Flight trajectory anomaly detection method and system based on dual-dynamics koopman operator

Abstract

The application discloses a kind of dual-kinetics Koopman operator flight trajectory anomaly detection method and system.The method is first to flight trajectory time series data is standardized and difference processing, and then based on singular value decomposition, system state variable is automatically separated into slow state subsystem and fast state subsystem;Local Koopman operator is independently constructed to two subsystems, and nonlinear dynamics is mapped into linear evolution process;From the operator, a variety of dynamically interpretable dynamics indexes are extracted, and the dynamics comprehensive distance is calculated by a weighted fusion module;Finally, based on statistical threshold, the anomaly is determined and the score is output.The application separates the mechanism of fast and slow dynamics, significantly improves the detection sensitivity of transient anomaly, while having good interpretability, high detection accuracy and low computing resource requirement, suitable for aviation safety monitoring and airborne real-time detection.

Description

Technical Field

[0001] This invention relates to the fields of aerospace engineering and data mining technology, and in particular to a method and system for detecting flight trajectory anomalies using a dual-dynamic Koopman operator. Background Technology

[0002] Multidimensional flight trajectory anomaly detection is a key technology for ensuring aviation safety and operational efficiency, and is widely used in fields such as UAV collision warning, civil aviation emergency identification, and pilot skills assessment. Flight trajectory data, as an observation sequence of a physical system, has significant nonlinear, high-dimensional, and multi-scale dynamic characteristics.

[0003] Existing technologies are mainly divided into two categories: one is traditional methods based on statistics and geometric features, such as cluster analysis. These methods usually rely on simplified trajectory data and are difficult to capture deep dynamic information. The other is deep learning-based methods, such as LSTM and VAE. Although they perform well in time series processing, they often lack physical interpretability and have a large demand for labeled anomalous data, while anomalous data in the aviation field is extremely scarce.

[0004] Koopman operator theory provides a powerful framework for linearizing nonlinear dynamic systems and has been used for trajectory prediction. However, traditional Koopman operators are typically global and time-invariant, tending to capture the overall behavior of the system but lacking sensitivity to local or transient anomalies (such as sudden mechanical failures or airflow disturbances). Furthermore, flight data exhibits a significant separation between fast and slow scales (e.g., rapid oscillations in attitude angles versus slow drift in the trajectory), and hybrid modeling can easily lead to fast-changing signals being masked by slow-changing trends, resulting in missed anomalies.

[0005] Therefore, a prominent problem in the existing technology is the lack of a flight trajectory anomaly detection scheme that can decouple multi-scale dynamic features, maintain physical interpretability, and be sensitive to local transient anomalies. Summary of the Invention

[0006] Purpose of the Invention: The purpose of this invention is to overcome the shortcomings of existing technologies and provide a method and system for detecting flight trajectory anomalies using a dual-dynamic Koopman operator, thereby addressing the problems of insufficient sensitivity and lack of interpretability in existing methods when dealing with multi-scale nonlinear systems. By constructing a fast-slow state separation and dual-Koopman operator architecture, this invention aims to achieve the following objectives:

[0007] By utilizing singular perturbation theory to decouple highly intertwined fast and slow time series in flight data, we can independently study their patterns; extract dynamic indicators with clear physical meaning, and achieve robust anomaly scoring through weighted fusion; and achieve high-precision anomaly detection in unsupervised or limited normal data scenarios.

[0008] Technical solution: To achieve the above objectives, the technical solution adopted by this invention is as follows:

[0009] A method for detecting flight trajectory anomalies using a dual-dynamic Koopman operator includes the following steps:

[0010] (a) Input multidimensional flight trajectory time series data;

[0011] (b) The data is preprocessed, and the preprocessed data is separated into fast and slow states based on singular value decomposition (SVD) to decouple the system dynamics;

[0012] (c) Construct Koopman operator models for each of the separated subsystems;

[0013] (d) Extract dynamic characteristic indicators;

[0014] (e) Weighted fusion of indicators to calculate the dynamic comprehensive distance;

[0015] (f) Determine anomalies based on thresholds and calculate scores.

[0016] Preferably, the data preprocessing in step (a) is performed to provide a standardized dynamic increment input for step (b), which is achieved by standardizing the original trajectory data with zero mean and unit variance and calculating the first-order time difference matrix to approximate the time derivatives of the state variables.

[0017] Preferably, the fast and slow state separation in step (b) specifically involves: receiving the difference matrix output from step (a) as input; performing singular value decomposition (SVD) on the difference matrix, setting a threshold based on the magnitude of the singular values, and dividing the dominant modes into a slow mode set (corresponding to low frequency / trend) and a fast mode set (corresponding to high frequency / transient); calculating the projection contribution of each original state variable onto the slow mode set and the fast mode set; and automatically classifying the original variables into the slow-varying state vector γ based on the magnitude of the contribution. slow or rapidly changing state vector γ fast This achieves multi-scale decoupling of physical perception, and the decoupled state vector is directly used as the input of step (c).

[0018] Preferably, the Koopman operator model construction in step (c) employs a parallel computing architecture, respectively processing γ slow and γ fastSolve independently; for any subsystem, the objective is to find a finite-dimensional linear operator K such that the linear evolution of the system state in the observation space satisfies g(x) t+1 )≈Kg(x t ), g(·) is the observation function; the operator K is solved by minimizing the prediction error, specifically by solving the linear matrix equation. The least squares solution is obtained, i.e., K = YX. † , where X † Representation matrix Moore-Penrose pseudo-inverse, X and These are matrices formed by the state vectors of the subsystem at adjacent time points.

[0019] Preferably, the dynamic characteristic indicators extracted in step (d) are a set of parameters that quantify the physical properties of the system, including at least the following six items:

[0020] Eigenvalue ratio (R) eig ): Reflects the distribution bias of eigenvalues; Lyapunov exponent ratio (R²) lyap ): Reflects the system's sensitivity to initial conditions and chaotic behavior; SVD energy ratio (R svd ): The relative contribution of fast and slow dynamic modes to the total energy; correlation ratio (R Corr ): The degree of coupling between variables within a measurement subsystem; prediction error ratio (R²) pred-error ): Reflects the adaptability of the linear approximation model to dynamics at different time scales; Stability index (Γ): Determined by the maximum eigenvalue modulus of the Koopman operator.

[0021] Preferably, the weighted fusion module in step (e) receives the multidimensional indicators output from step (d) and achieves fusion in the following way: after standardizing each indicator, it performs weighted summation through a set of learnable weight parameters to calculate the dynamic comprehensive distance (DCD); wherein, the weight parameters are optimized during the training phase through a multilayer perceptron (MLP) network with the goal of maximizing the distinguishability between normal samples and abnormal samples.

[0022] Preferably, the anomaly detection module in step (f) adopts a statistical thresholding strategy based on the interquartile range (IQR), that is, it uses the upper and lower quartiles of the DCD distribution in the training set to determine the strict threshold τ. strict and the relaxed threshold τ loose If the test sample DCD is greater than τ loose If the result is not found to be an anomaly, a normalized anomaly score will be output using a piecewise linear function.

[0023] The present invention also provides a flight trajectory anomaly detection system for implementing the method, the system comprising:

[0024] The data preprocessing module is used to standardize and differentially process multidimensional flight trajectory time series data;

[0025] The fast and slow state separation module is connected to the data preprocessing module and is used to perform singular value decomposition on the difference matrix to classify the original variables into slow-changing state subsystems and fast-changing state subsystems.

[0026] The dual-dynamic Koopman computation module is connected to the fast and slow state separation module and is used to compute the Koopman operator of the fast and slow subsystems in parallel.

[0027] The index extraction and fusion module is connected to the dual dynamics Koopman calculation module and is used to extract dynamic feature indices from the operators and weightedly fuse them to generate a comprehensive dynamic distance.

[0028] The anomaly detection module is connected to the indicator extraction and fusion module and is used to determine the trajectory status based on statistical thresholds and output anomaly scores.

[0029] The present invention also provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the method described thereon.

[0030] The present invention also provides an electronic device, including a memory, a processor, and a computer program stored in the memory, wherein the processor executes the computer program to implement the method described thereon.

[0031] Beneficial effects:

[0032] Compared with the prior art, the present invention has the following significant advantages:

[0033] 1. This invention innovatively proposes a fast-slow state separation mechanism based on physical characteristics. The slow-varying subsystem captures long-term trends (such as navigation drift), while the fast-varying subsystem captures transient responses (such as engine surge). This separation avoids the problem of global operators smoothing out high-frequency abnormal signals, significantly improving the detection sensitivity for different types of faults.

[0034] 2. Unlike the black-box nature of deep learning, this invention is based on dynamical system theory, and the extracted indicators (such as eigenvalue stability and energy distribution) have clear physical meanings, which can help domain experts understand the physical root causes of anomalies (such as system instability or changes in coupling relationships).

[0035] 3. Experiments show that the detection accuracy (94.44%) and F1 score of this invention on datasets such as CAFUC2 are significantly better than mainstream algorithms such as LSTM and DeepSAD. Furthermore, because the core computation is based on linear algebra operations, its training and inference speeds are extremely fast, making it suitable for deployment on edge devices.

[0036] 4. The method mainly relies on dynamic benchmark learning of normal flight data, which falls into the category of unsupervised or semi-supervised learning, effectively solving the problem of scarce abnormal samples in the aviation field. Attached Figure Description

[0037] Figure 1 : Overall process architecture diagram of the training phase of the method of this invention.

[0038] Figure 2 : Overall process architecture diagram of the testing phase of the method of this invention.

[0039] Figures 3-14 : A logic diagram of the fast and slow state separation module, wherein: Figure 3 : Eigenvalue spectrum (fast and slow state division); Figure 4 The contribution ratio of each variable to its fast or slow state; Figure 5 The contribution of variables in the dynamic model; Figure 6 Slow-state time evolution (the first 3 main modes); Figure 7 : Fast state time evolution (the first 3 main modes); Figure 8 Slow-state phase space (PCA dimensionality reduction); Figure 9 Fast state phase space (PCA dimensionality reduction); Figure 10 Frequency domain analysis (amplitude spectrum); Figure 11 : FFT amplitude spectrum (Hanning window); Figure 12 FFT phase spectrum; Figure 13 Power spectral density (Welch method); Figure 14 Comprehensive comparison of power spectral density.

[0040] Figure 15 : A comparison of the distribution of eigenvalues ​​of the Koopman operator in the complex plane, including two subplots: (a) normal state and (b) abnormal state.

[0041] Figure 16 : Modal diagram, containing six sub-graphs (a), (b), (c), (d), (e), and (f), which correspond to the fast and slow modal feature distributions of two groups of modes #1, #2, and #3, respectively.

[0042] Figures 17-22 The performance bubble chart is divided into two main categories: GPU performance analysis and CPU performance analysis. The size of the bubble represents memory usage. GPU performance analysis sub-chart: Figure 17 UCI dataset, Figure 18 :TEP dataset, Figure 19 CAFUC dataset, vertical axis represents average GPU utilization; CPU performance analysis subplot: Figure 20 UCI dataset, Figure 21:TEP dataset, Figure 22 The CAFUC dataset, with the vertical axis representing average CPU utilization. Detailed Implementation

[0043] The invention will now be further described with reference to the accompanying drawings.

[0044] Example 1

[0045] like Figure 1 and Figure 2 As shown, the overall process of the method of this invention is divided into two main modules: a training phase and a testing phase. In the training phase, the raw flight trajectory data is input and decoupled into two links—a slow state and a fast state—by a state separator. Both links are connected to a baseline module to obtain a normal flight dynamics benchmark. The slow state link undergoes Koopman dynamics decomposition to extract five types of dynamic feature indicators: Eigenvalue, Lyapunov exponent, SVD-Slope, Correlation, and Pred Error, generating a slow state stability index. and various ratio indicators (Eigenvalue ratio) (Lyapunov index ratio) (SVD energy ratio) (Correlation ratio) (Prediction error ratio); Fast-state links are analyzed by extracting similar indices through Koopman dynamics decomposition to generate fast-state stability indices. All dynamic metrics are input into the fusion module, aiming to maximize the distinction between normal and abnormal samples. The fusion weights are optimized using a multilayer perceptron network. A dynamic integrated distance calculation model was constructed to complete the benchmark modeling for the training phase.

[0046] The testing phase reuses the fast and slow dual-link processing logic from the training phase. The input flight trajectory data is processed by state decoupling and index extraction to generate slow state estimates. , , , , , With fast state estimate The optimized weights from the training phase are used to perform weighted fusion of the extracted metrics, and the overall dynamic distance for the testing phase is calculated. The dynamic comprehensive distance is compared with the preset statistical threshold, and the state judgment result of Normal or Anomaly is output. The dynamic comprehensive distance is mapped to Anomaly Score / abnormality score through a piecewise linear function (normalized to the [0,1] interval, the larger the value, the higher the probability of abnormality), thus completing end-to-end anomaly detection.

[0047] This embodiment details the core algorithm steps of a multidimensional flight trajectory anomaly detection method based on the dual dynamics Koopman operator (which can be referred to as the FlightKoopman2 framework).

[0048] 1. Separation of data preprocessing from fast and slow states

[0049] The input is the original multidimensional flight trajectory time series X. raw =[x1,x2,…,x N ] T ∈R N×d , where N is the sequence length and d is the variable dimension (such as longitude, latitude, altitude, airspeed, etc.).

[0050] (1) Standardization: Each dimension of the variable is processed to have zero mean and unit variance to obtain X. scaled .

[0051] (2) Time difference: Calculate the first-order time difference of the standardized data to approximate the time derivative (i.e., rate of change) of the state variables and capture the dynamic characteristics:

[0052] ΔX=[x2-x1,x3-x2,…,x N -x N-1 ] T ∈R (N-1)×d .

[0053] (3) SVD decomposition and mode partitioning: Perform singular value decomposition on the difference matrix ΔX ΔX=UΣV T Where Σ is the singular value matrix, and its diagonal elements are... Characterize energy distribution. Set a threshold. (For example, take the median of all singular values), and then take the singular value σ. i < The corresponding right singular vectors (i.e., columns of V) are assigned to the slow mode set S, and σ is... i ≥ The corresponding right singular vector is assigned to the fast mode set F.

[0054] (4) Variable decoupling: Calculate the projection contribution of each original variable j onto the fast mode set F and the slow mode set S. If the variable is mainly reconstructed by the right singular vector in the set F, then it is assigned to the fast-changing state subsystem to form the fast-changing state vector γ. fast Conversely, it is classified into a slowly varying state subsystem, forming a slowly varying state vector γ. slow .

[0055] like Figures 3-14 As shown, the fast-slow state separation module achieves precise decoupling from the difference matrix to the slow-changing and fast-changing state subsystems through multi-dimensional analysis. First, using the modal index as the horizontal axis, in... Figure 3 The text displays the eigenvalue distributions corresponding to fast and slow modes after singular value decomposition, providing a quantitative basis for mode speed classification; furthermore, it uses the mode index as the horizontal axis... Figure 4 , Figure 5 The reconstruction contribution of each original flight trajectory state variable on the fast and slow mode sets is quantified, and the criteria for determining the attribution of variable subsystems are clarified; then, taking PCA component 1 as the core dimension, in... Figure 8 , Figure 9 The phase space distribution characteristics of the slowly varying state subsystem are presented, reflecting the dynamic laws of the slowly varying system; finally, through frequency domain analysis with frequency as the horizontal axis, the phase space distribution characteristics of the slowly varying state subsystem are presented. Figure 10 , Figure 11 , Figure 12 The distribution characteristics of fast and slow states in the frequency domain are shown separately. The fast state corresponds to high frequency / transient, and the slow state corresponds to low frequency / trend. Figure 13 , Figure 14 The effectiveness of fast-slow separation was verified by comparing fused energy densities. Each analytical step progressed progressively, completing the entire process from modal partitioning to variable decoupling, achieving multi-scale dynamic decoupling based on physical perception.

[0056] Explanation of principles: Slow-changing subsystems typically correspond to long-period motions during flight (such as flight path drift and fuel consumption trends), while fast-changing subsystems correspond to short-period dynamics (such as attitude oscillations and engine transient response).

[0057] 2. Computation of the dual-dynamic Koopman operator

[0058] For γ slow and γ fast The following steps are executed in parallel (taking the slow subsystem as an example): (1) Constructing the data matrix: Construct the state matrix X at time t and the state matrix Y at time t+1. (2) Solving the operator: The goal is to find the optimal linear operator K such that Y≈XK in the observation space of the subsystem. In this embodiment, the analytical solution of the least squares method is used for fast calculation: K=YX † , where X †It is the Moore-Penrose pseudoinverse of X. (3) Spectral decomposition: Perform eigenvalue decomposition Kψ=λψ on the obtained Koopman operator matrix K to obtain the eigenvalue λ (which determines the growth / decrease rate and oscillation frequency of the system) and the corresponding eigenmode ψ.

[0059] 3. Extraction of kinetic indicators

[0060] Based on the Koopman operator matrices of the fast and slow subsystems obtained in step 2 and The following set of dynamic characteristic indicators with clear physical meaning were extracted respectively:

[0061] Eigenvalue ratio (R) eig ): Reflects the degree of concentration of eigenvalues ​​in the complex plane; Lyapunov exponent ratio (R lyap ): Approximately calculated from eigenvalues, reflecting the system's sensitivity to initial conditions and its tendency towards chaos; SVD energy ratio (R svd ): The relative contributions of the fast and slow subsystems to the total dynamic energy; correlation ratio (R Corr ): The degree of coupling between state variables within a measurement subsystem; prediction error ratio (R²) pred-error ): Reflects the fitting accuracy of the linear Koopman model to the dynamics at different time scales; stability index of the tachyon system ( ): Determined by the magnitude of the largest eigenvalue of the Koopman operator matrix of the fast subsystem, it directly characterizes the stability of the subsystem (magnitude < 1 indicates stability).

[0062] Slow subsystem stability index ( ): Determined by the magnitude of the largest eigenvalue of the Koopman operator matrix of the slow subsystem.

[0063] Therefore, a total of seven dynamic indices were extracted, the first five of which are proportionality indices, and the last two are stability indices. Among them, the stability indices of the tachyon system... With slow subsystem stability index The stability of each subsystem is directly characterized by the magnitude of the largest eigenvalue of the Koopman operator matrix for the fast and slow subsystems (magnitude < 1 indicates stability).

[0064] 4. Dynamic Integrated Distance and Anomaly Score

[0065] (1) Weighted fusion: The above indicators are standardized into Z-score (Z i The dynamic comprehensive distance (DCD) is calculated using a weighted summation model.

[0066]

[0067]

[0068] in, (i=1,...,5) correspond to five standardized indicators: eigenvalue ratio, Lyapunov exponent ratio, SVD energy ratio, correlation ratio, and prediction error ratio, respectively. and These are the stability indices of the fast and slow subsystems (i.e., the magnitude of the largest eigenvalue of the Koopman operator). (i=1,...,7) represent the corresponding fusion weights. For example... Figure 2 As shown in the testing phase, the weights optimized during the training phase are invoked. The estimated values ​​of the indicators extracted from the test samples , , Weighted fusion is performed to obtain the dynamic composite distance during the testing phase. .

[0069] Weight During the training phase, a three-layer multilayer perceptron (MLP) network was used for optimization. The input layer of the MLP receives a standardized 7-dimensional dynamic index vector, the hidden layer contains 32 nodes, and the output layer is a 7-dimensional weight vector. The training objective is to maximize the distribution difference between normal and abnormal samples in the dynamic comprehensive distance. The objective function includes a regularization term and a missed detection penalty term. (2) Threshold setting and scoring:

[0070] Calculate the interquartile range (IQR) using the DCD value distribution of all normal samples in the training set.

[0071] Set a strict threshold τ strict = Q3 + 1.5 × IQR, where the leniency threshold is τ loose = Q3 + 3.0 × IQR, where This is the third quartile. For the test sample:

[0072] If its It was determined to be "normal";

[0073] like It was determined to be "suspicious";

[0074] like It was determined to be "abnormal".

[0075] Finally, a piecewise linear function is used to map the DCD value to the interval [0, 1], and a normalized anomaly score is output, with a larger value indicating a higher probability of an anomaly.

[0076] 5. System Module Implementation

[0077] A system for implementing the above method comprises, in sequence:

[0078] Data preprocessing module: used to perform standardization and difference calculations in step 1.

[0079] Fast and slow state separation module: used to perform SVD decomposition and variable decoupling in step 1.

[0080] Dual-dynamics Koopman computation module: used to execute step 2 in parallel, calculating the Koopman operators for the fast and slow subsystems respectively.

[0081] Indicator extraction and fusion module: used to perform steps 3 and 4 (1), extract indicators and calculate DCD.

[0082] Anomaly detection module: used to execute step 4 (2), outputting the detection status and anomaly score based on the threshold.

[0083] 6. Experimental Results and Analysis

[0084] To verify the effectiveness of the method of this invention, extensive experiments were conducted on three datasets: the self-built CAFUC2 dataset (containing real general aviation flight data and synthetic anomalies), the TEP chemical process benchmark dataset, and the UCI machine learning dataset. Comparison algorithms included classic LSTM, Margin-LSTM, Auxiliary-LSTM, LSTM-VAE, and the latest DeepSAD algorithm.

[0085] like Figure 15 As shown, the dynamic stability of a system can be intuitively determined by the distribution of eigenvalues ​​of the Koopman operator in the complex plane. In normal flight conditions, eigenvalues ​​are tightly clustered within the unit circle, and the magnitudes of all eigenvalues ​​are less than 1, indicating stable and convergent dynamic characteristics of the flight system with no anomalous dynamic modes. In anomalous states, eigenvalues ​​are discretely distributed, with some eigenvalues ​​escaping the unit circle (magnitude > 1), indicating unstable divergent modes and abnormal changes in dynamic characteristics. Through a direct comparison of the eigenvalue distribution in the complex plane, the dynamic stability of the system can be directly determined. Furthermore, the dynamic scale (fast / slow) to which the anomaly belongs can be inferred from the frequency components (imaginary parts) of the eigenvalues, enabling physical interpretability analysis of the anomaly.

[0086] like Figure 16As shown, the feature distribution of each dominant mode after the separation of fast and slow states includes the fast and slow mode features of two sets of modes #1, #2, and #3. The numerical range of each mode is between [-0.1, 0.1], which clearly presents the distribution law and numerical characteristics of different fast and slow modes in the feature space, providing a modal basis for Koopman operator modeling and reflecting the intrinsic feature differences of different dynamic modes.

[0087] like Figures 17-22 As shown, the performance bubble chart is divided into two main categories: CPU performance analysis and GPU performance analysis. The size of the bubble represents the memory usage. The horizontal axis represents the total computation time (s), and the vertical axis represents the average hardware utilization rate (%). It intuitively compares the computation efficiency and hardware resource usage of the method of this invention on the three datasets TEP, CAFUC2, and UCI, clearly demonstrating the performance advantages of the invention in terms of low resource usage and fast inference speed, and providing quantitative performance basis for airborne devices and edge deployment.

[0088] 1) Comprehensive performance comparison analysis

[0089] The experimental results are shown in Table 1. On the multidimensional flight trajectory dataset CAFUC2, this invention (FlightKoopman2) demonstrates significant performance advantages.

[0090] Table 1: Comparison of quantitative performance of different algorithms

[0091]

[0092] This invention achieves a 100% recall rate, meaning that all real anomalous samples (including engine surge, heading deviation, etc.) are successfully detected without exception, which is crucial for aviation safety. Meanwhile, the F1 score reaches 94.12%, significantly outperforming deep learning benchmarks DeepSAD (90.91%) and traditional LSTM (22.22%).

[0093] On the TEP and UCI datasets, this invention also maintained a leading position (F1 score of 94.12% on TEP dataset and 82.35% on UCI dataset), demonstrating the framework's ability to generalize to different types of dynamic systems.

[0094] 2) Analysis of computational efficiency and resource consumption

[0095] With limited onboard computer resources, the real-time performance of the algorithms is crucial. Table 2 shows the training and inference times of each model on the CAFUC2 dataset.

[0096] Table 2: Comparison of computational efficiency

[0097]

[0098] The entire process of this invention takes only 22.04 seconds, which is more than 30 times faster than the reconstruction-based LSTM-VAE model (747.98 seconds) and about 40% faster than the lightweight DeepSAD.

[0099] The GPU utilization rate is only 10.43%, and the main computation relies on CPU matrix operations, making it suitable for deployment in airborne equipment or at the edge where high-end graphics cards are not available.

[0100] 3) The necessity of ablation experiments and separation of fast and slow ablation methods

[0101] To verify the contribution of the core step of separating fast and slow states, this invention conducted an ablation study. When the fast-slow state separation module was removed and the Koopman operator was applied directly to the global data, performance experienced a precipitous drop.

[0102] Table 3: Ablation Experiment Results of Fast and Slow Separation Module

[0103]

[0104] After removing the fast-slow signal separation, the accuracy dropped from 94.44% to 54.29%. This directly proves that mixing fast and slow signals causes the Koopman operator to smooth out fast-changing anomalous signals (such as transient jitter) in order to fit the slowly changing trend (dominant energy), leading to detection failure. The decoupling mechanism proposed in this invention is the key inventive point for achieving high-precision detection.

[0105] 4) Physical interpretability analysis

[0106] like Figure 15 As shown, the distribution of Koopman eigenvalues ​​in the complex plane is visualized:

[0107] Normal flight (a): Eigenvalues ​​are tightly clustered within the unit circle (modulus < 1), and the system exhibits stable convergence.

[0108] When an anomaly occurs (b): the eigenvalues ​​spread out of the unit circle (modulus > 1), indicating that the system has an unstable divergent mode. This visualization feature based on spectral theory allows pilots or engineers not only to know that "something is wrong," but also to infer "what kind of dynamics are wrong" through the frequency components of the eigenvalues, providing physical interpretability that deep learning models cannot match.

[0109] In summary, this invention provides profound physical insights while ensuring extremely high detection accuracy and real-time performance, effectively solving the pain points of existing technologies in multidimensional flight trajectory anomaly detection.

Claims

1. A method for detecting flight trajectory anomalies using a dual-dynamic Koopman operator, characterized in that, Includes the following steps: (a) Obtain multidimensional flight trajectory time series data, and perform standardization and time difference processing to obtain the difference matrix; (b) Perform singular value decomposition on the difference matrix, divide the dominant modes of the system into a slow mode set and a fast mode set according to the magnitude of the singular values, and classify the original variables into a slow-changing state subsystem and a fast-changing state subsystem based on the contribution of each original state variable to the slow mode set and the fast mode set; (c) Construct independent Koopman operator models for the slow-changing state subsystem and the fast-changing state subsystem respectively, and obtain the operator matrix describing the linear evolution law of each subsystem; (d) Based on the operator matrices of the slow-changing state subsystem and the fast-changing state subsystem, extract a set of dynamic characteristic indices respectively; (e) The dynamic characteristic indices are weighted and fused to obtain the comprehensive dynamic distance; (f) The dynamic integrated distance is compared with a preset threshold based on statistical distribution to determine the trajectory status and calculate the anomaly score.

2. The method according to claim 1, characterized in that, Step (b) specifically includes: (b1) Perform time difference on the standardized data to obtain the difference matrix of the approximate time derivatives of the state variables; (b2) Perform singular value decomposition on the difference matrix to obtain the right singular vector and the corresponding singular values; (b3) Based on a preset singular value threshold, modes with singular values ​​less than the threshold are classified into a slow mode set, and modes with singular values ​​greater than or equal to the threshold are classified into a fast mode set; (b4) Calculate the reconstruction contribution of each original state variable on the fast mode set and the slow mode set, and classify the variables whose contribution mainly comes from the fast mode set into the fast-changing state subsystem, and classify the variables whose contribution mainly comes from the slow mode set into the slow-changing state subsystem.

3. The method according to claim 1, characterized in that, In step (c), for any subsystem, the Koopman operator matrix K is constructed by solving the linear matrix equation Y≈XK using the least squares method, where X and Y are the state matrices of the subsystem at adjacent time points.

4. The method according to claim 1, characterized in that, The dynamic characteristic indicators extracted in step (d) include: eigenvalue ratio, Lyapunov exponent ratio, SVD energy ratio, correlation ratio, prediction error ratio, and stability index.

5. The method according to claim 1, characterized in that, The weighted fusion calculation in step (e) optimizes the fusion weights during the training phase using a multilayer perceptron network, which aims to maximize the discriminative power between normal and abnormal samples in the dynamic synthesis distance.

6. The method according to claim 1, characterized in that, Step (f) specifically includes: The interquartile range of the training set dynamics comprehensive distance distribution is used to determine a strict threshold and a lenient threshold, wherein the lenient threshold is greater than the strict threshold; If the overall dynamic distance of the test sample is greater than the lenient threshold, it is judged as abnormal; if it is between the strict threshold and the lenient threshold, it is judged as suspicious; if it is less than the strict threshold, it is judged as normal. The dynamic composite distance is then mapped to a normalized anomaly score using a piecewise linear function.

7. A flight trajectory anomaly detection system for implementing the method according to any one of claims 1 to 6, characterized in that, The system includes: The data preprocessing module is used to standardize and differentially process multidimensional flight trajectory time series data; The fast and slow state separation module is connected to the data preprocessing module and is used to perform singular value decomposition on the difference matrix to classify the original variables into slow-changing state subsystems and fast-changing state subsystems. The dual-dynamic Koopman computation module is connected to the fast and slow state separation module and is used to compute the Koopman operator of the fast and slow subsystems in parallel. The index extraction and fusion module is connected to the dual dynamics Koopman calculation module and is used to extract dynamic feature indices from the operators and weightedly fuse them to generate a comprehensive dynamic distance. The anomaly detection module is connected to the indicator extraction and fusion module and is used to determine the trajectory status based on statistical thresholds and output anomaly scores.

8. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by a processor, it implements the method as described in any one of claims 1 to 6.

9. An electronic device comprising a memory, a processor, and a computer program stored in the memory, characterized in that, When the processor executes the computer program, it implements the method as described in any one of claims 1 to 6.

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