Compilation Method of Flight Service Environment Spectrum Based on Altitude Features and Fuzzy Clustering

: The ﬂight service environment spectrum is essential to the evaluation of the life of components in aeroengines; however, real altitude, as an important ﬂight parameter, introduces considerable challenges when compiling the service environment spectrum because of its non-stationary and non-ergodic characteristics. In this article, by solving trend terms from original data and removing redundancy from peak–valley values, an altitude feature extraction method is developed for the Frequency of Climb–Descent Flight (FCDF). Then, taking the 2D vector, composed of the maximum ﬂight altitude and FCDF, as the input of fuzzy clustering, the service environment spectrum is compiled. Some examples are given to illustrate the presented methods. The results show that the FCDF does not increase with the maximum altitude; during high-altitude ﬂight, the FCDF is the largest, while during mid-to low-altitude ﬂight, the time proportion is the largest. On the other hand, mid-altitude or low-altitude ﬂight not only has a small training frequency for climb–descent actions but also a low time proportion. The research results will provide a reference for the compilation of a service environment spectrum that considers maneuvering ﬂight actions.


Introduction
The flight service environment spectrum refers to flight time proportions for various working environments in which an aircraft operates during service.In research related to the lifespan of aircraft components, the flight service environment can be characterized by the time history of various flight parameters; that is, some flight parameters' time history characteristics in single or multiple flights, such as altitude, airspeed, yaw angle, vertical overload coefficient, turning radius, or frequency.These time histories can be real data recorded by aircraft, or they can be some generalized parameters after data processing.The definition of the flight service environment spectrum is similar to the random load spectrum in fatigue testing, but a flight service environment has many parameters in one flight profile rather than one (such as only load).
It is necessary to consider the flight service environment spectrum when evaluating the life of components in aeroengines.For example, when calculating the load of the main bearing, real flight parameters under different service environments, as input variables, may save the establishment of strong coupling relationships on more than 13 flight parameters [1].However, the issue of compiling a flight service environment spectrum related to the main bearing load has not yet been addressed.
Pressure altitude, as an important indicator of aircraft maneuverability, is currently always used to compile the service environment spectrum of all engine components [2][3][4] because all combat or airdrop actions, such as air-air, medium-air-ground, and airground states, involve either fast or slow ascent and descent processes.
In NATO countries, several special research plans (ENSIP, LUCID, TURBISTAN, Flight-by-Flight, etc.) have been implemented for environment spectrum compilation [5][6][7][8], where the plan TURBISTAN [9,10] first decomposed maneuvering operations using altitude parameters to determine the load history of engine components.The flight service environments were divided into the initial segment (ground and climb), the middle segment (air stage), and the end segment (landing and ground taxiing stage) based on the altitude.However, no literature is available regarding the compilation of a detailed spectrum for the complex maneuvering motions in the middle segment.
In China, many scholars have carried out thorough research on feature extraction and clustering methods for use in the aircraft service environment [11][12][13][14][15]. References [13][14][15] used the maximum altitude of a flight profile as a parameter when compiling the environmental spectrum, where only the limit value of altitude worked and the process characteristics were ignored.However, the la er can more precisely reflect the time-varying service environment.
In addition, the method always used by major aircraft companies to classify flight service environments based on training tasks does not meet the requirements of the life analysis of some key components.For example, for main bearings, the compilation of the service environment spectrum needs to consider flight parameters related to them, such as engine rotor speeds, yaw angle, pitch angle, rolling angle, angle velocity in three directions, and overload coefficients, which are closely related not only to the training task but also to the changing details of altitude and Mach.
In fact, it is very difficult to extract altitude characteristics when compiling a flight service environment spectrum.There are many factors that affect flight altitude data, especially in complex flight training tasks such as those of military aircraft.
Mathematically, the stochastic process composed of altitude samples is non-stationary and non-ergodic.Due to the coupling relationship between flight parameters, most of the transient ascent-descent processes at altitude are accompanied by changes in the yaw angle.Therefore, with the help of yaw angle data, it is easier to identify certain changes in the service environment, such as a vertical dive or pull-up environment.However, for service environments such as air-air, medium-air-ground, and air-ground states, one flight profile can include many climb-descent flight sessions, and the time of a single climb-descent flight is so long that the environment cannot be distinguished using coupled flight parameters.It should be noted that climb-descent flight in this article is defined as the process of anticipation motion for some other training missions (such as rapid barrel rolling, turning with a small radius, inverted flight, and cruise).
Meanwhile, without the periodicity, methods such as rain flow filtering [15][16][17] and cycle counting [18], which are always used in the compilation of aircraft spectra, have difficulty extracting flight altitude features because the outputs of these methods are also data about cycles.Though the big data regarding altitude have non-ergodic properties, methods such as time-delay correlation [19] and empirical mode decomposition [20] used in the statistical field for non-ergodic data are highly dependent on frequency information and are unsuitable for extracting altitude features.The time-delay method is essentially a time domain form corresponding to the frequency domain method for data with periodicity, and empirical mode decomposition is also a theory that decomposes signals from low frequency to high frequency in turn.Due to the uncertainty of altitude time history, the methods that determine the distance between two maneuvering profiles, such as principal component analysis [4], neural networks [21], or other types of machine learning algorithms [22], cannot be directly applied to cluster altitude data.These methods often classify signals based on similar matrices, which means that the objects represented by the analyzed data have some similar implicit features.However, as the altitude time histories themselves do not have these features, we need to identify an algorithm that is able to look for parameters that can represent similar properties from the altitude data.
The abovementioned issues make it difficult to solve the problem of flight service environment classification relying on altitude features, so the proportions of the flight time of a certain service environment cannot be determined.Therefore, even if the time history of the main bearing load under a certain service environment could be worked out [23], a corresponding flight service environment spectrum cannot be compiled, making it impossible to evaluate the life of the components in aeroengines.
In this article, through filtering for trends and removing redundancies from peakvalley (PV) values, a flight altitude feature extraction method related to the Frequency of Climb-Descent Flight (FCDF) is presented.With the FCDF and the maximum flight altitude as input parameters, a clustering algorithm for the aircraft service environment is established based on fuzzy theories, and then the compilation flowchart of the service environment spectrum, which is related to flight altitude, is given.Through some practical examples, the effectiveness of the altitude feature extraction algorithm is verified, and the specific implementation procedure of the environment spectrum compilation is explained in detail.

Flight Altitude Feature Extraction Method
The time histories of flight altitude we often see are shown in Figure 1.These data come from real flight data with some sensitive information removed.Except for takeoff and landing, the altitude data for the two flight profiles differ significantly and do not have obviously similar features.The altitude history of each profile has typical non-stationary characteristics.Climb-descent flight training, often meaning that the maneuvering motion will change considerably from the previous action, is shown by the upper triangle or sine shapes with a chain line in Figure 1.
Two thresholds, namely, the shortest time and the maximum altitude difference, need to be set to determine whether the maneuvering actions form a climb-descent process, such as 15 s and 10% of the limit altitude of an aircraft.For the same aircraft and engine models, the thresholds can be chosen by training experiences and the definition of height levels; for example, the maximum altitude of an aircraft is 12,000 m, and the threshold of 10% represents 1200 m or two height levels when a height level is defined as 600 m.Those pitches with instantaneous property or climb-descent actions with a maximum altitude difference less than the threshold belong to other types of training and are not counted in the FCDF, as shown in the ellipses with a do ed and dashed line in Figure 1a.This section presents an altitude feature extraction algorithm that can obtain the FCDF within a flight profile, including trend filtering and PV difference de-redundancy methods.The purpose of trend filtering is to remove thin jumps from the altitude curve and weaken the impact of other flight actions on the climb-descent signal, such as hovering, rapid diving, or pulling up.The PV difference de-redundancy method further filters small fluctuations out of the data after trend filtering, which extracts the PV of the compressed altitude data and then removes the redundancy from the PV using a climb-descent threshold so as to retain the PV points that meet these thresholds.After conducting the PV difference method, the statistics of FCDF based on altitude features will be achieved.

Trend Filtering
The altitude signal is as follows: where is the length of X, and the subscript "hp" means the pressure height.
The process of trend filtering involves calculating the average of the k-adjacent data, which will replace the middle element of these k data.To signal X, assuming each interval composed of k-adjacent data is almost stable, if we use the mean to replace the middle element of the interval, then the data jumps in the interval will be filtered out.k is called the width of the sliding window.
The key step of trend filtering is to set a sliding window width k.If the minimum time for an aircraft to rapidly dive or jump is is an experience value determined by the aircraft's performance.After trend filtering, we can obtain the trend of X as: ˆˆˆˆˆˆˆˆ{ , , , , , } { (1), (2), , ( ), , ( ) where ˆˆ( ) Trend filtering is equivalent to lowpass filtering to X with 2fs/k, and trend extraction does not compress the total amount of data.For data with a sampling frequency hundreds of times more than the trend frequency, the effect of the trend filtering method is more pronounced than that of the lowpass filter when extracting trend terms, such as the Butterworth filter or Chebyshev filter.

PV Difference De-Redundancy Method
Due to the sliding window width in trend filtering being limited by the length of time of the maneuvering actions, after trend filtering, the data may fluctuate in a short time with smooth amplitude; very small noise of PV has been filtered out, but there are some pitch movements with small amplitude that have not been filtered out, so we cannot directly obtain the FCDF from the data after trend filtering.A PV difference de-redundancy algorithm is designed in this section for further processing of the altitude data to filter out small amplitude fluctuations (Figure 2).Firstly, the signal X from trend filtering is compressed at equal intervals n  , with n  representing the fewest or a small number of sampling points for aircraft in a climb- descent process rather than in a rapid action.We can set it based on the big training data from one model of aircraft with the same engine model.A compressed sequence X  will be drawn out from X at equal intervals, and we express it as: The data compression operation is equivalent to another lowpass filtering, which filters the components above frequency / s f n  out X , so that some rapidly changing altitude training actions will be removed, such as hovering with a small radius and rapid rolling.
Extract the polar values of the compressed signal X  to obtain the following sets p X  and v X  ; both subscripts "p" and "v" represent peak and valley, respectively.
where p n and v n are the numbers of the peak and valley values, respectively.
For the threshold c of the climb-descent height, such as 10% of the limit altitude of an aircraft, if one of the differences between a valley and its adjacent peaks on both sides is less than c, then the peak may be thought of as a redundant point and cannot represent one training.We need to remove it from p X  , and the principle of removal is: for and the adjacent peak value In the above, " " represents "any".Finally, the peak sequence after de-redundancy is wri en as:

Feature Extraction Flowchart
The general flowchart of the feature extraction for flight altitude is given in Figure 4.

Service Environment Clustering Method
The fuzzy c-means clustering algorithm (FCM) minimizes the objective function by calculating the distances of every sample to all cluster centers and then obtains the slave degrees of every sample to all clusters using the similarity relationship.Finally, the algorithm determines which cluster each sample belongs to according to its maximum slave degree.FCM is an adaptive machine learning clustering algorithm.

Fuzzy Cluster Center
The essence of FCM is the process of optimizing the cluster centers by means of iteration; the cluster centers will change with iterations. Assume  is a sample set composed of q samples.Let ij d be the Euclidean distance between the ith and jth samples.Then, the expression of ij d is as follows: To prevent a certain variable from significant impact on distances, the same variable in all samples should be normalized by its maximum before solving the distances.For example, a 3D set , 2 (120 0.5 0.004) S    ， ， , 3 (72 0.28 0.003) , 4 (114 0.45 0.008) If the sample set is normalized with the maximum values of each column, then the sample set becomes: The original distance between the samples .Normalization transforms the distances in the original sample set, of which the main component is the first column or first variable, into distances in the normalized set, of which the weight of each variable is basically the same, avoiding the clustering result that tends to come from one variable while the others are useless.Let the number of clusters be l; each cluster center can be randomly initialized with different samples in set { } i S , and the initialized cluster center set can be defined as

Fuzzy Similar Matrix
After se ing the initial value of each cluster center, a fuzzy similar matrix needs to be constructed to determine the similarity between samples.
For i S and j S , the similarity ij r between them is expressed as the following equation: where 0 1 ij r   ,  , and  are suitable real constants selected with 0 1    and 1   , for example,  = 0.9,  = 1.For more complex data, one can set parameters of fuzzy clustering with the aid of reference [24].

Cluster Center Updating Algorithm
If the distance between the sample ( 1, 2, , ) , calculate the similarity ik r and then obtain the similar matrix to be the slave degree of the ith sample i S in the kth cluster, and calculate the slave degree according to the following equation: From the above formula, we can verify that the following equation is valid: that is, the sum of the slave degrees of the samples in each cluster is 1.Update the kth cluster center: where b is a fuzzy constant with b > 1 for controllable clustering results, such as b = 2.
Update the cluster centers via iteration until the objective function J is less than a specified minimum threshold  : or until after a specified maximum number of iterations.
If the FCM converges, we can obtain all of the cluster centers as well as the slave degrees of every sample to each cluster.If a sample has the largest slave degree to the kth cluster, it is considered that the sample belongs to cluster k; thus, the FCM clustering is completed.

Flowchart of Environment Clustering
The general service environment clustering process based on FCM is shown in Figure 5. Make the serial number of each altitude cluster its cluster mark, and let the percentage of each cluster's sample number for the total sample number be the spectral result; then, the service environment spectrum based on altitude features can be compiled.

Examples
This section provides examples of altitude feature extraction, service environment clustering, and environment spectrum compilation to validate the effectiveness of the presented algorithms and explain the implementation processes further.All data in this section came from real flight data that had some sensitive information removed.These data also came from the same aircraft model with the same engine model and the same altitude limit, but the flight times differed due to the different flight missions.

Altitude Feature Extraction
Feature extraction was performed on the flight profile shown in Figure 1a, with a data sampling frequency of In Figure 2, due to the limitation of window width, the trend algorithm filters out shortterm altitude jump signals that do not denote climb-descent flight training but preserves and smooths cases with a climb-descent pattern and then obtains the trend of the altitude data.
Next, the data compression algorithm in Figure 3 resamples the trend signal according to the shortest time interval of the aircraft climb or descent, further filtering the relatively high-frequency signal in the trend data while retaining the climb-descent feature points.Based on the PV difference de-redundancy algorithm in Section 2.2, Figure 3 calculates the PV of the trend term after data compression and then uses a threshold to remove redundancies from the peak values, achieving the extraction of the FCDF.The number of feature points in Figure 3 indicates that the training frequency is 2.
Due to the non-ergodic nature of the altitude curve in the flight profile, the altitude feature extraction results of other flight profiles are listed, as shown in Figure 6.Compared to the large-span time interval between the two training sessions in Figure 3, Figure 6a shows two consecutive climb-descent training sessions, one short-term session, and one climb-descent session in sequence; Figure 6b shows the process of five consecutive training sessions that are inserted into other short-term training sessions during their executed progress; Figure 6c shows the process of intermittent climbing or other maneuvers during the second climbing training, and Figure 6d shows the process of trapezoidal climb-descent training only.From the number of feature points in Figures 3 and 6, the altitude feature extraction method presented in this article is effective: for the same aircraft model with the same engine model and different flight times, the presented methods can correctly obtain the FDCF in each profile.

The Effect from Parameters of Altitude Feature Extraction Method
To verify the robustness of the method, the impact of parameter se ings on FCDF was discussed, and the results are shown in Figure 7.For sliding window width k, a different point only occurs in k = 40 when the limitation of k is in interval [0, 200] or about 1/21 of the length of flight profile data; the others of the FCDF are the same, showing that parameter k affects the resulting li er.For intervals, when 60 n   , the FCDF is constant, so it will affect the resulting li er if large enough.Altitude threshold c affects the result greatly because it is a parameter related to the maneuvering motion; however, the altitude change is limited to the performance of the aircraft, so it is constant when larger than 14%.

Service Environment Clustering
The altitude data of 153 takeoff-to-landing flight profiles were clustered for the service environment.The number of clusters was 4 l  ; the fuzzy clustering results are shown in Figure 8 and Table 1.For only  Figure 8b shows the relationship between flight profile samples and each cluster center.The training frequency of the second and fourth clusters is significantly higher than that of the other two, indicating a significant correlation between the FCDF and the maximum flight altitude.The training frequency does not increase with the maximum altitude; that is, it is not the case that the larger the maximum altitude, the greater the training frequency.
It is worth noting that the determination of l relies on the effectiveness of the classification and training missions.The clustering result in this section is only from altitude, but it will be used to further classify the service environment with Mach, turning times, flight time, etc., so an unreasonable l may induce an empty sample in some clusters.The determination of the clustering number l is often made according to practical experience, and there is li le research on parameter determination using fuzzy clustering methods [25].

Environment Spectrum Compilation with Altitude Features
According to the fuzzy clustering results, the service environment spectrum was compiled using the altitude data of 153 flight profiles from aircraft.The results are shown in Figure 9.The time spent on flight training is not positively correlated with the FCDF.The time proportion of each cluster, namely, the sum of the flight time for all profiles in each cluster divided by the total flight time of the 153 profiles, is sorted in descending order at 2, 4, 3, and 1.Though the largest time proportion occurs in cluster 2, the FCDF is 2.78 in this cluster.This means that, although the FCDF is not the highest, the mid-to lowaltitude flights take more time for flight training.

Environment Spectrum Compilation with Altitude, Mach, Turning, and Time
To extend the application of the presented methods and improve the robustness of the service environment spectrum, other feature parameters of Mach, turning number, and time of flight profiles were introduced.If both the cluster numbers of Mach and turning number were 3, and the cluster number of time was 2, the clustering results are shown in Table 2.We also classified the flight data by the clustering method with Mach, turning number, and time, respectively.The "%" in Table 2 represents the percentage of the maximum design value of the variables, and "--" representing the empty value.
Then, we obtained 4 × 3 × 3 × 2 = 72 classes of service environments from the permutation of Table 2. Mark the classes with 4 × 1 vectors, such as (i, j, k, r), where i, j, k, r represent the cluster indexes of altitude, Mach, turning number, and time in turn.The 10 classes with the maximum time proportions in the 72 service environment classes are shown in Table 3, the time proportions are shown in Figure 10, and the service environment spectrum is finally obtained from multiple parameters of the aircraft.

Conclusions
This study highlights the problems related to flight service environment spectrum compilation.As an important indicator of aircraft maneuverability, altitude is currently always used to compile service environment spectra.Therefore, this study researched compilation methods that utilize altitude data.
A method of extracting the FCDF from flight altitude data was presented first, and then, taking the maximum flight altitude and the FCDF as fuzzy clustering input parameters, a service environment spectrum compilation method related to the FCDF and maximum flight altitude was presented.The effectiveness of this method was verified through flight examples, and the implementation process of the service environment spectrum compilation approach was also discussed.The results showed that there is no positively correlated relationship between the FCDF and maximum altitude; there are more training sessions for climb-descent actions at mid-to low-altitude and high-altitude flights, while low-altitude and middle-altitude not only account for a lower proportion of flight time but also have a lower number of training sessions for climb-descent actions during single takeoff-to-landing flights.
Although the FCDF was extracted, we ignored a number of features that were considered to be more easily identified with the help of turn data, Mach data, or other data from the aircraft.One can also study the feature extraction methods for other flight parameters and cluster the service environment using those.More clustering parameters can improve the robustness of the method, and one can further study the service environment spectrum compilation method using multiple flight parameters.Meanwhile, we only established a method that can compile the flight environment spectrum in more detail.There are many parameters that need to be determined using practical experience, and we did not identify any algorithms that can be set adaptively, meaning that the robustness of the method used in this article needs to be further improved.Furthermore, the cluster method established in this article can be extended to multiple flight parameters, the compilation of flight service environment spectra, and the classification of flight missions.When compiling the service spectrum, we can use the flight parameters of one flight profile in each cluster as the inputs of the main bearing model: the loads of the main bearing model for each cluster can be calculated and the life can be evaluated using the time proportion of the environment spectrum.These methods may not only be applied to the main bearings of aeroengines but also to other components of aircraft or engines.
rounding down.Variable k denotes the fewest sampling points taken by a rapid climb or descent action and min t 

n
is the length of the peak sequence * p X  , namely, the FCDF in a flight profile.Further explanation of the PV difference de-redundancy algorithm is shown in Figure 3.

Figure 4 .
Figure 4. General flowchart for altitude feature extraction.

1 sf
 Hz, a sliding window width of k = 60, and a data compression interval of 100 n   .Variable k denotes the fewest sampling points taken by a rapid climb or descent action and is usually closely related to the limit performance of an aircraft, which can be set using big flight data.Corresponding to k, n  denotes the fewest sampling points taken by one climb-descent process.There may be many rapid climb actions in the process because the altitude level can often change significantly in anticipation of some other training action.The results are shown in Figures 2 and 3.

Figure 7 .
Figure 7. Effect from parameters of altitude feature extraction method: (a) sliding window width k; (b) intervals of compressed data n  ; (c) altitude threshold c.

Figure 8 .
Figure 8. Cluster results of service environment based on maximum altitude and FCDF: (a) FCM with the variable of maximum flight altitude; (b) FCM with 2 variables of maximum flight altitude and FCDF.

Figure 9 .
Figure 9. Service environment spectrum of aircraft.

Figure 10 .
Figure 10.Service environment spectrum obtained from multiple parameters of aircraft.

Author
Contributions: Methodology, M.C.; investigation, M.C.; resources, X.W., H.W. and G.L.; writing-original draft preparation, M.C.; writing-review and editing, M.C.; funding acquisition, M.C.All authors have read and agreed to the published version of the manuscript.

Table 1 .
Cluster centers from the FCM method.

Table 2 .
Cluster centers of multiple parameters.

Table 3 .
The 10 classes with the maximum time proportion of the 72 service environments.