System kernel density residual life prediction method considering partition duration impact

Abstract

The invention belongs to the technical field of system life prediction, and discloses a system kernel density residual life prediction method considering partition duration impact, and the specific technical scheme is as follows: establishing a natural degradation model for a long-term natural degradation process; constructing a random impact model, determining an additional degradation amount caused by impact of a damage area and an additional degradation amount caused by impact of a probability damage area, and calculating a total degradation amount of the system; carrying out reliability analysis in a sudden failure mode, reliability analysis in a degradation failure mode and reliability analysis in a competitive failure mode on the system to obtain a probability density function of the residual life of the system; according to the method, parameters are set, a reliability function image is drawn, the sensitivity of the model parameters is analyzed, residual life prediction is carried out on a system, and the effectiveness of the model is verified through a numerical experiment and a C-MAPSS data set.

Description

Technical Field

[0001] The present invention belongs to the technical field of system life prediction, and particularly relates to a system kernel density remaining life prediction method considering partition duration shock. Background Art

[0002] With the progress of technology, modern industrial equipment is developing rapidly, mainly reflected in high automation and high integration. Since the equipment is vulnerable to various factors during operation, such as wear, vibration, corrosion, etc., these factors will have a greater impact on the performance of the equipment, and may ultimately lead to equipment failure and even serious consequences. Prognostics and Health Management (PHM) technology aims to accurately predict the life of the equipment, so as to achieve scientific management and maintenance of the equipment and ensure its operation in a safe and reliable state. Remaining Useful Life (RUL) prediction, as one of the core technologies of PHM, can provide efficient and credible theoretical support and practical basis for the maintenance management of the equipment. By predicting the remaining life of the equipment and taking effective maintenance measures, the occurrence of catastrophic accidents can be prevented, and goals such as cost reduction and operation safety improvement can be achieved.

[0003] Existing RUL prediction methods can be divided into physics-based failure model methods and data-driven methods according to their principles. The physics-based failure model prediction method realizes the prediction of RUL by analyzing the equipment failure mechanism and establishing its physical model. However, in the face of complex equipment, it is very difficult to obtain the physical model. Therefore, data-driven methods have emerged. This method extracts useful information from a large number of historical data that have been monitored to characterize the performance degradation of the equipment, uses the extracted monitoring data as samples, and uses statistical models, machine learning, deep network and other technologies to construct a prediction model that can characterize the degradation state of the equipment.

[0004] Li et al. proposed a Wiener-based remaining useful life (RUL) prediction method for multiple degradation modes, using a fitness evaluation algorithm that combines dynamic time warping and feature similarity to evaluate the performance of each degradation mode in real time. Finally, taking the simulation signals and experimental data of aeroengines and cutting tools as examples, the effectiveness of this method was verified. Dong Qing et al. proposed a two-stage adaptive Wiener process RUL prediction method. Since data-driven methods make assumptions about the degradation process by presetting degradation models, such as Wiener process, Gamma process, and inverse Gaussian process, etc., making assumptions about the model in advance may result in situations that do not conform to the actual degradation path, and parameter estimation cannot ensure global optimality. Improper selection of the degradation model will affect the prediction accuracy of the remaining useful life. Kernel density estimation (KDE), as a nonparametric estimation method, does not require model assumptions about the data. Zhang Jiangmin et al. proposed a real-time RUL prediction modeling method based on relative density kernel estimation. Li et al. proposed a KDE RUL prediction algorithm considering degradation state transition.

[0005] In engineering practice, as the working time increases, the performance of equipment will gradually decline, and degradation failure will occur when the degradation degree exceeds the natural degradation threshold. At the same time, equipment will inevitably be affected by some external environmental factors (such as shocks, collisions, and extreme weather, etc.), which can be regarded as random shocks. When the shock amplitude exceeds the threshold it can withstand, it will cause the equipment to have sudden failure. Under actual working conditions, it is inevitable for equipment to have degradation failure, while sudden failure is accidental. However, if both exist simultaneously, the final failure of the equipment is caused by the competition between the two. Equipment is often affected by multiple interdependent competing failure processes during operation. Bai et al. proposed a reliability analysis method for competing failure processes considering the coupling effect of multiple shock processes and degradation processes. Pan et al. studied a hybrid competing failure model considering multiple performance degradations and sudden failures, considering the correlated competition between multiple performance degradation failures. Gao et al. considered the influence of shocks on the failure threshold and degradation rate, and established a reliability model for a degradation shock-dependent system with multiple shocks.

[0006] For sudden failures caused by shocks, they often occur only after a single shock exceeds the sudden failure threshold. However, not every shock can cause catastrophic damage to the system. Jiang et al. found that due to the material strength, microengines have the ability to resist small shocks. Shocks are divided into a safe shock zone, a damage shock zone, and a fatal shock zone according to the shock intensity, and the different effects of regional shocks on the degradation process are considered. Song et al. divided shocks into different shock groups and developed a reliability model for multi-component systems affected by different shock groups. Che et al. considered the resistance of microelectromechanical systems to small shock loads, divided shocks into three shock zones according to their magnitudes: a safe zone, a damage zone, and a fatal zone, and established a reliability model of a microengine affected by natural degradation and related zonal shocks.

[0007] From the above research, it can be seen that shocks of different intensities have different effects on degradation. In order to ensure the long-term stable operation of the system, it is particularly important to conduct reasonable management and zonal research on shocks. In recent years, random shocks have been divided into a safe region, a damage region, and a fatal region according to their amplitudes, thus considering the zonal shock effect. In addition, Nakagawa et al. pointed out that there should be an intermediate region between the safe region and the damage region, and shocks in this region have a certain probability of causing damage to the system.

[0008] During the actual operation process, there are random correlation effects between shocks and natural degradation. On the one hand, shocks may lead to an increase in the degradation amount and an increase in the degradation rate, etc. Wang et al. established a reliability model for a competing failure process considering the change in the degradation rate under cumulative shocks, considering that a single shock can cause sudden degradation damage, and when the cumulative shock reaches a predetermined threshold, the degradation rate will change. Jiang et al. considered the increase in the degradation amount caused by shocks and proposed the variability of the shock failure threshold on this basis. Feng et al. established a reliability model in which the natural degradation rate changes under different types of random shock models.

[0009] The above-mentioned literature focuses on simulating the impact of random shocks on degradation. However, in engineering practice, the degree of natural degradation of the system will also have a certain impact on shocks. For example, for large-scale infrastructure such as bridges and dams, in the initial stage of operation, their reliability is very high, and they generally will not have accidents when encountering sudden external shocks such as strong winds, floods, and heavy rainfall. After long-term operation, the structural strength of the system weakens, and accidents may occur when encountering shocks of the same intensity as in the early stage of system operation. Summary of the Invention

[0010] To solve the technical problems existing in the prior art, the present invention provides a system kernel density remaining life prediction method considering zonal duration shocks, which has accurate prediction and good adaptability.

[0011] To achieve the above purpose, the technical solution adopted by the present invention is: a system kernel density remaining life prediction method considering partition duration impact. The overall degradation level of the system reflects the health status of the system to a certain extent. As the degree of system degradation continues to increase, its structural strength gradually weakens, resulting in a decrease in the system's resistance to external impacts. Based on the overall degradation level, the system is divided into two states: normal state and sensitive state. The normal state means that the system can effectively maintain its structural strength and function under the condition of a low overall degradation level, and has a strong ability to resist external impacts; in contrast, the sensitive state means that after the system has experienced a certain degree of degradation, its structural strength is significantly reduced, resulting in it being more prone to failure when facing conventional impacts.

[0012] Construct system description and model assumptions: When the overall degradation level X(t) of the system is between the predetermined sensitive state threshold H 1 and degradation failure threshold H 2 When the overall degradation level is less than H 1 The system is in normal state, t w Indicates the time it takes for the system to degrade from a normal state to a sensitive state.

[0013] Natural degradation modeling: Assume t k is the current monitoring time, assuming that at time [0,t k ]The degradation increment of the system ΔX 1 ,ΔX 2 ,...,ΔX k are independent and identically distributed random variables, f(Δx) is the density function of the degenerate increment, so the kernel density estimate of f(Δx) is:

[0014]

[0015] Where k is the time [0,t k ] degenerate increment ΔX i The number of k is the window width, and K(·) is the kernel function.

[0016] The choice of kernel function is likely to have less impact on the final estimate, so the Gaussian kernel function will be used because it has the obvious advantage of being simple to use:

[0017]

[0018] The error between the kernel density estimate and the probability density function of the degraded increment is measured using the mean integrated square error (MISE):

[0019]

[0020] The optimal window width in kernel density estimation is h k It can be obtained by solving the minimum of the integrated mean square error (MISE) as follows:

[0021]

[0022] Substitute the Gaussian kernel function K(Δxk) into equation (4) to obtain the optimal window width h k :

[0023]

[0024] where σ k is the variance of the k initial sample feature degradation increments ΔX 1 , ΔX 2 ,..., ΔX k :

[0025]

[0026] where is the sample mean of the known k degradation increments Δx 1 , Δx 2 ,..., Δx k , expressed as:

[0027]

[0028] Then the probability density function g(x) of the natural degradation process X c (t) at different times can be obtained by convolving the probability density functions of the degradation increments:

[0029]

[0030] Therefore, the reliability function of the system can be obtained in the natural degradation process of the system without considering the impact of shocks at time t as:

[0031]

[0032] where H 2 is the degradation failure threshold of the system.

[0033] Stochastic shock modeling:

[0034] The arrival of stochastic shocks: Assume that the arrival of stochastic shocks follows a homogeneous Poisson process with a constant rate of λ. Let N(t) denote the number of stochastic shocks occurring in the time interval (0, t]. Therefore, the probability of n shocks occurring by time t is:

[0035]

[0036] Assume the amplitude of the shock is \(W\). i , where \(i = 0, 1, 2,\cdots\) follows a normal distribution and the cumulative distribution function of independent and identically distributed random variables is \(F\) W (w), and the duration of the shock is \(T\) i s , where \(i = 0, 1, 2,\cdots\) follows a normal distribution The degradation increment caused by the shock is denoted as \(Y\) i , where \(i = 0, 1, 2,\cdots\) The random shock is divided into four regions according to its amplitude \(W\), namely, the safe region \((0, D\) 1 ), the probability damage region \([D\) 1 , D 2 ), the damage region \([D\) 2 , D 3 ) and the fatal region \([D\) 3 , +∞). The shock in the safe region is considered to have no impact on the system due to its relatively small amplitude. The shock in the probability damage region has a certain probability of causing damage to the system, and this probability is related to the amplitude of the shock and increases with the increase of the amplitude. The shock in the damage region will cause cumulative damage to the degradation process and thus accelerate the degradation of the system. The shock in the fatal region will cause the system to fail immediately because the amplitude is too large for the system to withstand. Generally speaking, when the system enters the sensitive state from the normal state, its resistance to external shocks decreases. A reasonable assumption is that the system in the sensitive state can only resist relatively small shocks. In the model proposed in the present invention, it is manifested as a decrease in the shock partition threshold.

[0037] The system degradation process includes long-term natural degradation and accelerated degradation caused by the duration of the shock. Therefore, there may be two competing failure modes in the system: one is the degradation failure when the overall degradation level exceeds the degradation failure threshold \(H\) 2 due to long-term natural degradation and additional degradation caused by the duration of the shock, and the other is the sudden failure that occurs when the shock amplitude exceeds the sudden failure threshold \(D\) (\(D = D\) 2 under normal conditions, \(D = D'\) 2 under sensitive conditions).

[0038] Assume that the probability that the \(i\)-th shock \(S\) i falls into the safe region under normal conditions is \(p\) 1 , assume that the probability that the \(i\)-th shock \(S\) i falls into the probability damage region under normal conditions is \(p\) 2 , assume that the probability that the \(i\)-th shock \(S\) i falls into the damage region under normal conditions is \(p\) 3 , assume that the probability that the \(i\)-th shock \(S\) i falls into the fatal region under normal conditions is \(p\) 4The above four different types of impacts can be regarded as independent Poisson processes with arrival rates of λp 1 , λp 2 , λp 3 and λp 4 .

[0039]

[0040] In the time interval (0, t], the number of impacts falling into four different regions is denoted by N 1 (t), N 2 (t), N 3 (t) and N 4 (t) respectively, which are independent homogeneous Poisson distribution processes with rates of o = 1, 2, 3, 4. Therefore, under normal conditions, the probability of n impacts occurring in different regions is:

[0041]

[0042] where n is the number of impacts, and o = 1, 2, 3, 4 represent that the regions where the impacts are located are the safe area, the probability damage area, the damage area, and the fatal area respectively.

[0043] Similarly, when the system is in a sensitive state, the probabilities that the i-th impact S i falls into the safe area, the probability damage area, the damage area, and the fatal area are p 1 ', p' 2 , p' 3 and p' 4 respectively, and their arrival rates are λp 1 ', λp' 2 , λp' 3 and λp' 4 . In the time interval [t w , t), the impact counts falling into the four regions are denoted by N 1 '(t), N' 2 (t), N 3 '(t) and N' 4 (t) respectively. Therefore, the probability of n impacts occurring in different regions under the sensitive state is:

[0044]

[0045] where n is the number of impacts, o = 1, 2, 3, 4 represent that the regions where the impacts are located are the safe area, the probability damage area, the damage area, and the fatal area, and t w is the time experienced by the system when changing from the normal state to the sensitive state.

[0046] Additional degradation caused by impacts in the damaged area: The cumulative damage caused by random impacts in the damaged area is as follows:

[0047]

[0048] where N 3 (t) represents the number of impacts reaching the damaged area under normal conditions, and W i represents the amplitude of the impact under normal conditions, and T i s represents the duration after the impact arrives, represents the damage caused by all impacts in the damaged area under normal conditions, and N 3 '(t) represents the number of impacts reaching the damaged area under sensitive conditions, and W j represents the amplitude of the impact under sensitive conditions, represents the duration after the impact arrives, represents the damage caused by all impacts in the damaged area under normal and sensitive conditions up to time t when the system is in a sensitive state; let Y 1 =W i ·T i s represents the additional increment caused by the i-th impact in the damaged area under normal conditions during the duration T i s ; represents the additional increment caused by the j-th impact in the damaged area under sensitive conditions, The probability density function of is:

[0049]

[0050] where is the probability density function of Y 1 .

[0051] Let We can obtain The probability density function of is:

[0052]

[0053] Therefore, The probability density function of can be obtained through convolution and its expression is:

[0054]

[0055] Additional degradation caused by impacts in the probabilistic damaged area: For impacts in the probabilistic damaged area, assuming that the probability of an impact causing additional degradation increases linearly with the increase in the impact amplitude, it is defined as:

[0056]

[0057] Wherein, W i represents the impact magnitude, D 1 is the safety zone threshold, D 2 is the probability damage zone threshold, which means that when the impact amplitude is closer to D 2 , the likelihood of causing damage increases.

[0058] The cumulative damage caused by random impacts in the probability damage zone is as follows:

[0059]

[0060] Wherein, N 2 (t) represents the number of impacts reaching the probability damage zone under normal conditions, W i represents the amplitude of the impact under normal conditions, T i s represents the duration after the impact arrives, represents the damage caused by all impacts in the probability damage zone under normal conditions; N' 2 (t) represents the number of impacts reaching the probability damage zone under sensitive conditions, W j represents the amplitude of the impact under sensitive conditions, represents the duration after the impact arrives, represents the damage caused by all impacts in the probability damage zone within the normal and sensitive states up to time t when the system is in a sensitive state. Let Z = a 0 W i T i s (a 0 = p(W i ) > 0), the cumulative distribution function F Z (z) of Z is:

[0061]

[0062] Wherein, is the cumulative distribution function of Y 1 ;

[0063] By taking the derivative of the cumulative distribution function, the probability density function f Z (z) of Z is:

[0064]

[0065] Using the same calculation method as for the damage zone, the probability density function of is:

[0066]

[0067] Let It can be obtained that The probability density function of is:

[0068]

[0069] Therefore,[[]] The probability density function of is:

[0070]

[0071] Overall system degradation: The overall system degradation X s (t) is the internal degradation X c (t) of the system and the additional damage and The sum of, that is:

[0072]

[0073] Let represent the additional damage caused by the damaged area and the probabilistic damage area in the normal state, represent the additional damage caused by the damaged area and the probabilistic damage area in the sensitive state, then X 1 and X 2 The probability density function of is:

[0074]

[0075] Reliability analysis of the system under the shock and degradation stochastic correlation model:

[0076] Reliability analysis under the sudden failure mode: In the normal state, when the shock amplitude exceeds the sudden failure threshold D 3 , sudden failure will occur; similarly, in the sensitive state, when the shock amplitude exceeds the sudden failure threshold D 3 ', sudden failure will occur. Therefore, when only considering sudden failure, the reliability function of the system at time t is:

[0077]

[0078] In the formula, m is the number of shock occurrences in the normal state, and n is the total number of shock occurrences including normal and sensitive states up to time t.

[0079] Reliability analysis under the degradation failure mode: When the system is under the combined action of long-term natural degradation and accelerated degradation caused by continuous shocks, when the overall degradation level reaches the degradation failure threshold H 2 , degradation failure will occur. Considering the influence of random shocks, the reliability function of the system is divided into the following two cases:

[0080] (1), N(t) = 0. No shock has reached the system up to time t. The system is only affected by long-term natural degradation. When only considering degradation failure, the reliability of the system at time t is:

[0081]

[0082] where g(x) is the probability density function of the natural degradation amount, and H 2 is the degradation failure threshold;

[0083] (2), N(t) = n > 0. The number of shocks reaching the system up to time t is n. Assume that the number of shocks occurring in the safe area under normal conditions is a, the number of shocks occurring in the probability damage area is b, and the number of shocks occurring in the damage area is c; the number of shocks occurring in the safe area under sensitive conditions is d, the number of shocks occurring in the probability damage area is h, and the number of shocks occurring in the damage area is r. And let a + b + c = j - 1, that is, the system has a total of j - 1 shocks under normal conditions,

[0084] and a total of n - j + 1 shocks under sensitive conditions. At this time, the reliability function of the system at time t is:

[0085]

[0086] where H 2 is the degradation failure threshold, and H 1 is the sensitive state threshold. When the system degradation amount is greater than H 1 the system is in a sensitive state;

[0087] Reliability analysis under competing failure modes: Competing failure is the result of the competition between sudden failure and degradation failure. The reliability functions of the two modes obtained respectively can be used to derive the competing failure reliability function of the system at time t as:

[0088]

[0089] Assume that t l is the current monitoring time. At this time, the probability density of the remaining life of the system is:

[0090]

[0091] Therefore, substituting equation (34) into equation (35) can obtain the probability density function of the remaining life of the system.

[0092] As the performance of the system affected by shocks continues to degrade, the PDF of RUL based on kernel density estimation becomes higher and narrower over time, which means that the predicted remaining life becomes more and more concentrated, with higher certainty, and the predicted value of RUL also gets closer and closer to the true value.

[0093] The present invention proposes a reliability analysis and remaining life prediction model for a system considering the impact of partition duration shock. The non-parametric estimation method of adaptive kernel density is used to model the natural degradation process of the system. The arrival of random shocks follows a homogeneous Poisson process with a constant rate of λ. The random shocks are divided into four regions according to their amplitudes W, namely the safe region, the probabilistic damage region, the damage region, and the fatal region. Reliability analysis and life prediction are carried out considering the impact of duration shock and the mutation failure threshold caused by system degradation. Finally, numerical experiments and the C-MPASS dataset are used for verification, verifying the necessity and accuracy of the model considering partitions, duration shock, and system state changes.

[0094] In the shock damage model proposed in the present invention, only the impact of extreme shocks where the shock amplitude exceeds a given threshold and causes damage is considered. Since the types of shocks received by the system during operation are uncertain, in future research, the change in the reliability of the system under the influence of other types of shocks (such as δ shocks, m shocks, k shocks) can be further considered. Description of the Drawings

[0095] Figure 1 is the system schematic diagram of the present invention.

[0096] Figure 2 is the system state and its degradation path diagram.

[0097] Figure 3 is the schematic diagram of the impact of partition duration shock on degradation.

[0098] Figure 4 is the curve graph of the reliability function image.

[0099] Figure 5 is the curve graph of the reliability function of the system under different λ.

[0100] Figure 6 is for the system under different H 2 is the curve graph of the reliability function.

[0101] Figure 7 is for the system under different μ W is the curve graph of the reliability function.

[0102] Figure 8 is for the system under different μ T is the curve graph of the reliability function.

[0103] Figure 9 is the remaining life prediction diagram of the system considering partition duration shock.

[0104] Figure 10 is the internal structure diagram of an aeroengine.

[0105] Figure 11 It is a curve graph of degradation data monitored by the T30 sensor.

[0106] Figure 12 It is a curve graph of the system reliability function obtained by the method proposed in the present invention.

[0107] Figure 13 It is a comparison graph of the reliability functions between different models.

[0108] Figure 14 It is a curve graph of the probability density function of the remaining life at different cycles. Specific implementation manners

[0109] In order to make the technical problems, technical solutions and beneficial effects to be solved by the present invention more clear and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0110] A system kernel density remaining life prediction method considering the impact of partition duration, a system reliability analysis and remaining life prediction model considering the impact of partition duration are proposed, and the basic principle of the model is as Figure 1 shown.

[0111] First, the system degradation process includes long-term natural degradation and additional degradation caused by the impact of duration. When the overall degradation level exceeds the degradation failure threshold, degradation failure will occur. Secondly, the arrival of random shocks follows a homogeneous Poisson process and is divided into four regions according to its amplitude. Among them, the shocks in the safe region will not affect the system, the shocks in the probability damage region have a certain probability of damaging the system, the shocks in the damage region will cause cumulative damage to the degradation process and thus accelerate the degradation of the system, and the shocks in the fatal region will cause sudden failure of the system because the amplitude exceeds the given sudden failure threshold. The occurrence of degradation failure or sudden failure will lead to system failure, and the two are competing. The reliability analysis of the system is carried out under the competing failure mode, and the remaining life of the system is predicted according to the reliability model.

[0112] Constructing System Description and Model Assumptions: The overall degradation level of the system reflects the system's health condition to a certain extent. As the degree of system degradation continuously increases, its structural strength gradually weakens, resulting in a decline in the system's resistance to external shocks. Based on the overall degradation level, the system is divided into two states: the normal state and the sensitive state. The normal state means that when the overall degradation level of the system is relatively low, it can effectively maintain its structural strength and functions and has a strong ability to resist external shocks. In contrast, the sensitive state means that after the system has experienced a certain degree of degradation, its structural strength significantly decreases, leading to a higher probability of failure when facing conventional shocks.

[0113] When the overall degradation level X(t) of the system is between the predetermined sensitive state threshold H 1 and the degradation failure threshold H 2 , the system is in the sensitive state. Similarly, when the overall degradation level is less than H 1 , the system is in the normal state, and t w represents the time duration during which the system degrades from the normal state to the sensitive state.

[0114] Natural Degradation Modeling: Let t k be the current monitoring time. Assume that the degradation increments ΔX k , ΔX 1 ,..., ΔX 2 ,..., ΔX k during the time [0, t k are independent and identically distributed random variables, and f(Δx) is the probability density function of the degradation increment. Therefore, the kernel density estimate of f(Δx) is:

[0115]

[0116] where k is the number of degradation increments ΔX k during the time [0, t i , h k is the window width, and K(·) is the kernel function.

[0117] The choice of the kernel function may have a relatively small impact on the final estimate. Therefore, the Gaussian kernel function will be used because it has the obvious advantage of being simple and easy to use:

[0118]

[0119] The integrated mean squared error (MISE) is used to measure the error between the kernel density estimate and the probability density function of the degradation increment:

[0120]

[0121] The optimal window width in the kernel density estimate is h kIt can be obtained by solving the minimum value of the integrated mean square error (MISE). as follows:

[0122]

[0123] Substitute the Gaussian kernel function K(Δxk) into equation (4) to obtain the optimal window width h k :

[0124]

[0125] where σ k is the variance of the k initial sample feature degradation increments ΔX 1 , ΔX 2 ,..., ΔX k :

[0126]

[0127] where is the sample mean of the known k degradation increments Δx 1 , Δx 2 ,..., Δx k , expressed as:

[0128]

[0129] Then the probability density function g(x) of the natural degradation process X c (t) at different times can be obtained by convolving the probability density functions of the degradation increments:

[0130]

[0131] Therefore, the reliability function of the system can be obtained in the natural degradation process of the system without considering the impact effect at time t as:

[0132]

[0133] where H 2 is the system degradation failure threshold.

[0134] Random shock modeling:

[0135] Arrival of random shocks: Assume that the arrival of random shocks follows a homogeneous Poisson process with a constant rate of λ. Let N(t) denote the number of random shocks occurring in the time interval (0, t]. Therefore, the probability of n shocks occurring by time t is:

[0136]

[0137] Assume that the magnitudes of the shocks W i , i = 0, 1, 2,... follow a normal distribution The cumulative distribution function of independent and identically distributed is F W Random variable of (w), the duration T of the impact i s , i = 0, 1, 2, … follows a normal distribution The degradation increment caused by the impact is denoted as Y i , i = 0, 1, 2, … The random impacts are divided into four regions according to their amplitudes W, that is, in the safe region (0, D 1 ), the probabilistic damage region [D 1 , D 2 ), the damage region [D 2 , D 3 ), and the fatal region [D 3 , +∞). The impacts in the safe region are considered not to affect the system due to their relatively small amplitudes. The impacts in the probabilistic damage region have a certain probability of damaging the system, and this probability is related to the amplitude of the impact and increases with the increase of the amplitude. The impacts in the damage region will cause cumulative damage to the degradation process and thus accelerate the degradation of the system. The impacts in the fatal region will cause the system to fail immediately because the amplitude is too large for the system to bear. Generally speaking, when the system enters the sensitive state from the normal state, its resistance to external impacts decreases. A reasonable assumption is that a system in the sensitive state can only resist relatively small impacts. In the model proposed in the present invention, it is manifested as a decrease in the impact partition threshold.

[0138] The system degradation process includes long-term natural degradation and accelerated degradation caused by duration impacts. Therefore, the system may have two competing failure modes. One is the degradation failure when the overall degradation level exceeds the degradation failure threshold H due to long-term natural degradation and additional degradation caused by duration impacts 2 When, and the other is the sudden failure that occurs when the impact amplitude exceeds the sudden failure threshold D (D = D 2 under normal conditions, D = D' 2 in the sensitive state), as shown in Figure 3 .

[0139] Assume that the probabilities of the i-th impact S i falling into the safe region, the probabilistic damage region, the damage region, and the fatal region under normal conditions are p 1 , p 2 , p 3 and p 4 , respectively. These different types of impacts can be regarded as independent Poisson processes, and their arrival rates are λp 1 , λp 2 , λp 3 and λp 4 .

[0140]

[0141] In the time interval (0, t], the number of shocks falling into four different regions are denoted by N 1 (t), N 2 (t), N 3 (t) and N 4 (t), respectively, which are independent homogeneous Poisson processes with rates o = 1, 2, 3, 4. Therefore, in the normal state, the probability of n shocks occurring in different regions is:

[0142]

[0143] where n is the number of shocks occurring, and o = 1, 2, 3, 4 represent that the regions where the shocks are located are the safe region, the probabilistic damage region, the damage region, and the fatal region, respectively.

[0144] Similarly, when the system is in the sensitive state, the probabilities that the i-th shock S i falls into the safe region, the probabilistic damage region, the damage region, and the fatal region are p 1 ', p' 2 , p' 3 and p' 4 , respectively, and their arrival rates are λp 1 ', λp' 2 , λp' 3 and λp' 4 . In the time interval [t w , t), the shock counts falling into the four regions are denoted by N 1 '(t), N' 2 (t), N 3 '(t) and N' 4 (t). Therefore, in the sensitive state, the probability of n shocks occurring in different regions is:

[0145]

[0146] where n is the number of shocks occurring, o = 1, 2, 3, 4 represent that the regions where the shocks are located are the safe region, the probabilistic damage region, the damage region, and the fatal region, and t w is the time elapsed for the system to transition from the normal state to the sensitive state.

[0147] Additional degradation caused by shocks in the damage region: The cumulative damage caused by random shocks in the damage region is as follows:

[0148]

[0149] where N 3 (t) represents the number of shock arrivals in the damage region in the normal state, and W iRepresents the amplitude of the impact under normal conditions, T i s Represents the duration after the impact arrives, Represents the damage caused by all impacts in the damaged area under normal conditions; N 3 '(t) represents the number of impacts arriving in the damaged area in the sensitive state, W j Represents the amplitude of the impact in the sensitive state, Represents the duration after the impact arrives, Represents the damage caused by all impacts in the damaged area in the normal state and the sensitive state up to time t when the system is in the sensitive state; Let Y 1 =W i ·T i s Represents the additional increment caused by the i-th impact in the damaged area when the system is in the normal state during the duration T i s under; Represents the additional increment caused by the j-th impact in the damaged area when the system is in the sensitive state. The probability density function of is:

[0150]

[0151] In the formula, is the probability density function of Y 1 of.

[0152] Let We can obtain The probability density function of is:

[0153]

[0154] Therefore, The probability density function of can be obtained through convolution and its expression is:

[0155]

[0156] Additional degradation caused by impacts in the probabilistic damage area: For impacts in the probabilistic damage area, assuming that the probability of the impact causing additional degradation increases linearly with the increase of the impact amplitude, it is defined as:

[0157]

[0158] In the formula, W i Represents the impact magnitude, D 1 is the safety zone threshold, D 2 is the probabilistic damage area threshold, which means that when the impact amplitude is closer to D 2 the possibility of causing damage will increase.

[0159] The cumulative damage caused by random impacts in the probability damage area is as follows:

[0160]

[0161] Where N 2 (t) represents the number of impacts reaching the probability damage area under normal conditions, and W i represents the amplitude of the impact under normal conditions, and T i s represents the duration after the impact arrives, represents the damage caused by all impacts in the probability damage area under normal conditions; N' 2 (t) represents the number of impacts reaching the probability damage area under sensitive conditions, and W j represents the amplitude of the impact under sensitive conditions, and T j s represents the duration after the impact arrives, represents the damage caused by all impacts in the probability damage area in the normal state and the sensitive state up to time t when the system is in the sensitive state. Let Z = a 0 W i T i s (a 0 = p(W i ) > 0), the cumulative distribution function F Z (z) of Z is:

[0162]

[0163] By differentiating the cumulative distribution function, the probability density function f Z (z) of Z is:

[0164]

[0165] Where is the cumulative distribution function of Y 1 ;

[0166] Using the same calculation method for the damage area, we can obtain the probability density function of is:

[0167]

[0168] Let we can obtain the probability density function of is:

[0169]

[0170] Therefore, The probability density function is as follows:

[0171]

[0172] Overall system degradation: The overall system degradation is X s (t) is the internal system degradation X c (t) and the additional damage and The sum of, that is:

[0173]

[0174] Let represent the additional damage brought by the damaged area and the probabilistic damage area under normal conditions, represent the additional damage brought by the damaged area and the probabilistic damage area under sensitive conditions, then X 1 and X 2 The probability density function is:

[0175]

[0176] Reliability analysis of the system under the random correlation model of shock and degradation:

[0177] Reliability analysis under the sudden failure mode: Under normal conditions, when the shock amplitude exceeds the sudden failure threshold D 3 , sudden failure will occur; similarly, under sensitive conditions, when the shock amplitude exceeds the sudden failure threshold D 3 ', sudden failure will occur. Therefore, when only considering sudden failure, the reliability function of the system at time t is:

[0178]

[0179] In the formula, m is the number of shock occurrences under normal conditions, and n is the total number of shock occurrences including normal and sensitive conditions up to time t.

[0180] Reliability analysis under the degradation failure mode: When the overall degradation level of the system reaches the degradation failure threshold H under the combined action of long-term natural degradation and accelerated degradation caused by continuous shocks 2 degradation failure will occur. Considering the influence of random shocks, the reliability function of the system is divided into the following two cases:

[0181] (1), N(t) = 0, no shock reaches the system up to time t, and the system is only affected by long-term natural degradation. When only considering degradation failure, the reliability of the system at time t is:

[0182]

[0183] where \(g(x)\) is the probability density function of the natural degradation amount, and \(H\) 2 is the degradation failure threshold;

[0184] (2) \(N(t)=n\gt0\), the number of shocks reaching by time \(t\) is \(n\). Assume that the number of shocks occurring in the safe area under normal conditions is \(a\), the number of shocks occurring in the probabilistic damage area is \(b\), and the number of shocks occurring in the damage area is \(c\); the number of shocks occurring in the safe area under sensitive conditions is \(d\), the number of shocks occurring in the probabilistic damage area is \(h\), and the number of shocks occurring in the damage area is \(r\), and let \(a + b + c=j - 1\), that is, the system has a total of \(j - 1\) shocks under normal conditions,

[0185] and a total of \(n - j + 1\) shocks under sensitive conditions. At this time, the reliability function of the system at time \(t\) is:

[0186]

[0187] where \(H\) 2 is the degradation failure threshold, \(H\) 1 is the sensitive state threshold, and the system is in a sensitive state when the system degradation amount is greater than \(H\) 1 ;

[0188] Reliability analysis under competing failure modes: Competing failure is the result of the competition between sudden failure and degradation failure. The reliability functions obtained for the two modes respectively can be used to derive the competing failure reliability function of the system at time \(t\) as:

[0189]

[0190] Assume that \(t\) l is the current monitoring time, and the probability density of the remaining life of the system at this time is:

[0191]

[0192] Therefore, substituting equation (34) into (35) can obtain the probability density function of the remaining life of the system.

[0193] Numerical experimental analysis:

[0194] I. Parameter settings:

[0195] Table 1 Parameter settings

[0196]

[0197] II. Plotting of the reliability function:

[0198] According to the parameter values in Table 1, the corresponding reliability function image is plotted as Figure 4 shown. From Figure 4It can be seen that in the initial stage, when the system is in a normal state, the reliability is relatively high and the degradation rate is relatively slow. In this stage, the system can effectively resist external shocks and internal wear, showing good stability and reliability. As time goes by, the system enters a sensitive state, the ability to resist shocks and wear decreases, the system performance degradation intensifies continuously, and the reliability function shows a sharp downward trend. By monitoring and analyzing the reliability function image, necessary maintenance or replacement when the reliability is lower than the acceptable threshold can ensure that the system maintains the best performance throughout its life cycle.

[0199] III. Sensitivity analysis: To guide the improvement of system reliability and optimal design, it is necessary to analyze the sensitivity of model parameters and explore their influence on system reliability. Therefore, this invention mainly studies the arrival rate λ of shocks, the system failure threshold H 2 , the mean value μ of shock amplitudes W and the mean value μ of shock durations T for sensitivity.

[0200] Considering the influence of different λ on system degradation, the corresponding system reliability curves are obtained as Figure 5 shown. It can be seen from the figure that the system reliability shifts to the left as λ increases, indicating that the higher the incidence rate of external shocks, the faster the system degrades and the shorter its life.

[0201] As Figure 6 shown, considering the influence of the failure threshold H 2 on system degradation, the system reliability curves for H 2 = 2, 3, 4 are obtained. It can be seen from the figure that the system reliability shifts to the right as H 2 increases, indicating that the larger the failure threshold, the longer the system can operate and the longer its life.

[0202] Regarding the influence of shock amplitudes on system degradation, the system reliability curves for the mean value μ W of amplitudes = 0.8, 1.0, 1.2, 1.4, 1.6 are obtained as Figure 7 shown. It can be seen from the figure that as μ W continuously increases, the larger the shock amplitude, the greater the impact of the shock on the system, resulting in faster system degradation and earlier failure.

[0203] As Figure 8 shown, considering the influence of the mean value μ T of shock durations on system degradation, the system reliability curves for μ T = 0.8, 1.0, 1.2, 1.4, 1.6 are obtained. It can be seen from the figure that as μ TAs it continues to increase, the impact lasts longer after reaching the system, has a greater impact on the system, and causes the system to degrade faster and fail earlier.

[0204] IV. Remaining Useful Life Prediction: Figure 9 It is the probability density function graph of the remaining useful life of the system considering the impact of the partition duration. As the performance of the system affected by the impact continues to degrade, the PDF of the RUL based on kernel density estimation gradually becomes higher and narrower over time, which means that the predicted remaining useful life becomes more concentrated, with higher certainty, and the predicted value of the RUL also gets closer to the true value.

[0205] The present invention uses the FD001 dataset from C-MPASS. Figure 10 It shows the internal structure of the aero-engine. The engine will be affected by some environmental factors during flight, such as extreme weather and random vibration, etc., which can all be regarded as sudden random impacts. FD001 contains all the data from the degradation to the failure of the engine, including 100 training datasets, 100 test datasets, and the RUL data of 100 test engines. Each dataset contains the monitoring data from 21 sensors. In this experiment, the monitoring data of the T30 sensor of the 42# engine with more obvious degradation characteristics is selected. The degradation data monitored by T30 is as Figure 11 shown, and the image of the system reliability function obtained based on the method proposed in the present invention is as Figure 12 shown.

[0206] To analyze the impact of the partition impact and the duration impact on the system reliability, the images of the reliability functions of the three models listed in Table 2 are plotted in Figure 13 .

[0207] Comparison of the main factors considered affecting the system failure process in Table 2

[0208]

[0209] From Figure 13It can be seen that: (1) The model without considering the partition shock (i.e., Model 1) has a higher reliability level than the model considering the partition shock (i.e., Model 3). This is because the model without considering the partition shock believes that the arrival of each shock will cause additional degradation to the system, without considering the impact of shocks with different intensities on the system. Especially when the shock intensity is relatively high, it may directly lead to system failure. Ignoring this impact may result in an overly optimistic estimation of the system's reliability, and thus a higher reliability level is obtained. (2) The model considering the duration shock (i.e., Model 3) has a lower reliability level than the model not considering the duration shock (i.e., Model 2). This is because when the shock duration is not considered, the shock only affects the system at the moment of shock occurrence, while when the shock duration is considered, the shock will cause continuous damage to the system within a short period, resulting in a lower reliability value of the system compared to the case where the shock duration is not considered.

[0210] Taking the monitoring time from 120 to 180 as an example, the probability density function distribution of RUL and its comparison with the actual RUL value are obtained as Figure 14 shown. It can be seen from the figure that as the engine performance continuously degrades, the sensor conducts real-time monitoring. The shape of the PDF of RUL based on kernel density estimation gradually becomes higher and narrower as time changes, and the predicted RUL value continuously approaches the true value.

[0211] Table 3 presents the root mean square error (RMSE) analysis of the system RUL prediction results. It can be seen that as the monitoring time increases, the RMSE gradually decreases, which means that the average difference between the model prediction value and the true value is decreasing, the prediction result is closer to the true value, and the prediction accuracy is improved.

[0212] Table 3 Remaining Useful Life Prediction Error Analysis

[0213]

[0214]

[0215] The present invention proposes a reliability analysis and remaining useful life prediction model for a system considering the impact of partition duration shock. The non-parametric estimation method of adaptive kernel density is used to model the natural degradation process of the system. The arrival of random shocks follows a homogeneous Poisson process with a constant rate of λ. The random shocks are divided into four regions according to their amplitudes W, namely the safety region, the probabilistic damage region, the damage region, and the fatal region. Reliability analysis and life prediction are carried out considering the impact of duration shock and the mutatable failure threshold caused by system degradation. Finally, numerical experiments and the C-MPASS dataset are used for verification, verifying the necessity and accuracy of the model considering partition, duration shock, and system state changes.

[0216] In the impact damage model proposed in the present invention, only the impact of extreme impacts that cause damage due to the impact amplitude exceeding a given threshold is considered. Since the type of impact received by the system during operation is uncertain, in future research, the change in the system reliability under the influence of other types of impacts (such as δ impact, m - time impact, k - time impact) can be further considered.

[0217] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included within the scope of the present invention.

Claims

1. A system kernel density remaining life prediction method considering partition duration impact, characterized in that: The specific steps are as follows: Step 1: Establish a natural degradation model for the long-term natural degradation process; Step 2: Construct a random impact model to determine the additional degradation caused by the impact on the damage area and the additional degradation caused by the impact on the probabilistic damage area, and calculate the overall degradation of the system: Step 3: In the natural degradation model and the random impact model, reliability analysis of the system under the sudden failure mode, reliability analysis under the degradation failure mode, and reliability analysis under the competition failure mode are performed to obtain the probability density function of the remaining life of the system; Step 4: Set parameters and draw reliability function graphs, analyze the sensitivity of model parameters, and predict the remaining life of the system.

2. The system kernel density remaining life prediction method considering partition duration impact according to claim 1 is characterized in that: In step 1, let t k is the current monitoring time, assuming that at time [0,t k ]The degradation increment of the system ΔX1, ΔX2, ..., ΔX k are independent and identically distributed random variables, f(Δx) is the density function of the degenerate increment, and the kernel density estimate of f(Δx) is: Where k is the time [0,t k ] degenerate increment ΔX i The number of k is the window width, K(·) is the kernel function; Use Gaussian kernel function as the kernel function, the specific expression is: The error between the kernel density estimate and the probability density function of the degraded increment is measured using the integrated mean square error: The optimal window width in kernel density estimation is h k By solving the minimum integrated mean square error (MISE) get: The Gaussian kernel function K(Δx k ) into formula (4) to obtain the optimal window width h k : In the formula, σ k is the feature degradation increment of k initial samples ΔX1, ΔX2, ..., ΔX k Variance of: In the formula, For k known degenerate increments Δx1, Δx2, ..., Δx k The sample mean of is expressed as: The natural degradation process X c (t) The probability density function g(x) at different times is obtained by convolving the probability density function of the degenerate increment: Therefore, the reliability function of the system obtained in the natural degradation process without considering the impact at time t is: Where H2 is the system degradation failure threshold.

3. The system kernel density remaining life prediction method considering partition duration impact according to claim 2 is characterized in that: In step 2, the specific steps are as follows: Step 21: Determine the arrival of random shocks: Assume that the arrival of random shocks follows a homogeneous Poisson process with a constant rate of λ, N(t) represents the number of random shocks occurring in the time interval (0, t], so the probability of n shocks occurring at time t is: Assume that the magnitude of the shock is W i , i=0,1,2,…is a normal distribution The independent and identically distributed cumulative distribution function is F W (w) is a random variable, and the duration of the shock is T i s ,i=0,1,2,…obeys normal distribution The degradation increment caused by the impact is recorded as Y i ,i=0,1,2,… Random shocks are divided into four regions according to their amplitude W, namely, the safety region (0,D1), the probability damage region [D1,D2), the damage region [D2,D3) and the fatal region [D3,+∞); The system degradation process includes long-term natural degradation and accelerated degradation caused by duration shock. There are two competing failure modes in the system. One is the additional degradation caused by long-term natural degradation and duration shock, which makes the overall degradation level exceed the degradation failure threshold H2. The other is the sudden failure that occurs when the shock amplitude exceeds the sudden failure threshold D. Assume that the ith shock S under normal conditions i The probabilities of falling into the safe zone, probabilistic damage zone, damage zone and fatal zone are p1, p2, p3 and p4 respectively. These different types of shocks are regarded as independent Poisson processes, and their arrival rates are λp1, λp2, λp3 and λp4 respectively. The specific expressions are: In the time interval (0, t], the number of impacts falling into the four different regions is represented by N1(t), N2(t), N3(t) and N4(t), which are the rates of In an independent homogeneous Poisson distribution process with o=1,2,3,4, the probability of n shocks occurring in different regions is: Where n is the number of impacts, o=1, 2, 3, 4 represent the impact areas as safe area, probabilistic damage area, damage area and fatal area respectively; Similarly, when the system is in a sensitive state, the i-th impact S i The probabilities of falling into the safe zone, the probabilistic damage zone, the damage zone and the fatal zone are p'1, ​​p'2, p'3 and p'4 respectively, and their arrival rates are λp'1, λp'2, λp'3 and λp'4 respectively. w ,t), the shock counts falling into the four regions are represented by N'1(t), N'2(t), N'3(t) and N'4(t), respectively. The probability of n shocks occurring in different regions under the sensitive state is: Where n is the number of impacts, o=1, 2, 3, 4 represent the impact areas, namely the safety area, the probability damage area, the damage area and the fatal area, t w The time it takes for the system to change from a normal state to a sensitive state; Step 22: Additional degradation caused by impact on the damaged area: The cumulative damage caused by random impact in the damage area is as follows: Where N3(t) represents the number of impacts arriving in the damaged area under normal conditions, and W i Indicates the amplitude of the shock under normal conditions, T i s It indicates the duration after the impact arrives. represents the damage caused by all impacts in the damage area under normal conditions, N'3(t) represents the number of impacts arriving in the damage area under sensitive conditions, and W j Indicates the amplitude of the shock in the sensitive state, It indicates the duration after the impact arrives. It means that when the system is in a sensitive state, up to time t, the damage caused by the impact of all damaged areas in the normal state and sensitive state; let Y1 = W i ·T i s It means that the duration T of the ith impact in the damage area when the system is in a normal state i s The additional increment caused by It represents the additional increment caused by the j-th impact in the damaged area when the system is in a sensitive state. The probability density function of is: In the formula, is the probability density function of Y1; make get The probability density function of is: therefore, The probability density function of is obtained by convolution and its expression is: Step 23: Determine the additional degradation caused by the impact of the probabilistic damage zone: For shocks in the probabilistic damage region, the probability of additional degradation due to shock is assumed to increase linearly with the shock amplitude, defined as: Where W i Indicates the impact size, D1 is the safety zone threshold, and D2 is the probability damage zone threshold; The cumulative damage caused by random impact in the probabilistic damage area is as follows: Where N2(t) represents the number of impacts arriving in the probabilistic damage area under normal conditions, and W i Indicates the amplitude of the shock under normal conditions, T i s It indicates the duration after the impact arrives. represents the damage caused by all shocks in the probabilistic damage area under normal conditions, N'2(t) represents the number of shocks arriving in the probabilistic damage area under sensitive conditions, and W j Indicates the amplitude of the shock in the sensitive state, It indicates the duration after the impact arrives. It means that when the system is in a sensitive state, up to time t, the damage caused by the impact of all probability damage zones in the normal state and sensitive state, let Z = a0W i T i s (a0=p(W i )>0), the cumulative distribution function F of Z Z (z) is: In the formula, is the cumulative distribution function of Y1; By taking the derivative of the cumulative distribution function, we get the probability density function f of Z Z (z) is: The calculation method is the same as that of the damage area. The probability density function of for: make have to The probability density function of is: therefore, The probability density function of is: Step 24: Calculate the overall degradation of the system: Overall system degradation X s (t) is the internal degradation of the system X c (t) and additional damage and The sum of, that is: make It indicates the additional damage caused by the damage area and the probabilistic damage area under normal conditions. Represents the additional damage caused by the damage area and the probabilistic damage area in the sensitive state, then the probability density functions of X1 and X2 are:

4. The system kernel density remaining life prediction method considering partition duration impact according to claim 3 is characterized in that: In step three, the specific steps are as follows: Step 31: reliability analysis under sudden failure mode; In the normal state, when the impact amplitude exceeds the sudden failure threshold D3, a sudden failure will occur; similarly, in the sensitive state, when the impact amplitude exceeds the sudden failure threshold D'3, a sudden failure will occur. Therefore, when only considering sudden failures, the reliability function of the system at time t is: Where m is the number of shocks in the normal state, and n is the number of shocks in the normal and sensitive states up to time t. Step 32: Reliability analysis under degradation failure mode: When the overall degradation level of the system reaches the degradation failure threshold H2 under the combined effect of long-term natural degradation and the accelerated degradation process caused by the duration shock, degradation failure will occur. Considering the influence of random shocks, the system reliability function is divided into the following two cases: Case 1: N(t) = 0. No shock reaches the system until time t. The system is only affected by long-term natural degradation. When only degradation failure is considered, the reliability of the system at time t is: Where g(x) is the probability density function of the natural degradation amount, and H2 is the degradation failure threshold; Case 2: N(t)=n>0. The number of shocks arriving by time t is n. Assume that the number of shocks in the safe zone under normal conditions is a, the number of shocks in the probabilistic damage zone is b, and the number of shocks in the damage zone is c; the number of shocks in the safe zone under sensitive conditions is d, the number of shocks in the probabilistic damage zone is h, and the number of shocks in the damage zone is r. Let a+b+c=j-1, that is, the system has j-1 shocks in normal conditions and n-j+1 shocks in sensitive conditions. At this time, the reliability function of the system at time t is: In the formula, H2 is the degradation failure threshold, H1 is the sensitive state threshold, and when the system degradation amount is greater than H1, the system is in a sensitive state; Step 33: reliability analysis under competition failure mode; Competition failure is the result of competition between sudden failure and degradation failure. According to the reliability functions of the two modes obtained in step 31 and step 32, the competition failure reliability function of the system at time t is derived as follows: Assume t l is the current monitoring moment, and the probability density of the remaining life of the system is: Therefore, substituting equation (34) into (35) can obtain the probability density function of the remaining life of the system.

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