# Semigroup of Finite-State Deterministic Intuitionistic Fuzzy Automata with Application in Fault Diagnosis of an Aircraft Twin-Spool Turbofan Engine

Authors: Sajida Kousar; Farah Aslam; Nasreen Kausar; Yaé Ulrich Gaba
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The twin-spool turbofan engine is an important component of almost every modern aircraft. Fault detection at an early stage can improve engine performance and health. The current research is based on the construction of an inference system for fault diagnosis in a generalized fuzzy environment. For such an inference system, finite-state deterministic intuitionistic fuzzy automata (FDIFA) are established. A semigroup of FDIFA and its algebraic properties including substructures and structure-preserving maps are studied. The FDIFA semigroups are used as variables for the inference system, and FDIFA semigroup homomorphisms are used to indicate the relation between variables. The newly established model is then applied to diagnose the possible fault and their nature in aircraft twin-spool turbofan engines by modelling the performance of the supercharger and air cooler.


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## Body

## 1. Introduction

During the last several decades, significant investigation and research have been conducted to increase aircraft safety by enhancing the performance of diagnostic systems for aircraft components, especially the gas turbine engines. Improved safety [1], consistency, and operational costs are just a few of the advantages that such systems can give to aircraft operators. With the considerable expansion in air traffic forecast in the coming years, the demand for improved diagnostic methods is expected to continue to rise. In general, aircraft engine performance diagnostics are achieved by predicting the performance, health, and strength parameters of certain components of the engine based on the accessible sensor measurements [2–4]. Productivity, efficiency, and flow capability of engine components are some of the health parameters that indicate the performance effectiveness or decline of the component. The sensor readings comprise gas path pressures and temperatures, fuel flows, and spool speeds and provides essential data on the engine’s health. The semigroup is an important algebraic structure, serving as the theoretical foundation for a variety of scientific fields with several applications [5]. The semigroup’s role in theoretical computer science is inevitable in particular; the semigroup and automata are widely studied and applied in artificial intelligence, game theory, dynamical system, system biology, and fault diagnosis. A finite-state machine (FSM) or finite-state automata (FSA) are a mathematical model of computation that can be changed from one state to another state in connection to suitable inputs. There are two types of FSMs: deterministic (DFSM) the one that accepts or rejects a given string of inputs, following a state sequence uniquely obtained from the string, and nondeterministic (NDFSM) which does not obey these restrictions. Moreover, for each NDFSM, an equivalent DFSM can be constructed. The relation between the semigroup and FSA was established by Krohn and Rhodes [6], and they discussed semigroup decomposition employing decomposition of FSA.During the investigation, it was discovered that to diagnose a machine’s problem situation, it is necessary to look at its operating state, the fault degree, the accuracy degree, and expected changes between states at different stages. That can be done by establishing an appropriate FSA. The uncertainty and ambiguity in the diagnosis process enforce to carry out mathematical computation in a fuzzy environment. Fuzzy automata are used to handle system uncertainties more accurately where classical automata fail to cater to the circumstances. Fuzzy automata have been frequently employed since the introduction of fuzzy technology and neural networks [7–13]. Furthermore, there were a variety of problems to be resolved, for example, a car anticrash radar, freeway management, urban road traffic control, and obstacle recognition in front of a vehicle which required flexible, quick, and accurate decisions. Fuzzy neural network automata (FNNA) [14, 15] are an excellent choice. FNNA had an increasingly prominent role, particularly in data communications. In FNNA, fuzzy technology is used to compare with ordinary identification and control devices utilizing several features and techniques of the neural network including fast and accurate numerical calculation of large numbers and self-organization learning ability. As a result, FNNA was able to solve a variety of problems in a flexible, quick, and precise manner and it had several advantages such as the compact size, being lightweight, and stability. The research findings also revealed that it has a significant ability for self-organization learning and the experimental results demonstrated fuzzy automata’s superiority. Lvzhou and Daowen [16] introduced the technique of minimizing fuzzy automata and constructed a new automaton system that dealt accurately with many states at a time. They used an ordered lattice to reduce a fuzzy automaton to another fuzzy automaton with m states that were functionally correspondent to the original fuzzy automata. Several language-preserving methods for minimizing deterministic fuzzy automata were established. The Moghari and Zahedi [17] method preserves both language and behavior in the minimization process. De Mendivil and Garitagoitia [18] described factorization of fuzzy states that is used in the determination of fuzzy automata. Stamenkovic et al. [19] studied fuzzy automata and the reduction of fuzzy automata states and relational equations of the fuzzy system by considering the solution of the fuzzy equivalent equation. These solutions are then used to reduce the number of acceptable states. The solution of the fuzzy system of the quasiorder number was also used to reduce the states of the fuzzy automaton.Fuzzy inference is a method of formulating a rule or a mapping from a given set of inputs to a given set of outputs utilizing the fuzzy logic. The fuzzy inference system is widely used in control systems [20], artificial intelligence [21], and game theory [22]. The fuzzy set over a nonempty set X as defined by Zadeh [23] is a function (membership function) from X to the closed interval 0,1 so the representation of uncertain data by using closed interval is equivalent to the representation by using fuzzy sets [24, 25]. The fuzzy inference of fuzzy automata is introduced and utilized in machine defect diagnosis as the impact of the combination of fuzzy automaton rules and the method of the fuzzy set is greater than the classical automata. For the derivation of automata, some researchers used neural networks [24, 26, 27]. Although neural networks and fuzzy systems are not the same, fuzzy inference has been widely applied in the disciplines of control and intelligence [28]. The fuzzy set proposed by Zadeh is based on the formulation of the membership function from X to 0,1, where the images are termed as membership grades or degrees of membership of elements of X. Atansassov [29] proposed the notion of an intuitionistic fuzzy set (IFS) which is an extension of the perception of the fuzzy set where the degree of nonmembership is also considered along with the degree of membership. Fuzzy inference is a method of formulating a rule or a mapping from a given set of inputs to a given set of outputs utilizing the fuzzy logic. The fuzzy inference system and fuzzy automata and algebraic structures on fuzzy automata are investigated and successfully applied in computer science and engineering [30]. IFS provides a more effective framework to handle imprecision and uncertainties more accurately than the fuzzy set and logic. The work is based on the development of inference systems and automata in an intuitionistic fuzzy environment. Intuitionistic fuzzy automata have several states at a certain time or stage, each equipped with certain grade of reliability and nonreliability. Thus, the intuitionistic fuzzy automata are more realistic and close to the objective. To achieve the goal firstly, finite-state deterministic intuitionistic fuzzy automata (FDIFA) are defined. Secondly, a semigroup on intuitionistic fuzzy automata is designed and investigated its algebraic properties and intuitionistic fuzzy inference rules on the FDIFA semigroup are established. The mathematical computations are supported and justified by conduction fault diagnosis in the aircraft twin-spool turbofan engine.


## 2. Finite-State Deterministic Intuitionistic Fuzzy Automata (FDIFA)

The septupleMT=R,Z,ζ,r0,H,P,E is termed as finite-state deterministic intuitionistic fuzzy automata (FDIFA), where R is a finite collection of states, Z is a finite collection of input characters, r0∈R is the initial state, H is an intuitionistic fuzzy set of final states, P,E are sets of fuzzy grades of the membership and nonmembership at the state transition and P,E⊆0,1, and ζ:R×Z×P×E⟶R is the transition mapping with ζri,z,μ,ν=rj, where ri,rj∈R,z∈Z,μ∈Pandν∈E. These states rj∈R exist in some accepted states. The extended transition function ζ∗:R×Z∗×P×E⟶R is defined as follows:
(1)ζ∗r,ε,μ,ν=r,ζ∗r,ϖz,μ,ν=ζζ∗r,ϖ,μ′,ν′,z,μ,ν,whereε,ϖ∈Z∗ (set of finite strings/words/sequences of input characters), ∀z∈Z,μ,μ′∈P, and ν,ν′∈E. Let H be a intuitionistic fuzzy set of final states; the cut set Hα,γ of H is given as follows:
(2)Hα,γ=x∈R:μHx≥α,νHx≤γ,where α,γ∈0,1 are fixed numbers, such that α+γ≤1. The set
(3)Hα=x,μHx,νHx:x∈RandμHx≥αis called the set of the level of membership α generated by H. Also,
(4)Hγ=x,μHx,νHx:x∈RandνHx≤γis called the set of the level of nonmembership γ generated by H. Clearly,
(5)Hα,γ⊂Hα,Hγ⊂H.Example 1.
Consider(1)
R=r0,r1,r2,r3,r4,r5,r6,r7,r8 the set of input states with r0 as the initial state(2)
Z=z1,z2,z3,z4,z5 the set of input characters(3)
P=1,0.9,0.7,0.5,0.4,0.3,0.2 the set of state transition membership grades(4)
E=0,0.1,0.2,0.3,0.5,0.7 the set of state transition nonmembership gradesDefine the transition mappingζ:R×Z×P×E⟶R for the accepting states r1,r2,r3,r4,r5,r6,r7 as follows:
(6)ζr0,z4,0.4,0.2=r1,ζr0,z2,0.2,0.5=r4,ζr1,z1,0.7,0.1=r2,ζr2,z5,0.9,0=r3,ζr4,z3,0.3,0.7=r5,ζr5,z5,0.5,0.3=r6,ζr6,z2,1,0=r7,with the transition diagram in Figure 1.Figure 1
State transition diagram.Then,H=r1,0.4,0.2,r4,0.2,0.5,r2,0.7,0.1,r3,0.9,0,r5,0.3,0.7,r6,0.5,0.3,r7,1,0 is an intuitionistic fuzzy set of final states. Thus, MT=R,Z,ζ,r0,H,P,E is an FDIFA.


## 3. Semigroup of the Finite-State Deterministic Intuitionistic Fuzzy Automata

To define the semigroup structure of FDIFA, first, we need a binary operation that involves state transition membership and nonmembership grades.Definition 1.
LetMT=R,Z,ζ,r0,H,P,E be an FDIFA. For any input string or sequence ϖ∈Z∗, each state q∈R is transited to another state r∈H with membership μ and nonmembership ν. As a result, the triplet ϖ,μ,ν induces a state transition ζri,ϖ,μ,ν=rj from R to H. If ri and rk are two states in R such that ζri,ϖ,μ,ν=rj and ζrk,ϖ,μ,ν=rl, respectively, the overall transition through states ri and rk is denoted by rj∘rl=rn. Clearly, ∘ is a binary operation on FDIFA such that for any mt1,mt2andmt3∈MT with
(7)mt1=R1,Z1,ζ1,r01,H1,P1,E1,ζ1ri1,a1,μ1,ν1=rj1,mt2=R2,Z2,ζ2,r02,H2,P2,E2,ζ2ri2,a2,μ2,ν2=rj2,mt3=R3,Z3,ζ3,r03,H3,P3,E3,ζ3ri3,a3,μ3,ν3=rj3,where for i=1,2,3, the set Ri is the collection of states, Zi is the collection input symbols, Hi is the intuitionistic fuzzy set of final states, and r0i is the initial state. We have
(8)mt1∘mt2∘mt3=mt1∘mt2∘mt3,which turns FDIFA MT into a semigroup, called an FDIFA semigroup. The expression mt1∘mt2 represents the overall state transition through ζ1 and ζ2 in the intuitionistic fuzzy automata mt1 and mt2, respectively.The FDIFAMT,∘ is a called a monoid (semigroup with identity) if there exist a fuzzy automata e∈MT such that mt1∘e=e∘mt1=mt1 for all mt1∈MT. Clearly, the identity element is unique in MT. Being a semigroup, the elements of MT must satisfy the power axioms of semigroups that are as follows:
(1)
mtn=mt∘mt∘,⋯,∘mt (n times)(2)
mtl∘mtn=mtl+n(3)
mtln=mtln(4)
mt0=efor all l,n∈ℤ+. FDIFA semigroup MT is termed to be commutative if the binary operation ∘ is commutative. FDIFA semigroup MT is called Boolean if all of its elements are idempotent, that is, mt∘mt=mt for all mt∈MT. If MT is a monoid, then, the identity element is an essential idempotent in MT. An FDIFA monoid MT,∘ is called a cyclic if there is an element mt∈MT such that MT=mtn:n∈ℤ+. That is, all the elements of MT are expressed as nonnegative powers of mt.Proposition 2.
Every FDIFA cyclic monoid automatonMT is commutative.Proof.
LetMT,∘ be a FDIFA cyclic monoid automaton generated by mt. Now, for mt1,mt2∈MT, there exist l,k∈ℤ+ such that mt1=mtl and mt2=mtk. Thus, mt1∘mt2=mtl∘mtk=mtl+k=mtk+l=mt2∘mt1 imply that MT is commutative.Definition 3.
LetMT,∘ be an FDIFA semigroup, where
(9)MT=Rn,Zn,ζn,r0n,Hn,Pn,En:n∈N,ζnri,a,μ,ν=rj.
Consider that(10)MTi=Ri,Zi,ζi,r0i,Hi,Pi,Ei:i∈N,ζirk,`a,`μ,`ν=rl,where rj∈Hn and rl∈Hi, Ri⊆Rn,Zi⊆Zn,ζirk,`a,`μ,`ν⊆ζnri,a,μ,ν, r0i=r0n,Hi⊆Hn,andPi⊆Pn. If MTi is closed under the binary operation ∘; symbolically, mt1∘mt2∈MTi∀mt1andmt2∈MTi. Equivalently, rj∘rl=rn⊆Hi, then, MTi,∘ is known as a subsemigroup automaton of MT,∘. If MT is a monoid with identity e and e belongs to the submonoid MT1, then, MT1 is known as a submonoid automaton of MT.Proposition 4.
For any FDIFA commutative monoidMT,∘, the set of idempotent MTI is a submonoid automaton.Proof.
Assume thatMTI is set of all idempotents of MT. Clearly, the identity element e∈MTI. As MTI is a set of idempotents, so, mtk1∘mtk1=mtk1,mtk2∘mtk2=mtk2,∀mtk1,mtk2∈MTI. The commutativity of MT imply that mtk1∘mtk22=mtk1∘mtk2∘mtk1∘mtk2=mtk1∘mtk1∘mtk2∘mtk2=mtk1∘mtk2. Thus, mtk1∘mtk2∈K and MTI,∘ is a submonoid.Definition 5.
Any two FDIFA semigroups can be related to each other employing of structure-preserving maps, commonly known as homomorphisms. Such maps not only preserve the binary operations used to design semigroups on FDIFA but also the components involved in the formulation of FDIFAs. Formally, ifMT,∘ and MT′,∘′ are any two FDIFA semigroups, then, a map θ:MT⟶MT′ is called an FDIFA semigroup homomorphism, if ∀mt1,mt2∈MT, θmt1∘mt2=θmt1∘′θmt2. If θ is a homomorphism that also establishes a one-to-one correspondence between MT and MT′, then, it is termed to be an isomorphism and the FDIFA semigroups are denoted by MT≃MT′.Theorem 6.
LetMT,∘, MT′,∘′, and MT′′,∘′′ be any three FDIFA semigroups and θ:MT⟶MT′ and θ′:MT′⟶MT′′ be an FDIFA semigroup homomorphism.
(1)
Ifmt∈MTI, then, θmt∈MTI′(2)
Ife is the identity element of MT, then, θe is the identity element of θMT. If θ is an isomorphism, then, θe is the identity element of MT′(3)
IfMT1 is a subsemigroup automaton of MT, the θMT1 is a subsemigroup automaton of MT′(4)
θ′∘θ:MT⟶MT′′ is an FDIFA semigroup homomorphismProof.
(1)
mt∘mt=mt imply that θmt=θmt∘mt=θmt∘′θmt. Thus, θmt is idempotent(2)
mt∘e=mt=e∘mt imply that θmt∘′θe=θmt∘e=θmt=θe∘mt=θe∘′θmt. Thus, θe is the identity element of θMT(3)
mt1∘mt2∈MT1 imply that θmt1∘′θmt2∈θMT1. Thus, θMT1 is an FDIFA subsemigroup of MT′(4)
Asθ′∘θmt1∘mt2=θ′θmt1∘′θmt2=θ′θmt1∘′′θ′θmt2=θ′∘θmt1∘′′θ′∘θmt2, we get that θ′∘θ is an FDIFA semigroup homomorphism


## 4. Intuitionistic Fuzzy Inference Rule on the FDIFA Semigroup

On the semigroup of FDIFA, intuitionistic fuzzy inference is a cognitive procedure that proceeds to a new decision or statement based on one or more previous decisions or propositions. In general, the FDIFA inference is divided into two components. The premise is a well-known decision used as a starting point for inference. The second is the conclusion, which is a fresh decision generated by the proposition. The following inference rules are offered based on the characteristics of homomorphic mapping:(Premise1)
Ifa∈X semigroup, then, b∈Y must be a semigroup(Premise2)
Ifa∈X′ is a semigroupThe conclusion is thatb∈Y′=X′∘X⟶Y, where ∘ is a binary operation and X⟶Y is a homomorphism.The intuitionistic fuzzy inference system is made up of certain inference rules that must obey certain arithmetic requirements. The relation betweenX and Y for premise1 is X⟶Y, which is a homomorphism, grades of membership, and nonmembership corresponding to an intuitionistic fuzzy relation matrix G, which is defined as follows:
(11)Gp,q=μX⟶Yp,q,νX⟶Yp,q=μXp∧μYq∨1−μXp,νXp∨νYq∧1−νXp−ε,where μX⟶Yp,q,νX⟶Yp,q are the grades of membership and nonmembership for the intuitionistic inference rule, ε is a error of hesitancy, μYq,νYq are the grades of membership and nonmembership for the assumption that q∈Y is a semigroup, and μXp,νXp are the grades of membership and nonmembership for the assumption that p∈X is a semigroup and ∨ is an “or” operation and ∧ is an “and” operation:
(12)Y′=X′∘X⟶Y.The inference relationship betweenX′ and from X to Y can be used to synthesize the conclusion Y′. G can be used to obtain the conclusion’s membership and nonmembership functions:
(13)μY′q,νY′q=μX′p,νX′p∘Gp,q.The following is a description of∘, the synthetic binary operator:
(14)μY′q,νY′q=μX′p,νX′p∘Gp,q=∨p∈XμX′p∧μGp,q,∧p∈XνX′p∨νGp,q,such that 0≤μY′q+νY′q≤1. In the classical fuzzy inference system, the grade on the membership of the fuzzy subset is a variable and variables are related through the operations defined on membership grades. However, in the inference system of the FDIFA semigroup, the FDIFA semigroup serves as a variable and the relation between variables is exhibited by utilizing the FDIFA semigroup homomorphism. In this case, state transitions are carried out with different membership and nonmembership grades that are further combined to get the overall state transition.


## 5. Application

The search for the fastest traveling and trading routes and mechanisms turns aviation into a multimillion dollar industry. The first aircraft was launched in 1939; since then, several aircraft models are introduced with numerous modifications in the original designs. Air traffic accidents caused either by mechanical fault or human error are the biggest threats to the aviation industry. Proper investigation of such accidents will be beneficial in the improvement of aircraft models and mechanisms. Fault detection at an early level can reduce the risk of air crashes saving the lives of hundreds on board. Engines are the most important components of any aircraft, not only that they give enough lift to fly the plane but also that major accidents are caused by engine failure, fuel leakage, or sudden power shut down. The first aircraft Heinkel He 178 engines use a gas turbine to keep the plane in the air. With a slight modification of the design, the first jetliner de Havilland DH.106 Comet was launched in 1952 where the turbojet engines were placed near the wing roots. Four types of gas turbine engines are used in the aircraft, a turbojet engine that uses its exhaust, a turbofan engine that uses its fan, and turboprop that uses the propulsion system and converts heat energy into shaft power to create thrust. A twin-spool turbofan engine contains three compressors, two turbines, a burner, and two nozzles interconnected through various duct channels. A two-spool engine is designed to maintain high pressure and low temperature by two concentric shafts rotating at two different speeds, connecting the high-pressure turbine with the high-pressure compressor stages and the low-pressure turbine to the low-pressure compressor and fan. The structural characteristics of the twin-spool engine make it suitable for engine fuel efficiency and the ultimate choice of jetliners [31]. Here, we will create an intuitionist fuzzy inference system for fault detection in a twin-spool turbofan engine by considering the working principle of the supercharger and air cooler [32].


### 5.1. Fault Detection Rules

The fault detection problem is stated as follows: assign a value between0 and 1 that indicates the supercharger fault case, with 1 indicating a serious fault. The air cooler’s fault scenario is represented by a number between 0 and 1. In the same way, the number 1 signals a major flaw. Then, we use three rules to determine the state of the machine fault. The following are the three rules:
(1)
The engine has a light fault if the supercharger has a weak fault or the air cooler is faulty(2)
The engine is faulty if the supercharger has a major issue(3)
The engine defect is serious if the supercharger and the air cooler are both seriously faultyThe degree of the FDIFA semigroup membership and nonmembership can show the engine’s degree of fault and accuracy. As a result, the degree of membership is utilized to indicate the severity of the defect. Allow 5% to represent a minor issue with the engine and 10% to denote a weak malfunction. 15% denotes an evident problem with the engine, while 15% denotes a medium-level defect. If the percentage is greater than 25%, the engine has a major problem and it should be considered a serious flaw. As indicated in Figure2, the dual-input and dual-output systems will be built in this example. The dual inputs are a decrease in supercharger efficiency and a decrease in air cooler heat transfer, while the outputs are the engine fault level and engine accuracy level.Figure 2
Intuitionistic inference system of the FDIFA semigroup for the engine fault.To put the first rule in the aforementioned inference system’s rule editor, various selections are updated as follows:(1)
Select “weak” for the supercharger fault variable(2)
Select “fault” for the air cooler fault variable(3)
In the connection box, select the wireless button “or”(4)
Under the output variable of the engine fault, select “weak”The following are the rules that are generated:(1)
Weak fault: the supercharger fault is weak or the air cooler is faulty(2)
Medium fault: the supercharger fault is medium(3)
Serious fault: the supercharger fault is serious or the air cooler fault is seriousFigure3 depicts these three rules. The FDIFA semigroup’s inference system has now been completely specified, comprising variables, membership functions, nonmembership functions, and the essential rules for diagnosing engine faults. The FDIFA semigroup rule viewer permits us to totally comprehend the entire intuitionistic fuzzy inference process at once. It also demonstrates the influences of the membership and nonmembership functions on the overall outcomes of the intuitionistic fuzzy inference. If the problem being examined is complex in nature subject to several conflicting factors, then, FDIFA semigroup inference is an appropriate tool for its solution.Figure 3
Fuzzy rules.


### 5.2. FDIFA Semigroup

Now, we construct an FDIFA semigroupMT,∘ fault diagnosis model for the twin-spool turbofan engine. Let MT=R,Z,ζ,r0,H,P,E be an FDIFA, where R is a collection of states that the twin-spool turbofan engine can be in, such as normal, medium, or serious faults; Z is an input character signal set; r0∈R is the initial state of the processing signal; H is a fault state set and a subset of R; P is a fuzzy degree of the membership set; E is a intuitionistic fuzzy degree of the nonmembership set; ζ:R×Z×P×E⟶R is the transition function, that is, ζri,a,μ,ν=rj, where ri,rj∈R, a∈Z, μ∈P,andν∈E. Because r0∈R, R denotes the entire states, H⊆R, and ζ is a transition procedure from one state to another; H, Z, P, and E are mostly treated in the following order:
(i)
Fault-state or output variable setH: the normal working condition is represented by W1, fan efficiency drop is represented by W2, fan flow reduction is denoted by W3, booster (BST) flow reduction is denoted by W4, high-pressure turbine (HPT) efficiency drop is denoted by W5, HPT flow reduction is denoted by W6, high-pressure compressor (HPC) efficiency drop is denoted by W7, HPC flow reduction is denoted by W8, low-pressure turbine (LPT) efficiency drop is denoted by W9, and LPT flow reduction is denoted by W10. The range of values from W1 to W10 is included inside the 0,1 interval. The 0 in the interval denotes the absence of such a flaw. 1 denotes a major flaw [33](ii)
Processing signal or input variable setZ: the following 12 felt parameters are chosen as input variables: z1 is the low-pressure spool speed, z2 is the high-pressure spool speed, z3 is the HPC inlet temperature, z4 is the combustor inlet temperature, z5 is the bypass duct static pressure, z6 is the fan exit static pressure, z7 is the booster tip pressure, z8 is the combustor inlet static pressure, z9 is the interturbine pressure, z10 is the fuel flow, z11 is the nozzle area, and z12 is the variable bypass duct area [33](iii)
Fault degreeP and accuracy degree E: the degrees to which each fault and accuracy parameter correspond to the categories of “normal,” “serious,” and “medium” are presented in terms of the membership and the intuitionistic degree of nonmembership grades as follows:(15)μnormal,νnormal=1,00.2,0.40.1,0.60.2,0.50.1,0.80,0.90,0.90.1,0.70.1,0.90,0.6,(16)μserious,νserious=0,0.80.2,0.70.3,0.70.4,0.50.6,0.20.8,0.10.6,0.20.7,0.10.1,0.90.3,0.5,(17)μmedium,νmedium=1,00.3,0.60.2,0.80.2,0.70.4,0.50.3,0.60.3,0.70.2,0.80.1,0.90,0.7.


#### 5.2.1. Intuitionistic Fuzzy Inference Model

Faults are divided into three categories to make diagnosis easier: serious fault, medium fault, and no fault. The result is processed as follows, based on the output state of the intuitionistic fuzzy inference model of the FDIFA semigroup:If0.60≤μWi≤1, 0≤νWi≤0.40, Wi is level 1, recognized as the serious fault.When0.25<μWi<0.60, 0.40<νWi<0.75, Wi is level 2, recognized as the medium defect.IfμWi≤0.25, νWi≤0.75, Wi is normal, recognized as no fault.The seriousness of the defect can be obtained by utilizing this information. The semigroup’s intuitionistic fuzzy inference model is put to the test with a collection of real data to see if it can correctly identify a twin-spool turbofan engine issue. The intuitionistic fuzzy inference model can be employed if the fault can be appropriately diagnosed. The corresponding parameters are monitored and their values are acquired whenW5,W6,W7, and W8 are defective. The data is utilized as input vectors in simulation functions to compute the output of an intuitionistic fuzzy inference model. The fault criterion is used to determine whether or not a flaw exists. Table 1 shows the test results.Table 1
Test data.
Input vectorCause of fault0.2,0.70.1,0.70.2,0.70.6,0.30,0.80,0.90.1,0.70.1,0.80,0.9W5: serious fault0.2,0.60.1,0.70.2,0.60.1,0.70.4,0.50,0.80.1,0.70.1,0.80,0.9W6: medium fault0.2,0.60.1,0.80.2,0.70.1,0.80,0.80.6,0.30.1,0.80.1,0.70,0.8W7: serious fault0.2,0.70.1,0.80.2,0.70.1,0.80,0.80,0.80.7,0.20.1,0.70,0.9W8: serious fault5, 6, 7, and 8 inWi−out signify distinct types of faults, as illustrated in Table 1.Suppose thatMT1,MT2 belongs to the FDIFA semigroup MT,∘ constructed above and h:MT1⟶MT2 is an inference rule in terms of FDIFA homomorphism. In the aforesaid application, the inference model for fault diagnostics can be built as follows:
(Premise1)
IfMT1′s parameter a is normal, MT2′s output b is serious(Premise2)
IfMT1′s parameter a′ is normalThe conclusion is to try to figure out what the fault level of the outputb′ in MT.IfMT3=MT1∘MT1⟶MT2, then, MT3∈MT. The degree of membership and nonmembership that corresponds to each inference step of the model can be determined using the intuitionistic fuzzy inference system, starting with the known condition.
(18)Ga,b=μMT1⟶MT2a,b,νMT1⟶MT2a,b=μMT1a∧μMT2b∨1−μMT1a,νMT1a∨νMT2b∧1−νMT1a−ε,A=μMT1a,νMT1a∧,∨μMT2b,νMT2b=0,0.80.2,0.70.3,0.70.4,0.50.6,0.20.8,0.10.6,0.20.7,0.10.1,0.90.3,0.50,0.80.2,0.70.2,0.70.2,0.50.2,0.40.2,0.40.2,0.40.2,0.40.1,0.90.2,0.50,0.80.1,0.70.1,0.70.1,0.60.1,0.60.1,0.60.1,0.60.1,0.60.1,0.90.1,0.60,0.80.2,0.70.2,0.70.2,0.50.2,0.50.2,0.50.2,0.50.2,0.50.1,0.90.2,0.50,0.80.1,0.80.1,0.80.1,0.80.1,0.80.1,0.80.1,0.80.1,0.80.1,0.90.1,0.80,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.80.1,0.70.1,0.70.1,0.70.1,0.70.1,0.70.1,0.70.1,0.70.1,0.90.1,0.70,0.90.1,0.90.1,0.90.1,0.90.1,0.90.1,0.90.1,0.90.1,0.90.1,0.90.1,0.90,0.80,0.70,0.70,0.60,0.60,0.60,0.60,0.60,0.90,0.6,B=1−μMT1a,1−νMT1a=0,0.90.8,0.50.9,0.30.8,0.40.9,0.11,01,00.9,0.20.9,01,0.3,Ga,b=Gμ,νa,b=A∨,∧B=0,0.80.2,0.70.3,0.70.4,0.50.6,0.20.8,0.10.6,0.20.7,0.10.1,0.90.3,0.50.8,0.50.8,0.50.8,0.50.8,0.50.8,0.40.8,0.40.8,0.40.8,0.40.8,0.50.8,0.50.9,0.30.9,0.30.9,0.30.9,0.30.9,0.30.9,0.30.9,0.30.9,0.30.9,0.30.9,0.30.8,0.40.80.40.8,0.40.8,0.40.8,0.40.8,0.40.8,0.40.8,0.40.8,0.40.8,0.40.9,0.10.9,0.10.9,0.10.9,0.10.9,0.10.9,0.10.9,0.10.9,0.10.9,0.10.9,0.11,01,01,01,01,01,01,01,01,01,01,01,01,01,01,01,01,01,01,01,00.9,0.20.9,0.20.9,0.20.9,0.20.9,0.20.9,0.20.9,0.20.9,0.20.9,0.20.9,0.20.9,00.9,00.9,00.9,00.9,00.9,00.9,00.9,00.9,00.9,01,0.31,0.31,0.31,0.31,0.31,0.31,0.31,0.31,0.31,0.3,where ∧ is an “and” operation that can be used in the “min” operation and ∨ is an “or” operation that can be used in the “max” operation.Step 1.
The intuitionistic fuzzy relation matrixG can be calculated using premise1 and formula (11), as well as the intuitionistic fuzzy degree of membership and nonmembership given by the aforementioned equations (13) and (14).Step 2.
The defect level of outputb′ for the conclusion is as follows, based on premise2 and formula (14), as well as the above expression (17). The Fuzzy degree membership and intuitionistic degree nonmembership of b′=a′ismedium∘aisnormal,thenbisserious are as follows:
(19)μMT3b′,νMT3b′=μMT1a′,νMT1a′∘Ga,b=1,00.3,0.60.2,0.80.2,0.70.4,0.50.3,0.60.3,0.70.2,0.80.1,0.90,0.7∘Ga,b=0.4,0.50.4,0.50.4,0.50.4,0.50.6,0.20.8,0.10.6,0.20.7,0.10.4,0.50.4,0.5.
When compared to the degreeμserious,νserious0,0.80.2,0.70.3,0.70.4,0.50.6,0.20.8,0.10.6,0.20.7,0.10.1,0.90.3,0.5 of “serious” in MT2, which corresponds to the fault parameters W1,W2,⋯,W10, the degree of b′ in MT3 is 0.4,0.50.4,0.50.4,0.50.4,0.50.6,0.20.8,0.10.6,0.20.7,0.10.4,0.50.4,0.5 which signifies “more serious.” Since MT3∈MT, if a′ is a medium fault, the inference result for the output b′∈MT is a more serious fault, which corresponds to objective reality.


## 5.1. Fault Detection Rules

The fault detection problem is stated as follows: assign a value between0 and 1 that indicates the supercharger fault case, with 1 indicating a serious fault. The air cooler’s fault scenario is represented by a number between 0 and 1. In the same way, the number 1 signals a major flaw. Then, we use three rules to determine the state of the machine fault. The following are the three rules:
(1)
The engine has a light fault if the supercharger has a weak fault or the air cooler is faulty(2)
The engine is faulty if the supercharger has a major issue(3)
The engine defect is serious if the supercharger and the air cooler are both seriously faultyThe degree of the FDIFA semigroup membership and nonmembership can show the engine’s degree of fault and accuracy. As a result, the degree of membership is utilized to indicate the severity of the defect. Allow 5% to represent a minor issue with the engine and 10% to denote a weak malfunction. 15% denotes an evident problem with the engine, while 15% denotes a medium-level defect. If the percentage is greater than 25%, the engine has a major problem and it should be considered a serious flaw. As indicated in Figure2, the dual-input and dual-output systems will be built in this example. The dual inputs are a decrease in supercharger efficiency and a decrease in air cooler heat transfer, while the outputs are the engine fault level and engine accuracy level.Figure 2
Intuitionistic inference system of the FDIFA semigroup for the engine fault.To put the first rule in the aforementioned inference system’s rule editor, various selections are updated as follows:(1)
Select “weak” for the supercharger fault variable(2)
Select “fault” for the air cooler fault variable(3)
In the connection box, select the wireless button “or”(4)
Under the output variable of the engine fault, select “weak”The following are the rules that are generated:(1)
Weak fault: the supercharger fault is weak or the air cooler is faulty(2)
Medium fault: the supercharger fault is medium(3)
Serious fault: the supercharger fault is serious or the air cooler fault is seriousFigure3 depicts these three rules. The FDIFA semigroup’s inference system has now been completely specified, comprising variables, membership functions, nonmembership functions, and the essential rules for diagnosing engine faults. The FDIFA semigroup rule viewer permits us to totally comprehend the entire intuitionistic fuzzy inference process at once. It also demonstrates the influences of the membership and nonmembership functions on the overall outcomes of the intuitionistic fuzzy inference. If the problem being examined is complex in nature subject to several conflicting factors, then, FDIFA semigroup inference is an appropriate tool for its solution.Figure 3
Fuzzy rules.


## 5.2. FDIFA Semigroup

Now, we construct an FDIFA semigroupMT,∘ fault diagnosis model for the twin-spool turbofan engine. Let MT=R,Z,ζ,r0,H,P,E be an FDIFA, where R is a collection of states that the twin-spool turbofan engine can be in, such as normal, medium, or serious faults; Z is an input character signal set; r0∈R is the initial state of the processing signal; H is a fault state set and a subset of R; P is a fuzzy degree of the membership set; E is a intuitionistic fuzzy degree of the nonmembership set; ζ:R×Z×P×E⟶R is the transition function, that is, ζri,a,μ,ν=rj, where ri,rj∈R, a∈Z, μ∈P,andν∈E. Because r0∈R, R denotes the entire states, H⊆R, and ζ is a transition procedure from one state to another; H, Z, P, and E are mostly treated in the following order:
(i)
Fault-state or output variable setH: the normal working condition is represented by W1, fan efficiency drop is represented by W2, fan flow reduction is denoted by W3, booster (BST) flow reduction is denoted by W4, high-pressure turbine (HPT) efficiency drop is denoted by W5, HPT flow reduction is denoted by W6, high-pressure compressor (HPC) efficiency drop is denoted by W7, HPC flow reduction is denoted by W8, low-pressure turbine (LPT) efficiency drop is denoted by W9, and LPT flow reduction is denoted by W10. The range of values from W1 to W10 is included inside the 0,1 interval. The 0 in the interval denotes the absence of such a flaw. 1 denotes a major flaw [33](ii)
Processing signal or input variable setZ: the following 12 felt parameters are chosen as input variables: z1 is the low-pressure spool speed, z2 is the high-pressure spool speed, z3 is the HPC inlet temperature, z4 is the combustor inlet temperature, z5 is the bypass duct static pressure, z6 is the fan exit static pressure, z7 is the booster tip pressure, z8 is the combustor inlet static pressure, z9 is the interturbine pressure, z10 is the fuel flow, z11 is the nozzle area, and z12 is the variable bypass duct area [33](iii)
Fault degreeP and accuracy degree E: the degrees to which each fault and accuracy parameter correspond to the categories of “normal,” “serious,” and “medium” are presented in terms of the membership and the intuitionistic degree of nonmembership grades as follows:(15)μnormal,νnormal=1,00.2,0.40.1,0.60.2,0.50.1,0.80,0.90,0.90.1,0.70.1,0.90,0.6,(16)μserious,νserious=0,0.80.2,0.70.3,0.70.4,0.50.6,0.20.8,0.10.6,0.20.7,0.10.1,0.90.3,0.5,(17)μmedium,νmedium=1,00.3,0.60.2,0.80.2,0.70.4,0.50.3,0.60.3,0.70.2,0.80.1,0.90,0.7.


### 5.2.1. Intuitionistic Fuzzy Inference Model

Faults are divided into three categories to make diagnosis easier: serious fault, medium fault, and no fault. The result is processed as follows, based on the output state of the intuitionistic fuzzy inference model of the FDIFA semigroup:If0.60≤μWi≤1, 0≤νWi≤0.40, Wi is level 1, recognized as the serious fault.When0.25<μWi<0.60, 0.40<νWi<0.75, Wi is level 2, recognized as the medium defect.IfμWi≤0.25, νWi≤0.75, Wi is normal, recognized as no fault.The seriousness of the defect can be obtained by utilizing this information. The semigroup’s intuitionistic fuzzy inference model is put to the test with a collection of real data to see if it can correctly identify a twin-spool turbofan engine issue. The intuitionistic fuzzy inference model can be employed if the fault can be appropriately diagnosed. The corresponding parameters are monitored and their values are acquired whenW5,W6,W7, and W8 are defective. The data is utilized as input vectors in simulation functions to compute the output of an intuitionistic fuzzy inference model. The fault criterion is used to determine whether or not a flaw exists. Table 1 shows the test results.Table 1
Test data.
Input vectorCause of fault0.2,0.70.1,0.70.2,0.70.6,0.30,0.80,0.90.1,0.70.1,0.80,0.9W5: serious fault0.2,0.60.1,0.70.2,0.60.1,0.70.4,0.50,0.80.1,0.70.1,0.80,0.9W6: medium fault0.2,0.60.1,0.80.2,0.70.1,0.80,0.80.6,0.30.1,0.80.1,0.70,0.8W7: serious fault0.2,0.70.1,0.80.2,0.70.1,0.80,0.80,0.80.7,0.20.1,0.70,0.9W8: serious fault5, 6, 7, and 8 inWi−out signify distinct types of faults, as illustrated in Table 1.Suppose thatMT1,MT2 belongs to the FDIFA semigroup MT,∘ constructed above and h:MT1⟶MT2 is an inference rule in terms of FDIFA homomorphism. In the aforesaid application, the inference model for fault diagnostics can be built as follows:
(Premise1)
IfMT1′s parameter a is normal, MT2′s output b is serious(Premise2)
IfMT1′s parameter a′ is normalThe conclusion is to try to figure out what the fault level of the outputb′ in MT.IfMT3=MT1∘MT1⟶MT2, then, MT3∈MT. The degree of membership and nonmembership that corresponds to each inference step of the model can be determined using the intuitionistic fuzzy inference system, starting with the known condition.
(18)Ga,b=μMT1⟶MT2a,b,νMT1⟶MT2a,b=μMT1a∧μMT2b∨1−μMT1a,νMT1a∨νMT2b∧1−νMT1a−ε,A=μMT1a,νMT1a∧,∨μMT2b,νMT2b=0,0.80.2,0.70.3,0.70.4,0.50.6,0.20.8,0.10.6,0.20.7,0.10.1,0.90.3,0.50,0.80.2,0.70.2,0.70.2,0.50.2,0.40.2,0.40.2,0.40.2,0.40.1,0.90.2,0.50,0.80.1,0.70.1,0.70.1,0.60.1,0.60.1,0.60.1,0.60.1,0.60.1,0.90.1,0.60,0.80.2,0.70.2,0.70.2,0.50.2,0.50.2,0.50.2,0.50.2,0.50.1,0.90.2,0.50,0.80.1,0.80.1,0.80.1,0.80.1,0.80.1,0.80.1,0.80.1,0.80.1,0.90.1,0.80,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.80.1,0.70.1,0.70.1,0.70.1,0.70.1,0.70.1,0.70.1,0.70.1,0.90.1,0.70,0.90.1,0.90.1,0.90.1,0.90.1,0.90.1,0.90.1,0.90.1,0.90.1,0.90.1,0.90,0.80,0.70,0.70,0.60,0.60,0.60,0.60,0.60,0.90,0.6,B=1−μMT1a,1−νMT1a=0,0.90.8,0.50.9,0.30.8,0.40.9,0.11,01,00.9,0.20.9,01,0.3,Ga,b=Gμ,νa,b=A∨,∧B=0,0.80.2,0.70.3,0.70.4,0.50.6,0.20.8,0.10.6,0.20.7,0.10.1,0.90.3,0.50.8,0.50.8,0.50.8,0.50.8,0.50.8,0.40.8,0.40.8,0.40.8,0.40.8,0.50.8,0.50.9,0.30.9,0.30.9,0.30.9,0.30.9,0.30.9,0.30.9,0.30.9,0.30.9,0.30.9,0.30.8,0.40.80.40.8,0.40.8,0.40.8,0.40.8,0.40.8,0.40.8,0.40.8,0.40.8,0.40.9,0.10.9,0.10.9,0.10.9,0.10.9,0.10.9,0.10.9,0.10.9,0.10.9,0.10.9,0.11,01,01,01,01,01,01,01,01,01,01,01,01,01,01,01,01,01,01,01,00.9,0.20.9,0.20.9,0.20.9,0.20.9,0.20.9,0.20.9,0.20.9,0.20.9,0.20.9,0.20.9,00.9,00.9,00.9,00.9,00.9,00.9,00.9,00.9,00.9,01,0.31,0.31,0.31,0.31,0.31,0.31,0.31,0.31,0.31,0.3,where ∧ is an “and” operation that can be used in the “min” operation and ∨ is an “or” operation that can be used in the “max” operation.Step 1.
The intuitionistic fuzzy relation matrixG can be calculated using premise1 and formula (11), as well as the intuitionistic fuzzy degree of membership and nonmembership given by the aforementioned equations (13) and (14).Step 2.
The defect level of outputb′ for the conclusion is as follows, based on premise2 and formula (14), as well as the above expression (17). The Fuzzy degree membership and intuitionistic degree nonmembership of b′=a′ismedium∘aisnormal,thenbisserious are as follows:
(19)μMT3b′,νMT3b′=μMT1a′,νMT1a′∘Ga,b=1,00.3,0.60.2,0.80.2,0.70.4,0.50.3,0.60.3,0.70.2,0.80.1,0.90,0.7∘Ga,b=0.4,0.50.4,0.50.4,0.50.4,0.50.6,0.20.8,0.10.6,0.20.7,0.10.4,0.50.4,0.5.
When compared to the degreeμserious,νserious0,0.80.2,0.70.3,0.70.4,0.50.6,0.20.8,0.10.6,0.20.7,0.10.1,0.90.3,0.5 of “serious” in MT2, which corresponds to the fault parameters W1,W2,⋯,W10, the degree of b′ in MT3 is 0.4,0.50.4,0.50.4,0.50.4,0.50.6,0.20.8,0.10.6,0.20.7,0.10.4,0.50.4,0.5 which signifies “more serious.” Since MT3∈MT, if a′ is a medium fault, the inference result for the output b′∈MT is a more serious fault, which corresponds to objective reality.


## 5.2.1. Intuitionistic Fuzzy Inference Model

Faults are divided into three categories to make diagnosis easier: serious fault, medium fault, and no fault. The result is processed as follows, based on the output state of the intuitionistic fuzzy inference model of the FDIFA semigroup:If0.60≤μWi≤1, 0≤νWi≤0.40, Wi is level 1, recognized as the serious fault.When0.25<μWi<0.60, 0.40<νWi<0.75, Wi is level 2, recognized as the medium defect.IfμWi≤0.25, νWi≤0.75, Wi is normal, recognized as no fault.The seriousness of the defect can be obtained by utilizing this information. The semigroup’s intuitionistic fuzzy inference model is put to the test with a collection of real data to see if it can correctly identify a twin-spool turbofan engine issue. The intuitionistic fuzzy inference model can be employed if the fault can be appropriately diagnosed. The corresponding parameters are monitored and their values are acquired whenW5,W6,W7, and W8 are defective. The data is utilized as input vectors in simulation functions to compute the output of an intuitionistic fuzzy inference model. The fault criterion is used to determine whether or not a flaw exists. Table 1 shows the test results.Table 1
Test data.
Input vectorCause of fault0.2,0.70.1,0.70.2,0.70.6,0.30,0.80,0.90.1,0.70.1,0.80,0.9W5: serious fault0.2,0.60.1,0.70.2,0.60.1,0.70.4,0.50,0.80.1,0.70.1,0.80,0.9W6: medium fault0.2,0.60.1,0.80.2,0.70.1,0.80,0.80.6,0.30.1,0.80.1,0.70,0.8W7: serious fault0.2,0.70.1,0.80.2,0.70.1,0.80,0.80,0.80.7,0.20.1,0.70,0.9W8: serious fault5, 6, 7, and 8 inWi−out signify distinct types of faults, as illustrated in Table 1.Suppose thatMT1,MT2 belongs to the FDIFA semigroup MT,∘ constructed above and h:MT1⟶MT2 is an inference rule in terms of FDIFA homomorphism. In the aforesaid application, the inference model for fault diagnostics can be built as follows:
(Premise1)
IfMT1′s parameter a is normal, MT2′s output b is serious(Premise2)
IfMT1′s parameter a′ is normalThe conclusion is to try to figure out what the fault level of the outputb′ in MT.IfMT3=MT1∘MT1⟶MT2, then, MT3∈MT. The degree of membership and nonmembership that corresponds to each inference step of the model can be determined using the intuitionistic fuzzy inference system, starting with the known condition.
(18)Ga,b=μMT1⟶MT2a,b,νMT1⟶MT2a,b=μMT1a∧μMT2b∨1−μMT1a,νMT1a∨νMT2b∧1−νMT1a−ε,A=μMT1a,νMT1a∧,∨μMT2b,νMT2b=0,0.80.2,0.70.3,0.70.4,0.50.6,0.20.8,0.10.6,0.20.7,0.10.1,0.90.3,0.50,0.80.2,0.70.2,0.70.2,0.50.2,0.40.2,0.40.2,0.40.2,0.40.1,0.90.2,0.50,0.80.1,0.70.1,0.70.1,0.60.1,0.60.1,0.60.1,0.60.1,0.60.1,0.90.1,0.60,0.80.2,0.70.2,0.70.2,0.50.2,0.50.2,0.50.2,0.50.2,0.50.1,0.90.2,0.50,0.80.1,0.80.1,0.80.1,0.80.1,0.80.1,0.80.1,0.80.1,0.80.1,0.90.1,0.80,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.90,0.80.1,0.70.1,0.70.1,0.70.1,0.70.1,0.70.1,0.70.1,0.70.1,0.90.1,0.70,0.90.1,0.90.1,0.90.1,0.90.1,0.90.1,0.90.1,0.90.1,0.90.1,0.90.1,0.90,0.80,0.70,0.70,0.60,0.60,0.60,0.60,0.60,0.90,0.6,B=1−μMT1a,1−νMT1a=0,0.90.8,0.50.9,0.30.8,0.40.9,0.11,01,00.9,0.20.9,01,0.3,Ga,b=Gμ,νa,b=A∨,∧B=0,0.80.2,0.70.3,0.70.4,0.50.6,0.20.8,0.10.6,0.20.7,0.10.1,0.90.3,0.50.8,0.50.8,0.50.8,0.50.8,0.50.8,0.40.8,0.40.8,0.40.8,0.40.8,0.50.8,0.50.9,0.30.9,0.30.9,0.30.9,0.30.9,0.30.9,0.30.9,0.30.9,0.30.9,0.30.9,0.30.8,0.40.80.40.8,0.40.8,0.40.8,0.40.8,0.40.8,0.40.8,0.40.8,0.40.8,0.40.9,0.10.9,0.10.9,0.10.9,0.10.9,0.10.9,0.10.9,0.10.9,0.10.9,0.10.9,0.11,01,01,01,01,01,01,01,01,01,01,01,01,01,01,01,01,01,01,01,00.9,0.20.9,0.20.9,0.20.9,0.20.9,0.20.9,0.20.9,0.20.9,0.20.9,0.20.9,0.20.9,00.9,00.9,00.9,00.9,00.9,00.9,00.9,00.9,00.9,01,0.31,0.31,0.31,0.31,0.31,0.31,0.31,0.31,0.31,0.3,where ∧ is an “and” operation that can be used in the “min” operation and ∨ is an “or” operation that can be used in the “max” operation.Step 1.
The intuitionistic fuzzy relation matrixG can be calculated using premise1 and formula (11), as well as the intuitionistic fuzzy degree of membership and nonmembership given by the aforementioned equations (13) and (14).Step 2.
The defect level of outputb′ for the conclusion is as follows, based on premise2 and formula (14), as well as the above expression (17). The Fuzzy degree membership and intuitionistic degree nonmembership of b′=a′ismedium∘aisnormal,thenbisserious are as follows:
(19)μMT3b′,νMT3b′=μMT1a′,νMT1a′∘Ga,b=1,00.3,0.60.2,0.80.2,0.70.4,0.50.3,0.60.3,0.70.2,0.80.1,0.90,0.7∘Ga,b=0.4,0.50.4,0.50.4,0.50.4,0.50.6,0.20.8,0.10.6,0.20.7,0.10.4,0.50.4,0.5.
When compared to the degreeμserious,νserious0,0.80.2,0.70.3,0.70.4,0.50.6,0.20.8,0.10.6,0.20.7,0.10.1,0.90.3,0.5 of “serious” in MT2, which corresponds to the fault parameters W1,W2,⋯,W10, the degree of b′ in MT3 is 0.4,0.50.4,0.50.4,0.50.4,0.50.6,0.20.8,0.10.6,0.20.7,0.10.4,0.50.4,0.5 which signifies “more serious.” Since MT3∈MT, if a′ is a medium fault, the inference result for the output b′∈MT is a more serious fault, which corresponds to objective reality.


## 6. Conclusion

The intuitionistic fuzzy set (IFS) is a well-known generalization of fuzzy sets widely studied, and significant research has been carried out to investigate IFS set theocratic properties and application in other fields. Due to the involvement of nonmembership grades, IFS can handle uncertainties better than fuzzy sets. In this article, effectiveness of IFS is utilized for fault detection in a turbofan engine. Finite deterministic intuitionistic fuzzy automata FDIFA is defined, and then, the semigroup over FDIFA is designed. The algebraic properties of the FDIFA semigroup are discussed and employed in the formulation of inference systems over the FDIFA semigroup. The proposed method is superior to the previously defined fuzzy inference method. The fuzzy inference can be derived from intuitionistic fuzzy inference by considering only the membership values. Moreover, the maximum value of the membership and nonmembership grades can be used to diagnose the maximum default and current running state of any machine. The present approaches must build equations or expressions to deal with the target based on the discussed target properties. The approach described, on the other hand, can perform characteristic processing using its state transition mapping and just requires the selection of system parameters. There are several practically useful generalizations of fuzzy sets [29, 34, 35]; the same methodology can be employed to set inference rules for these generalizations. The real-life example indicates the supremacy of the proposed methodology as it detects the fault accurately and efficiently. The proposed methodology can be used for the detection of default in any machine by identifying its crucial components, the parameters that can cause any fault in the crucial components, and the fault caused by these parameters. Membership and nonmembership grades can be assigned based on the nature and seriousness of the fault.



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*Source: 1994732-2021-11-26.xml*