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introduction

Hydraulic system has the advantages of large power, small size, light weight, fast response, high precision and anti-load rigidity. It is often at the core of control and power transmission in all kinds of equipment and systems. The hydraulic system has a high failure rate If not dealt with timely after the failure, it will affect the production, resulting in greater economic losses. Therefore, the study of its effective reliability analysis and fault diagnosis methods are often the key to be perfect in industrial technology [1].

Fault Tree Analysis (FTA) method is to establish the relationship between these events based on the relationship between the direct and indirect causes of system failure and failure, and to determine the cause of the system failure A variety of possible combinations, to estimate the incidence of system events and the importance of the bottom event of an analytical method.

In the early 1960s, Bell Labs first used the FTA method to predict the random failure of the militia missile launch control system. Since then, the United States Boeing developed FTA computer program for aircraft design improvements. In the early 1970s, Massachusetts Institute of Technology (MIT) conducted a nuclear safety analysis using FTA and event tree analysis, and concluded that nuclear energy is a very safe energy source. The publication of this report aroused great repercussions in various fields and promoted the fault tree analysis method from aerospace and nuclear energy to the industrial sectors of electronics, chemical industry and machinery [2].

At present, the FTA method has been applied to all fields of national economy, playing an important role in improving the reliability and safety of the system, and has a wide range of development prospects [3]. FTA has become one of the effective methods for the reliability, safety prediction and analysis, fault analysis and diagnosis of hydraulic system.

1 traditional FTA

1.1 basic characteristics

Based on Boolean algebra and probability theory, FTA uses "events" to represent the probabilities of faults and "logic gates" to describe the relationships between faults of components. The event is a description of the state of the system and its components. Commonly used logic gates and AND, OR, voting door, forbidden doors and XOR gates.

The FTA method needs to solve the minimum cut set in the qualitative and quantitative analysis. According to the combination of logic gates in the system fault tree, the structure function is written out, and the probability of occurrence of the top event is calculated by disjoint processing to further calculate the importance of each event .

The cut sets (road sets) are a collection of some of the bottom events in the fault tree. The top events must happen (not occur) when these bottom events occur at the same time (not occurring). If the cut set (road set) contained in the bottom event is arbitrarily removed from a cut set (road set), such a cut set (road set) is the minimum cut set (minimum road set).

A structure function is a Boolean function that represents the state of a system. If the system top event status using state variables, the structure function is the end of event state variables function. In general, when the fault tree is given, the structure function can be written directly according to the fault tree. However, the expression is complicated and lengthy. Therefore, in the actual calculation, the structure function is expressed by the minimum cut set or the minimum path set.

1.2 FTA in the hydraulic system

Most hydraulic systems can be categorized as tandem systems. The fault trees are often composed of OR gates. The occurrence of a single event generally results in a top event [4]. But the actual system can not simply start from improving the reliability of each hydraulic component, which will result in the waste of time and resources. The weak links of the hydraulic system have a significant impact on the system reliability. The reliability of the system depends on whether the location of the weak links and the degree of influence are accurately predicted. The FTA method can help to find out the failure modes of the system and find out the weak links of the system. The qualitative and quantitative analysis and calculation of the failure probability of the system and other reliability indexes are provided to provide basis for improving and evaluating the reliability of the hydraulic system [5].

For example, some fault symptoms and fault sources are not one-to-one correspondence, often with the phenomenon of staggering and overlapping, and the fault diagnosis is more difficult. The FTA method identifies all the failure modes of the top event by looking for the cause of the top event and the combination of causes, which can help to identify potential faults in the hydraulic system in order to guide fault diagnosis and improve the design and maintenance solution [6].

The traditional FTA method has the following shortcomings: Firstly, when analyzing the reliability of the system, the traditional FTA method considers that the part has only two states of work or failure, and can not make an exact evaluation of the reliability of the system. Secondly, the traditional FTA method uses Based on Boolean algebra, it is necessary to know precisely the relationship between the probability of failure of a part and the failure event, and the probability value of a part takes a lot of statistical data to obtain the probability value. Environmental ambiguity and inaccuracy of data will affect the probability of occurrence of parts, and treat the probability of occurrence of parts as exact value, which brings great error to the quantitative calculation of fault tree. Finally, when the fault tree is simplified , There are a large number of non-intersecting process, the calculation is very huge, and sometimes it is difficult to obtain the minimum cut set of the fault tree.

2 Fuzzy FTA

Hydraulic system is a complex non-linear system of mechanical, electrical and fluid coupling. The failure forms and failure mechanisms are complex and diverse. It is difficult to accurately determine the cause of the failure and the degree of failure [7]. The application of fuzzy set theory to hydraulic system FTA, not only reflects the fuzziness of probability itself, but also allows the probability assignment to a certain degree of error, but also the scene and the experimental data with the experience of engineers and technicians can be combined, you can It can solve the ambiguity and uncertainty of fault probability better, reduce the difficulty of obtaining the exact value of fault probability, and has greater flexibility and adaptability.

The fuzzy FTA method obscures the probability of occurrence of basic events in the fault tree, adopts the fuzzy numbers to replace the exact probability values, and still uses the AND and OR gates of the traditional fault tree, but introduces the fuzzy operator, instead of the traditional logic operation, Set the fuzzy probability of occurrence of the top event and its membership function distribution, and quantitative analysis by calculating the degree of fuzzy importance.

Fuzzy numbers are the uncertainties caused by the conceptual fuzziness or the influence of various fuzzy factors. The fuzzy numbers describe the probability values and emphasize the subjective role of people in the FTA. There are many forms of fuzzy numbers, such as triangular fuzzy numbers, trapezoidal fuzzy numbers, L-R fuzzy numbers, normal fuzzy numbers, interval fuzzy numbers and language values [8]. In the hydraulic system engineering practice, when a large number of statistical data, you can determine the exact probability of occurrence of the bottom event probability; when the lack of statistical data, according to the actual situation by a variety of fuzzy numbers and language values to represent and combine Expert survey to assess the probability of occurrence of the end of the incident [9]. In order to facilitate the FTA, various forms of the probability of occurrence of the bottom event should be normalized. Since the trapezoidal fuzzy number is a piecewise linear distribution membership function, the algebraic operation is relatively simple. It is intuitive and easy to convert other forms of fuzzy numbers into trapezoidal fuzzy numbers [10].

The process of using extension principle to determine the membership function of the top event fuzzy probability is actually a mathematical programming problem, often encounter various fuzzy operations, such as the four arithmetic of fuzzy numbers. For complex systems, the structure function dimension Very high, the optimal solution to a programming problem generally encounters math problems. Then it will produce the fuzzy calculation results are credible and credible degree that is "diffusible" and the different types of membership function fuzzy probability crossover calculation and so on. For this reason, [11] adopted a method based on convolution fuzzy operator, which led to the gradual disappearance of edge membership of output fuzzy number. By neglecting the unlikely elements on the edge, the extension of finite branch set could be effectively compensated, That is, "diffusivity" narrows. In order to solve the coupling problem of different types of fuzzy probabilities, Ref. [12] adopted the method of splitting the membership degree of the target domain after the target domain first, and then weighted the intersection by the extended principle and fuzzy the operator. In [13], the interval operation for each λ cutoff of fuzzy numbers equivalent to the extended principle is adopted. By taking different values of λ, the probability interval of system failure under different confidence levels can be obtained.

Due to the traditional logic gates, the above fuzzy FTA method still needs to find out the fault mechanism and find the event connection. In practice, the mechanism of failure and the connection of events are often uncertain. In addition, the different degree of failure will bring different consequences, the traditional fuzzy FTA can not describe the impact of the degree of failure on the system. In order to solve these problems, the literature [14] introduced the TS fuzzy model into FTA, described the fault probability of components as fuzzy likelihood, described the relation between events as TS gate, and described the degree of fault as fuzzy number, respectively according to Part Possibility of Fog Fuzzy and Degree of Failure Calculate the fuzzy likelihood of a superordinate event. Literature [15] applied this T-S fuzzy FTA method to the hydraulic system and achieved good results.

3 Importance analysis

Importance is an important index for quantitative analysis of fault tree. It can not only be used for the reliability analysis of the system, but also can be used in the system optimization design and guidance system for maintenance and diagnosis. Importance describes the contribution to the top event in the event of a component failure. There are mainly three kinds of importance of traditional fault tree: structural importance, probability importance and critical importance. Structural importance is defined as the proportion of the component's key vectors in the total number of key components in the remaining components reflected in the importance of the location of the event in the logical structure of the fault tree, regardless of the probability of occurrence of the underlying event. The probability importance is defined as the partial derivative of the occurrence probability of a top event to the probability of occurrence of a bottom event, which reflects the degree of influence of each bottom event state on the system state. The critical importance is defined as the ratio of the rate of change of the failure probability of a part to the rate of change of the probability of failure of the top event caused by it. It also reflects the influence of the probability of the bottom event on the top event and the unreliability of the bottom event.

The traditional fault tree importance analysis is based on the two-state assumption, but the actual system is often manifested as a variety of failure modes and a variety of fault levels. In order to meet the reliability requirements of multi-state systems, literature [16] extends the importance of traditional two-state system components to multi-state systems, and presents a multi-state system based on a system horizontal event or a state event The general definition of structural importance and probability importance and its calculation method are in accordance with the importance of components of two-state system.

In order to reveal the impact of component states on the state itself and the entire multi-state system failure, literature [17] based on the assumption that system components can not be repaired, divide the failure modes into state faults and state transition faults, expanding the traditional probability importance Degree and critical importance analysis method, the importance is equally divided into state importance and transfer importance.

In order to reflect the influence of the critical state and the non-critical state of all components on the probability of failure of the whole system, Literature [18] proposed a concept of equivalent failure probability and its calculation method, using probability decomposition method to analyze all existing states of components and systems, Using Markov chain method and probability theory to calculate the expected number of work of the system, and then obtain the equivalent failure probability.

In order to reflect the interaction of two components in the system on system reliability, literature [19] proposed the concept of joint importance, which is defined as the ratio of two components to improve the reliability of the system. The importance of joint structure reflects the relationship between two components when the reliability is invalid. The importance of joint reliability reflects the relationship between two components when the reliability is valid. Reference [20] extends the joint importance of two components to multiple components and investigates the notion of the importance of conditional reliability when a component's operating condition is known.

When a single element represents a different failure mode or is not valid, one needs to consider all relevant bottom events as a combination in order to determine the importance of the element. To solve the above problem, differential importance is proposed as a first-order sensitivity method. Considering the interaction between components, literature [21] proposed the second-order differential importance degree by using joint importance as second-order supplementary information.

In [22], two Fussell-Vesely-based importance methods are used, namely component importance and cut importance, component importance is used to identify the most likely component failure, and cut importance importance reflects component failure combination may cause Symptoms of system failures are generated, taking into account the components themselves and their impact on the system.

Above all the importance is defined at the component level, for a fault tree is the basic event level, and for the door event level, the basic events in different door events may be repeated, making the failure probability of each event Has a certain relevance, the literature [23] derives the importance of the door event from the importance of the basic event.

The traditional fault tree importance degree analysis method is based on probability hypothesis, fuzzy and randomness often exist in practical systems, probability hypothesis is replaced by probability hypothesis gradually, and fuzzy importance degree analysis method comes into being. For example, with the aid of the definition of traditional importance concept, that is, the mathematical expectation of the difference between the fuzzy probability of the top event and the failure state of the bottom event [24] The difference between the median value of the fuzzy event and the median event number of the top event in the normal state [25]; the Hamming distance method, which is the difference between the similarities of the actual failure mode and the ideal failure mode [26].

Based on the importance of traditional fault tree, literature [27] proposed the importance algorithm of TS fuzzy fault tree and defined TS probability importance degree, TS critical importance degree and TS fuzzy importance degree, and verified the feasibility of this algorithm Sex. This method can be regarded as a simple and reliable method when the failure rate is uncertain or unknown.

FTA-based fault diagnosis optimization

The knowledge needed to diagnose the hydraulic system depends to some extent on the practical experience of experts in the field. Therefore, the expert system fault diagnosis method plays an important role in the hydraulic system. Knowledge acquisition is recognized as the "bottleneck" problem of expert system. The knowledge acquisition is realized by using fault tree. The logic relationship between each fault is clear and the diagnostic rules are intuitive, which reduces the difficulty of knowledge acquisition of expert system. The top event of the fault tree corresponds to the task to be analyzed and solved by the expert system. A minimal cut set is a final result. The logical relationship of the fault tree from top to bottom corresponds to the reasoning process of the expert system. The branches correspond to the rules in the knowledge base, The number of branches is equal to the number of rules, the knowledge in the knowledge base comes from the fault tree.

However, the traditional fault tree is not conducive to the computer storage and retrieval, especially when the hydraulic system is more complex, the commonly used storage takes up more storage space, the retrieval process is complex, the diagnosis can not be quickly inference, and not conducive to system maintenance. Binary tree storage structure and retrieval process is relatively simple, easy to computer expression and processing, the fault tree can be transformed into a binary tree to solve the above problems

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