Electromechanical equipment reliability evaluation method based on switching Markov process and fractional Brownian motion

Abstract

The invention relates to the technical field of reliability management of electromechanical equipment, in particular to an electromechanical equipment reliability evaluation method based on a switching Markov process and fractional Brownian motion, which comprises the following steps: constructing a switching Markov process model, the description module is used for describing a multi-stage degradation process experienced by equipment in a life cycle and state transition thereof; based on the external working condition parameters and the switching Markov process, constructing a stage dependency model; establishing a nonlinear degradation model of the electromechanical equipment based on the fractional Brownian motion process, and introducing a random effect into the model to represent individual differences; and constructing a two-stage parameter estimation method, and sequentially estimating unknown parameters in the switching Markov process and the degradation model. The performance degradation of the electromechanical equipment is accurately modeled, and the nonlinearity, randomness, individual difference, long-term memorability and stage dependence of the multi-stage degradation of the electromechanical equipment are effectively captured, so that the reliability of the electromechanical equipment is accurately evaluated.

Description

Technical Field

[0001] This invention relates to the field of electromechanical equipment reliability management technology, and more specifically to a method for evaluating the reliability of electromechanical equipment based on switching Markov processes and fractional Brownian motion. Background Technology

[0002] Reliability assessment of electromechanical equipment is a key technology for ensuring the safe and efficient operation of industrial systems. Reliability-centered maintenance avoids resource waste caused by over-maintenance while effectively preventing sudden failures due to insufficient maintenance, thus significantly reducing downtime losses and safety risks. This is of strategic importance to the stable operation of core industries such as energy, transportation, and manufacturing. Therefore, high-precision reliability assessment has become a rigid requirement for the operation and maintenance of industrial electromechanical equipment.

[0003] The degradation process of electromechanical equipment exhibits characteristics of long-term memory and multi-stage degradation, with dependencies between stages. Traditional stochastic process models based on Markov properties neglect long-term memory; while multi-stage degradation models can describe the stage-specific characteristics, they sever the physical connections between stages and fail to address the randomness of the changing points; and while single fractional Brownian motion models can characterize long-term memory, they cannot accommodate multi-stage degradation. These shortcomings result in insufficient accuracy and low computational efficiency in degradation models, making it difficult to support engineering decisions with high reliability requirements. Summary of the Invention

[0004] In view of this, the present invention provides a method for reliability assessment of electromechanical equipment based on switching Markov processes and fractional Brownian motion, aiming to solve the above-mentioned technical problems.

[0005] To achieve the above objectives, the present invention adopts the following technical solution: A reliability assessment method for electromechanical equipment based on switching Markov processes and fractional Brownian motion includes the following steps: S1. Construct a switching Markov process model to describe the multi-stage degradation process and state transitions experienced by the device during its life cycle. S2. Based on external operating parameters and switching Markov processes, construct a stage dependency model; S3. Establish a nonlinear degradation model for electromechanical equipment based on fractional Brownian motion, and introduce random effects into the model to characterize individual differences; S4. Construct a two-stage parameter estimation method to estimate the unknown parameters in the switching Markov process and the degradation model in turn. S5. Based on the definition of first arrival time, conduct reliability and mean time between failures assessment of electromechanical equipment.

[0006] Furthermore, S1 includes: S1.1, through discrete random processes Describe the state of the degradation stage of electromechanical equipment, among which This indicates the degradation stage at time t, and that there are m degradation stages in the device's lifecycle. , and These represent the initial degradation stage and the final degradation stage, respectively. S1.2 Describing stochastic time-varying processes using Markov processes The state transitions, where the next degenerate stage depends only on the current stage, are defined by the transition rate matrix of the degenerate stage: in Indicates from arrive The state transition rate, and Based on the initial degradation state probability vector The probabilities of the device being in various degradation states at time t can be obtained by solving the Kolmogorov differential equation. .

[0007] Furthermore, S2 includes: S2.1, The stress level of the external working conditions Normalized to: S2.2, Based on the stress level and stage state probability at different stages, the comprehensive impact of different degradation stages on equipment degradation is calculated as follows:

[0008] in , It is considered as the weight of the current stage of degradation in the overall degradation.

[0009] Furthermore, S3 includes: S3.1 Establish a nonlinear degradation model that considers long-term memory:

[0010] in This is the initial degradation value. The diffusion coefficient is... The drift coefficient, and These are parameters used to describe the overall characteristics. Represents a time scale function. For fractional Brownian motion, The Hearst index is used to capture individual differences. Follows a normal distribution ; S3.2. Characterize the memory characteristics by the Hurst exponent H: when 0 < H < 0.5, it has short-term memory; when H = 0.5, the fractional Brownian motion degenerates into Brownian motion; when 0.5 < H < 1, it has long-term memory.

[0011] Furthermore, the fractional Brownian motion is defined as:

[0012] where is the standard Brownian motion, is the gamma function; its autocorrelation function is expressed as:

[0013] The set of unknown parameters to be estimated is .

[0014] Furthermore, it is characterized in that the S4 includes: S4.1. Conduct statistical inference based on the historical data of n degradation samples ( , ). S4.2. Determine the change point using the Schwarz information criterion:

[0015] where is the maximum likelihood estimate of , is the number of parameters in the model, is the sequence length, is the maximum likelihood function without considering multi-stage degradation; Determine the change point interval by comparing the SIC values of adjacent data segments, take the left endpoint of the interval as the change point, and then calculate the average residence time of the device at each degradation stage; S4.3. Determine the unknown parameters of the non-linear degradation model by maximum likelihood estimation, and the log-likelihood function is:

[0016] where , , , is the detection time, [ , is the number of samples, is the number of detections; Use a multi-dimensional search algorithm to solve the optimal value of the unknown parameter .

[0017] Furthermore, the S5 includes: ​​​​​S5.1, Define equipment lifespan based on first arrival time:

[0018] in It is the failure threshold.

[0019] The probability density function of equipment life T in S5.2 is:

[0020] in , , ; The probability density function of the equipment lifespan T, considering individual differences, is:

[0021] in yes The probability density distribution function; The formula for calculating the reliability of equipment is:

[0022] Mean time between failures (MTBF) is: .

[0023] Furthermore, the method is specifically applied to the reliability assessment of electromechanical equipment, wherein: degradation data is collected through sensors; multi-stage degradation characteristics caused by varying operating conditions; the change points determined by the Schwarz information criterion are used to divide different degradation stages; and degradation parameters are determined based on maximum likelihood estimation and multidimensional search algorithms.

[0024] Compared with the prior art, the present invention has the following positive effects: This method accurately describes the state transitions of a device during multi-stage degradation by switching Markov processes. It can identify the dynamic switching behavior between different degradation stages and uses fractional Brownian motion to characterize the long-term memory and nonlinearity of the degradation process. At the same time, it introduces random effects to effectively characterize the differences between individual devices, thus accommodating multiple degradation characteristics such as nonlinearity, multi-stage degradation, memory, randomness, and individual differences within a unified framework. Attached Figure Description

[0025] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0026] Figure 1 This is a flowchart of the electromechanical equipment reliability assessment method based on switching Markov processes and fractional Brownian motion according to the present invention.

[0027] Figure 2 A schematic diagram of the basic structure of a hydraulic pump provided in an embodiment of the present invention.

[0028] Figure 3 This is a schematic diagram of hydraulic pump return oil flow data provided in an embodiment of the present invention.

[0029] Figure 4 This is a schematic diagram illustrating the estimated occurrence time of a hydraulic pump change point, provided in an embodiment of the present invention.

[0030] Figure 5 This is a schematic diagram illustrating the state probabilities of different degradation stages of a hydraulic pump, as provided in an embodiment of the present invention.

[0031] Figure 6 This is a schematic diagram of the reliability curve of the hydraulic pump provided in an embodiment of the present invention. Detailed Implementation

[0032] The technical solutions in the embodiments of the present invention will be clearly and completely described below. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0033] See appendix Figure 1-6 To make the above-mentioned objectives of the present invention more apparent and understandable, and to demonstrate the features and advantages of the inventive method, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0034] This invention provides a reliability assessment method for electromechanical equipment based on switching Markov processes and fractional Brownian motion. First, a stage-dependent Markov switching process model is established to describe the transitions between different states during multi-stage degradation. Second, a nonlinear degradation model is established using fractional Brownian motion to accurately characterize the long-term memory characteristics of the equipment during degradation. Considering the differences between individual equipment units, random effects are introduced to describe inter-unit variability. Then, a two-stage parameter estimation method is established, and equipment reliability assessment is conducted. This invention can accurately capture the comprehensive characteristics of nonlinearity, randomness, individual variability, long-term memory, and multi-stage dependence in the equipment degradation process. Applying this method can achieve accurate equipment reliability assessment.

[0035] This invention is achieved through the following techniques: Step 1: Construct a switching Markov process model to describe the multi-stage degradation process and state transitions experienced by the device during its lifecycle, including the following steps: Step 2: Based on external operating parameters and the switching Markov process, construct a stage dependency model; Step 3: Establish a nonlinear degradation model of electromechanical equipment based on fractional Brownian motion process, and introduce random effects into the model to characterize individual differences; Step 4: Construct a two-stage parameter estimation method to estimate the unknown parameters in the switching Markov process and the degradation model in turn. Step 5: Based on the definition of first arrival time, conduct reliability and mean time between failures assessment of electromechanical equipment.

[0036] The specific steps for step 1) are as follows: Step 1.1, Indicates in The degradation stage of time, using discrete stochastic processes This describes the state of mechanical and electrical equipment during its degradation stage. Mechanical and electrical equipment exists throughout its life cycle. Each stage of degradation ,in and These represent the initial degradation stage and the final degradation stage, respectively.

[0037] Step 1.2: Describe the stochastic time-varying process using a Markov process. According to the Markov property, the state transition in the next time step depends only on the current time step and is independent of all previous time steps. This can be represented as: (1) In the formula, and These represent the current and subsequent stages of degradation, respectively.

[0038] The transition process of component degradation stages is a unidirectional degradation process, occurring sequentially until reaching the "final degradation stage." Therefore, the transition rate matrix of degradation stages can be defined as follows: (2) In the formula, Indicates from arrive The state transition rate, and .

[0039] Based on the given initial degradation state probability vector The time-dependent operation of the device can be obtained by solving the Kolmogorov differential equation. The probability of being in each degenerate state: (3) In the formula, Indicates time The state probability vector.

[0040] The specific steps of step 2 are as follows: Step 2.1, external operating conditions cause a dependency on degradation stages; more precisely, the correlation between degradation stages is related to measurable external operating conditions; different degradation stages stress level This indicates that the stress data is then normalized to... (4) Step 2.2: Based on the stress level and stage state probability at different stages, the comprehensive impact of different degradation stages on equipment degradation can be calculated as follows: (5) In the formula, This can be considered as the weight of the current stage of degradation within the overall degradation. .

[0041] The specific steps of step 3 are as follows: Step 3.1, fractional Brownian motion is used to describe the long-term memory property of device degradation. The nonlinear degradation model considering long-term memory is defined as follows: (6) In the formula, Let be the degenerate value at initial time 0 (without loss of generality, let be) ), The diffusion coefficient is... The drift coefficient, and These are parameters used to describe the overall characteristics. Represents a time scale function. Represents a fractional Brownian motion process. This is the Hearst exponent.

[0042] To describe the differences between units, let Satisfies a normal distribution Let . It is a widely used form, for The degradation trajectory is linear, and equation (6) can be further written as (7) Step 3.2, in fractional Brownian motion, Within the range of 0-1, it is used to characterize memory properties. For Fractional Brownian motion exhibits short-term memory, and future states are negatively correlated with past states. For Fractional Brownian motion degenerates into Brownian motion, and there is no correlation between future states and past states. For Fractional Brownian motion has long-term memory, and there is a positive correlation between future increments and past states.

[0043] Fractional Brownian motion is a typical zero-mean Gaussian process, possessing long-term memory and self-similarity. Therefore, it can describe non-Markovian degenerate processes, specifically fractional Brownian motion. Can be defined as (8) In the formula, It is standard Brownian motion. for: (9) In the formula, It is a gamma function.

[0044] The autocorrelation function of the standard fraction Brownian motion given in equation (8) can be expressed as: (10) Analysis of the switched Markov model and the nonlinear degradation model shows that, These are unknown parameters that need to be estimated.

[0045] The specific steps of step 4 are as follows: Step 4.1: Perform statistical inference based on existing historical degradation data to estimate unknown parameters; A degraded sample, historical degradation data ( , ) indicates that the device is in The degradation value at that time, This is the total number of tests. Furthermore, for simplicity, let... , , .

[0046] Step 4.2: The moment from the current degradation stage to the next degradation stage is called the change point. The Schwarz information criterion is used to determine the change point, which can be expressed as: (11) In the formula, for Maximum likelihood estimation, The number of parameters in the model. For sequence length, The maximum likelihood function without considering multi-stage degradation: (12) in (13) (14) use Indicates degraded data , and It can be calculated using formulas (11) and (12). If , and They belong to the same degenerative stage; otherwise, they are two different degenerative stages. ( There exists a changing point on the interval. Repeat the above process until the first changing point interval is found. If the interval of the first changing point is... Next, we need to find the next changing interval, using... Indicates degraded data It can be calculated using the same method. and The second changeover interval is determined by comparing the magnitudes of the two. This process is repeated until all changeover intervals are found.

[0047] Hypothesis Sample of The interval of each variable point is The left endpoint of the interval of change is considered the change point. The change points of each sample are denoted as Therefore, the average residence time of the equipment in each degradation stage is: (15) In the formula, express The sample exists in the first The number of variable points.

[0048] In a Markov process, the state transition time follows an exponential distribution. Based on the properties of the exponential distribution, the state transition rate matrix can be calculated as follows: (16) Step 4.3, Determine the parameters back, These are unknown parameters to be determined. For simplicity, let... It's not hard to see that... Therefore, the overall log-likelihood function can be expressed as: (17) In the formula, , , .

[0049] Command (17) regarding and The partial derivatives of are all equal to zero, so we can obtain (18) (19) According to equations (18) and (19), and They cannot be obtained directly; they are related to Related. and Substituting into (17), we can obtain the conditional log-likelihood function as follows: (20) The conditional log-likelihood function can be found using a multidimensional search algorithm to determine the maximum likelihood estimate of the unknown parameters. The maximum likelihood estimate can be solved directly using the "fminsearch" function in MATLAB. Substituting into equations (18) and (19) respectively, we can determine and The optimal value.

[0050] The specific steps of step 5 are as follows: Step 5.1, according to the definition of first arrival, for the failure threshold The lifespan of equipment can be defined as (twenty one) Since fractional Brownian motion is neither a Markov process nor... The semimartingale process makes it difficult to obtain an analytical expression for the lifetime probability density function. We approximate the fractional Brownian motion using Brownian motion through the dynamic diffusion coefficient: (twenty two) In the formula, .

[0052] and Following the same distribution, let Therefore, equation (7) can be further written as (twenty three) Step 5.2, according to equation (23), equipment lifespan The probability density function can be expressed as: (twenty four) because Follows a normal distribution, lifespan The probability density function is (25) In the formula, .

[0053] Based on the probability density function of lifetime, the formula for calculating equipment reliability is as follows: (26) Therefore, the mean time between failures (MTBF) of the equipment is (27) The invention will now be further described with reference to the accompanying drawings: Figure 1 The flowchart of the reliability assessment method for electromechanical equipment based on switching Markov processes and fractional Brownian motion is as follows: The performance degradation data and operating parameters of multiple hydraulic pumps were obtained; a multi-stage degradation model based on the switching Markov process was established for the hydraulic pumps; a nonlinear degradation model based on fractional Brownian motion was established for the hydraulic pumps, and individual differences were introduced into the model; the unknown parameters in the switching Markov process and the nonlinear degradation model were determined by a two-stage parameter estimation method; the reliability and mean time between failures of the hydraulic pumps were calculated, and the simulation results were obtained.

[0054] Figure 2 The basic structure of a hydraulic pump is shown. Wear is the primary form of hydraulic pump degradation. In the initial wear stage, the surface roughness peak becomes smooth, and the wear rate decreases rapidly over a short period. In the steady wear stage, the wear rate is low and stable, lasting for a long time. In the rapid wear stage, the wear rate increases rapidly, resulting in noticeable vibration and noise. Generally, hydraulic pumps are repaired when they reach the rapid wear stage. Therefore, the degradation failure threshold of a hydraulic pump can be defined as the time required to reach the rapid wear stage. Thus, during normal use of a hydraulic pump, only the initial and steady wear stages exist.

[0055] The return oil flow rate of a hydraulic pump can be used to describe its performance degradation. Therefore, a flow sensor is used to record the return oil flow rate data of the hydraulic pump.

[0056] Figure 3 Normalized return oil flow rate data for four hydraulic pumps are presented, with a normalized fault threshold of 1 and a normalized discharge pressure coefficient of [0.02, 0.98]. In the initial stage, the pressure is low, the oil flow is stable, and the return oil flow rate increases rapidly. In the steady stage, as the load increases, the return oil flow rate begins to decrease.

[0057] Figure 4 The inflection points of the four hydraulic pump degradation stages are given.

[0058] like Figure 5As shown, based on the estimated change points, the state probabilities of the hydraulic pump being in the initial wear state and the steady wear state can be determined. Figure 6 The obtained hydraulic pump reliability curve is presented.

[0059] The evaluation yielded a mean time between failures (MTBF) of 1223.7 h for the hydraulic pump. Note that the actual MTBF of the hydraulic pump used is 1200 h, representing a relative error of only 1.975%.

[0060] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of the present invention.

[0061] The various embodiments described in this specification are presented in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A reliability assessment method for electromechanical equipment based on switching Markov processes and fractional Brownian motion, characterized in that, It includes the following steps: S1. Construct a switching Markov process model to describe the multi-stage degradation process and its state transition experienced by the device during its life cycle; S2. Based on the external operating conditions parameters and the switching Markov process, construct a stage dependence model; S3. Establish a non-linear degradation model for the electromechanical device based on the fractional Brownian motion process, and introduce random effects into the model to characterize individual differences; S4. Construct a two-stage parameter estimation method to sequentially estimate the unknown parameters in the switching Markov process and the degradation model; S5. Based on the definition of the first passage time, conduct the reliability and mean time between failures assessment of the electromechanical device.

2. The method for reliability assessment of electromechanical equipment based on switching Markov processes and fractional Brownian motion according to claim 1, characterized in that, The S1 includes: S1.1, through discrete random processes Describe the state of the degradation stage of electromechanical equipment, among which This indicates the degradation stage at time t, and that there are m degradation stages in the device's lifecycle. , and These represent the initial degradation stage and the final degradation stage, respectively. S1.2 Describing stochastic time-varying processes using Markov processes The state transitions, where the next degenerate stage depends only on the current stage, are defined by the transition rate matrix of the degenerate stage: ； in Indicates from arrive The state transition rate, and Based on the initial degradation state probability vector The probabilities of the device being in various degradation states at time t can be obtained by solving the Kolmogorov differential equation. .

3. The method for reliability assessment of electromechanical equipment based on switching Markov processes and fractional Brownian motion according to claim 1, characterized in that, The S2 includes: S2.1, The stress level of the external working conditions Normalized to: ； S2.

2. According to the stress levels and stage state probabilities in different stages, calculate the comprehensive impact of different degradation stages on the device degradation as: ； in , It is considered as the weight of the current stage of degradation in the overall degradation.

4. The method for reliability assessment of electromechanical equipment based on switching Markov processes and fractional Brownian motion according to claim 1, characterized in that, The S3 includes: S3.

1. Establish a non-linear degradation model considering long-term memory: ； in This is the initial degradation value. The diffusion coefficient is... The drift coefficient, and These are parameters used to describe the overall characteristics. Represents a time scale function. For fractional Brownian motion, The Hearst index is used to capture individual differences. Follows a normal distribution ; S3.

2. Characterize the memory characteristics through the Hurst exponent H: when 0 < H < 0.5, it has short-term memory; when H = 0.5, the fractional Brownian motion degenerates into Brownian motion; when 0.5 < H < 1, it has long-term memory.

5. The method for reliability assessment of electromechanical equipment based on switching Markov processes and fractional Brownian motion according to claim 4, characterized in that, The fractional Brownian motion Defined as: ； in It is standard Brownian motion. It is the gamma function; its autocorrelation function is expressed as: ； The set of unknown parameters that need to be estimated is .

6. The method for reliability assessment of electromechanical equipment based on switching Markov processes and fractional Brownian motion according to claim 5, characterized in that, The S4 includes: S4.1 Historical data based on n degraded samples ( , Perform statistical inference; S4.

2. Use the Schwarz information criterion to determine the change point: ； in for Maximum likelihood estimation, The number of parameters in the model. For sequence length, This is the maximum likelihood function that does not consider multi-stage degradation; Determine the change point interval by comparing the SIC values of adjacent data segments, use the left endpoint of the interval as the change point, and then calculate the average residence time of the device in each degradation stage; S4.

3. Determine the unknown parameters of the non-linear degradation model through maximum likelihood estimation, and the log-likelihood function is: ； Where, , , , It's the detection time. , It is the sample size. It refers to the number of tests; Solving for unknown parameters using a multidimensional search algorithm The optimal value.

7. The method for reliability assessment of electromechanical equipment based on switching Markov processes and fractional Brownian motion according to claim 1, characterized in that, The S5 includes: S5.

1. Define the device life based on the first passage time; ； in It is the failure threshold; The probability density function of the device life T in S5.2 is: ； in , ， ； The probability density function of the device life T considering individual differences is ； in yes The probability density distribution function; The calculation formula for the reliability of the device is: ； The mean time between failures is: 。 8. The method for reliability assessment of electromechanical equipment based on switching Markov processes and fractional Brownian motion according to claim 1, characterized in that, The method is specifically applied to the reliability assessment of electromechanical devices, where: the degradation data is collected by sensors; the multi-stage degradation characteristics caused by variable operating conditions; the change points determined by the Schwarz information criterion are used to divide different degradation stages; the degradation parameters are determined based on maximum likelihood estimation and the multi-dimensional search algorithm.

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