 10,” etc., it is not easy to tell who belongs and who does not. Fuzzy sets, first suggested by [3] in that year, may help us get around this problem by characterising these ideas by assigning a degree of membership to each member.

When dealing with complex issues in the domains of economics, engineering, the environment, medicine, and the social sciences, it is probable that traditional mathematical techniques will not be helpful. Mathematical modelling of the aforementioned uncertainties is essential, but no area of mathematics presently offers a suitable parametrization tool. In light of the need for a broad mathematical tool to deal with ambiguities, [4] proposed the soft set. Theories and techniques for dynamically updating three-way regions arising from multidimensional change of object values, removal of object set, and simultaneous increase of object set in the hybrid incomplete decision system are investigated in this study (HIDS)-Hybrid Incomplete Decision System. First, the hybrid neighbourhood information granule and the hybrid neighbourhood relation are built for the hybrid incomplete decision system with Boolean, categorical, numerical, set-valued, and interval-valued attributes by combining with the hybrid distance metric, which fuses the distances of the different data types. We present a matrix-based method for calculating HIDS rough estimations, which is shown below. It examines matrix-based incremental techniques for generating three-way HIDS areas for simultaneous object value update, object insertion, and object deletion. Following that, we show how to update three-way sections of HIDS with concurrent object value updates, object deletion, and object insertion. Following that, a series of tests are run, with the results proving that the suggested dynamic method efficiently updates the three-way regions of HIDS and outperforms the static strategy.

1.2 Motivation

Specifically, the topological notions of fuzzy sets, rough sets, and soft sets will be explored. Here, we lay up the groundwork for our study of fuzzy topological space. We provide two generalisations of the idea of a fuzzy closed set: fuzzy closed sets and fuzzy weakly-closed sets. Fuzzy closed sets are introduced, and we then explore various topological concepts as closure, interior, compactness, closed space, continuity, and many generalisations of continuity. A new fuzzy set boundary definition in a fuzzy topological space is also proposed and analysed. Several strategies for investigating rough set topology are given. The idea of a “covering based rough set,” which we introduce and investigate, has also been examined. Soft set topological structures are presented in Section 3 as a final section. In order to generalise various topological concepts, we construct semi open and semi closed soft sets for a soft topological space. In addition, a topological study of fuzzy sets and soft sets is carried out.

The following subheadings describe each aspect of the planned work: There will be six sections total: (2) an introduction, (3) a literature review, (4) a proposed technique, (5) a results and analysis section, (6) a discussion section, and (7) a conclusion and recommendations for future research.

2 Review of literature

Data mining has been singled out as a fruitful subject for interdisciplinary research due to the breadth of its possible applications. Data mining is an important and active subject of research with both theoretical difficulties and practical applications due to the challenge of detecting (or extracting) relevant and previously discovered information from unusually huge real-world datasets. Data mining has been the subject of research in many different academic fields. [5] The comparison of previous efforts is shown in Table 1. Therefore, it is evident that by just sliding down in Table- -, a subset of the original data set may be built using a smaller number of characteristics while maintaining the same precision of approximation. At the heart of rough sets is the idea of attribute reduction. A redact is a reliable set of characteristics that keeps information separate. According to this theory, a redact is the smallest possible set of criteria that may nevertheless be used to organise the universe into meaningful categories. There is no way to reduce the complexity of the cosmic phenomena known as super flows to a simple equation.

Table 1
Comparison of existing work

Author and reference Application field Study category Study contribution

[18] Feature selection FR Set To address issues with the propositional satisfiability viewpoint, the authors propose a model that makes use of rough sets.

[19] Feature selection Fuzzy - rough feature selection “proposed an approach based on dimensionality reduction together with sample reduction for a heuristic process of fuzzy- “ @” rough feature selection.”)

[20] Evaluated features FR Set Developed rough sets to mitigate the effect of noise, which led to the proposal of soft fuzzy-rough sets.

[21] Machine learning FR Set Incomplete quantitative data sets based on rough sets were used to propose a learning method.

[22] Machine learning FR Set For the purpose of selecting “@” fuzzy-rough instances, we have introduced a novel classification strategy based on the fuzzy-rough closest neighbour technique.

[23] Number of features FR Set A novel feature selection– instance selection hybrid technique was presented for data reduction.

[24] Selected features FR Set Using granular computing, a novel rough fuzzy method for pattern categorization was proposed.

Author and reference	Application field	Study category	Study contribution
[18]	Feature selection	FR Set	To address issues with the propositional satisfiability viewpoint, the authors propose a model that makes use of rough sets.
[19]	Feature selection	Fuzzy - rough feature selection	“proposed an approach based on dimensionality reduction together with sample reduction for a heuristic process of fuzzy- “ @” rough feature selection.”)
[20]	Evaluated features	FR Set	Developed rough sets to mitigate the effect of noise, which led to the proposal of soft fuzzy-rough sets.
[21]	Machine learning	FR Set	Incomplete quantitative data sets based on rough sets were used to propose a learning method.
[22]	Machine learning	FR Set	For the purpose of selecting “@” fuzzy-rough instances, we have introduced a novel classification strategy based on the fuzzy-rough closest neighbour technique.
[23]	Number of features	FR Set	A novel feature selection– instance selection hybrid technique was presented for data reduction.
[24]	Selected features	FR Set	Using granular computing, a novel rough fuzzy method for pattern categorization was proposed.

Since its introduction to topological space by [6] in 1963, semiopen sets and semicontinuous functions have been the subject of much study. The concept of generalised closed sets [7] was introduced later by him. To comprehend why certain topological spaces have closed sets and others do not, this provides the framework. In [8], semiopen and semiclosed sets were combined to form sg-closed sets, which were then investigated. In [9], the semi closure operator was introduced to generate gs-closed sets. Weakly continuous functions, nearly continuous functions, and semicontinuous functions in the fuzzy set were constructed and analysed [10]. These generalisations have an impact on the topological space of fuzzy sets [11].

The concept of definability is explored for ten different types of covering-based rough sets in [12]. Studies [13] have investigated the use of topological methods for covering-based rough sets. Many new and intriguing conclusions regarding the topological structures of rough sets are presented in [14], where the concept of transferring the expression of a link is developed. The minimum and maximum estimations serve as the foundation for the first group. Rough sets have a special feature that is used in a creative way in [15], where we see a topological rough set with this attribute. Several characteristics of topological rough sets, such as homeomorphism, compactness, Hausdorffness, etc., have been studied by the authors. [16] suggested using a soft fuzzy-rough set-based approach for segmenting brain MRIs in order to get over the problems of imprecision and vagueness. In [17], we see a potent fuzzy-rough set technique for feature selection.

Few models for the fuzzy generalisation of rough set theory have been suggested and developed over the last few decades. Therefore, we think it’s important to thoroughly examine the most current and pertinent studies in this field. Therefore, the purpose of this paper is to provide a review of the literature that has either added to or clarified the idea of FR Set [25]. This review greatly advances our understanding of fuzzy-rough set theory by offering a new viewpoint on the previous studies. Among them include categorising the articles based on the authors’ ages, countries of origin, fields of study, study designs, contribution kinds, and journals in which they were published.

2.1 Hybrid incomplete rough-set

However, hybrid statistics are widespread in genuine worldwide record sets. These hybrid facts frequently contain Boolean, category, numerical, set-valued, c programming language-valued, and various record kinds. Furthermore, the hybrid data is supplied by missing information. These data sets are sometimes referred to as hybrid incomplete facts units.

Definition 1. For a decision system DS = (U, C∪D, V, f), if C = C^B∪C^C∪C^N∪C^S∪C^I. The decision-making process is known as the hybrid decision system (HDS), where C^B, C^C, C^N, C^S, and C^I denote for Boolean, Categorical, Numerical, Set-Valued, and Interval-Valued attribute sets, respectively. The hybrid incomplete decision system (HIDS) is the name given to a hybrid decision system when it contains incomplete data.

Example 1. A hybrid incomplete decision system HIDS = (U, C ∪ D, V, f), where U = {x₁, x₂, x₃, x₄, x₅, x₆}, C = {a₁, a₂, a₃, a₄, a₅}, D = {D₁, D₂}. For example, f (x₅, a₁) = ′ * ′, f (x₄, a₂) = ′ * ′, f (x₆, a₂) = ′ * ′, f (x₅, a₃) = ′ * ′, f (x₆, a₄) = ′ * ′, and f (x₂, a₅) = ′ * ′.

2.2 In the rough set model, three-way area

A pair of thresholds (α, β) , (0 ⩽ β ⩽ α ⩽ 1), to create the probabilistic positive, borderline, and negative areas of the rough set model, which offers a plausible semantic explanation for the three fields. Three decisions acceptance, rejection, and non-acceptance can be seen as the causes of the positive, negative, and threshold fields.

Definition 2. In this item is separated into three distinct, non-overlapping sections as a consequence of the three-way choice, which is a result framework of the three-part procedure. Which zone the item belongs to for a specific metric is determined by a predetermined evaluation function.

Example 2. For a 4-tuple DS (Decision System) and A⊆C, the equivalence relation IND(A) on U can be denoted as, $\begin{matrix} IND (A) & = {(x, y) U \times U | f (x, a) \\ = f (y, a), \forall a \in A} (1 a) \end{matrix}$ (1a)

The equivalence (IND)-indiscernibility relation IND(A) can induce a partition of U. For an object x∈U, [X]_A={y∈U|(x,y) ∈U*U|f(x,a) = f(y,a), ∀a∈A}. The equivalence relation IND(A) can induce a partition of U. For an object x ∈ U, [x] _A ={y∈U|(x,y)∈IND(A)} denotes the equivalence class induced by x in regard of A. Suppose a subset of objects X ⊆ U represents a concept. The condition probability can be expressed as: $Pr (X | {[x]}_{A}) = \frac{| X \cap {[x]}_{A} |}{| {[x]}_{A} |}$ where, |·| denotes the cardinality of a set.

For a DS, X ⊆ U and A ⊆ C, a pair of thresholds (α,β) (0 ≤ β < α ≤ 1), the probabilistic lower and upper approximations of X in regard of A can be defined as follows, respectively. $\underline{{AP}_{A}} (X) = {x \in U | Pr (X | {[x]}_{A}) ⩾ α},$ $\bar{{AP}_{A}} (X) = {x \in U | Pr (X | {[x]}_{A}) 〉 β} .$

Given X ⊆ U and a pair of thresholds α and β (0 ≤ β < α ≤ 1), the probabilistic positive, boundary and negative regions of X can be described as follows, respectively. ${PR}_{A}^{(α, β)} (X) = {Pr (X | {[x]}_{A}) ⩾ α},$ ${BR}_{A}^{(α, β)} (X) = {x \in U | β < Pr (X | {[x]}_{A}) < α},$

When the pair of thresholds (α, β) are determined, all objects in X can be divided into positive, boundary, and negative regions. In three-way decision theory, three regions represent three different decisions, that is, (Positive Region) PR (denotes the acceptance, BR (Boundary Region) represents the delay (no commitment), and NR (Negative ${NR}_{A}^{(α, β)} (X) = {x \in U | Pr (X | {[x]}_{A}) ⩽ β}$ Region) stands for the rejection.

3 Proposed methodology

Definition 3.1. Intuitionistic fuzzy rough relation (IFRr-topology) on R is a subfamily of IFR(R) satisfying the following axioms: $\begin{matrix} \begin{matrix} [O_{1}] R_{Φ}, R \in τ, \\ [O_{2}] {R_{i} : i \in I} \subseteq τ \Rightarrow \cup_{i \in I} R_{i} \in τ, \\ [O_{3}] {IfR}_{1}, R_{2} \in τ, {thenR}_{1} \cap R_{2} \in τ . \end{matrix} (1 b) \end{matrix}$ (1b)

In such case, we refer to (R,) as an IFRr-topological space. Open IFR-relations (or IFRr-open sets or just open sets) are the sets that satisfy the requirements [O₁], [O₂], and [O₃].

Example 3.1.1. In this example, we’ll pretend that U = a,b,c and that the equivalence relation = (a,a),(b,b),(c,c),(a,b),(b,a) is defined on U. Let X = {a, c}⊆U, then the upper approximation of X×X is provided by (X×X)^- ={(a,a),(b,b),(c,c),(a,b),(b,a),(a,c),(c,a),(b,c),(c,b)} and the lower approximation is _(X×X) ={(c,c)}. Let

$\begin{matrix} R_{Φ} = {〈 (a, a), (0, 1) 〉, 〈 (a, b), (0, 1) 〉, 〈 (a, c), (0, 1) 〉, 〈 (b, a), (0, 1) 〉, \\ 〈 (b, b), (0, 1) 〉, 〈 (b, c), (0, 1) 〉, 〈 (c, a), (0, 1) 〉, 〈 (c, b), (0, 1) 〉, 〈 (c, c), (0, 1) 〉}, \end{matrix}$ (2)

$\begin{matrix} R = {〈 (a, a), (0.8, 0.1) 〉, 〈 (a, b), (0.7, 0.1) 〉, 〈 (a, c), (0.6, 0.2) 〉, 〈 (b, a), (0.5, 0.2) 〉, \\ 〈 (b, b), (0.7, 0.2) 〉, 〈 (b, c), (0.5, 0.3) 〉, 〈 (c, a), (0.6, 0.2) 〉, 〈 (c, b), (0.6, 0.3) 〉, 〈 (c, c), (1, 0) 〉}, \\ R_{1} = {〈 (a, a), (0.3, 0.2) 〉, 〈 (a, b), (0.1, 0.2) 〉, 〈 (a, c), (0.4, 0.2) 〉, 〈 (b, a), (0.1, 0.2) 〉, \\ 〈 (b, b), (0.3, 0.2) 〉, 〈 (b, c), (0.3, 0.3) 〉, 〈 (c, a), (0.4, 0.2) 〉, 〈 (c, b), (0.3, 0.3) 〉, 〈 (c, c), (1, 0) 〉}, \\ R_{2} = {〈 (a, a), (0.1, 0.3) 〉, 〈 (a, b), (0.4, 0.2) 〉, 〈 (a, c), (0.3, 0.3) 〉, 〈 (b, a), (0.4, 0.2) 〉, \\ 〈 (b, b), (0.3, 0.2) 〉, 〈 (b, c), (0.1, 0.2) 〉, 〈 (c, a), (0.3, 0.3) 〉, 〈 (c, b), (0.1, 0.2) 〉, 〈 (c, c), (1, 0) 〉}, \\ R_{3} = R_{1} \cup R_{2} \\ = {〈 (a, a), (0.3, 0.2) 〉, 〈 (a, b), (0.4, 0.2) 〉, 〈 (a, c), (0.4, 0.2) 〉, 〈 (b, a), (0.4, 0.2) 〉, \\ 〈 (b, b), (0.3, 0.2) 〉, 〈 (b, c), (0.3, 0.3) 〉, 〈 (c, a), (0.4, 0.2) 〉, 〈 (c, b), (0.3, 0.3) 〉, 〈 (c, c), (1, 0) 〉}, \\ R_{4} = R_{1} \cap R_{2} \\ = {〈 (a, a), (0.1, 0.3) 〉, 〈 (a, b), (0.1, 0.2) 〉, 〈 (a, c), (0.3, 0.3) 〉, 〈 (b, a), (0.1, 0.2) 〉, \\ 〈 (b, b), (0.3, 0.2) 〉, 〈 (b, c), (0.1, 0.2) 〉, 〈 (c, a), (0.3, 0.3) 〉, 〈 (c, b), (0.1, 0.2) 〉, 〈 (c, c), (1, 0) 〉} . \end{matrix}$ (3)

Observe that the subfamily $τ_{1} = {R_{Φ}, R, R_{1}, R_{2}, R_{3}, R_{4}}$ of IFRr( $R$ ) is an IFRr-topology on $R$ since it satisfies the necessary three axioms [O₁] , [O₂] and [O₃] and ( $R, τ_{1}$ ) is an IFRr-topological space. But the subfamily $τ_{2} = {R_{Φ}, R, R_{1}, R_{2}, R_{3}}$ of $R$ is not an IFRr-topology on $R$ since the union R₁ ∪ R₂ = R₃ of two members of τ₂ does not belong to τ₂, i.e. τ₂ does not satisfy the axiom [O₂].

Also, the subfamily $τ_{3} = {R_{Φ}, R, R_{1}, R_{2}, R_{4}}$ of $IFR (R)$ is not an IFRrtopology topology on $R$ since the intersection R₁ ∩ R₂ = R₄ of two members of τ₃ does not belong to τ₃, i.e. τ₃ does not satisfy the axiom [O₃].

Example 3.1.2. Let R be an equivalence relation on the universal set U. Then the pair (U, R) is called a Pawlak approximation space. An equivalence class of R containing ’x’ will be denoted by [x] _R. Let τ be a fuzzy set of U. Now for each α-level set α_r, the lower and upper approximation of α_r with respect defined by:

$\begin{matrix} {\bar{apr}}_{x} (α_{r}) = {x \in U : [x]_{n} \subseteq α_{r}} \\ {\bar{apr}}_{n} (α_{r}) = {x \in U : [x]_{n} \cap α_{r} \neq φ} . \end{matrix}$ (4)

Then the pair $({\underline{apr}}_{s} (α_{r}), {\bar{apr}}_{R} (α_{e}))$ is called a rough set with reference set α_r. For the family of α-level sets, we have a family of lower and upper approximations ${({\underline{apr}}_{R}^{R} (α_{z}))}_{α}$ and ${({\bar{apr}}_{R} (α_{x}))}_{o}$ respectively for α ∈ [0, 1]. These families define a pair of pair of fuzzy sets ${\underline{apr}}_{n} (τ)$ and ${\bar{apv}}_{n} (τ)$ such that ${({\underline{apr}}_{s} (τ))}_{e} = {\underline{apr}}_{n} (α_{r})$ and ${({\bar{apr}}_{s} (r))}_{Q} = {\bar{apv}}_{n} (α_{r})$ . They are defined by the following membership functions:

$\begin{matrix} μ_{{ax}_{x} (τ)} (x) = \sup {α \in [0, 1] : x \in {({\underline{apr}}_{s} (τ))}_{e}} \\ μ_{- \underset{α}{α} (r)} (x) = \sup {α \in [0, 1] : x \in {({\bar{apr}}_{R} (f))}_{α}} forx \in U . \end{matrix}$ (5)

Combination of rough sets with fuzzy sets and intuitionistic fuzzy sets Based on the equivalence relation on the universe of discourse, [24] introduced the lower and upper approximation of fuzzy sets in a Pawlak’s approximation space and obtained a new notion called rough fuzzy sets.

Example 3.1.3. Let (U, R) be a Pawlak approximation space and τ ∈ FS^U. Then the lower and upper rough approximations of τ ∈ FS^∪ in (U, R) are denoted by $\underline{R} (τ)$ and $\bar{R} (τ)$ , respectively, which are fuzzy subsets in U defined by

$\underline{R} (τ) (x) = \land_{ye | r | x} μ_{τ} (y) and \bar{R} (τ) (x) = v_{ye ∣ r, z} μ_{x} (y) for all x \in U .$ (6)

The operators $\underline{R}$ and $\bar{R}$ are called the lower and upper rough fuzzy approximation operators on fuzzy sets. τ is said to be definable in U if $\underline{R} (τ) = \bar{R} (τ)$ ; otherwise $(\underline{R} (τ), \bar{R} (τ))$ is called a rough fuzzy set.

Example 3.1.4. Let $(U, M)$ be a fuzzy approximation space where U is a non-empty universe set and $R$ is a fuzzy similarity relation on U × U. Then for τ ∈ FS^U, the lower and upper rough approximations of τ w.r.t (U, r) are denoted by ↓R (τ) and ↑R (τ) respectively, which are fuzzy subsets in U defined by

$\begin{matrix} ↓ R (τ) (x) = \land_{y \in U} ((1 - ℘ (x, y)) \lor μ_{r} (y)) and \\ ↑ R (τ) (x) = \lor_{yaU} (R (x, y) \land μ_{τ} (y)) for x \in U . \end{matrix}$ (7)

The pair R (τ) (x) is referred to as a fuzzy rough set, and the operators R (τ) (x) are the lower and upper fuzzy rough approximation operators on fuzzy sets, respectively.

Example 3.1.5. Take the equivalence relation R on the empty set U as an example. A Pawlak approximation space is defined as the pair (U,R). The notation [x]_R will be used to indicate that R contains the equivalence class’x ’. Let (f, A) also represent a fuzzy soft set over U. Lower and higher approximation of _((f,A))s with regard to (U,R) are thus represented by $R (x, y)$ respectively, and defined as follows for any-level set μ_τ (y):

$\begin{matrix} {\underline{apr}}_{R} (α_{(f, A)}^{s}) = {x \in U : [x]_{R} \subseteq α_{(f, A)}^{s}} \\ {\bar{apr}}_{R} (α_{(t, A)}^{s}) = {x \in U : [x]_{R} \cap α_{(L, A)}^{s} \neq φ} . \end{matrix}$ (8)

Then the pair $({\underline{apr}}_{R} (α_{(f, A)}^{S}), {\bar{apr}}_{R} (α_{(f, A)}^{S}))$ is called a rough set with reference set $α_{(t, A)}^{s}$ .

Example 3.1.6. For the family of α-level sets, we have a family of lower and upper approximations ${({\underline{apr}}_{R} (α_{(L, A)}^{s}))}_{α}$ and ${({\bar{apr}}_{R} (α_{(f, A)}^{s}))}_{α}$ respectively respectively for α ∈ [0, 1]. These families define a pair of fuzzy sets ${\underline{apr}}_{R} ((f, A))$ and ${\bar{apr}}_{R} ((f, A))$ such that

${({\underline{apr}}_{R} ((f, A)))}_{α} = {\underline{apr}}_{R} (α_{(LA)}^{S}) and {({\bar{apr}}_{R} ((f, A)))}_{α} = {\bar{apr}}_{R} (α_{(f, A)}^{s}) .$ (9)

They are defined by the following membership functions:

$\begin{matrix} = \sup {α \in [0, 1] : x \in {apr}_{R} (α_{(t, A)}^{5})} \\ = \sup {α \in [0, 1] :] x]_{R} \subseteq α_{(t, A)}^{s}}, \\ μ_{\bar{{apr}_{A}} (([, A))} (x) = \sup {α \in [0, 1] : x \in {({\bar{apr}}_{R} ((f, A)))}_{α}} \\ = \sup {α \in [0, 1] : x \in {\bar{apr}}_{R} (α_{(t, A)}^{s})} \\ = \sup {α \in [0, 1] :] x]_{R} \cap α_{(L, A)}^{s} \neq φ} for x \in U . \end{matrix}$ (10)

So for the α-level set $α_{(i, A)}^{S}$ , we get a rough set $({\underline{apr}}_{R} (α_{(f, A)}^{S}), {\bar{apr}}_{R} (α_{(f, A)}^{s}))$ . Type-2 interval valued intuitionistic fuzzy sets are presented, and their fundamental features are studied. After that, type-2 interval valued intuitionistic fuzzy sets and rough sets based on -level cuts are introduced. By applying the -level cut on fuzzy soft sets, we can now create rough sets. Numerous disciplines, such as machine learning, pattern recognition, image analysis, information retrieval, and bioinformatics, make use of topology and sequence as vital mathematical tools for exploratory data mining and common methodologies for statistical data analysis. In this chapter, we offer a new series of IFR-relations in IFRr-topological spaces, along with certain fundamental features over a defined parameter set, and use them to create topological structure (IFRr-topological spaces). Some theorems are shown to hold, just as they do for actual sequences. Some variations from actual situations, however, are mentioned.

4 Results and analysis of proposed work

Theorem 1 Given a scalar γ, there exist matrices P₁ > 0, Q₁ > 0, Q₂ > 0, Q₃ > 0, R₁ > 0, R₂ > 0, T₁ > 0, T₂ > 0, Z₁ > 0, Z₂ > 0, diagonal matrices A_1i > 0, A_2i > 0, symmetric matrices X₁, X₃, Y₁, Y₃ ∈ R^3n×3n, and any matrices X₂, Y₂ ∈ R^3nx3n, N₁, N₂, M₁, M₂ ∈ R^3n×n, and S, G, L with appropriate dimensions such that the following LMIs hold for all i ∈ S:

$\begin{matrix} ω_{0} (δ) + τ ω_{1} < 0, \\ Φ_{0} (δ) + τ ω_{2} < 0, \\ Γ_{1} = [\begin{matrix} X_{1} & X_{2} & N_{1} \\ * & X_{3} & N_{2} \\ * & * & G_{1} \end{matrix}] ⩾ 0, \\ Γ_{2} = [\begin{matrix} Y_{1} & Y_{2} & M_{1} \\ * & Y_{3} & M_{2} \\ * & * & G_{1} \end{matrix}] ⩾ 0, \end{matrix}$ (11)

Where, $\begin{matrix} Φ_{0} (δ) & = Π_{1} + Π_{2} + Π_{3} + Π_{4} + Π_{5} + Π_{6} + Π_{7} + Π_{8} + Π_{9} \\ Φ_{1} & = Ξ, Φ_{2} = r \end{matrix}$ and $\begin{matrix} \begin{matrix} Π_{1} = 2 e_{1} P_{1} e_{12}^{T} + e_{1} \sum_{j = 1}^{N} q_{ij} (δ) P_{j} e_{1}^{T} \\ I_{2} = e_{1} Q_{1} e_{1}^{T} - e_{2} Q_{1} e_{2}^{T} (1 - μ) + e_{4} Q_{2} e_{4}^{T} - e_{5} Q_{2} e_{5}^{T} (1 - μ) + e_{1} Q_{3} e_{1}^{T} - e_{3} Q_{3} e_{3}^{T}, \\ Π_{3} = e_{12} τ R_{1} e_{12}^{T} + e_{1} τ R_{2} e_{1}^{T} \\ + Sym {[\begin{matrix} e_{1} & e_{2} & ɛ_{10} \end{matrix}] N_{1} {(e_{1} - e_{2})}^{T} + [\begin{matrix} e_{1} & e_{2} & e_{10} \end{matrix}] N_{2} {(2 e_{10} - e_{1} - e_{2})}^{T}} \\ + Sym {[\begin{matrix} e_{2} & e_{3} & e_{11} \end{matrix}] M_{1} {(e_{2} - e_{3})}^{T} + [\begin{matrix} e_{2} & e_{3} & e_{11} \end{matrix}] M_{2} {(2 e_{11} - e_{2} - e_{3})}^{T}}, \\ Π_{4} = \frac{τ^{-}}{4} ɛ_{1} T_{1} e_{1}^{T} - ɛ_{8} T_{1} e_{8}^{T}, Π_{5} = \frac{τ^{6}}{36} ɛ_{1} T_{2} e_{1}^{T} - ɛ_{9} T_{2} e_{9}^{T}, \end{matrix} \end{matrix}$

$\begin{matrix} Π_{6} = e_{1} Z_{1} e_{1}^{T} - e_{7} Z_{1} e_{7}^{T} + h^{2} e_{12} Z_{2} e_{12}^{T} + [\begin{matrix} e_{1} & e_{6} & e_{7} \end{matrix}] Ω^{*} {[\begin{matrix} e_{1} & e_{6} & e_{7}^{7} \end{matrix}]}^{T}, \\ I_{7} = - 2 e_{1} {Ge}_{12}^{T} - 2 e_{1} {GD}_{1} e_{1}^{T} + 2 e_{1} {GA}_{i} e_{4}^{T} + 2 e_{1} {GB}_{1} e_{5}^{T} + 2 e_{1} {Le}_{13}^{T} + 2 e_{1} {Le}_{6}^{T} \\ - 2 γ e_{12} {Ge}_{12}^{T} - 2 γ e_{12} {GD}_{1} e_{1}^{T} + 2 γ e_{12} {GA}_{i} e_{4}^{T} + 2 γ e_{12} {GB}_{1} e_{5}^{T} + 2 γ e_{12} {Le}_{13}^{T} + 2 γ e_{12} {Le}_{B}^{T}, \\ Π_{8} = - e_{1} F_{1} A_{11} e_{1}^{T} + e_{1} F_{2} A_{11} e_{4}^{T} + e_{4} F_{2} A_{11} e_{1}^{T} - e_{4} A_{11} e_{4}^{T}, \\ Π_{9} = - e_{2} F_{1} A_{21} e_{2}^{T} + e_{2} F_{2} A_{2 t} e_{5}^{T} + e_{5} F_{2} A_{2 i} e_{2}^{T} - e_{5} A_{2 i} e_{5}^{T}, \\ Ξ = [\begin{matrix} e_{1} & e_{2} & e_{10} \end{matrix}] (X_{1} + \frac{1}{3} X_{3}) {[\begin{matrix} e_{1} & e_{2} & e_{10} \end{matrix}]}^{T} - e_{10} R_{2} e_{10}^{T} \\ r = [\begin{matrix} e_{2} & e_{3} & e_{11} \end{matrix}] (Y_{1} + \frac{1}{3} Y_{3}) {[\begin{matrix} e_{2} & e_{3} & e_{11} \end{matrix}]}^{T} - e_{11} R_{2} e_{11}^{T} \\ ɛ_{i} = {[\begin{matrix} e_{n \times (1 - 1) n} & I_{n \times n} & e_{n \times (12 - 1) n} \end{matrix}]}^{T}, \hat{i} = 1, 2, \dots, 12 . \end{matrix}$ (12)

In addition, the gain matrix is given by K = G^-1L.

Proof: Construct the LKF for error system (23) as:

$\begin{matrix} V (e_{t}, t, r (t)) & = V_{1} (e_{t}, t, r (t)) + V_{2} (e_{t}, t, r (t)) + V_{3} (e_{t}, t, r (t)) \\ . & + V_{4} (e_{t}, t, r (t)) + V_{5} (e_{4}, t, r (t)) + V_{6} (e_{t}, t, r (t)), \end{matrix}$ (13) where

$V_{1} (e_{t}, t, r (t)) = e^{T} (t) P_{i} e (t)$ (14)

$\begin{matrix} V_{2} (e_{t}, t, r (t)) = \int_{t - τ (t)}^{t} e^{T} (s) Q_{1} e (s) d s + \int_{t - τ (t)}^{t} g^{T} (e (s)) Q_{2} g (e (s)) ds \\ + \int_{t - τ}^{t} e^{T} (s) Q_{s} e (s) ds, \\ V_{3} (e_{t}, t, r (t)) = \int_{- τ}^{0} \int_{t + θ}^{t} e^{T} (s) R_{1} e (s) d d d θ + \int_{- τ}^{0} \int_{t + θ}^{t} e^{T} (s) R_{2} e (s) d s d θ, \\ V_{4} (e_{1}, t, r (t)) = \frac{τ^{2}}{2} \int_{- τ}^{0} \int_{θ}^{0} \int_{t + λ}^{t} e^{T} (s) T_{1} e (s) dsd λ d θ, \\ V_{5} (e_{t}, t, r (t)) = \frac{τ^{3}}{6} \int_{- τ}^{0} \int_{a}^{0} \int_{0}^{0} \int_{t + λ}^{t} e^{T} (s) T_{2} e (s) dsd λ d θ d α, \\ V_{6} (e_{t}, t, r (t)) = \int_{t - h_{3}}^{t} e^{T} (s) Z_{1} e (s) ds + h_{M} \int_{- h_{M}}^{0} \int_{t + 9}^{t} e^{T} (s) Z_{2} \dot{e} (s) dsd θ . \end{matrix}$ (15)

For the infinitesimal generator $C$ , we can have

$\begin{matrix} L V (e_{t}, t, i) & = L V_{1} (e_{t}, t, i) + L V_{2} (e_{t}, t, i) + L V_{3} (e_{t}, t, i) \\ + L V_{4} (e_{t}, t, i) + C V_{5} (e_{t}, t, i) + L V_{6} (e_{t}, t, i) \end{matrix}$ (16) where

$\begin{matrix} L V_{1} (e_{t}, t, i) = 2 e^{T} (t) P_{i} e (t) + e^{T} (t) \sum_{j = 1}^{N} q_{ij} (δ) P_{j} e (t), \\ = η^{T} (t) Π_{1} η^{'} (t) . \\ L V_{2} (e_{t}, t, i) = e^{T} (t) Q_{1} e (t) - e^{T} (t - τ (t)) Q_{1} e (t - τ (t)) (1 - μ) \\ + g^{T} (e (t)) Q_{2} g (e (t)) - g^{T} (e (t - r (t))) Q_{2} \\ \times g (e (t - τ (t))) (1 - μ) + e^{T} (t) Q_{3} e (t) \\ - e^{T} (t - τ) Q_{s} e (t - τ), \\ = η^{T} (t) Π_{2} η (t), \\ L V_{3} (e_{t}, t, i) = τ e^{T} (t) R_{1} \dot{e} (t) - \int_{t - τ}^{t} e^{T} (s) R_{1} \dot{e} (s) ds \\ + τ e^{T} (t) R_{2} e (t) - \int_{t - τ}^{t} e^{T} (s) R_{2} e (s) d s, \\ = 1}^{T} (t) {e_{12} {TR}_{1} e_{12}^{T} + e_{1} {TR}_{2} e_{1}^{T}} η (t) \\ - \int_{t - τ}^{t} {\dot{e}}^{T} (s) R_{1} \dot{e} (s) d s - \int_{t - τ}^{t} e^{T} (s) R_{2} e (s) d s \end{matrix}$ (17)

By using Lemma 1.6, it is clear that if (16) and (17) hold, the estimation of the R₁ -dependent integral term becomes

$\begin{matrix} - \int_{t - τ}^{t} {\dot{e}}^{T} (s) R_{1} \dot{e} (s) d s = - \int_{t - τ (t)}^{t} e^{T} (s) R_{1} \dot{e} (s) d s - \int_{t - τ}^{t - τ (t)} e^{T} (s) R_{1} \dot{e} (s) ds \\ ⩽ η^{T} (t) {τ (t) [\begin{matrix} e_{1} & e_{2} & e_{10} \end{matrix}] (X_{1} + \frac{1}{3} X_{3}) {[\begin{matrix} e_{1} & e_{2} & e_{10} \end{matrix}]}^{T} \\ + (τ - τ (t)) [\begin{matrix} e_{2} & e_{3} & e_{11} \end{matrix}] (Y_{1} + \frac{1}{3} Y_{3}) {[\begin{matrix} e_{2} & e_{3} & e_{11} \end{matrix}]}^{T} \\ + Sym [\begin{matrix} e_{1} & e_{2} & e_{10} \end{matrix}] N_{1} {(e_{1} - e_{2})}^{T} \\ + [\begin{matrix} e_{1} & e_{2} & e_{10} \end{matrix}] N_{2} {(2 e_{10} - e_{1} - e_{2})}^{T} \\ + Sym [\begin{matrix} e_{2} & e_{3} & e_{11} \end{matrix}] M_{1} {(e_{2} - e_{3})}^{T} \\ + [\begin{matrix} e_{2} & e_{3} & e_{11} \end{matrix}] M_{2} {(2 e_{11} - e_{2} - e_{3})}^{T}} \end{matrix}$ (18)

By Lemma 1.5, the integral term in (18) becomes

$\begin{matrix} - \int_{t - τ}^{t} e^{T} (s) R_{2} e (s) ds = - \int_{t - τ (t)}^{t} e^{T} (s) R_{2} e (s) d s - \int_{t - τ}^{t - τ (t)} e^{T} (s) R_{2} e (s) d s \\ ⩽ - \frac{1}{τ (t)} {(\int_{t - τ (t)}^{t} e (s) ds)}^{T} R_{2} (\int_{t - τ (t)}^{t} e (s) d s) \\ - \frac{1}{τ - τ (t)} {(\int_{t - τ}^{t - τ (t)} e (s) ds)}^{T} R_{2} (\int_{t - τ}^{t - τ (t)} e (s) ds) \\ = η^{T} (t) {- τ (t) e 10 R_{2} e_{10}^{T} - (τ - τ (t)) e_{11} R_{2} e_{11}^{T}} η (t) . \end{matrix}$ (19)

Thus, it can be known that if Γ₁ ⩾ 0 and Γ₂ ⩾ 0 hold, an estimation of $L V_{3} (ɛ_{i}, t, i)$ can be obtained as

$\begin{matrix} L V_{3} (e_{t}, t, i) ⩽ η^{T} (t) {Π_{3} + τ (t) \equiv + (τ - T (t)) Γ} η (t) \\ C V_{4} (e_{t}, t, i) ⩽ \frac{T^{4}}{4} e^{T} (t) T_{1} e (t) \\ - {(\int_{- τ}^{0} \int_{t + θ}^{t} e (s) d s d θ)}^{T} T_{1} (\int_{- τ}^{0} \int_{t + θ}^{t} e (s) d s d θ) \\ = η^{T} (t) Π_{4} η (t), \end{matrix}$ (20)

$\begin{matrix} L V_{5} (e_{t}, t, \dot{t}) ⩽ \frac{τ^{6}}{36} e^{T} (t) T_{2} e (t) - {(\int_{- τ}^{0} \int_{a}^{0} \int_{t + θ}^{t} e (s) d s d θ d α)}^{T} T_{2} \\ \times (\int_{- τ}^{0} \int_{a}^{0} \int_{t + θ}^{t} e (s) dsd θ d α) \\ = r^{T} (t) Π_{5} η (t), \\ L V_{6} (e_{t}, t, i) = e^{T} (t) Z_{1} e (t) - e^{T} (t - h_{M}) Z_{1} e (t - h_{M}) \\ + h^{2} {\dot{e}}^{T} (t) Z_{2} \dot{e} (t) - h_{M} \int_{t - h_{M}}^{t} {\dot{e}}^{T} (s) Z_{2} \dot{e} (s) d s, \\ ⩽ η^{T} (t) [e_{1} Z_{1} ɛ_{1}^{T} - e_{7} Z_{1} e_{T}^{T} + h_{M}^{2} ɛ_{12} Z_{2} e_{12}^{T}] η (t) \\ - h_{M} \int_{t - h_{M}}^{t} {\dot{e}}^{T} (s) Z_{2} \dot{e} (s) d s . \end{matrix}$ (21)

Let us consider,

${\bar{κ}}_{1} (t) = \int_{t - h (t)}^{t} \dot{e} (s) ds, {\bar{κ}}_{2} (t) = \int_{t - ℏ_{M}}^{t - h (t)} \dot{e} (s) d ds$ (22) with 0 < h (t) < h

$\begin{matrix} h_{M} \int_{t - h_{M}}^{t} e^{T} (s) Z_{2} \dot{ɛ} (s) ds = h_{M} \int_{t - h (t)}^{t} e^{T} (s) Z_{2} \dot{e} (s) ds + h_{M} \int_{t - h_{M}}^{t - h (t)} {\dot{e}}^{T} (s) Z_{2} \dot{e} (s) d s \\ ⩾ \frac{h_{M}}{h_{h} (t)} {\bar{κ}}_{1}^{T} (t) Z_{2} {\bar{K}}_{1} (t) + \frac{h_{M}}{h_{M} - h (t)} {\bar{κ}}_{2}^{T} (t) Z_{2} {\bar{κ}}_{2} (t) \\ ⩾ {[\begin{matrix} {\bar{κ}}_{1} (t) \\ {\bar{κ}}_{2} (t) \end{matrix}]}^{T} [\begin{matrix} Z_{2} & S \\ * & Z_{2} \end{matrix}] [\begin{matrix} {\bar{K}}_{1} (t) \\ {\bar{K}}_{2} (t) \end{matrix}] . \end{matrix}$ (23)

In particular, when h (t) = 0 or h (t) = h_M, we have ${\bar{κ}}_{1} (t) = 0$ or ${\bar{κ}}_{2} (t) = 0$ , respectively. Thus, (24) becomes

$- h \int_{t - h}^{t} {\dot{e}}^{T} (s) Z_{2} \dot{e} (s) ds ⩽ ϖ^{T} (t) Ω^{*} ϖ (t)$ (24) where

$\begin{matrix} w (t) & = {[\begin{matrix} e^{T} (t) & e^{T} (t - h (t)) & e^{T} (t - h_{M}) \end{matrix}]}^{T} \\ Ω^{*} & = [\begin{matrix} - Z_{2} & Z_{2} - S & S \\ * & - 2 Z_{2} + S + S^{T} & - S + Z_{2} \\ * & * & - Z_{2} \end{matrix}] . \end{matrix}$ (25)

Therefore,

$C V_{6} (e_{i}, t, i) ⩽ η^{T} (t) Π_{6} η (t) .$ (26)

For any matrix G and scalar γ, the following holds:

$\begin{matrix} 0 = & 2 [e^{T} (t) G + γ e^{T} (t) G] [- \dot{e} (t) - D_{i} e (t) + A_{i} g (e (t)) \\ + B_{i} g (e (t - τ (t))) + Kz (t) + Ke (t - h (t))], \\ = & η^{T} (t) Π_{ı} η (t) . \end{matrix}$ (27)

Form Assumption 3.1, for positive diagonal matrices ${\tilde{A}}_{1 i}$ and ${\tilde{A}}_{2 i}$ , we have

$\begin{matrix} 0 ⩽ - e^{T} (t) F_{1} {\tilde{Λ}}_{1 i} + e^{T} (t) F_{2} {\bar{A}}_{i i} g (e (t)) + g^{T} (e (t)) F_{2} {\bar{A}}_{ki} e (t) \\ - g^{T} (e (t)) {\bar{A}}_{ti} g (e (t)) \\ = η^{T} (t) Π_{8} η (t), \\ 0 ⩽ - e^{T} (t - τ (t)) F_{1} {\bar{Λ}}_{2 i} e (t - τ (t)) + e^{T} (t - τ (t)) F_{2} {\bar{A}}_{2 i} \\ \times g (e (t - τ (t))) + g^{T} (e (t)) F_{2} {\bar{A}}_{2 i} e (t - τ (t)) \\ - g^{T} (e (t - τ (t))) {\bar{A}}_{1 i} g (e (t - τ (t))) \\ = η^{T} (t) Π_{9} η (t) . \end{matrix}$ (28)

By Equations (36)–(37) and adding (32)–(36) with $L V (e_{t}, t, t)$ , one can have

$E {C V (e_{t}, t, i)} ⩽ E {η^{T} (t) [ω_{0} (δ) + τ (t) ω_{1} + (τ - τ (t)) ω_{2}] η (t)} ⩽ 0$ (29) where Φ₀, Φ₁ and Φ₂. Based on convex combination approach, we can know that if (20) - (28) bold, then $C V (e_{t}, t, i) < 0$ . From the above, we say that the slave system (29) is globally asymptotically synchronized with the master system (39) by Lyapunov stability theory. This completes the proof.

An IT1 TS fuzzy system (20) with three fuzzy rules is considered in this example. The associated system parameters are selected as

$\begin{matrix} M_{1} = [\begin{matrix} - 0.2140 & 0.1741 \\ 0.0649 & 0.9960 \end{matrix}], M_{2} = [\begin{matrix} - 0.2591 & 0.0265 \\ 2.0526 & 0.5940 \end{matrix}], \\ M_{3} = [\begin{matrix} - 0.2176 & 0.6312 \\ - 1.0196 & 0.6512 \end{matrix}], M_{d 1} = [\begin{matrix} 0 & 0.014 \\ 0 & 0.011 \end{matrix}], M_{d 2} = [\begin{matrix} 0 & 0.013 \\ 0 & 0.002 \end{matrix}], \\ M_{d 3} = [\begin{matrix} 0 & 0.011 \\ 0 & 0.012 \end{matrix}], N 1 = [\begin{matrix} - 0.1012 \\ - 0.0001 \end{matrix}], N = [\begin{matrix} - 0.0003 \\ 0.0010 \end{matrix}], \\ N_{3} = [\begin{matrix} - 1.1010 \\ - 0.0201 \end{matrix}], E_{1} = [\begin{matrix} - 0.0568 \\ - 0.0732 \end{matrix}], E_{2} = [\begin{matrix} 0.0532 \\ 0.0961 \end{matrix}], E_{3} = [\begin{matrix} - 0.0660 \\ - 0.0703 \end{matrix}], \\ S_{1} = S_{2} = S_{3} = [\begin{matrix} 1 & 0 \end{matrix}], S_{d 1} = S_{d 2} = S_{d 3} = [\begin{matrix} 0.01 & 0.19 \end{matrix}], F_{1} = 0.0834, \\ F_{2} = 0.1043, and F_{3} = 0.0088 . \end{matrix}$ (29)

Figure 1 is proposed controller of Simulink structure that is adopted fuzzy set theory of proposed dataset. The input from the external disturbance is set at w(k)=0.1 rand, and the parameters for the feasibility test are set as follows: γ_min = 0.56, α = 0.5, ζ₀ = 0.7, ζ₁ = 0.1. Further, we choose a time-varying delay of 0.6 ⩽ Σ ⩽ 0.8 and 2 ⩽ μ (k) ⩽ 4.

Fig. 1

Fuzzy controller using Simulink structure.

The corresponding nonlinear weight coefficient functions are taken as ${\underline{a}}_{1} (s) = {\underline{a}}_{3} (s) = {\underline{b}}_{1} (s) = \cos^{2} (s_{1}), {\underline{a}}_{2} (s) = {\underline{b}}_{2} (s) = \sin^{2} (s_{1})$ , ${\bar{a}}_{p} (s) = 1 - {\underline{a}}_{p} (s)$ and ${\bar{b}}_{q} (s) = 1 - {\underline{b}}_{q} (s) (p = 1, 2, 3 & q = 1, 2)$ given in Tables 2 and 3. In addition, we divide the state vector of the system into 10 equal-sized sub-states using the formula s₁ (k) = s₁ ∈ [- 81, 81]. Further, the lower and upper limits of sub-states of the system state vector s₁ may be written as s_1,1 = 16.2 × (1 - 6) and ${\bar{s}}_{1, t} = 16.2 \times (1 - 5)$ , respectively and define $κ_{1 h} (s_{1}) = 1 - \frac{s_{1} - ɛ_{1, 1}}{s_{1, 1} - ɛ_{1, 2}}, κ_{12 t} (s_{1}) =$ 1 - κ₁₁ (s₁).

In addition, the lower and upper membership functions of the scalars may be calculated as ${\underline{σ}}_{pq 1 l} = {\underline{β}}_{p} ({\underline{s}}_{1, 1}) {\underline{η}}_{q} ({\underline{s}}_{1, 1}), {\underline{σ}}_{pq 2_{1}} = {\underline{β}}_{p} ({\bar{s}}_{1, 1}) {\underline{η}}_{q} ({\bar{s}}_{1, 1})$ , ${\bar{σ}}_{pq 1 l} = {\bar{β}}_{p} ({\underline{s}}_{1, 1}) {\bar{η}}_{q} ({\underline{s}}_{1, 1})$ and ${\bar{σ}}_{pq 2 L} = {\bar{β}}_{p} ({\bar{s}}_{1, 1}) {\bar{η}}_{q} ({\bar{s}}_{1, 1}), \forall l = 1, 2, \dots, 10$ . We may acquire three sets of workable solutions based on the aforementioned parameter values: (i) Fault-tolerant memory state feedback control, (ii) Memoryless state feedback control $(B_{q} = 0)$ and (iii) Memory state feedback control (u^f (k) = u (k)). The control parameters for the possible solutions are then identified and stated in a clear fashion, as follows:

Table 2

Bounds of membership functions of the plant

Lower bounds	Upper bounds
$Ψ_{V_{1}^{1}} (s_{1}) = 1 - \exp (\frac{- s_{1}^{1}}{1.2})$	$Ψ_{V_{1}} (s_{1}) = 1 - 0.13 \exp (\frac{- 5_{1}^{5}}{0.3})$
$Ψ_{V_{1}^{2}} (s_{1}) = 0.5 \exp (\frac{- s_{1}^{1}}{0 .})$	$Ψ_{V_{1}^{2}} (s_{1}) = \exp (\frac{- s_{1}^{2}}{2.5})$
${\underline{Ψ}}_{V_{1}^{3}} (s_{1}) = 1 - Ψ_{V_{1}^{2}} (s_{1})$	$Ψ_{V_{1}^{s}} (s_{1}) = 1 - {\underline{Ψ}}_{V_{1}^{2}} (s_{1})$

Table 3

Bounds of membership functions the controller

Lower bounds	Upper bounds
${\underline{φ}}_{W_{1}^{1}} (s_{1}) = \exp (- \frac{s_{1}^{2}}{0.35})$	${\bar{φ}}_{W_{1}^{1}} (s_{1}) = \exp (\frac{- s_{1}^{1}}{0.35})$
${\underline{φ}}_{W_{1}^{2}} (s_{1}) = 1 - {\bar{φ}}_{W_{1}^{1}} (s_{1})$	${\bar{φ}}_{W_{1}^{2}} (s_{1}) = 1 - {\underline{φ}}_{W_{1}^{2}} (s_{1})$

(i) Parameters for controlling the feedback condition of fault-tolerant memories

$\begin{matrix} A_{1} & = [\begin{matrix} - 0.1874 & 0.2026 \end{matrix}], \\ A_{2} = [\begin{matrix} - 0.2178 & 0.2327 \end{matrix}], \\ B_{1} = [\begin{matrix} - 0.1738 & - 0.0299 \end{matrix}], \\ B_{2} = [\begin{matrix} - 0.1961 & - 0.0309 \end{matrix}] . \end{matrix}$ (40)

(ii) Parameters for feedback control in a state with no memory

$A_{1} = [\begin{matrix} - 0.6254 & 0.6256 \end{matrix}], A_{2} = [\begin{matrix} - 0.6291 & 0.6284 \end{matrix}] .$ (41)

(iii) Parameters for regulating the memory state’s feedback

$\begin{matrix} A_{1} = [\begin{matrix} - 0.7128 & 0.6002 \end{matrix}], A_{2} = [\begin{matrix} - 0.7181 & 0.6025 \end{matrix}] \\ B_{1} = [\begin{matrix} - 0.4614 & 0.0067 \end{matrix}], B_{2} = [\begin{matrix} - 0.4719 & 0.0068 \end{matrix}] \end{matrix}$ (42)

Figure 2 shown as lower and higher membership line of proposed system. Performance of the closed-loop and open-loop systems are presented in Fig. 5 by replacing the studied IT1 TS fuzzy system (42) with the aforementioned three sets of gain values and taking into account the beginning condition s (0) = [- π/2π/2] ^T.

Fig. 2

Lower and upper membership lines.

A comparison of the system’s performance with and without a memoryless state feedback controller, as well as a memory state feedback controller, is shown graphically. Figures 3 and 4 depict the input and output of the control system. Figure 5 depicted as control output performance of the proposed model.

Fig. 3

Performance of closed and open loop system.

Fig. 4

Control input performance.

Fig. 5

Control output performance.

The investigated IT1 TS fuzzy system (23) is asymptotically stable with the required extended passive performance index, despite the presence of time-varying latency and actuator failures.

Taking into account the state delay in the feedback control law, we examine the state feedback stabilisation issue for the T-S fuzzy time-varying delay system. The T-S fuzzy time-delay system’s stabilisation criteria in the LMI frameworks have been improved. Convex combination of LMIs and the HOPBRII method are used to calculate the fuzzy controller gains for a given delay upper limit in the proposed TSFLF LMI based stabilisation conditions. The usefulness of the suggested TSFLF stabilising conditions to enhance the delay upper limit and reduce the control effort is shown through many numerical examples. In addition, a delayed state feedback controller is constructed in this chapter for a highly nonlinear pendulum system utilising the T-S fuzzy control approach. The simulation results validate the efficacy of the T-S fuzzy model with delay feedback control in balancing the inverted pendulum system at the equilibrium points, demonstrating the simplicity and effectiveness of this approach to solving a previously unanswered problem. Six UCI data units as shown in Table 4, were chosen from the University of California at Irvine’s device studying data repository. The static algorithm like MSTW (Minimum spanning Tree Weight) and incremental algorithm (Minimum Incremental Three Way-Objective Variant) MITW-OV algorithms are run on a personal computer equipped with an Intel CPU i7 4790, 3.6GHZ, and 8 GB of memory; the operating system is Windows 10, and the programming environment is C++ of Visual Studio 2007.

Table 4

Description of the data sets

S.No	Datasets	* No. of objectives	* No. of Attributes	* No. of Classes	Completion status
1	Dermatology	360	32	8	NO
2	Annealing	790	34	8	NO
3	Mice	1085	79	7	NO
4	Statlog	6432	39	7	YES
5	Mushroom	8130	23	4	YES
6	Bankruptcy	10621	64	3	YES

We divide the original records set into two amounts for each records set of nine information units shown in Table 4: (1) we randomly choose 75% of the authentic facts set as the simple facts set; and (2) the final 25% of the whole facts set are utilised as the additional gadgets. The first one-third of the objects are utilised as the deleted item set, the second one-third of the objects appear as the modified object set, and the last one-third of the objects stay untouched. These three parts make up the basic statistics set. Figure 6 shows the distribution schematic diagram of the operated object blocks. (in which, ODS denotes the authentic information set and BDS denotes the bottom information set).

Fig. 6

Distribution schematic diagram of operated objects.

As a consequence, 10%, 20%,…, 100% of the objects in the upload gadgets block are handled as incremental objects, 10%, 20%,…, 100% of the objects in the deleted items block are treated as things to get rid of, and 10%, 20%,…, 100% of the objects in the altered object block are treated as items to modify. The size (i.e., 10%, 20%,…, 100%) within as a consequence. Instead of being tied to the actual statistics set, Fig. 6 shows the percentage of the revised item in relation to the operational information block.

In the subsection, the percentage of modified and added items increases sequentially within the same range (i.e., 10%, 20%,…, 100% items in the modified object block and the brought object block are simultaneously chosen in turn to be modified and deleted from HIDS), whereas the percentage of deleted items maintains four distinct ranges (i.e., 10%, 40%, 70%, 100% objects of the deleting object block). The abbreviation for this situation is MA-D. For the scenario MA-D, we set up tests to assess the effectiveness of the incremental MITW-OV with the static set of rules MSTW. The x-axis in the sub-graph as displayed in Fig. 7. Three represents the ratios of the changed items and the delivered items (i.e., ‘Ratio of MA-objects’ for quick), the y-axis represents the ratios of the deleted items (i.e., ‘Ratio of D-items’ for quick), and the z-axis represents the algorithm strolling time. Additionally, hollow circle traces show the MITW-OV running time, and asterisk lines reveal the MSTW jogging duration.

Fig. 7

Comparison of computational time of algorithms MSTW and MITW-OV for MA-D situation.

The subsection’s modified items, delivered items, and deleted items all increase in share in the same order (specifically, 10%, 20%, …, 100% of the items in the modified item block, added item block, and deleted item block are simultaneously chosen to be modified, added to HIDS, and deleted from HIDS, respectively). The case in question is known as MAD in short. In order to compare the overall performance of the static method MSTW with the incremental MITW-OV for the example MAD as shown in Fig. 8. As shown in each sub-graph of 8, from the positive direction of the x-axis, it shows that the computational time of algorithms MSTW and MITW-OV monotonously decreases with the increase of increment ratio of the modified and added objects. From the reverse direction of the y-axis, it shows that the computational time of algorithms MSTW and MITW-OV increases with the increase of updating ratio of the added objects.

Fig. 8

Comparison of computational time of algorithms MSTW and MITW-OV for MAD situation.

5 Conclusion and future work

Through the development of a novel TSFLF and an original integral inequality, a novel LF with extended dissipative properties was produced. To demonstrate the practicality of the offered solutions, several numerical simulations are provided to demonstrate the efficiency and value of the proposed method using a real-world benchmark issue. Few fascinating and hard subjects exist outside of the findings reported in this Proposed work. Discrete time-delay systems with challenging nonlinear activation functions may also benefit from the enhanced DRD stability criteria suggested in this Proposed work. It will be worthwhile to think about the issues of state estimation and synchronisation for T-S fuzzy Discrete time-delay systems when these systems are subjected to external disturbances. Adaptive control, sliding mode control, output tracking control, and many more may be used to address the stability issue within a data-driven framework, yielding outcomes with greater relevance to industry. There is still a discrepancy between the time-delay T-S fuzzy model and the original nonlinear Discrete time-delay systems, which are represented by the time-delay T-S fuzzy model. Considering the approximation error between the underlying nonlinear systems and the fuzzy models makes the challenges considerably more complicated and difficult. Applications in practical control systems are expected to benefit greatly from this line of inquiry, since it has the potential to bridge the gap between nonlinear Discrete time-delay systems and time-delay T-S fuzzy systems.

Footnotes

The Appendices are available in the electronic version of this article: .

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