A time-constrained weighted lagrangian extrapolated trajectory prediction method and device

Abstract

The application discloses a kind of time sequence constraint weighted Lagrange extrapolation trajectory prediction method and device, it is related to numerical extrapolation field, solve the problem such as high-order oscillation caused by equal weight processing in existing Lagrange extrapolation method, insufficient contribution of near time point, difficult to solve reverse weight and so on.The method comprises the following steps: obtaining the known flight trajectory observation data of the continuous time sequence of the aircraft, and determining the extrapolation target point;Calculate the Lagrange basis function value of each observation point at the extrapolation target point;Based on the structure of the weighting coefficient set by time sequence constraint, the weighting coefficient satisfies the monotone increasing constraint and only allocates non-zero weight to the latest observation point;According to the preset extrapolation result and the Lagrange basis function value, the weighting coefficient that meets the constraint condition is obtained by reverse solving.The application realizes the unique and stable solution of weighting coefficient by structured weight design, and considers the extrapolation accuracy and engineering practicability, suitable for short-term time sequence extrapolation scene.

Description

Technical Field

[0001] This invention relates to the field of numerical extrapolation technology, specifically to a method and apparatus for predicting time-constrained weighted Lagrange extrapolation trajectories. Background Technology

[0002] Lagrange interpolation and extrapolation are classic function approximation methods in numerical analysis. They construct polynomials using known observation points to predict unknown points and are widely used in engineering fields such as navigation and positioning, inertial measurement, time series prediction, and precision measurement.

[0003] Traditional Lagrange extrapolation methods apply equal weights to all known observation points, failing to consider the inherent characteristics of time-series data. Specifically, observation points closer to the target point (nearer time points) have higher data value and greater influence on the extrapolation results; while observation points farther from the target point (farther time points) have high data redundancy and may even introduce interference, leading to decreased extrapolation accuracy. When the number of observation points is large, traditional equal-weighted Lagrange extrapolation constructs high-order polynomials, easily resulting in Runge's phenomenon, which manifests as violent oscillations in the extrapolation results, failing to meet the accuracy requirements of engineering scenarios.

[0004] In practical engineering applications, there is often a need to reverse-engineer the weighting coefficients based on the known extrapolation results of the target. Examples include reproducing specific extrapolation accuracy metrics, solidifying weight parameters in algorithms, and adapting to the characteristics of observation data in different scenarios. Existing research on weighted Lagrange extrapolation largely focuses on forward weighted interpolation (extrapolating the results from known weights), with less emphasis on weight inversion based on the target extrapolation results. Furthermore, existing weight inversion methods generally suffer from drawbacks such as too many unknowns, non-unique solutions, and lack of physical constraints (e.g., non-negative weights, temporal consistency), resulting in weights that lack engineering practicality and cannot be directly applied to real-world systems.

[0005] Furthermore, for extrapolation scenarios with a small number of observation points (such as 5 sets of time-series data), traditional methods either employ equal weighting or arbitrarily set weights, failing to balance extrapolation accuracy and physical plausibility, thus making it difficult to meet the high-precision requirements of navigation and positioning scenarios. Therefore, there is an urgent need for a weighted Lagrange extrapolation trajectory prediction method and device that can solve the aforementioned technical problems and address the time-series constraints. Summary of the Invention

[0006] The purpose of this invention is to provide a time-constrained weighted Lagrange extrapolation trajectory prediction method and apparatus, which solves the technical problems of high-order oscillations caused by equal weighting in existing Lagrange extrapolation methods, insufficient near-time contribution, and difficulty in solving reverse weights, and achieves unique and stable solution of weighting coefficients, while taking into account extrapolation accuracy and engineering practicality.

[0007] To achieve the above objectives, the present invention adopts the following technical solution:

[0008] A time-constrained weighted Lagrange extrapolation trajectory prediction method includes the following steps:

[0009] Step 1: Obtain known flight trajectory observation data for n consecutive time sequences of aircraft. The flight trajectory observation data is... The corresponding timing is , The latest trajectory observation data of the spacecraft is used as the preset extrapolation result, and the corresponding preset extrapolation time series is as follows: ;

[0010] Step 2: Calculate the time sequence corresponding to each flight trajectory observation data in the preset extrapolation time sequence. Lagrange basis function value at ... The formula for calculating the Lagrange basis functions is as follows:

[0011] ;

[0012] Step 3: Set the weighting coefficients for the n flight trajectory observation data. The weighting coefficients in the given structure satisfy the following constraints:

[0013] Only the latest k flight trajectory observation data are assigned non-zero weighting coefficients, and the weighting coefficients for the remaining flight trajectory observation data are 0, where k is a positive integer and ;

[0014] The non-zero weighting coefficients satisfy a monotonically increasing constraint, meaning that the closer the flight trajectory observation data is to the preset extrapolation time series in terms of time series, the larger its weighting coefficients are.

[0015] The sum of all weighted coefficients is 1, and each weighted coefficient is non-negative;

[0016] Step 4: Based on the preset extrapolation results Combined with the Lagrange basis function values By using structural constraints and weighting coefficients, the weighting coefficients of each flight trajectory observation data are obtained through reverse engineering.

[0017] Step 5: Obtain the latest known flight trajectory observation data for the n consecutive time series of aircraft, and combine it with the Lagrange basis function values. By combining the weighting coefficients, the predicted flight trajectory data for the next moment is calculated.

[0018] Furthermore, the structure of the weighting coefficients in step 3 is as follows:

[0019] The k non-zero weighting coefficients are respectively , And k is an integer; where, when When all weighting coefficients are non-zero, hour, arrive All nk weighting coefficients are 0;

[0020] Where a and b are parameters to be determined, and the constraints are satisfied. .

[0021] Furthermore, the specific process of inversely solving for the weighting coefficients in step 4 is as follows: substituting the structural expression of the weighting coefficients into the weighted Lagrange extrapolation formula. Combined with constraints Let a be represented as Substituting into the extrapolation formula yields a linear equation in one variable for b, solving for the value of b, and then substituting back to obtain the value of a, finally determining all weighting coefficients. .

[0022] Furthermore, in step 4, if the weighted coefficients obtained by solving have negative values, then let k = k-1, and repeat steps 3 and 4 until all weighted coefficients are non-negative.

[0023] A time-constrained weighted Lagrange extrapolation trajectory prediction device, used to perform any one of the methods described above, comprising:

[0024] The data acquisition module acquires known flight trajectory observation data of n aircraft in consecutive time sequences. The flight trajectory observation data is... The corresponding timing is , The latest trajectory observation data of the spacecraft is used as the preset extrapolation result, and the corresponding preset extrapolation time series is as follows: ;

[0025] The basis function calculation module is used to calculate the time sequence corresponding to each flight trajectory observation data in the preset extrapolation time sequence. Lagrange basis function value at ... ;

[0026] The weighting module is used to set the weighting coefficients for n flight trajectory observation data. The weighting coefficients in the given structure satisfy the following constraints:

[0027] Only the latest k flight trajectory observation data are assigned non-zero weighting coefficients, and the weighting coefficients for the remaining flight trajectory observation data are 0, where k is a positive integer and ;

[0028] The non-zero weighting coefficients satisfy a monotonically increasing constraint, meaning that the closer the flight trajectory observation data is to the preset extrapolation time series in terms of time series, the larger its weighting coefficients are.

[0029] The sum of all weighted coefficients is 1, and each weighted coefficient is non-negative;

[0030] The weight calculation module is used to extrapolate based on preset results. Combined with the Lagrange basis function values By using structural constraints and weighting coefficients, the weighting coefficients of each flight trajectory observation data are obtained through reverse engineering.

[0031] The trajectory prediction module is used to acquire the latest known flight trajectory observation data of n consecutive time series of aircraft, and combine it with the Lagrange basis function values. By combining the weighting coefficients, the predicted flight trajectory data for the next moment is calculated.

[0032] Furthermore, the weighting coefficient structure set by the weight setting module is as follows:

[0033] The k non-zero weighting coefficients are respectively , And k is an integer; where, when When all weighting coefficients are non-zero, hour, arrive All nk weighting coefficients are 0;

[0034] Where a and b are parameters to be determined, and the constraints are satisfied. .

[0035] Furthermore, the specific process of the weight calculation module in reverse solving the weighting coefficients is as follows: substituting the structural expression of the weighting coefficients into the weighted Lagrange extrapolation formula. Combined with constraints Let a be represented as Substituting into the extrapolation formula yields a linear equation in one variable for b, solving for the value of b, and then substituting back to obtain the value of a, finally determining all weighting coefficients. .

[0036] Furthermore, the weight solving module is also used to: if the weighted coefficients obtained by solving have negative values, then let k=k-1, and rerun the weight setting module and the weight solving module until all the weighted coefficients output by the weight solving module are non-negative.

[0037] Due to the adoption of the above technical solution, the beneficial effects of the present invention are as follows:

[0038] 1. This invention, by setting a weighted coefficient structure with time-series constraints, assigns non-zero weights only to the latest few observation points, and the weights monotonically increase, which conforms to the physical characteristic of time-series data being "larger in the near and smaller in the far," effectively avoiding interference from distant time points, reducing the risk of high-order polynomial oscillations, and improving extrapolation accuracy.

[0039] 2. This invention realizes the reverse solution of weighted coefficients based on preset extrapolation results. Through structured design, it transforms the high-dimensional unconstrained solution into the low-dimensional constrained solution, and obtains unique and stable weighted coefficients, thus solving the problems of non-unique solutions and poor practicality of existing weight inversion methods.

[0040] 3. The method of the present invention has low computational cost, does not require complex iteration, can be solved directly analytically, is suitable for engineering scenarios such as embedded systems, and the weights satisfy non-negativity and normalization constraints, with clear physical meaning;

[0041] 4. This invention uses a weight validity verification mechanism to adjust the weight structure when negative weights occur, ensuring the engineering practicality of the weighting coefficients and further improving the stability and reliability of the extrapolation method. Detailed Implementation

[0042] The present invention will be further described in detail below with reference to specific embodiments, so as to facilitate understanding by those skilled in the art.

[0043] Example 1:

[0044] This embodiment provides a time-constrained weighted Lagrange extrapolation trajectory prediction method, taking 5 sets of continuous time-series observation data as an example, specifically including the following steps:

[0045] Step S1: Obtain 5 sets of known flight trajectory observation data of the aircraft in continuous time sequence. The flight trajectory observation data is... , , , , The corresponding timing is , The latest trajectory observation data of the spacecraft is used as the preset extrapolation result, and the corresponding preset extrapolation time series is as follows: ;

[0046] That is, the preset extrapolation results and the known flight trajectory observation data of 5 sets of continuous time series of aircraft constitute 6 sets of known flight trajectory observation data of continuous time series of aircraft.

[0047] Step S2: Calculate the time sequence corresponding to each flight trajectory observation data in the preset extrapolation time sequence. Lagrange basis function value at ... According to the Lagrange basis function calculation formula:

[0048] ;

[0049] The calculations yielded the following results:

[0050] ;

[0051] ;

[0052] ;

[0053] ;

[0054] ;

[0055] Step S3: Set the weighting coefficients corresponding to the n flight trajectory observation data. The structure satisfies the following constraints:

[0056] a) Select k=5, that is, assign non-zero weighting coefficients to all 5 sets of flight trajectory observation data, and there are no observation points with a weight of 0; where, in this embodiment, n=5 and k=5;

[0057] b) The non-zero weighting coefficients satisfy the monotonically increasing constraint, i.e. ;

[0058] c) The sum of all weighted coefficients is 1, and each weighted coefficient is non-negative;

[0059] The specific weighting coefficient structure is set as follows: ,

[0060] Where a and b are parameters to be determined, and considering the constraints... Substituting the values, we get:

[0061]

[0062] get .

[0063] Step S4: Based on the preset extrapolation result By combining the structural constraints of the Lagrange base function values ​​and the weighting coefficients, the weighting coefficients are solved in reverse.

[0064] The weighted Lagrange extrapolation formula is:

[0065] ;

[0066] Will ~ Let a and b represent the values ​​of a and b respectively. Substituting the flight trajectory observation data and basis function values, we obtain a linear equation in one variable about b. Solving for b, we get the value of b. Substituting back into the equation, we get the value of a, and finally determine the value of a. ~ ;

[0067] For example, if the extrapolation result is preset (Reasonable positive weight extrapolation value), after substituting and solving, we get: b≈0.12, a=0.2-2×0.12=-0.04 (a negative value appears, which does not meet the constraint);

[0068] At this point, the value of k is reduced to 4, retaining the last 4 observation points. Assign non-zero weights and reset the weight structure. , , , Constraints Solving for b, we get b ≈ 0.20 and a = -0.05 (negative values ​​appear, which does not satisfy the constraints).

[0069] Further reduce k to 3, retaining only the last 3 observation points. Assign non-zero weights and reset the weight structure. , , Constraints The solution is then recalculated to obtain: b≈0.13, a=0.20, and the non-negative weights satisfy the constraints.

[0070] Finally, the latest known flight trajectory observation data of the five consecutive time series of the aircraft were obtained, and combined with the Lagrange basis function values. By combining the weighting coefficients, the predicted flight trajectory data for the next moment is calculated.

[0071] Example 2:

[0072] This embodiment provides a time-constrained weighted Lagrange extrapolation trajectory prediction device, corresponding to the method in Embodiment 1, including:

[0073] The data acquisition module is used to acquire five sets of continuous time-series known flight trajectory observation data of the aircraft. , , , , The corresponding timing sequence is... , The latest trajectory observation data of the spacecraft is used as the preset extrapolation result, and the corresponding preset extrapolation time series is as follows: ;

[0074] The basis function calculation module is used to calculate the time sequence corresponding to each flight trajectory observation data in the preset extrapolation time sequence. Lagrange basis function value at ... , , , , ;

[0075] The weight setting module is used to set the weighting coefficient structure as follows: Constraints Furthermore, the weights are non-negative and monotonically increasing;

[0076] The weight calculation module is used to extrapolate based on preset results. Substitute the values ​​of a and b into the extrapolation formula to solve for the weighting coefficients. If negative weights are found, rerun the weight setting module and the weight solving module until all weighting coefficients output by the weight solving module are non-negative.

[0077] The trajectory prediction module is used to acquire the latest known flight trajectory observation data of the five consecutive time series of the aircraft, and combine it with the Lagrange basis function values. By combining the weighting coefficients, the predicted flight trajectory data for the next moment is calculated.

[0078] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Any simple modifications, equivalent changes and alterations made by those skilled in the art to the above embodiments based on the technical essence of the present invention without departing from the scope of the technical solution of the present invention shall fall within the protection scope of the technical solution of the present invention.

Claims

1. A time-constrained weighted Lagrange extrapolation trajectory prediction method, characterized in that, Includes the following steps: Step 1: Obtain known flight trajectory observation data for n consecutive time sequences of aircraft. The flight trajectory observation data is... The corresponding timing is , The latest trajectory observation data of the spacecraft is used as the preset extrapolation result, and the corresponding preset extrapolation time series is as follows: ; Step 2: Calculate the time sequence corresponding to each flight trajectory observation data in the preset extrapolation time sequence. Lagrange basis function value at ... The formula for calculating the Lagrange basis functions is as follows: ； Step 3: Set the weighting coefficients for the n flight trajectory observation data. The weighting coefficients in the given structure satisfy the following constraints: Only the latest k flight trajectory observation data are assigned non-zero weighting coefficients, and the weighting coefficients for the remaining flight trajectory observation data are 0, where k is a positive integer and ; The non-zero weighting coefficients satisfy a monotonically increasing constraint, meaning that the closer the flight trajectory observation data is to the preset extrapolation time series in terms of time series, the larger its weighting coefficients are. The sum of all weighted coefficients is 1, and each weighted coefficient is non-negative; Specifically, the structure of the weighting coefficients is as follows: The k non-zero weighting coefficients are respectively , And k is an integer; where, when When all weighting coefficients are non-zero, hour, arrive All nk weighting coefficients are 0; Where a and b are parameters to be determined, and the constraints are satisfied. ; Step 4: Based on the preset extrapolation results Combined with the Lagrange basis function values With structural constraints on the weighting coefficients, the weighting coefficients of each flight trajectory observation data are obtained by inverse solving; specifically: Substituting the structural expression of the weighting coefficients into the weighted Lagrange extrapolation formula Combined with constraints Let a be represented as Substituting into the extrapolation formula yields a linear equation in one variable for b, solving for the value of b, and then substituting back to obtain the value of a, finally determining all weighting coefficients. ; Step 5: Obtain the latest known flight trajectory observation data for the n consecutive time series of aircraft, and combine it with the Lagrange basis function values. By combining the weighting coefficients, the predicted flight trajectory data for the next moment is calculated.

2. The time-constrained weighted Lagrange extrapolation trajectory prediction method according to claim 1, characterized in that, In step 4, if the weighted coefficients obtained by solving have negative values, let k = k-1, and repeat steps 3 and 4 until all weighted coefficients are non-negative.

3. A time-constrained weighted Lagrange extrapolation trajectory prediction device, characterized in that, For performing the method according to any one of claims 1-2, comprising: The data acquisition module acquires known flight trajectory observation data of n aircraft in consecutive time sequences. The flight trajectory observation data is... The corresponding timing is , The latest trajectory observation data of the spacecraft is used as the preset extrapolation result, and the corresponding preset extrapolation time series is as follows: ; The basis function calculation module is used to calculate the time sequence corresponding to each flight trajectory observation data in the preset extrapolation time sequence. Lagrange basis function value at ... ; The weighting module is used to set the weighting coefficients for n flight trajectory observation data. The weighting coefficients in the given structure satisfy the following constraints: Only the latest k flight trajectory observation data are assigned non-zero weighting coefficients, and the weighting coefficients for the remaining flight trajectory observation data are 0, where k is a positive integer and ; The non-zero weighting coefficients satisfy a monotonically increasing constraint, meaning that the closer the flight trajectory observation data is to the preset extrapolation time series in terms of time series, the larger its weighting coefficients are. The sum of all weighted coefficients is 1, and each weighted coefficient is non-negative; The weighting coefficient structure set by the weight setting module is as follows: The k non-zero weighting coefficients are respectively , And k is an integer; where, when When all weighting coefficients are non-zero, hour, arrive All nk weighting coefficients are 0; Where a and b are parameters to be determined, and the constraints are satisfied. ; The weight calculation module is used to extrapolate based on preset results. Combined with the Lagrange basis function values By using structural constraints and weighting coefficients, the weighting coefficients of each flight trajectory observation data are obtained through reverse engineering. The specific process of the weight calculation module in reverse solving the weighting coefficients is as follows: substituting the structural expression of the weighting coefficients into the weighted Lagrange extrapolation formula. Combined with constraints Let a be represented as Substituting into the extrapolation formula yields a linear equation in one variable for b, solving for the value of b, and then substituting back to obtain the value of a, finally determining all weighting coefficients. ; The trajectory prediction module is used to acquire the latest known flight trajectory observation data of n consecutive time series of aircraft, and combine it with the Lagrange basis function values. By combining the weighting coefficients, the predicted flight trajectory data for the next moment is calculated.

4. The time-constrained weighted Lagrange extrapolation trajectory prediction device according to claim 3, characterized in that, The weight calculation module is also used to: if the weighted coefficients obtained by the calculation have negative values, let k=k-1, and rerun the weight setting module and the weight calculation module until the weighted coefficients output by the weight calculation module are all non-negative.

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