A Multi-Dimensional Health Status Assessment Method for Equipment Based on Bulldozer Distance Measurement

By using a bulldozer distance measurement method, the problem of failing to fully utilize the temporal characteristics of equipment parameters in existing technologies was solved. A multi-dimensional health status assessment process for equipment was constructed, enabling accurate assessment and systematic improvement of the performance status of aerospace vehicles.

CN115841260BActive Publication Date: 2026-06-30BEIHANG UNIV

Patent Information

Authority / Receiving Office
CN · China
Patent Type
Patents(China)
Current Assignee / Owner
BEIHANG UNIV
Filing Date
2021-09-18
Publication Date
2026-06-30

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Abstract

This invention discloses a multi-dimensional health status assessment method for equipment based on bulldozer distance measurement, which uses monitoring parameters (also known as monitoring data) that characterize the equipment's performance status as input. The method includes: first, preprocessing the full-lifecycle data and real-time data to improve data usability; then, performing a weighted average of degradation-sensitive parameters with the same direction of inversion, determining a standard health set using the health status sample sets of the full-lifecycle data and real-time data, and determining a standard fault set using the fault status sample set of historical data; then, calculating the bulldozer distance through a time-series sliding window to obtain a performance degradation sample set; and finally, linearly generating the equipment health status set.
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Description

Technical Field

[0001] This invention relates to a method for assessing the health status of complex equipment systems, and more particularly to a multi-dimensional health status assessment method for equipment based on bulldozer distance measurement. Background Technology

[0002] In the aerospace field, large and complex systems with high reliability, such as spacecraft, are constantly improving in function and performance, as well as in complexity and integration. To ensure the reliable and stable operation of spacecraft during cruise and in orbit, monitoring and assessing the health status of complex systems using degradation information obtained from sensors is an effective health management activity. Currently, data-driven health assessment methods are widely used in the field of spacecraft health status assessment. With the continuous development of sensor technology, the degradation sequences of monitoring parameters of complex spacecraft equipment over time are becoming increasingly easier to obtain, further promoting the development of data-driven health assessment methods.

[0003] In existing data-driven equipment condition assessment methods, the invention patent with application number CN201610036846.X, entitled "A Method for Assessing the Health Status of a Planetary Gearbox," assesses the health status of a planetary gearbox by calculating the energy value of the original vibration signal through empirical mode decomposition. However, this method fails to fully utilize the temporal distribution characteristics of equipment parameter data. The invention patent with application number CN201610173851.5, entitled "A Method for Assessing the Health Status of a Solid Rocket Engine Based on Test Data," utilizes membership degrees under a triangular fuzzy function to achieve target allocation for different health status levels. This method is mostly based on nonlinear and non-stationary vibration signals, neglecting the description of equipment performance status by sensor parameters with obvious degradation trends. Furthermore, none of the above-mentioned existing technologies have formed a systematic and complete condition assessment process. The application effects of existing technologies vary significantly depending on the complexity of the equipment and its multidimensional monitoring parameters. These shortcomings affect the accuracy of equipment condition assessment. Summary of the Invention

[0004] Based on the aforementioned technical problems in the existing technology, this invention proposes a complete multi-dimensional health status assessment method for equipment by fully utilizing the temporal trend characteristics of equipment performance degradation parameter data, based on the property of similarity distribution of bulldozer distance measurement.

[0005] According to one aspect of the present invention, a multi-dimensional equipment condition assessment method based on bulldozer distance measurement is proposed. The method includes: acquiring monitoring parameters of the equipment, wherein the monitoring parameters are monitoring data characterizing equipment performance degradation; performing data preprocessing on the monitoring data to improve data usability; performing a weighted average of degradation-sensitive parameters in the same direction on the preprocessed monitoring data; determining a standard health set using a health state sample set of sample data and data to be assessed; determining a standard fault set using a fault state sample set of historical data; the standard health set and the standard fault set constitute a condition assessment standard sample set; performing time-series sliding window bulldozer distance calculation based on the condition assessment standard sample set to obtain bulldozer distance and performance degradation sample set; and linearly generating a health score based on the bulldozer distance and the performance degradation sample set.

[0006] Preferably, the monitoring parameters are derived from a condition assessment sample set, which includes lifecycle data and real-time data.

[0007] Preferably, the data preprocessing includes data pruning, data imputation, and data denoising. The data pruning operation removes data points that do not meet the requirements or outliers with obvious errors. The data imputation operation addresses the problem of missing data after outlier removal by using interpolation to fill in missing data values. The data denoising operation smooths and reduces noise in the data after pruning and imputation to further improve data quality.

[0008] Preferably, the weighted average of the decay-sensitive parameters with the same direction of reversal includes: normalization of the decay-sensitive parameters, reversal of the decay-sensitive parameters with the same direction of reversal, weighted average of the decay-sensitive parameters, and determination of the healthy sample set and the faulty sample set.

[0009] Preferably, the normalization of the decay-sensitive parameters involves normalizing each dimension of the parameters to the range of 0-1, removing the dimensions of the parameters, and providing a data basis for subsequent steps; the same-direction flipping of the decay-sensitive parameters involves unifying the matrix sequences of the training data and the test data so that each column of the matrix sequences of the training data and the test data is parameterized to have the same trend of change.

[0010] Preferably, the parameters of the standardization process are weighted and averaged to determine the initial health state H = {H1, H2, ..., Ht} of each training device for the first t cycles. t},in

[0011]

[0012] Among them, w j The weight value corresponding to the j-th sensitive parameter is used; the engine life failure state value F = {F} is determined by weighted averaging after t cycles for each training device.n-t ,F n-t+1 ,…,F n},in:

[0013]

[0014] Preferably, all health status values ​​H of the test and training equipment are sorted from smallest to largest, and the initial status of the top-ranked equipment is selected as the health set sample. All fault status values ​​F of the training equipment are sorted from largest to smallest, and the life-end status of the top-ranked equipment is selected as the fault set sample. The selection ratio is 10%.

[0015] Preferably, the time-series sliding window bulldozer distance calculation includes: time-series sliding window bulldozer distance calculation for healthy sample sets and faulty sample sets; and time-series sliding window bulldozer distance calculation for training and test data.

[0016] Preferably, the time-series sliding window bulldozer distance calculation of the training and testing data includes: calculating the healthy state bulldozer distance sequence S. h Calculate the distance sequence S of the bulldozer in fault state. f The bulldozer distance calculation specifically involves: dividing the equipment parameter sample to be evaluated into discrete segments by using a sliding window with a time length of t and a step size of 1; calculating the bulldozer distance for each parameter within each discrete segment sample and its corresponding parameter data distribution in the healthy sample set; then, taking the arithmetic mean of the bulldozer distance calculation results for all parameters; and normalizing this arithmetic mean to obtain the final healthy state bulldozer distance sequence S. h .

[0017] Preferably, the step of generating health status includes: generating a variable-weight sequence, reducing the weight value of the tail data in the faulty bulldozer sample set, increasing the weight value of the front data, and finally generating a healthy variable-weight sequence and a faulty variable-weight sequence; linearly generating health status; and calculating the health status sequence for each piece of equipment based on the health and faulty bulldozer distance datasets, denoted as:

[0018] CV = {CV1, CV2, ..., CV} l},

[0019] This sequence quantitatively characterizes the equipment health status using variable weights:

[0020] CV i =s fi *w hi +(1-s hi )*w fi

[0021] Among them, CV i To equip the i-th cycle health value, shi s fi These are the distance measurement results for the bulldozers in the i-th cycle of each piece of equipment, representing the healthy and faulty samples, respectively.

[0022] The method of this invention requires sufficient equipment monitoring parameter data to support it, and the data needs to have the following characteristics:

[0023] (1) The parameters tend to degrade with engine use (i.e., they are non-constant parameters);

[0024] (2) The parameters can be collected during each working cycle (i.e. the parameters are continuously distributed with the number of cycles).

[0025] The present invention will now be described in detail with reference to the accompanying drawings. Attached Figure Description

[0026] Various embodiments or examples (“Examples”) of this disclosure are disclosed in the following detailed description and accompanying drawings. It is not necessary to draw the drawings to scale. Generally, unless otherwise specified in the claims, the operations of the components and methods disclosed in this invention can be performed in any order. In the drawings:

[0027] Figure 1 This is a flowchart of a method for assessing the multi-dimensional health status of equipment based on bulldozer distance measurement according to the present invention;

[0028] Figure 2 A flowchart of the bulldozer distance calculation method according to the present invention is shown;

[0029] Figure 3 This is a schematic diagram of bulldozer distance window opening calculation according to the present invention;

[0030] Figure 4 This is a parameter state curve characterizing the engine health state according to an example of the present invention;

[0031] Figure 5 yes Figure 4 The parameters shown are the state curves after data preprocessing.

[0032] Figure 6 This is a schematic diagram showing the bulldozer distance measurement results for training engine No. 1 and test engine No. 1.

[0033] Figure 7 This is a sequence diagram of the quantitative health status representation of training engine No. 1 and test engine No. 1. Detailed Implementation

[0034] Before explaining one or more embodiments of this disclosure in detail, it should be understood that the embodiments are not limited to the construction details in their specific applications, and the steps or methods presented in the following embodiments or drawings. The schematic diagrams in the drawings are merely illustrative and do not have a specific drawing scale and size. Embodiments with the technical features of the drawings of this invention should be within the protection scope of this invention.

[0035] According to the method of the present invention, it takes monitoring parameters (also known as monitoring data) that characterize the performance status of equipment as input, wherein the monitoring data can be from various signal sources such as operational data and simulation data. The method generally includes: firstly, preprocessing the full-lifecycle data and real-time data to improve data usability; then, performing a weighted average of degradation-sensitive parameters with the same direction of inversion, determining a standard health set using the health status sample set of the full-lifecycle data and real-time data, and determining a standard fault set using the fault status sample set of historical data; then, calculating the bulldozer distance through a time-series sliding window to obtain the performance degradation sample set; finally, linearly generating the equipment health status set. A flowchart of a multi-dimensional equipment health status assessment method based on bulldozer distance measurement according to the present invention is shown below. Figure 1 As shown, the detailed steps are as follows:

[0036] Step 1: Perform parameter data preprocessing

[0037] Step 1.1: Data culling

[0038] Aerospace vehicles have complex internal structures and collect massive amounts of diverse parameters. Different types of sensors monitor the equipment in real time and return multi-dimensional parameter data. During actual operation, due to sensor acquisition or data transmission failures, the acquired parameters may suffer from packet loss, data aliasing, and data distortion. Therefore, to improve data usability, it is necessary to remove data points that do not meet the requirements or outliers with obvious errors.

[0039] Step 1.2: Data Completion

[0040] To address the issue of missing data after outlier removal, interpolation is used to fill in the missing values. By calculating the slope between the upper and lower adjacent values ​​of the missing value, the missing value is linearly calculated, thus completing the data and improving its usability.

[0041] Step 1.3: Data Denoising

[0042] Due to the complex structure of aerospace vehicles, the data collected by sensors will have data fluctuations such as acquisition errors. In order to remove other noise features while preserving the original degradation trend of parameter data, common noise reduction methods such as wavelet noise reduction are used to smooth and reduce noise on the data after the removal and completion process, thereby further improving the data quality.

[0043] Step 2: Weighted average of the same-direction flipped parameters of the degradation sensitivity parameters to obtain the standard sample set for state assessment.

[0044] Step 2.1: Normalization of decay-sensitive parameters

[0045] We selected parameter data that showed significant degradation during equipment operation as degradation-sensitive data. We then used linear normalization to normalize each parameter to the 0-1 range, removing the dimensions of the parameters and providing a data foundation for subsequent steps.

[0046] Step 2.2 Reversal of the same direction of the decay sensitivity parameter

[0047] Assume the database contains h pieces of equipment, each with n flight cycles throughout its lifespan. Each operational cycle collects historical data on k decay-sensitive parameters throughout the equipment's lifespan as training data, denoted as a matrix sequence.

[0048]

[0049] Assume the database contains w pieces of equipment, each equipment's entire lifecycle includes m flight cycles, and real-time data of k degradation-sensitive parameters are collected during each operational cycle as test data, denoted as a matrix sequence.

[0050]

[0051] Each column of the original dataset represents a parameter. Different parameters exhibit different characteristics depending on the number of work cycles. Therefore, it is necessary to parameterize each column of both the Train and Test datasets to reflect the same trend of change. The data standardization formula is as follows:

[0052]

[0053] Where, x ij To equip the j-th sensitive parameter value for the i-th cycle.

[0054] Step 2.3: Weighted average of recession-sensitive parameters

[0055] Step 2.2 involves uniformly flipping the degradation-sensitive parameters into a parameter list that increases over time. A weighted average is then used to determine the initial health status value H = {H1, H2, ..., H} for each piece of equipment over the first t cycles. t}

[0056]

[0057] Among them, w j For the j-th sensitive parameter, the weight value is determined by weighted averaging. The subsequent t cycles for each training unit are then used as the engine's life-end fault state value F = {F}. n-t ,Fn-t+1 ,…,F n}

[0058]

[0059] Step 2.4: Determine the healthy sample set and the faulty sample set

[0060] Sort all health status values ​​H of the test and training equipment from smallest to largest, and select the initial status of the top-ranked equipment as the health set sample. Sort all fault status values ​​F of the training equipment from largest to smallest, and select the life-end status of the top-ranked equipment as the fault set sample. It is recommended to select 10-20%, preferably 10%.

[0061] Step 3: Calculate the distance between the sliding window bulldozer and the time sequence.

[0062] Figure 2 A schematic diagram illustrating the bulldozer distance calculation method of the present invention is shown. According to... Figure 2 As shown, the bulldozer distance calculation of the present invention includes the bulldozer distance calculation for each parameter of the sample to be evaluated and the bulldozer distance calculation for each parameter of the standard sample set, specifically including:

[0063] Step 3.1: Calculation of the time-series sliding window bulldozer distance between the healthy sample set and the faulty sample set.

[0064] The bulldozer distance is calculated for each parameter in the sliding window of the healthy sample set and the corresponding parameter in the sliding window of the fault sample set. The obtained distances are then standardized. The mean of the standardized distances is then taken to obtain the bulldozer distance between the standard healthy set and the standard fault set for each parameter. The bulldozer distances of all parameters are then standardized to the 0-1 interval, and the average value of the bulldozer distances of each parameter is taken as the health-fault normalization standard S.

[0065] Step 3.2: Calculation of bulldozer distance using time-series sliding window for training and testing data

[0066] Step 3.2.1: Calculate the distance sequence S of the bulldozer in healthy state. h

[0067] Figure 3 A schematic diagram illustrating the window calculation of the bulldozer distance for the sample to be evaluated according to the present invention is shown. Figure 3As shown, the equipment parameter sample to be evaluated is divided into discrete segments by a sliding window with a time length of t and a step size of 1. The bulldozer distance is calculated for each parameter within each discrete segment and its corresponding parameter data distribution in the healthy sample set. Then, the bulldozer distance calculation results for all parameters are averaged, preferably using the arithmetic mean. The bulldozer distance S between the healthy and faulty sample sets from step 3.1 is then normalized to obtain the final healthy state bulldozer distance sequence S. h This sequence characterizes the process of changes in equipment health by quantifying the distribution distance between the equipment to be evaluated and the healthy sample set.

[0068] Step 3.2.2: Calculate the distance sequence S of the bulldozer in the fault state. f

[0069] Similar to step 3.2.1, the equipment parameter sample to be evaluated is divided into discrete segments by a sliding window with a time length t and a step size of 1. The bulldozer distance is then calculated for each parameter within each discrete segment, comparing it to the corresponding parameter data distribution in the fault sample set. The bulldozer distance calculation results for all parameters are then averaged, preferably using the arithmetic mean. The bulldozer distance S between the healthy sample set and the fault sample set from step 3.1 is then normalized to obtain the final bulldozer fault state distance sequence S. f This sequence characterizes the process of change in the degree of equipment failure by quantifying the distribution distance between the equipment to be evaluated and the failure sample set.

[0070] Step 4: Generate health status based on bulldozer distance and performance degradation sample set

[0071] Step 4.1: Generate a variable-weight sequence

[0072] Regarding the bulldozer distance measurement process, the overlapping of data distributions as the bulldozer approaches a fault state leads to lower accuracy of the data at the tail of the faulty bulldozer sample set. To improve the accuracy of the assessment, the weight values ​​of the data at the tail of the faulty bulldozer sample set are reduced, while the weight values ​​of the data at the front are increased. This ultimately generates a healthy variable-weight sequence and a faulty variable-weight sequence, denoted as follows:

[0073] W h -{w h1 w h2 , ...w hi ..., wh l}, W f ={w f1 w f2 , ...w fi ..., w fl}

[0074] Among them, w hi with w fi These are the bulldozer distance weight values ​​for healthy samples and faulty samples, respectively, for the i-th flight cycle.

[0075] The calculation formula is as follows:

[0076]

[0077] w fi =1-w hi

[0078] Among them, w hi +w fi =1, where l is the distance of each equipped bulldozer from the dataset length.

[0079] Step 4.2: Linearly generate health score

[0080] Based on the above dataset of healthy and faulty bulldozers, calculate the health sequence for each piece of equipment, denoted as:

[0081] CV = {CV1, CV2, ..., CV} l}

[0082] This sequence quantitatively characterizes the health status of equipment using variable weights.

[0083] CV i =s fi *w hi +(1-s hi )*w fi

[0084] Among them, CV i To equip the i-th cycle health value, s hi s fi These are the distance measurement results for the bulldozers in the i-th cycle of each piece of equipment, representing the healthy and faulty samples, respectively.

[0085]

Example

[0086] To better demonstrate the method of this invention, engine data from the 2008 PHM Data Challenge was selected for case analysis. This data was obtained through C-MAPSS (Commercial Modular Aero-Propulsion System Simulation), which included 100 training engines (engines operating throughout their entire lifespan) with a total of 20,631 cycles, and 100 test engines (engines to be evaluated). This invention uses Engine 1 (operating throughout its entire lifespan) and Engine 1 (to be evaluated) as examples to conduct a health status assessment case analysis. Engine 1 (operating throughout its entire lifespan) has a historical cycle count of 192 cycles. Engine 1 (to be evaluated) has a historical cycle count of 31 cycles.

[0087] Based on the operational characteristics of aero-engines, eight parameters characterizing engine health were identified: low-pressure compressor outlet temperature, high-pressure compressor outlet temperature, low-pressure turbine outlet temperature, high-pressure compressor outlet pressure, fan speed, high-pressure compressor outlet static pressure, fuel flow ratio, and combustor air-fuel ratio. Taking the No. 1 full-life engine as an example, the raw data for these eight parameters are as follows: Figure 4 As shown.

[0088] Step 1: Identify 8 parameters in the dataset as degradation-sensitive parameters. After performing data preprocessing including data removal, imputation, noise reduction, and normalization, the following results are obtained: Figure 5 The parameter state curves shown.

[0089] Step 2: Based on the data degradation trend, the parameters of high-pressure compressor outlet pressure and low-pressure turbine outlet temperature are reversed so that the degradation direction of all eight parameters is from small to large. The engine numbers corresponding to the healthy sample set and the faulty sample set are determined through the above steps, as shown in Table 1.

[0090] Table 1 Engine Numbers Corresponding to Healthy and Faulty Sample Sets

[0091] Engine number Health Sample Set 77#、82#、94#、14#、8#、1#、46#、60#、27#、81# Fault Sample Set 55#、61#、21#、83#、7#、39#、90#、72#、65#、15#

[0092] Step 3: Set the bulldozer distance sliding window width to t=5. Calculate the bulldozer distance using the sliding window between the test engine and the health and fault sample sets. Take the average of the 8 parameters and normalize the results for display. Taking training engine No. 1 and test engine No. 1 as examples, the distance measurement results are as follows: Figure 6 As shown.

[0093] Step 4: Taking the No. 1 training engine and test engine as examples, with a bulldozer distance dataset length of 188, calculate the health value for each flight cycle to obtain a quantitative representation sequence of engine health status, and visualize the results. The calculation results are as follows: Figure 7As shown in the results, the method proposed in this invention can effectively evaluate the engine performance status. The health status of the No. 1 full-life training engine continuously declines with the increase of the number of operating cycles. The No. 1 test engine is in the initial stage of operation, and its health status has not shown significant decline.

[0094] The above embodiments of the present invention are merely exemplary and do not constitute any limitation on the scope of the present invention. Those skilled in the art should understand that modifications or substitutions can be made to the details and form of the technical solutions of the present invention without departing from the spirit and scope thereof, but all such modifications and substitutions fall within the protection scope of the present invention.

Claims

1. A method for multi-dimensional health status assessment of equipment based on bulldozer distance measurement, the method comprising: Acquire monitoring parameters used to characterize equipment performance degradation, and perform data preprocessing on the monitoring data to improve data usability; The monitoring data after data preprocessing is subjected to a weighted average of the same direction flipping of the decay-sensitive parameters. A standard health set is determined using the health status sample set of the sample data and the data to be evaluated, and a standard fault set is determined using the fault status sample set of historical data. The standard health set and the standard fault set constitute the standard sample set for status assessment. Based on the aforementioned condition assessment standard sample set, time-series sliding window bulldozer distance calculations are performed to obtain bulldozer distance and performance degradation sample sets; The time-series sliding window bulldozer distance calculation includes: time-series sliding window bulldozer distance calculation for healthy sample sets and faulty sample sets; time-series sliding window bulldozer distance calculation for training and test data; the time-series sliding window bulldozer distance calculation for training and test data includes: calculating the bulldozer distance sequence S in the healthy state. h Calculate the distance sequence S of the bulldozer in fault state. f The bulldozer distance calculation specifically involves: dividing the equipment parameter sample to be evaluated into discrete segments by using a sliding window with a time length of t and a step size of 1; calculating the bulldozer distance for each parameter within each discrete segment sample and its corresponding parameter data distribution in the healthy sample set; then, taking the arithmetic mean of the bulldozer distance calculation results for all parameters; and normalizing this arithmetic mean to obtain the final healthy state bulldozer distance sequence S. h ; Health is generated linearly based on the bulldozer distance and the performance degradation sample set; The aforementioned weighted average with simultaneous reversal of decay-sensitive parameters includes: normalization of decay-sensitive parameters, simultaneous reversal of decay-sensitive parameters, weighted averaging of decay-sensitive parameters, and determination of healthy and faulty sample sets. Normalization of decay-sensitive parameters involves normalizing each dimension of the parameters to the range of 0-1, removing the dimensions of the parameters, and providing a data foundation for subsequent steps. Simultaneous reversal of decay-sensitive parameters involves unifying the matrix sequences of training and testing data so that each column of the matrix sequences of training and testing data exhibits the same trend of change. The parameters of the standardization process are weighted and averaged to determine the initial health state H={H1, H2,…,H} of each training device based on the first t cycles. t },in ; Among them, w j The weight value corresponding to the j-th sensitive parameter is used; the engine life failure state value F = {F} is determined by weighted averaging after t cycles for each training device. n-t , F n-t+1 ,…,F n }, x ij The value of the j-th sensitive parameter is set for the i-th cycle; k is the number of sensitive parameters; where: 。 2. The equipment multi-dimensional health status assessment method according to claim 1, characterized in that, The monitoring parameters are derived from a condition assessment sample set, which includes lifecycle data and real-time data.

3. The equipment multi-dimensional health status assessment method according to claim 1, characterized in that, The data preprocessing includes data pruning, data imputation, and data denoising. The data pruning operation removes data points that do not meet the requirements or outliers with obvious errors. The data imputation operation addresses the problem of missing data after outlier removal by using interpolation to fill in missing data values. The data denoising operation smooths and reduces noise in the data after pruning and imputation to further improve data quality.

4. The equipment multi-dimensional health status assessment method according to claim 3, characterized in that, Sort all health status values ​​H of the test and training equipment from smallest to largest, and select the initial status of the top-ranked equipment as the health set sample. Sort all fault status values ​​F of the training equipment from largest to smallest, and select the life-reaching status of the top-ranked equipment as the fault set sample. The selection ratio is 10%.

5. The equipment multi-dimensional health status assessment method according to claim 1, characterized in that, The steps for generating health status include: A variable-weight sequence is generated by reducing the weight value of the tail data in the faulty bulldozer sample set and increasing the weight value of the front data, ultimately generating a healthy variable-weight sequence and a faulty variable-weight sequence. Linearly generate health scores; based on the health and faulty bulldozer distance datasets, calculate the health score sequence for each piece of equipment, denoted as: , This sequence quantitatively characterizes the equipment health status using variable weights: Among them, CV i To equip the i-th cycle health value, These represent the distance measurement results for the bulldozers in the i-th cycle of each piece of equipment, for both healthy and faulty samples; w hi with w fi These represent the bulldozer distance weight values ​​for healthy samples and faulty samples, respectively, for the i-th flight cycle; where w hi +w fi =1, where l is the distance of each equipped bulldozer from the dataset length.