CN103413023A - Multi-state system dynamic reliability assessment method - Google Patents

Abstract

The invention discloses a multi-state system dynamic reliability assessment method. According to state information monitored by multi-state systems and constitution logic of units in the multi-state systems, current state probabilities of the units in the multi-state systems are calculated by building a Bayes recursive model, and therefore the state probabilities and reliability of the multi-state systems in the surplus service period are deduced. The method makes full use of the state information monitored by each system to update a reliability assessment model. Compared with an existing reliability assessment method, the multi-state system dynamic reliability assessment method can achieve the fact that more accurate and dynamically updated reliability assessment values of each system can be obtained, and therefore each system can be effectively prevented from losing efficacy, and more accurate maintaining strategies are generated in a guided mode.

Description

A Dynamic Reliability Evaluation Method for Multi-state Systems

technical field

The invention belongs to the technical field of reliability evaluation of complex systems, and in particular relates to a dynamic reliability evaluation method of a multi-state system.

Background technique

Reliability assessment techniques have been widely used. It helps engineers understand and master the life and health status of engineering systems. With the development of modern systems and equipment towards large-scale, complex and precise directions, and the gradual in-depth exploration of the failure mechanism and potential laws of the system, it is found that the failure evolution process of the system and its components during the life cycle is often It presents the characteristics of multi-state, and the failure law and mechanism, work performance and efficiency of each state are different. In this case, if the conventional two-state reliability theory is used to roughly divide the system into two states of "normal" and "failure", it is obviously not in line with the actual situation, and it is impossible to ignore the multi-state characteristics of the system itself. To accurately describe the complex failure process of systems and units, it is urgent to carry out research on multi-state reliability theory to solve the reliability problems of large and complex equipment and systems in modern engineering. In recent years, with the increasing demand for engineering, multi-state reliability theory has become a hot topic of common concern in academia and industry, and has developed rapidly in many fields, such as: mechanical engineering, computer and network systems, network grid, communication system, energy system, supply system, urban infrastructure, strategy and defense, etc.

The traditional multi-state system reliability evaluation method is to obtain the instantaneous state probability of the unit in the life cycle according to the state transition rate of the system component unit, and according to the unit structure, the system's instantaneous state probability is obtained from the system structure function. Therefore, for any two identical multi-state systems, their reliability functions are completely consistent. In fact, the actual service life of any two multi-state systems is different, and this difference is affected by many reasons, such as: the magnitude of the load experienced by each multi-state system during the service phase, environmental factors, frequency of use, Differences in factors such as failure causes. Therefore, for each independent multi-state system, if a more accurate reliability function and model can be established by obtaining useful information about its use and reliability, it will greatly prevent the potential risks of the system and help to improve System reliability and reduced life cycle costs.

Because the traditional multi-state system reliability evaluation method only utilizes the reliability data of the same multi-state system units and systems, such as the state transition time of units and systems, and the failure time of units and systems, but ignores the reliability of the system during use. The status information monitored in . These important information can realize the dynamic update mechanism of reliability evaluation, greatly improve the accuracy of single system reliability evaluation, effectively avoid the occurrence of faults, and help realize the fault prediction and health management of the system.

So far, the research on the reliability evaluation of multi-state systems using the monitored state information is still blank at home and abroad, and the accurate evaluation of the reliability of each system cannot be realized only by using the reliability history data, and the reliability evaluation cannot be realized. dynamic updates.

Contents of the invention

The purpose of the present invention is to solve the above-mentioned problems existing in the existing multi-state system reliability evaluation method, and propose a multi-state system dynamic reliability evaluation method.

The technical solution of the present invention is: a method for evaluating the dynamic reliability of a multi-state system, specifically comprising the following steps:

S1. According to the composition structure of the multi-state system and the state of the units in the system, specify the unit state combination corresponding to each state of the system:

，其中，N_l为单元l的所有可能状态数，s_l,i代表单元l的i状态，且

为单元l的最好状态，s_l,1为最差状态；The multi-state system is composed of a plurality of units, each unit has two or more discrete states, and the discrete states describe all the states that a unit may experience from normal operation to complete failure, expressed as

, where N _l is the number of all possible states of unit l, s _l,i represents the i state of unit l, and

is the best state of unit l, s _l,1 is the worst state;

为系统的最好状态，S₁为最差状态；若X_S(t)表示系统在t时刻的状态，且有X_S(t)∈S；X_l(t)表示单元l在t时刻的状态，且有X_l(t)∈s_l,则有X_S(t)=φ(X₁(t),X₂(t),...,X_M(t))，即系统在任意时刻的状态由单元的状态以及系统组成结构函数φ(·)决定；The state of the system is expressed as Among them, N _S is the number of all possible states of the system, S _i represents the i state of the system,

is the best state of the system, and S ₁ is the worst state; if X _S (t) represents the state of the system at time t, and there is X _S (t)∈S; X _l (t) represents the state of unit l at time t state, and there is X _l (t)∈s _l , then there is X _S (t)=φ(X ₁ (t),X ₂ (t),...,X _M (t)), that is, the system is in any The state at any moment is determined by the state of the unit and the system composition structure function φ(·);

用

表示使系统处于状态i的所有单元状态组合的集合，其中，L_i表示使系统处于状态i的单元状态组合总数，S_i,m表示第m种单元状态组合；S_i,m(l)表示在系统处于状态i的第m种单元状态组合下，单元l的状态，且有S_i,m(l)∈s_l；M is the total number of units that make up the system, and the number of state combinations of units in the system is

use

Indicates the set of all unit state combinations that make the system in state i, where L _i represents the total number of unit state combinations that make the system in state i, S _i,m represents the mth kind of unit state combination; S _i,m (l) represents When the system is in the mth unit state combination of state i, the state of unit l, and S _i,m (l)∈s _l ;

S2. Collect status information of the multi-state system during use:

For each multi-state system in use, it is assumed that the state of the system at a certain moment can be monitored, expressed as X _S (t _k )={X _S (t ₁ ),...,X _S (t _i ), ...,X _S (t _k )}, where, t ₁ <...<t _i <...<t _k , X _S (t _i ) means that the _observed system is at X _S (t _i ) state, and there is X _S (t _i )∈S, the state of the unit in the system is unobservable at any time;

S3. Determine the state of the unit in the multi-state system according to the state information:

According to the state information obtained in step S2, the conditional probability value Pr{X _l (t _k )=s _i |X _S (t _k )}, s _i ∈ _{s l} of each unit of the system in each state at the last state monitoring is obtained, Among them, X _l (t _k ) represents the state of unit 1 at time t _k , and X _S (t _k ) represents the state information of the system collected up to time t _k ;

S4. Calculate the reliability of the multi-state system within the remaining life according to the current unit state:

According to the state probability of each unit obtained in step S3 and the known unit state transition rate, the state probability and system reliability of the system at any time within the remaining life are calculated.

Further, the Bayesian recursive model corresponding to the conditional probability of each unit in each state in the system described in step S3 is specifically:

$\mathrm{<span\; class="notranslate">\; PR</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>$

$=\frac{\underset{m\&Element;u_j}{\mathrm{\&Sigma;}}\mathrm{PR}\{x_S\left(t_k\right)=S_{i,\mathrm{no}}|x_S\left(t_{k-1}\right)=S_{j,m}\}\&Center\; Dot;\mathrm{PR}\{x_S\left(t_{k-1}\right)=S_{j,m}\left|x_S\left(t_{k-1}\right)\right\}}{\underset{\mathrm{no}\&Element;u_i}{\mathrm{\&Sigma;}}\underset{m\&Element;u_j}{\mathrm{\&Sigma;}}\mathrm{PR}\{x_S\left(t_k\right)=S_{i,\mathrm{no}}|x_S\left(T_{K-1}\right)=S_{j,m}\}\&CenterDot;\mathrm{PR}\{x_S\left(t_{k-1}\right)=S_{j,m}\left|x_S\left(t_{k-1}\right)\right\}}$ Formula 1)

代表系统在N_S状态且各单元均处于最好状态组合；Pr{X_S(t_k)=S_i,n|X_S(t_k-1)=S_j,m}表示在t_k-1时刻系统处于j状态并且处于第m种单元状态组合，而在t_k时刻系统处于i状态且为第n种单元状态组合的概率，具体为：Among them, the set U _i is the possible state combination of its units when the system is in state i, and the set U _j is the possible state combination of its units when the system is in state j; Pr{X _S (t _k-1 )=S _j,m | X _S (t _k-1 )} is the conditional probability, which means that the system observes the system state information X _S (t _k-1 )={X _S (t ₁ ),...,X _S (t _i ),. ..,X _S (t _k-1 )}, the probability that the system is in the mth unit state combination; formula (1) is a Bayesian recursive model, and its initial condition is in,

Represents that the system is in the _NS state and each unit is in the best state combination; Pr{X _S (t _k )=S _i,n |X _S (t _k-1 )=S _j,m } means that at t _k-1 The probability that the system is in state j and is in the m-th unit state combination at time t _k , and the system is in state i and is in the n-th unit state combination at time t k is specifically:

$\mathrm{PR}\{x_S\left(t_k\right)=S_{i,\mathrm{no}}|x_S\left(t_{k-1}\right)=S_{j,m}\}=\underset{l=1}{\overset{m}{\mathrm{\&Pi;}}}\mathrm{PR}\{x_1\left(t_k\right)=S_{i,\mathrm{no}}\left(l\right)|x_l\left(t_{k-1}\right)=S_{j,m}\left(l\right)\}$ Formula (2)

Among them, X _l (t _k )=S _i,n (l) represents the state of unit l at time t _k , S _i,n (l) represents the state of unit l when the system is in state i and the unit is the nth state combination probability, and there is S _i,n (l)∈s _l .

Here, for the unit whose state degradation process satisfies the Markov process, Pr{X _l (t _k )=S _i,n (l)|X _l (t _k-1 )=S _j,m (l)} is established by The Markov model corresponding to the state degradation process of the unit is obtained by solving the Kolmogorov state transition differential equations corresponding to the Markov model; for the unit whose state degradation process does not satisfy the Markov characteristic, it can be obtained through Monte Carlo simulation or Petri net and other general stochastic simulation methods to obtain the state transition probability.

Further, the calculation of the multi-state system reliability within the remaining life according to the current unit state in step S4 is specifically:

$R\left(t^{`}\right)=\underset{j={\mathrm{no}}_{\mathrm{SF}}}{\overset{x_S\left(t_k\right)}{\mathrm{\&Sigma;}}}\mathrm{PR}\{x_S(t^{`}+t_k)=S_j,.|x_S\left(t_k\right)\}$ Formula (3)

Among them, n _SF is the threshold state. If the state of the multi-state system is lower than the state n _SF , it is considered to be a failure; t' represents the time after t _k time; the conditional probability Pr{X _S (t'+t _k )= S _j ,.|X _S (t _k )} represents the probability that the system is in state j at time t _k +t' under the observed state information X _S (t _k ), specifically:

$\mathrm{<span\; class="notranslate">\; PR</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}^{<span\; class="notranslate">\; `</span>}{<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span>}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>$

$=\underset{\mathrm{no}\&Element;u_i}{\mathrm{\&Sigma;}}\mathrm{PR}\{x_S(t^{`}+t_k)=S_{j,.}|x_S\left(t_k\right)=S_{i,\mathrm{no}}\}\&CenterDot;\mathrm{PR}\{x_S\left(t_k\right)=S_{i,\mathrm{no}}\left|x_S\left(t_k\right)\right\}$ Formula (4)

$<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; \&Element;</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}\underset{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; \&Element;</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}\mathrm{<span\; class="notranslate">\; PR</span>}<span\; class="notranslate">\; \{</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}^{<span\; class="notranslate">\; `</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \}</span><span\; class="notranslate">\; \&Center\; Dot;</span>\mathrm{<span\; class="notranslate">\; PR</span>}<span\; class="notranslate">\; \{</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \}</span>$

in,

$\mathrm{PR}\{x_S(t^{\&prime;}+t_k)=S_{j,m}|x_S\left(t_k\right)=S_{i,\mathrm{no}}\}=\underset{l=1}{\overset{m}{\mathrm{\&Pi;}}}\mathrm{PR}\{x_l\left({t^{\&prime;}+t}_k\right)=S_{j,m}\left(l\right)|x_l\left(t_k\right)=S_{i,\mathrm{no}}\left(l\right)\}$ Formula (5)

Beneficial effects of the present invention: the multi-state system dynamic reliability evaluation method of the present invention calculates the location of the units in the current multi-state system by constructing a Bayesian recursive model according to the state information monitored by the multi-state system and the composition logic of the units in the system. The state probabilities of the multi-state system are deduced from the state probability and reliability of the multi-state system in the remaining service period. The method of the present invention makes full use of the monitored state information of each system to update the reliability evaluation model, and compared with the existing reliability evaluation method, it can obtain a more accurate and dynamically updated reliability evaluation value for each system, In this way, the failure of each system can be effectively avoided, and a more lean maintenance strategy can be guided.

Description of drawings

FIG. 1 is a schematic flowchart of the multi-state system dynamic reliability evaluation method of the present invention.

Fig. 2 is a schematic diagram of the system structure of the water supply pipeline system according to the embodiment of the present invention.

Fig. 3 is a schematic diagram of system state probability obtained according to state monitoring information in an embodiment of the present invention.

Fig. 4 is a schematic diagram of system dynamic reliability obtained according to status monitoring information in an embodiment of the present invention.

Detailed ways

The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments.

As shown in Figure 1, the multi-state system dynamic reliability evaluation method of the present invention includes steps: according to the multi-state system structure and unit state, determine the unit state combination corresponding to each state of the system; collect the state information of the multi-state system during use ; Determine the state of the unit in the multi-state system according to the state information; calculate the reliability of the multi-state system in the remaining life according to the current unit state.

，其中N_l为单元l的所有可能状态数，s_l,i代表单元l的i状态，且

为单元l的最好状态，s_l,1为最差状态。系统的状态表示为

其中N_S为该系统的所有可能状态数，S_i代表系统的i状态，为系统的最好状态，S₁为最差状态。系统的状态可表示成

即系统的状态由单元的状态以及系统组成结构φ决定，M为组成系统的单元总数。系统中单元的状态组合数为

一般情况下，有

即系统的某些状态对应于多个单元状态组合。这里用

表示使系统处于i状态的所有单元状态组合的集合，其中，L_i表示是系统处于i状态的单元状态组合总数，S_i,m表示第m个单元状态组合。The multi-state system is composed of multiple units, and each unit has more than two discrete states. This discrete state describes all possible states of a unit from normal operation to complete failure, expressed as

, where N _l is the number of all possible states of unit l, s _l,i represents the i state of unit l, and

is the best state of unit l, and s _l,1 is the worst state. The state of the system is expressed as

Among them, N _S is the number of all possible states of the system, S _i represents the i state of the system, is the best state of the system, and S ₁ is the worst state. The state of the system can be expressed as

That is, the state of the system is determined by the state of the unit and the system composition structure φ, and M is the total number of units that make up the system. The number of state combinations of units in the system is

Generally, there are

That is, certain states of the system correspond to combinations of multiple unit states. use here

Represents the set of all unit state combinations that make the system in state i, where L _i represents the total number of unit state combinations that the system is in state i, and S _i,m represents the mth unit state combination.

Here, the form of the φ function is related to the unit structure in the system. For a series flow system (such as: a series pipeline system), X _S (t)=φ(X ₁ (t),X ₂ (t),. ..,X _M (t))=min{X ₁ (t),X ₂ (t),...,X _M (t)}; while the parallel flow type system (such as: multiple pipelines in the pipeline system parallel transfers) then there are

At the same time, assuming that the state transition rate of each unit in the system is known, and the state decay process between units is independent of each other, the probability of the system being in a certain state at any time can be obtained by calculating the probability of each unit being in each state at any time.

According to the state information of the collected multi-state system in use, the Bayesian recursive model constructed by the present invention can be used to calculate the various state probabilities of the unit in the multi-state system at the moment of obtaining state information; according to the state probability, it can be proposed by the present invention The multi-state system reliability model within the remaining life of the multi-state system is updated to obtain the dynamic reliability of the multi-state system.

Take a water supply pipeline system as an example below to illustrate the specific working process of the method.

In this embodiment, the water supply pipeline system is composed of three pipeline units, as shown in FIG. 2 . Among them, pipeline 1 is a unit with two states, and pipeline 2 and pipeline 3 are units with three states. Pipeline 1 and Pipeline 2 are connected in parallel and then connected in series with Pipeline 3. The performance of the entire water supply pipeline system is completely determined by the performance of the three pipeline units. The state of each pipeline unit is unobservable, but the performance status of the entire water supply pipeline system can be known through irregular state monitoring.

The specific implementation steps of the inventive method are as follows:

Step 1: According to the composition structure of the multi-state system and the state of the units in the system, clarify the unit state combination corresponding to each state of the system.

In this embodiment, the state transition of each pipeline unit obeys the Markov model, and the transition rate between states is shown in Table 1. in, Indicates the transition rate of pipeline unit l from state i to state j.

Table 1

The performance of each pipeline unit in different states is shown in Table 2.

Table 2

unit number state 1 state 2 state 3 #1 0.0 tons/second 2.5 tons/second / #2 0.0 tons/second 2.0 tons/second 3.5 tons/second #3 0.0 tons/second 4.0 tons/second 6.0 tons/second

Then the performance of the entire water supply pipeline system depends entirely on the performance of the three pipeline units. According to the connection method between the pipeline units and the structure of the system, the performance of the system is G _S =min{G ₁ +G ₂ ,G ₃ }, where , G _i represents the performance of pipeline unit i. The number of states of the system is determined by the different performance levels it may take. The system has seven states in total, and the state combinations of pipeline units corresponding to each state are shown in Table 3.

table 3

It can be seen from Table 2 that when the system is in state 5, 4, 3, 2 and 1, there are two or more unit state combinations that can make the system in this state. For example: when the system is in state 3, the states of pipeline units 1 and 2 are state 2 and state 1 respectively, and the state of pipeline unit 3 can be either state 3 or state 2. Comparing these two combinations, it is obvious that when the pipeline unit 3 is in state 2, the whole system fails faster and has lower reliability in the remaining life period. However, since the state of the pipeline unit is unobservable, when the system is monitored to be in state 3, it is impossible to judge whether the pipeline unit 3 is in state 3 or state 2, and the state of the pipeline unit 3 can be further judged correctly. Update the reliability evaluation model of the system.

Step 2: Collect state information of the multi-state system during use.

In this embodiment, a brand new water supply pipeline system is put into use at time t=0. Table 4 lists the time series information of system state evolution observed by the system at different times.

Table 4

Step 3: Determine the state of the unit in the multi-state system according to the state information.

According to the state transition rate of the above system units and the observed system state, according to formula (1), when the state of the system at time _t1 is state 5, the probability that the system is in the first unit state combination is:

$\mathrm{<span\; class="notranslate">\; PR</span>}<span\; class="notranslate">\; \{</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="5"\; label="S"\; filenames="HDA00003496364400021.png,HDA00003496364400022.png"\; state="\{\{state\}\}"><figure-callout\; id="1"\; label="S"\; filenames="HDA00003496364400012.png,HDA00003496364400021.png"\; state="\{\{state\}\}">S</figure-callout></figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 5,1</span>}}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="5"\; label="S"\; filenames="HDA00003496364400021.png,HDA00003496364400022.png"\; state="\{\{state\}\}">S</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; ,</span>}<span\; class="notranslate">\; .</span><span\; class="notranslate">\; \}</span>$

$=\frac{\underset{m\&Element;{\mathrm{<figure-callout\; id="7"\; label="u"\; filenames="HDA00003496364400021.png,HDA00003496364400022.png"\; state="\{\{state\}\}">u</figure-callout>}}_7}{\mathrm{\&Sigma;}}\mathrm{PR}\{x_S\left(t_1\right)={\mathrm{<figure-callout\; id="5"\; label="S"\; filenames="HDA00003496364400021.png,HDA00003496364400022.png"\; state="\{\{state\}\}"><figure-callout\; id="1"\; label="S"\; filenames="HDA00003496364400012.png,HDA00003496364400021.png"\; state="\{\{state\}\}">S</figure-callout></figure-callout>}}_{5,1}|x_S\left(t_0\right)={\mathrm{<figure-callout\; id="7"\; label="S"\; filenames="HDA00003496364400021.png,HDA00003496364400022.png"\; state="\{\{state\}\}">S</figure-callout>}}_{7,m}\}\&Center\; Dot;\mathrm{PR}\{x_S\left(t_0\right)={\mathrm{<figure-callout\; id="7"\; label="S"\; filenames="HDA00003496364400021.png,HDA00003496364400022.png"\; state="\{\{state\}\}">S</figure-callout>}}_{7,m}\}}{\underset{\mathrm{no}\&Element;{\mathrm{<figure-callout\; id="5"\; label="u"\; filenames="HDA00003496364400021.png,HDA00003496364400022.png"\; state="\{\{state\}\}">u</figure-callout>}}_5}{\mathrm{\&Sigma;}}\underset{m\&Element;{\mathrm{<figure-callout\; id="7"\; label="u"\; filenames="HDA00003496364400021.png,HDA00003496364400022.png"\; state="\{\{state\}\}">u</figure-callout>}}_7}{\mathrm{\&Sigma;}}\mathrm{PR}\{x_S\left(t_1\right)={\mathrm{<figure-callout\; id="5"\; label="S"\; filenames="HDA00003496364400021.png,HDA00003496364400022.png"\; state="\{\{state\}\}">S</figure-callout>}}_{5,\mathrm{no}}|x_S\left(t_0\right)={\mathrm{<figure-callout\; id="7"\; label="S"\; filenames="HDA00003496364400021.png,HDA00003496364400022.png"\; state="\{\{state\}\}">S</figure-callout>}}_{7,m}\}\&CenterDot;\mathrm{PR}\{x_S\left(t_0\right)={\mathrm{<figure-callout\; id="7"\; label="S"\; filenames="HDA00003496364400021.png,HDA00003496364400022.png"\; state="\{\{state\}\}">S</figure-callout>}}_{7,m}\}}$ Formula (5)

Among them, U ₇ ={{2,3,3}}, then Pr{X _S (t ₀ )=S _7,m }=Pr{X _S (t ₀ )=S _7,1 }=1.0; U ₅ ={{2,3,2},{2,2,2}}; Since the state transition of each pipeline unit obeys the Markov model, then Pr{X _S (t ₁ )=S _5,1 |X _S (t ₀ )=S _7,1 } can be expressed as:

$\mathrm{PR}\{x_S\left(t_1\right)={\mathrm{<figure-callout\; id="5"\; label="S"\; filenames="HDA00003496364400021.png,HDA00003496364400022.png"\; state="\{\{state\}\}"><figure-callout\; id="1"\; label="S"\; filenames="HDA00003496364400012.png,HDA00003496364400021.png"\; state="\{\{state\}\}">S</figure-callout></figure-callout>}}_{\mathrm{5,1}}|x_S\left(t_0\right)={\mathrm{<figure-callout\; id="7"\; label="S"\; filenames="HDA00003496364400021.png,HDA00003496364400022.png"\; state="\{\{state\}\}">S</figure-callout>}}_{\mathrm{7,1}}\}=\underset{l=1}{\overset{3}{\mathrm{\&Pi;}}}\mathrm{PR}\{x_1\left(t_1\right)={\mathrm{<figure-callout\; id="5"\; label="S"\; filenames="HDA00003496364400021.png,HDA00003496364400022.png"\; state="\{\{state\}\}">S</figure-callout>}}_{\mathrm{5,1}}\left(l\right)|x_1\left(t_0\right)={\mathrm{<figure-callout\; id="7"\; label="S"\; filenames="HDA00003496364400021.png,HDA00003496364400022.png"\; state="\{\{state\}\}">S</figure-callout>}}_{\mathrm{7,1}}\left(l\right)\}$ Formula (6)

Among them, Pr{X _l (t ₁ )=S _5,1 (l)|X _l (t ₀ )=S _7,1 (l)} means that the unit l is in the first type of state 7 at t ₀ The state of the unit state combination, and the probability of the system being in the first unit state combination of state 5 at t ₁ (that is, the pipeline unit 1 is in the 2 state, the pipeline unit 2 is in the 3 state, and the pipeline unit 3 is in the 2 state) . Since the state degradation process of the pipeline unit obeys the Markov process, the probability can be obtained by solving the Kolmogorov state transition differential equations corresponding to the Markov model of each unit. Therefore, Pr{X _S (t ₁ )=S _5,1 |X _S (t ₀ )=S _7,1 } can be further expressed as:

$\mathrm{<span\; class="notranslate">\; PR</span>}<span\; class="notranslate">\; \{</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="5"\; label="S"\; filenames="HDA00003496364400021.png,HDA00003496364400022.png"\; state="\{\{state\}\}"><figure-callout\; id="1"\; label="S"\; filenames="HDA00003496364400012.png,HDA00003496364400021.png"\; state="\{\{state\}\}">S</figure-callout></figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 5,1</span>}}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="7"\; label="S"\; filenames="HDA00003496364400021.png,HDA00003496364400022.png"\; state="\{\{state\}\}">S</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 7,1</span>}}<span\; class="notranslate">\; \}</span>$

$<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; PR</span>}<span\; class="notranslate">\; \{</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; \}</span><span\; class="notranslate">\; \&Center\; Dot;</span>\mathrm{<span\; class="notranslate">\; PR</span>}<span\; class="notranslate">\; \{</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; \}</span><span\; class="notranslate">\; \&Center\; Dot;</span>\mathrm{<span\; class="notranslate">\; PR</span>}<span\; class="notranslate">\; \{</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; \}</span>$

$<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; PR</span>}<span\; class="notranslate">\; \{</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="t"\; filenames="HDA00003496364400012.png,HDA00003496364400021.png"\; state="\{\{state\}\}">t</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1.5</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; \}</span><span\; class="notranslate">\; \&Center\; Dot;</span>\mathrm{<span\; class="notranslate">\; PR</span>}<span\; class="notranslate">\; \{</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="t"\; filenames="HDA00003496364400012.png,HDA00003496364400021.png"\; state="\{\{state\}\}">t</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1.5</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; \}</span><span\; class="notranslate">\; \&CenterDot;</span>\mathrm{<span\; class="notranslate">\; PR</span>}<span\; class="notranslate">\; \{</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="t"\; filenames="HDA00003496364400012.png,HDA00003496364400021.png"\; state="\{\{state\}\}">t</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1.5</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; \}</span>$

Formula (7)

Solving the Kolmogorov state transition differential equation of the Markov model of each unit, we can get:

$\mathrm{PR}\{x_1({\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="HDA00003496364400012.png,HDA00003496364400021.png"\; state="\{\{state\}\}">t</figure-callout>}}_1=1.5)=2|x_2\left(0\right)=2\}=\exp (-{\mathrm{\&lambda;}}_{\mathrm{2,1}}^1{t}_1)=\exp (-1.5{\mathrm{\&lambda;}}_{\mathrm{2,1}}^1)$ Formula (8)

$\mathrm{PR}\{x_2({\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="HDA00003496364400012.png,HDA00003496364400021.png"\; state="\{\{state\}\}">t</figure-callout>}}_1=1.5)=3|x_2\left(0\right)=3\}=\exp -({\mathrm{\&lambda;}}_{\mathrm{3,1}}^2+{\mathrm{\&lambda;}}_{\mathrm{3,2}}^2){\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="HDA00003496364400012.png,HDA00003496364400021.png"\; state="\{\{state\}\}">t</figure-callout>}}_1=\exp (-1.5\&times;({\mathrm{\&lambda;}}_{\mathrm{3,1}}^2+{\mathrm{\&lambda;}}_{\mathrm{3,2}}^2))$ Formula (9)

$\mathrm{<span\; class="notranslate">\; PR</span>}<span\; class="notranslate">\; \{</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="t"\; filenames="HDA00003496364400012.png,HDA00003496364400021.png"\; state="\{\{state\}\}">t</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1.5</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; \}</span>$

$=\frac{{\mathrm{\&lambda;}}_{\mathrm{3,2}}^3(\exp (-({\mathrm{\&lambda;}}_{\mathrm{3,2}}^3+{\mathrm{\&lambda;}}_{\mathrm{3,2}}^3)t_1)-\exp (-{\mathrm{\&lambda;}}_{\mathrm{3,2}}^3)t_1)}{{\mathrm{\&lambda;}}_{\mathrm{2,1}}^3-{\mathrm{\&lambda;}}_{\mathrm{3,2}}^3-{\mathrm{\&lambda;}}_{\mathrm{3,1}}^3}$ Formula (10)

$<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; 3,2</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1.5</span>}<span\; class="notranslate">\; \&times;</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; 3,2</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; 3,1</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; exp</span>}\mathrm{<span\; class="notranslate">\; 1.5</span>}<span\; class="notranslate">\; \&times;</span><span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; 2,1</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; 2,1</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; 3,2</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; 3,1</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}}}$

All conditional probabilities in formula (5) can be obtained similarly.

Step 4: Calculate the multi-state system reliability in the remaining life according to the current unit state.

In this embodiment, if the system is in state 1, it is regarded as a system failure, and the reliability of the pipeline system within the remaining life can be obtained according to formula (3), expressed as:

$R\left(t^{`}\right)=\underset{j=2}{\overset{5}{\mathrm{\&Sigma;}}}\mathrm{PR}\{x_S(t^{`}+t_1)=S_{j,}.|x_S\left(t_1\right)=5\}$ Formula (11)

Among them, Pr{X _S (t'+t ₁ )=S _j ,|X _S (t ₁ )=5} represents the conditional probability that the system is in state j at time t'+t ₁ , which can be further expressed as:

$\mathrm{<span\; class="notranslate">\; PR</span>}<span\; class="notranslate">\; \{</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}^{<span\; class="notranslate">\; `</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>}<span\; class="notranslate">\; .</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; \}</span>$

$=\underset{\mathrm{no}\&Element;{\mathrm{<figure-callout\; id="5"\; label="u"\; filenames="HDA00003496364400021.png,HDA00003496364400022.png"\; state="\{\{state\}\}">u</figure-callout>}}_5}{\mathrm{\&Sigma;}}\mathrm{PR}\{x_S(t^{`}+t_1)=x_S\left(t_1\right)={\mathrm{<figure-callout\; id="5"\; label="S"\; filenames="HDA00003496364400021.png,HDA00003496364400022.png"\; state="\{\{state\}\}">S</figure-callout>}}_{5,\mathrm{no}}\}\&Center\; Dot;\mathrm{PR}\{x_S\left(t_1\right)=S_{i,\mathrm{no}}|x_S\left(t_1\right)=5\}$ Formula (12)

$<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; \&Element;</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}\underset{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; \&Element;</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="5"\; label="u"\; filenames="HDA00003496364400021.png,HDA00003496364400022.png"\; state="\{\{state\}\}">u</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 5</span>}}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}\mathrm{<span\; class="notranslate">\; PR</span>}<span\; class="notranslate">\; \{</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}^{<span\; class="notranslate">\; `</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; \}</span><span\; class="notranslate">\; \&Center\; Dot;</span>\mathrm{<span\; class="notranslate">\; PR</span>}<span\; class="notranslate">\; \{</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; \}</span>$

Among them, Pr{X _S (t ₁ )=S _i,n |X _S (t ₁ )=5} has been obtained by step 3; Pr{X _S (t'+t ₁ )=S _j,m |X _S (t ₁ )=S _i,n } Similarly, it can be calculated by formula (2) and the Markov model of each unit.

Figure 3 shows the state probability change curves of the system within the remaining lifespan when the monitored system state information is not obtained and obtained. It can be seen from the figure that when the state information of the system is obtained at t ₁ =1.5 months, the state probability of the system in the future time has been updated. The probability that the system is in state 1 at month time is 0.85, after the update, its probability is 0.35. When the system state is obtained again at t ₂ =3.5 months, the state of the system in the future time is updated again, for example, before the update, the probability of being in system state 3 at t=4.0 months is 0.09 (not updated) and 0.05 ( updated at t ₁ =1.5 months), when updated at t ₂ =3.5 months, its probability is 0.5.

Figure 4 shows the change curves of the system reliability in the remaining service life without and after the monitored system status information is obtained. Similarly, it can be seen from the figure that when the system state information is obtained at t ₁ =1.5 months, the system reliability curve is updated, and when the system state information is obtained again at t ₂ =3.5 months, the system reliability curve is updated again , realizes the dynamic update mechanism of system reliability, which helps to obtain more accurate and dynamic update reliability evaluation value of each system, so as to effectively avoid failure of each system and guide the formulation of a more lean maintenance strategy.

Those skilled in the art will appreciate that the embodiments described here are to help readers understand the principles of the present invention, and it should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical revelations disclosed in the present invention without departing from the essence of the present invention, and these modifications and combinations are still within the protection scope of the present invention.

Claims (3)

1. A multi-state system dynamic reliability assessment method specifically comprises the following steps:

s1, according to a composition structure of a multi-state system and states of units in the system, defining unit state combinations corresponding to the states of the system:

the multi-state system is composed of a plurality of units, each unit has two or more discrete states, the discrete states describe all possible states of a unit from normal operation to complete failure and are represented as

Wherein N is_lIs the number of all possible states of the cell l, s_l,iRepresents the i state of cell l, and

is the best state of cell l, s_l,1Is in the worst state;

the state of the system is represented as

Wherein N is_SIs the number of all possible states, S, of the system_iRepresenting the i-state of the system,is the best state of the system, S₁Is in the worst state; if X_S(t) represents the state of the system at time t and has X_S(t)∈S；X_l(t) represents the state of cell l at time t and has X_l(t)∈s_lThen there is X_S(t)=φ(X₁(t),X₂(t),...,X_M(t)), i.e. the state of the system at any moment in time is determined by the state of the cell and the system composition structure function phi (·);

m is the total number of units constituting the system, and the number of state combinations of the units in the system isBy usingRepresents the set of all cell state combinations that put the system in state i, where L_iRepresenting the total number of combinations of cell states, S, that put the system in state i_i,mRepresents the m type unit state combination; s_i,m(l) Represents the state of cell l in the mth cell state combination of the system in state i, and has S_i,m(l)∈s_l；

S2, collecting state information of the multi-state system in the using process:

for each multi-state system in use, it is assumed that the state of the system at a certain time can be monitored, denoted as X_S(t_k)={X_S(t₁),...,X_S(t_i),...,X_S(t_k) Where t is₁<...<t_i<...<t_k，X_S(t_i) Is shown at t_iThe time observation system is at X_S(t_i) State and has X_S(t_i) E S, the state of the unit in the system is not observable at any moment;

s3, determining the state of a unit in the multi-state system according to the state information:

according to the state information obtained in the step S2, obtaining the conditional probability value Pr { X) of each state of each unit of the system during the last state monitoring_l(t_k)=s_i|X_S(t_k)},s_i∈s_lWherein X is_l(t_k) Is shown at t_kThe state of time cell l, X_S(t_k) Represents to t_kStatus information of the system collected up to the time;

s4, calculating the reliability of the multi-state system in the remaining service life according to the current unit state:

the state probability of the system at any time during the remaining life and the system reliability are calculated from the state probabilities of the respective cells obtained in step S3 and the known cell state transition rates.

2. The method for evaluating the dynamic reliability of a multi-state system according to claim 1, wherein the bayesian recursive model corresponding to the conditional probability of each state of each unit in the system in step S3 is specifically:

formula (1)

Wherein, the set U_iSet U for the possible state combinations of the units of the system in state i_jIs the possible state combination of the unit when the system is in the state j; pr { X_S(t_k-1)=S_j,m|X_S(t_k-1) The conditional probability represents that the system observes the system state information X_S(t_k-1)={X_S(t₁),...,X_S(t_i),...,X_S(t_k-1) The probability that the system is in the mth unit state combination; equation (1) is a Bayesian recursive model with initial conditions ofWherein

representing the system in N_SThe state and each unit are in the best state combination; pr { X_S(t_k)=S_i,n|X_S(t_k-1)=S_j,mDenotes at t_k-1At time the system is in the j state and in the m cell state combination, and at t_kThe probability that the time system is in the i state and is the nth unit state combination specifically is as follows:

formula (2)

Wherein, X_l(t_k)=S_i,n(l) Indicating unit l at time t_kState of (1), S_i,n(l) Represents the probability of the cell l when the system is in the i state and the cell is in the nth state combination, and has S_i,n(l)∈s_l。

3. The method for evaluating the dynamic reliability of a multi-state system according to claim 1 or 2, wherein the step S4 of calculating the reliability of the multi-state system in the remaining life according to the current cell state specifically comprises:

wherein n is_SFIs a threshold state, if the state of the multi-state system is lower than state n_SFIf so, the failure is judged; t' is represented at t_kA time after the time; conditional probability Pr { X_S(t'+t_k)=S_j,·|X_S(t_k) Denotes that the state information X is observed_S(t_k) At t, below_kThe probability that the system is in the state j at the moment + t' is specifically as follows:

wherein,

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