Research on knowledge representation and modeling of health management based on fuzzy ontology

Abstract

The long-term maintenance of good condition for equipment is the basis of carrying out combat missions under the high technology and fast pace of modern war. However, the knowledge in the health management field at present has the characteristics of distribution, multi-source, heterogeneity and uncertainty, which seriously affects the efficiency of knowledge sharing and reuse. In order to improve the utilization of health management knowledge, an ontology-based knowledge representation method is proposed to describe knowledge in a unified and standardized way, and the classical ontology is extended to express the uncertain knowledge in the field of health management. In addition, to improve the maintenance and knowledge updating efficiency, a global ontology model and a hierarchy, time and activity (HTA) ontology model are constructed. This paper takes the guidance subsystem of a missile as an example to illustrate the process of knowledge modeling. The results show this method realizes knowledge sharing in the health management field and can provide decision support for health management of equipment.

Keywords

Knowledge representation fuzzy ontology HTA model knowledge modeling

1 Introduction

With the rapid development of network, the knowledge and intellectualization of equipment health management are increasingly required. The ability to provide users with high-quality health assessment, fault prediction, knowledge reasoning and other services largely depends on the completeness of equipment health management knowledge [1]. The health management is a dynamic process of applying knowledge and producing knowledge [2], and the knowledge resources mainly come from manufacturers, research institutes, domain experts, et al. Sometimes the process needs knowledge from different life cycle stages to provide decision-making support for the maintenance of security work. Therefore, it is significant to realize the sharing and reuse of health management knowledge to improve the maintenance and support level of equipment.

However, the health management knowledge has characteristics of distribution, multi-source, heterogeneity and uncertainty. The knowledge resources integration and interoperability are poor. At the same time, due to the lack of sharing platform, the fruitful experience knowledge of experts is hard to share and reuse, which results in the lack of knowledge resources and low utilization rate of knowledge. Once an equipment fails, it is difficult for basic maintenance and support personnel to obtain relevant knowledge timely and accurately, which seriously affects the maintenance and support efficiency and is difficult to meet the requirements of health management service in the networked and service-oriented environment. To solve these problems, it is necessary to realize the unified expression of knowledge [3], the processing of uncertain knowledge [4] and the modeling of knowledge [5].

As early as 1983, the International Organization for Standardization (ISO) began to develop standards to describe and share data information throughout the product life cycle. Subsequently, ISO successively promulgated ISO15926 to achieve information interaction. In addition, the National Institute of Standards and Technology (NIST) has published an information exchange standard that enables knowledge interoperability. Each standard has its corresponding specification and unified description language to express. At present, the common expression methods of knowledge are logic-based, rule-based, framework-based, semantic network-based methods. The logic-based method can accurately express the law of human thinking and activity, but it is difficult to express uncertain knowledge and difficult to manage [6]. Rule-based method has been widely used, but each rule can only represent fragments of knowledge, and it is easy to cause combinatorial explosion problem [7]. Framework-based method modularizes knowledge into knowledge base and inherits it from high to low. But it is inflexible and difficult to express and deal with uncertain knowledge [8]. Semantic network-based method has strong descriptive ability, abstract ability and flexibility, but due to the lack of modular design of knowledge representation, knowledge management in semantic networks is difficult [9].

Ontology is the carrier of knowledge work in the new era, and it is a recognized concept for specific domain [10]. As a knowledge management model, ontology has been widely used in the field of artificial intelligence and knowledge engineering. It has gradually become the research focus and core of knowledge engineering such as knowledge acquisition, knowledge representation and knowledge reasoning. This paper studies the knowledge representation and modeling of health management field based on ontology. Firstly, an ontology-based knowledge representation method is proposed to describe health management knowledge in a unified and standardized manner. Secondly, in order to express the uncertain knowledge, the classical ontology is fuzzy extended. Moreover, the global ontology model of health management knowledge and a HTA local ontology model are constructed based on the hybrid ontology method. Finally, the guidance subsystem of a certain missile is taken as an example to illustrate the process of knowledge modeling.

2 Knowledge representation based on ontology

2.1 Composition of ontology

Ontology is a clear and normative expression of concepts. Its goal is to acquire, describe and represent knowledge in related fields, provide a common understanding of knowledge and a clear definition of terms [11]. In the actual research, the formal definition of ontology is different. Perez [12] concluded that ontology includes five basic modeling elements: concept, relation, function, axiom and instance. Combined with object-oriented expression methods. The definitions of each element are as follows:

Concept, also called class, is a set of abstractions for some instances of the real world that have common characteristics. In ontology, concept is the basic unit of knowledge storage and the standard of instance classification.

The subset C₁ × C₂ × ·· ·· C_n of n-dimensional Cartesian product is used to represent the relation between concepts. There are generally four basic relationships between concepts: part-of, kind-of, instance-of and attribute-of.

Function represents a special class of relations. The first n-1 elements of the function can uniquely determine the nth element, which is formally defined as a subset of the n-dimensional Cartesian product: F : C₁ × C₂ × ·· ·· C_n-1 → C_n.

Axiom is a collection of eternal-truth assertions used in domain knowledge to constrain concepts, properties, and instances, usually used to test the problem of ontology consistency.

Instance is a collection of unique and independent entities and the extension of concepts. An instance can be subordinate to one or more concepts, and the subordinate relationship between the instance and the concept can be divided into direct and indirect subordinate relations. In the actual health management ontology, function and relation are contained in properties to realize the association. $O = (C, P, I, A)$ (1)

where O is the health management domain ontology. C represent domain concepts such as system, failure mode. P refer to the properties of concepts, which are usually divided into object properties and datatype properties. Object properties describe the relationship between domain concepts, where the domain and the range are both concepts such as system and subsystem. Datatype properties refer to the characteristics and parameters of concepts, and the domain is class while the range is an integer, string, et al. For example, the datatype property hasValue represents a specific value corresponding to a parameter. I refer to specific entities such as guidance subsystem and infrared detector. A represent the eternal truth assertion about concepts, properties and instances.

2.2 Fuzzy extension of ontology

In the health management process of equipment, there is some fuzzy and inaccurate information such as too high and very poor, which is not suitable for classical ontology to represent and process. However, as the combination of classical ontology and fuzzy set theory, fuzzy ontology introduces fuzzy set theory into ontology, inherits all the characteristics of classical ontology, and is an effective means to deal with uncertain information of ontology.

In classical set theory, elements and sets have definite membership. The formal definition of set concept can be described by eigenfunction method [13].

Let the nonempty set U be the domain, for ∀x ∈ U, there exists a mapping $\begin{matrix} f_{A} : U \to {0, 1} \\ x | \to f (x) \end{matrix}$ (2) where $f_{A} (x) = {\begin{matrix} 1 1 pt, x \in A \\ 0 1 pt, x \notin A \end{matrix}$ , A is the set on U and f_A is an eigenfunction of A.

In fuzzy set theory, the membership relationship between an element and a fuzzy set is measured by a membership degree with a range of [1]. Based on the set definition in classical set theory, the formal definition of fuzzy set is as follows:

Let the non-empty set U be the domain. For a fuzzy set A on U, ∀x ∈ U, there exists a mapping $\begin{matrix} F_{A} : U \to [0, 1] \\ x | \to F (x, A) \end{matrix}$ (3) where F (x, A) is called membership function and represents the membership degree of x to A.

Fuzzy relation refers to the relationship between fuzzy semantics and fuzzy description. Let R be the fuzzy set on M×N, reflecting the relationship between elements M and N, and the fuzzy relationship can be expressed as $\begin{matrix} R : M \times N \to [0, 1] \\ (m, n) | \to R (m, n) \end{matrix}$ (4)

For instance, Temperature of solenoid valve is too high. According to fuzzy set theory, high temperature is a fuzzy expression and can be mapped to a fuzzy value in the range of [1]. $F (Temperature, a little high) = 0.65$ (5)

According to the formula, the reliability of temperature belonging to high is 0.65 However, it does not specify the belonging of the temperature. In order to express fuzzy information accurately, we introduce the instance into ontology as $\begin{matrix} F (Solenoid valve, Temperature, \\ a little high) = 0.65 \end{matrix}$ (6)

In this expression, Solenoid valve is an instance, temperature is a property, a little high is a property value. From the perspective of ontology, the set of membership functions can be understood as an assertion related to concepts, properties and instances. Fuzzy extension of classical ontology can be realized by assigning fuzzy assertion values.

Based on the formal definition of classical ontology, researchers proposed the definition of fuzzy ontology in various forms [8]. Based on the analysis and combined with the realistic requirements of health management, fuzzy ontology is expressed as follows: $O_{F} = (C, I, P_{c}, P_{R}, A_{F}, F)$ (7) where C and I represent the concepts and instances of domain respectively. (P_c, P_R, A_F, F) is the fuzzy extension of classical ontology.

P_c are concept properties. Each concept property can be described as a 3-tuple (c, v_F, q_F). c is an ontology concept. v_F represents the property value. q_F is a language qualifier that describes the strength of a property value. The language qualifier q_F is normally, a little, relatively or very. When the property value v_F is a specific value, it is normally described by normally.

P_R are relational properties. Each relation property can be described as a 4-tuple (c₁, c₂, r, s_F). c₁ and c₂ are ontology concepts. r represents the relationship between c₁ and c₂. s_F represents the strength of the relationship between concept pairs.

A_F are fuzzy rules. Different from classical ontology, fuzzy rule set is not a collection of eternal true assertions. Each rule has the credibility of [1] and needs to be updated constantly during the construction of ontology knowledge base.

F are membership functions. It mainly maps fuzzy language variables into a range of [1].

Web Ontology Language (OWL) is the current mainstream ontology description language. However, OWL cannot effectively describe fuzzy ontology, while FUZZYOWL2 [14] is an extended application of OWL language. Fuzzy logic in ontology can be extended through annotation attribute FuzzyLabel to encode fuzzy ontology. Each annotation is separated by a start tag < fuzzyOwl2 > and an end tag < /fuzzyOwl2 > . The fuzzy element of the tag is specified by attribute fuzzyType.

In fuzzy ontology, common types of membership function include trapezoidal, left-shoulder, right-shoulder and linear, which are shown in Fig. 1.

The number of parameters and semantic restrictions of membership functions are different. The specific expressions are shown in Table 1. Through the fuzzy extension of classical ontology, the fuzzy ontology can express the uncertain knowledge in the domain of health management, which can effectively improve the quality of knowledge service and support the formal knowledge modeling.

Fig. 1

Membership functions.

Table 1

Representation of membership functions

Number	Function	Formation	Semantic restriction
1	Trapezoidal	Trapezoidal (k₁, k₂, a, b, c, d)	k₁ ⩽ a ⩽ b ⩽ c ⩽ d ⩽ k₂
2	Triangular	Triangular (k₁, k₂, a, b, c)	k₁ ⩽ a ⩽ b ⩽ c ⩽ k₂
3	Left-shoulder	Left (k₁, k₂, a, b)	k₁ ⩽ a ⩽ b ⩽ k₂
4	Right-shoulder	Right (k₁, k₂, a, b)	k₁ ⩽ a ⩽ b ⩽ k₂
5	Linear	Linear (k₁, k₂, a, b)	0 ⩽ a ⩽ 1, 0 ⩽ b ⩽ 1

3 Knowledge modeling based on ontology

Ontology-based methods for realizing multi-source and heterogeneity knowledge integration can be divided into three types: global ontology, local ontology and hybrid ontology, as shown in Fig. 2.

Fig. 2

Ontology-based knowledge integration.

The global ontology method integrates multi-source and heterogeneity knowledge into a global ontology and provides the same ontology view for knowledge sources. However, the global ontology method is sensitive to the change of knowledge structure, and any change of knowledge structure will require readjust the global ontology with huge structure, which is difficult to maintain.

For the local ontology method, each knowledge source constructs the local ontology according to the characteristics of domain, and realizes the information interaction through the mapping mechanism among the local ontologies. The advantage is that each local ontology is not affected by other knowledge sources. When a certain knowledge source needs to add or subtract information, it only needs to adjust the corresponding local ontology with small-scale. However, the problem of heterogeneity among local ontologies is prominent. When new knowledge sources need to be added, the workload and difficulty of mapping update are quite large.

The hybrid ontology method is the combination of the first two methods as shown in Fig. 3. Each knowledge source expands its local ontology based on global ontology. The global ontology contains the most basic knowledge structure and terms in each local ontology, and each local ontology adds each knowledge source information on this basis. The hybrid ontology method supports the integration of multi-source and heterogeneity knowledge and can be used for global query. When the knowledge source changes, it only needs to adjust the corresponding local ontology while the global ontology will not be greatly affected, and the interoperability of local ontology is greatly improved. This paper adopts hybrid ontology method as the knowledge representation and modeling scheme of health management domain.

Fig. 3

The flow chart of hybrid ontology model.

3.1 Global ontology modeling

3.1.1 Selection of knowledge sources

Since the design, develop, use of equipment, the relevant technical documents are rich and complicated including equipment technical specifications, Failure Mode, Effects and Criticality Analysis (FMECA) reports and test reports. In the life cycle of equipment, its test data, maintenance records, failure cases, information and knowledge resources are increasingly accumulated. How to select appropriate knowledge sources and extract beneficial knowledge for maintenance and support is the primary task of ontology modeling.

Structure knowledge. Equipment is a complicated system integrating mechanical, electronic optics, automatic control and artificial intelligence. In this paper, the structure knowledge is extracted by using hierarchical structure model, which is divided into several subsystems including guidance subsystem, warfare subsystem, propulsion subsystem, electrical subsystem, missile body subsystem. Each subsystem can be subdivided into assembly, sub assembly and part. Part is defined as the smallest component in the knowledge of ontology structure.

Health knowledge. Equipment accumulates an ocean of knowledge resources related to health management after being deployed. These resources are important references for the maintenance and support work such as fault diagnosis, health evaluation and maintenance decision making. But in the actual process, health management knowledge is usually scattered in different equipment independently lacking effective interaction with outside. Such a reality situation brings great challenges to the health management of equipment. Therefore, it is a major purpose of ontology modeling to make full use of the abundant health knowledge accumulated in the daily maintenance and support process.

Maintenance knowledge. There are abundant sources of maintenance knowledge, among which FMECA analysis is an important means to acquire maintenance knowledge [15]. FMECA technology was first applied in the fault analysis of the master control system of American combat aircraft in the middle of the 20th century, and has been widely used in engineering circles for a long time due to its advantages of economy, practicality and full life cycle of equipment [16]. As a reliability analysis method, FMECA analyzes various potential faults and harmfulness of system components through inductive logic, and proposes possible preventive or maintenance measures to improve system reliability. In the FMECA report, abnormal test signals can be mapped to all possible fault modes, and the harmfulness of each fault mode can be determined according to its probability and severity.

3.1.2 Conceptualization of global ontology

Global ontology is a common conceptual model for global semantic description of systems, which contains the most basic knowledge structure and terminology concepts in each local ontology. Besides, it can describe the heterogeneous knowledge within each knowledge source to be integrated in a standardized way. The basic concepts in the knowledge source are transformed into basic classes in the global ontology, as shown in Table 2.

Table 2
Basic concepts in global ontology

Concept Note

EquipmentComponent The description of all equipment unit names.

DegradationMode The description of degradation mode for equipment components.

TemporalEntity The description of time point and time period.

Parameter The description of parameters for equipment components.

HealthState The description of performance status for equipment components.

FailureMode The description of failure modes for equipment components.

DetectionMethod The description of fault detection methods for equipment components.

FailurePrediction The description of failure prediction for equipment components.

MaintenanceMeasure The description of maintenance measures for fault components.

Concept	Note
EquipmentComponent	The description of all equipment unit names.
DegradationMode	The description of degradation mode for equipment components.
TemporalEntity	The description of time point and time period.
Parameter	The description of parameters for equipment components.
HealthState	The description of performance status for equipment components.
FailureMode	The description of failure modes for equipment components.
DetectionMethod	The description of fault detection methods for equipment components.
FailurePrediction	The description of failure prediction for equipment components.
MaintenanceMeasure	The description of maintenance measures for fault components.

The basic classes in a global ontology are clearly related to each other. Take MaintenanceMeasure as an example, time, parameters, failure mode and other basic concepts are needed to support. Time provides the information such as the failure time. Parameter provides failure data, maintenance data, et al. Fault mode provides the location of failure and fault forms. In order to formally describe the relationships among various types in the global ontology, the object properties are defined as shown in Table 3. Each object property has a corresponding definition and value range, which is used to constrain the mapping relationship of the basic class and ensure the correctness of the syntax.

Table 3

Object properties of the global ontology

Object property	Domain	Range	Note
isTestedBy	FailureMode	DetectionMethod	The relation between failure mode and its detection method.
isMaintainedBy	FailureMode	MaintenanceMeasure	The relation between failure mode and its maintenance measure.
hasParameter	EquipmentComponent	Parameter	The relation between equipment component and its parameter.
hasHealthState	EquipmentComponent	HealthState	The relation between equipment component and its health state.
hasRUL	EquipmentComponent	FailurePrediction	The relation between equipment component and its failure prediction.
isHappenedAt	FailureMode/DegradationMode	EquipmentComponent	The relation between failure mode/degradation mode with equipment component.
hasTimeLabel	HealthState/Parameter	TemporalEntity	The relation between health state/parameter and temporal entity.

3.1.3 Construction of global ontology

The realization of ontology uses machine-recognized language to complete the coding of ontology. Based on the strong reading and writing ability, logical description and logical reasoning ability, this paper uses OWL to encode ontology and chooses Protege as the modeling tool of ontology. After the global ontology is conceptualized, the basic model structure of the global ontology can be constructed, as shown in Fig. 4. Thing represents the superclass and is the parent of all basic classes, which are connected through object properties. For example, class HealthState belongs to the superclass Thing. In the whole global ontology, hasTimeLabel relationship is established with TemporalEntity, which is used to describe the health status of objects at different time stages. DegradationMode establishes isHappenedAt relationship with EquipmentComponent to describe the degradation location of an object. Each base class also contains subclasses that are related to each other through object properties.

Fig. 4

Global ontology model.

3.2 Local ontology modeling

3.2.1 HTA model

Based on the description of global ontology in the previous section, a HTA knowledge model is established in this section, and local ontology model of health management is constructed from three dimensions of hierarchy domain, time domain and activity domain, as shown in Fig. 5.

Fig. 5

HTA knowledge model.

To build the HTA knowledge model, a health management task can be described as $HMT = (H, T, A)$ (8) where H is the hierarchy domain. It mainly describes the structural relations of subsystems and components. The system hierarchy is relatively stable, the domain can be regarded as a fixed dimension of a particular health management task. T is the time domain. The time domain is used to describe the life cycle of the health management task. A is the activity domain. It mainly includes signal acquisition, signal processing, fault diagnosis, fault prediction, optimal resource allocation and knowledge update. Each activity requires the participation of one or more dimensions.

3.2.2 Ontology model of hierarchy domain

The ontology model of hierarchy domain is a knowledge model that describes hierarchical relationships and connections among subsystems and components. For the system with simple structure, ontology can be directly constructed to describe the system. Nevertheless, the physical structure is complicated and the components are various, which makes ontology difficult to manage and maintain. Therefore, it is necessary to decompose the physical structure to reduce the construction complexity of the ontology model.

The hierarchy can be described in the form of hierarchy tree. As for the hierarchical decomposition of the system, different application scenarios and maintenance system have different hierarchical classification methods [17, 18]. This paper takes a missile as an example, divides it into five levels: system, subsystem, assembly, subassembly and part, as shown in Fig. 6. The system layer only includes the missile as an undecomposed whole. Subsystem layer is the structural disassembly of system layer and usually includes guidance subsystem, warfare subsystem, propulsion subsystem, electrical subsystem, missile body subsystem, et al. Each subsystem can be further subdivided into assembly layer, subassembly layer and part layer.

Fig. 6

Hierarchy model.

A structural unit is a set of classes at a specific level including concepts such as system, subsystem, assembly, subassembly and parts. Each structural unit may be a subunit of a unit at the next level or a parent unit at the next level.

In order to formally describe the specific relationships among different levels in the domain ontology, some object properties are defined as shown in Table 4. isPartOf is used to describe the inclusion relationship between two structural units. For example, isPartOf (S₁, S₂) indicates that S₁ is a subunit of S₂. HasSameLevel describes the relationship between two structural units at the same level. For example, HasSameLevel (S₁, S₂) indicates that S₁ and S₂ are at the same level. ConnectsWith describes the directionality of the connection between two units. For example, ConnectsWith (S₁, S₂) indicates that S₁ is connected to S₂. The ontology model of hierarchy domain is shown in Fig. 7.

Table 4

Object properties of the structure domain ontology

Object property	Domain	Range	Note
isPartOf	StructureUnit	StructureUnit	The affiliation structure relation between equipment components.
hasSameLevel	StructureUnit	StructureUnit	The equal structure relation between equipment components.
ConnectsWith	StructureUnit	StructureUnit	The connection structure relation between equipment components.

Fig. 7

Ontology model of hierarchy domain.

3.2.3 Ontology model of time domain

Time domain ontology is a clear formal specification of time conceptual model, used to describe time entities and relations, and provides descriptions of measurement, calculation and representation methods of time [19]. At present, the main time ontology includes DARPA Agent Markup Language (DAML) [20] ontology, OWL ontology [21], Semantic Web Chinese ontology [22], and official standard of OWL ontology published in 2017 [23]. These time ontologies provide vocabularies and models of standard events that describe time points, time periods and relationships between time. In terms of the content and formal expression of time ontology, Hobbs [24] studied the topological relationship of time and the association between time and events based on first-order predicate logic. Pan [25] deepened his research on the calculation of duration and labeling of time information. Zhang [22] divided time ontology into three parts from three different levels: topological layer, measurement and expression layer and semantic layer, and made a comparative analysis and summary of the current mainstream time ontology.

Health management extends to the whole life cycle. Time, as the basic dimension of health management, records and expresses the time information of activities and state changes. If a component fails, the start time and duration of the fault is an important composition of the fault information, which can provide support for later fault diagnosis, maintenance decisions and other maintenance support work. Therefore, it is necessary to introduce time domain ontology model when constructing knowledge model of health management.

The top-level concepts that distinguish time ontologies are usually abstract and general. TemporalEntity is used to represent time elements in the health management process. TemporalEntity has two subcategories: Instant and Interval, which represent time points and time periods respectively. The time entity can be formally expressed as $\begin{matrix} (\forall T, T_{ins}, T_{int}) [TemporalEntity \equiv \\ Instant (T_{ins}) \lor Interval (T_{int})] \end{matrix}$ (9)

Instant can be understood as a time period with the same start time and finish time. Instant can be described as $\begin{matrix} (\forall T_{ins}) {Instant (T_{ins}) \supset [timeStart (t, T_{ins}) = \\ timeFinish (t, T_{ins})]} \end{matrix}$ (10)

Interval indicates the duration of an activity. It consists of two sequential Instant. The form of the time period is defined as follows: $\begin{matrix} (\forall T_{int}, t_{1}, t_{2}) {Interval (T_{int}) \supset [timeStart (t_{1}, T_{int}) \\ \land timeFinish (t_{2}, T_{int}) \land (t_{1} < t_{2})]} \end{matrix}$ (11)

The distinction between Instant and Interval is related to the granularity of time and the purpose of the research. For example, the time entity of one hour can be treated as an Instant based on the whole life cycle of health management, while it should be treated as an Interval from the perspective of daily maintenance.

In health management domain, there is a special relationship intEquals between time points. The equality of time points can be understood as that the time of two events coincide on the time axis and have the same time value in a certain time granularity. Table 5 shows the topological relationship of time points.

Table 5

Topology relationship of Instant

Number	Object	Inherited	Expression
	property	paternity
1	insEquals	equal	insEquals (t₁, t₂)

The relationship between Interval is the default description of time laps in the ontology of time domain, which basically includes all the topological relationships of time. Generally, there are 14 temporal relationships as shown in Table 6. For example, if the end time of noise and refrigeration flow of a certain missile are the same, then the relationship between the two periods can be explained $\begin{matrix} (\forall T_{1}, T_{2}) {intStarts (T_{1}, T_{2}) \supset (\exists t_{0}, t_{1}, t_{2}) \\ [timeStart (t_{0}, T_{1}) \land timeFinish (t_{1}, T_{1}) \land \\ timeFinish (t_{2}, T_{2}) \land insEquals (t_{1}, t_{2}) \land \\ Before (t_{0}, t_{2})]} \end{matrix}$ (12)

Table 6

Topology relationship of Interval

Number	Object	Inherited	Expression
	property	paternity
1	intEquals	equal	intEquals (T₁, T₂)
2	intBefore	separate	intBefore (T₁, T₂)
3	intAfter	separate	intAfter (T₂, T₁)
4	intContains	contain	intContains (T₁, T₂)
5	intDuring	contain	intDuring (T₂, T₁)
6	intStarts	contain	intStarts (T₁, T₂)
7	intStartedBy	contain	intStartedBy (T₂, T₁)
8	intFinishs	contain	intFinishs (T₁, T₂)
9	intFinishedBy	contain	intFinishedBy (T₂, T₁)
10	intMeets	connect	intMeets (T₁, T₂)
11	intMetBy	connect	intMetBy (T₂, T₁)
12	intOverlaps	intersect	intOverlaps (T₁, T₂)
13	intOverlapedBy	intersect	intOverlapedBy (T₂, T₁)
14	intNonOverlap	separate	intNonOverlap (T₁, T₂)

The special topological relationship between Instant and Interval mainly includes insIntContains, insIntInside, timeStarts and timeFinishs as shown in Table 7.

Table 7

Topology relationship between Instant and Interval

Number	Object	Inherited	Expression
	property	paternity
1	insIntContains	contain	insIntContains (T, t)
2	insIntInside	contain	insIntInside (t, T)
3	timeStarts	contain	timeStarts (t, T)
4	timeFinishs	contain	timeFinishs (t, T)

Time domain ontology has many built-in datatypes for describing absolute time, Gregorian calendar and 24-hour system, which can effectively record the whole process of maintenance and support. However, in health management activities, relatively time participation is often needed. For example, in the degradation process of system components, the absolute time description is not intuitive. It is more appropriate to use the time when relative components are put into use for the first time. Therefore, it is necessary to construct time coordinate system to lay a foundation for quantitative description of time information, time relation and data time correlation degree, and then provide support for health management activities. The time coordinate system in the time domain ontology can be expressed as $TCS = (O, U, D)$ (13)

Where TCS stands for time coordinate system, O, U and D respectively stand for time reference datum, scale datum and positive time direction. O and D jointly determine the time and sequence of events. U determines the size of the time distance between time entities. Any time in the time coordinate system is called a TimePosition, which is a mapping of time entities in the time coordinate system. In the time domain ontology, object property hasTimeLabel is defined to describe the time-related information of the object. To use TimePosition, we first define the time coordinate system and then reference the time entity through hasTimeLabel. Ontology model of time domain is constructed as shown in Fig. 8.

Fig. 8

Ontology model of time domain.

In the knowledge model of health management, the domain ontology describes the hierarchical relationships and the connections between subsystems and components by decomposes the physical structure, which reduces the complexity of the ontology model. As the basic dimension of health management, the time domain ontology model records and expresses the time information of various activities and state changes of life cycle. The core of health management is to make full use of knowledge resources to carry out health assessment, failure prediction, maintenance and life extension activities. Therefore, ontology model of activity domain should be introduced when constructing knowledge model of health management.

3.2.4 Activity domain ontology model

Generally, the performance of equipment degrades with the time of operation and storage. An important way to judge whether performance degrades is to observe the measured values of system parameters. If the measured values are within a reasonable error range, the system performance is considered normal. Otherwise, degradation is considered to have occurred. The measured value of parameters is not always exactly equal to the ideal value of parameters, and there is usually a certain offset which directly reflects the health status of the system. Therefore, based on the original parameter properties, datatype properties hasIdealValue and hasDeviationValue are defined to describe the ideal value and the offset of the parameter respectively. The offset of the parameter can be expressed as $△ p^{k} {_{i}}_{j} = M u i j - I u i j$ (14)

where represents the offset of the jth parameter in the ith structural unit. Mv_ij is the measured value of the parameter while Iv_ij is the ideal value of the parameter. For fuzzy semantics, the mapping of quantitative information to fuzzy semantics can be realized by membership function, and the extension of fuzzy logic can be realized by annotation attribute FuzzyLabel in ontology.

Ideally, the off set is 0. However, with the operation and storage of equipment, the deviation will gradually increase. While exceeds the maximum offset, the fault may have taken place in the structure unit SU_i. Each failure that occurs is called a failure mode and is represented by . The set of all degradation modes is DM_i ={ dm_i1, dm_i2, ⋯ , dm_in }, each degradation mode has degradation degrees of different levels such as good, poor, bad, fault, and the degradation degree is determined by parameter offset. The health status of structural unit SU_i is related to the degradation degree of all degradation modes and can be expressed as $HS (S U_{i}) = f_{ev} (D (d m_{i 1}), D (d m_{i 2}), \dots D (d m_{in}))$ (15)

It is assumed that the occurrence of each degradation mode is irreversible and that the health status is mainly determined by the degradation mode with the most severe degradation degree.

To describe degradation mode and its mapping to failure mode, object properties hasDegradationMode, hasFailureMode, and resultIn are defined. Besides, a set of health states {good,..., poor, bad} can be defined, and use the property hasHealthState to extend them vaguely. In addition, the degradation process and health state of structural units are closely related to time, so it is necessary to introduce time domain to provide time information related to degradation and health state. The ontology model of performance degradation is shown in Fig. 9.

Fig. 9

Ontology model of degradation.

From the perspective of time characteristics, faults can be divided into abrupt faults and gradual faults [26]. Abrupt fault refers to the occurrence of an unpredictable jump in the offset of parameters, which leads to the failure of a structural unit and affects the normal implementation of functions. Abrupt fault is unpredictable during service. Gradual fault is a gradual process. The absolute value of parameter offset increases gradually, and finally exceeds the maximum fault offset, resulting in the occurrence of faults.

Fault prediction is a process in which the offset of parameters is predicted based on degradation model to evaluate the future health status and predict the remaining life of structural units. When the parameter offset exceeds this failure threshold, the structural unit SU_i is considered to have failed. The remaining life of the structural unit can be expressed as $RUL (S U_{i}, t_{s}) = FL (S U_{i}, t_{s}) - t_{s}$ (16)

where t_s is the start time of fault prediction, and FL is the earliest time when the parameter offset exceeds the failure threshold.

When describing the remaining life of structural units, experts generally use natural languages such as short and long. Therefore, the ontology needs to be transformed into fuzzy logic description by fuzzy extension. A set of semantic values for the remaining life Semantics (RUL) ={ S₁, S₂, . . . , S_n } can be defined, and there will be a corresponding set of failure time points Instant (RUL) ={ t₀, t₁, . . . , t_n } in time domain. The fuzzy values assigned to the semantic values can be obtained by integrating the probability density function $F (S U_{i}, RUL) = \int_{k - 1}^{k} P (RUL (S U_{i})) dt$ (17)

For example, the semantic value set of the remaining life is {very short, short, long, very long}, and the corresponding set is {0, t₁, t₂, t₃, + ∞ }. The probability distribution of remaining life and the distribution of fuzzy values are shown in Fig. 10.

Fig. 10

Probability distribution of remaining life.

In the ontology model of fault prediction, object property hasRUL is to describe the fault prediction and fuzzy expansion is carried out. The ontology model of fault prediction is shown in Fig. 11.

Fig. 11

Ontology model of fault prediction.

4 Experiments and discussion

A certain missile mainly undertakes the close air combat task, and has the characteristics of large off-axis launch and strong mobility. Maintaining the readiness state of good combat is necessary to ensure its normal wartime effectiveness, and is of great significance to the formation, maintenance and growth of combat effectiveness. Guidance subsystem is the subsystem with the highest failure rate in the service process of the missile, which integrates mechanical, electronic, optical, automatic control, et al, and has a complex structure. In this paper, the guidance subsystem of the missile is taken as an example to illustrate the health management model. The software Prot $\overset{´}{e}$ g $\overset{´}{e}$ , an ontology editor developed by Stanford University that supports multiple inheritance, is chosen. The ontology development interface of the guidance subsystem is shown in Fig. 12.

Fig. 12

Ontology development interface.

According to the division of missile structure hierarchy, the concepts of System, Subsystem, Assembly, Subassembly and Part are defined under the concept of structural unit. Among them, System has the highest level, and Part has the lowest level, and Part is defined as the minimum replaceable unit in the maintenance process. Each structural unit establishes the inclusion relationship between the upper and lower levels through the object property isPartOf. Two units at the same level are described by the object property hasSameLevel, and the physically connected relationship between the two units is described by the object property ConnectsWith. Figure 13 takes infrared detector as an example to show the ontology model of hierarchical structure. In the figure, the hierarchical structure from top to bottom is successively reduced, which are system level, subsystem level, component level, component level and component level respectively. Through the effective decomposition of the complex structure, maintenance and support personnel can better understand the structure of the missile, and provide important reference for fault diagnosis and maintenance decisions.

Fig. 13

Hierarchical ontology model of guidance subsystem.

Figure 14 shows the health ontology model of a solenoid valve in the seeker including degradation mode, failure mode, degradation model, health state. As can be seen from the figure, the main degradation modes of the solenoid valve (SV) include SealLeakage, MaterialAging, Remanence, CoilInsulatingMelt. The degradation mode of coil insulation melting can cause the failure mode of coil short-circuit. The changes of working temperature and working voltage will have an impact on the health status of SV1. Therefore, the time ontology is needed to construct the ontology model of performance degradation. The figure shows that there are three parameters in the degradation model of SV1. The health status of solenoid valve at a certain time can be divided into four kinds: good, poor, bad and fault. Based on fuzzy expansion, each health status will be assigned a corresponding confidence degree to indicate whether the seeker solenoid valve SV1 needs to be repaired at this time.

Fig. 14

Health ontology model of the solenoid valve.

In the domain ontology of missile health management, there are an ocean of fuzzy and uncertain knowledge, such as degradation degree, health state, residual life. However, the classical ontology cannot accurately represent and process the knowledge. Therefore, fuzzy extension of classical ontology is needed to introduce. In this paper, Fuzzy OWL2 is introduced into Protege to solve this problem.

Figure 15 shows the editing interface for fuzzy datatypes, which is responsible for defining membership functions. In fuzzy ontology, common membership function types include linear membership function, left shoulder membership function, right shoulder membership function, trapezoidal membership function and trigonometric membership function. The number of parameters and semantic restrictions of different membership functions are different. In Fig. 14, a datatype of SVHighTemperature is defined. The fuzzy datatype is the right shoulder membership function, which has four parameters A, B, K1 and K2. A and B are two parameters of the membership function.

Fig. 15

Editing interface of fuzzy datatype.

Figure 16 shows the editing interface of fuzzy attribute assertion. The health states of solenoid valve SV1 are defined as Good, Poor, Bad and Fault. Based on fuzzy expansion, each health state will be assigned a corresponding confidence level. For example, SV1 is in poor health has a confidence level of 0.6. In the health management activities, the confidence value is based on the measured data and combined with the membership function defined by experts, and generally will not be directly assigned by experts according to their empirical knowledge. Through the extension of classical ontology, the fuzzy ontology can successfully express the express the uncertain knowledge in the field of health management.

Fig. 16

Editing interface of fuzzy attribute assertion.

5 Conclusions and future enhancements

The knowledge representation method in the field of health management based on ontology is proposed in this paper. By virtue of OWL, the health management knowledge is formally described in a standardized way and the classical ontology is extended to fuzzy ontology to express the uncertain health management knowledge. Based on the hybrid ontology method, a global ontology model and a HTA model of health management knowledge are constructed. This method achieves knowledge sharing and reuse between equipment design enterprises and users, it is suitable to provide supports to fault diagnosis, fault prediction and maintenance decision. The conclusions of this research are summarized as follows:

The ontology has the advantages of both explicitness in structure and accurateness in expression. The OWL can perfectly represent the irregular knowledge in the field of health management by its flexibility in the expression of domain knowledge, and this is convenient for knowledge sharing and reuse.

The HTA model, which is constructed from three dimensions of hierarchy domain, time domain and activity domain, has advantages of good interoperability between local ontologies that can improve the maintenance and knowledge updating efficiency.

The structure of health management knowledge is described by using ontology classes and their relations, and the domain knowledge are filled into the ontology model in the form of instances. Thus, this method is easy to be extended to the health management of other products by just changing the instances in the ontology model.

There are also some disadvantages in this method which need future researches:

Due to the constraints of the experimental condition, the global ontology model and the HTA local model established in the article is relatively simple, thus the instance amount and the relations among instances cannot completely reflect the complicacy and dynamics of the actual situation, and this need to be perfected in practical engineering in the future.

The study preliminarily realized knowledge representation and modeling, but the performance of OWL in rule reasoning is inadequate. In the future research, we will try to introduce SRWL to realize knowledge inference of ontology.

The study is still in its initial phase, and the final purpose of this study is to provide reference for fault diagnosis, fault prediction and maintenance decision. Till now, enough resources have not been collected to evaluate the performance of the method quantitatively. Thus, there are not any quantitative indices to evaluate the performance in this article. In the future research, we will try to evaluate the performance of our method with quantitative indices.

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