A method for analytically predicting hypersonic trajectories based on matrix operator homotopy analysis

Abstract

The application discloses a hypersonic trajectory analytical prediction method based on matrix operator homotopy analysis, and belongs to the technical field of aerospace. The method comprises the following steps: constructing a reentry dynamics model and extracting a system Jacobian linearization matrix at an initial time; constructing a homotopy auxiliary linear operator based on the Jacobian linearization matrix to obtain a physical driven zero-order analytical solution; solving a high-order deformation equation to calculate an analytical solution of a high-order correction term based on a polynomial approximation; and superimposing the zero-order analytical solution and the high-order correction term to output a full trajectory high-precision prediction trajectory. The application improves the convergence of the homotopy series, avoids the divergence problem of a traditional method under a large step length condition, avoids the inversion error caused by the fact that the Jacobian linearization matrix is close to singular in flight dynamics, converts the solution of a complex nonlinear differential equation into simple algebraic operation, and the calculation speed is improved by an order of magnitude compared with a traditional numerical integration method.

Description

Technical Field

[0001] This invention relates to an analytical prediction method for hypersonic trajectories based on matrix operator homotopy analysis, belonging to the field of aerospace technology. Background Technology

[0002] Hypersonic vehicles face a complex dynamic environment with strong nonlinearity and strong coupling during the reentry and gliding phase. When performing online trajectory planning, guidance command generation, or closed-loop correction tasks, they need to frequently and rapidly make high-precision predictions of future flight trajectories. The accuracy and computational efficiency of trajectory prediction directly determine the performance of the vehicle's online guidance system.

[0003] Existing trajectory prediction methods face the following bottlenecks in engineering applications:

[0004] (1) Numerical integration method:

[0005] While numerical integration methods, such as the Runge-Kutta method, have high prediction accuracy, they require extremely small integration steps to ensure numerical stability during the integration process, resulting in huge computational overhead and making it difficult to meet the millisecond-level online real-time requirements of onboard computers.

[0006] (2) Traditional analytical method:

[0007] Traditional analytical methods based on the quasi-equilibrium gliding assumption are fast, but they introduce strong simplification assumptions in the modeling process. When the aircraft performs large-scale lateral maneuvers or long-endurance flight missions, the prediction error will accumulate significantly, which cannot support the requirements of high-precision guidance.

[0008] (3) Traditional HAM (Hotopy Analysis):

[0009] HAM, with its characteristic of not relying on small parameter assumptions, has shown certain advantages in solving nonlinear dynamic problems. However, traditional HAM usually uses simplified general differential operators (such as the highest-order derivative operator), which fails to effectively utilize the local dynamic characteristics of the dynamic system. As a result, its zeroth-order solution lacks corresponding physical meaning, and it often needs to derive and calculate a large number of high-order correction terms to meet the accuracy requirements, which seriously restricts the online computation efficiency of the method.

[0010] Therefore, designing a high-fidelity, fast trajectory prediction method that can both eliminate the tedious numerical integration process and accurately capture the full nonlinear dynamic characteristics of hypersonic vehicles is a key technical problem that urgently needs to be solved in the field of hypersonic vehicle guidance. Summary of the Invention

[0011] To address the problems existing in the background technology, this invention provides an analytical prediction method for hypersonic trajectories based on matrix operator homotopy analysis.

[0012] To achieve the above objectives, the present invention adopts the following technical solution: a hypersonic trajectory analytical prediction method based on matrix operator homotopy analysis, the method comprising the following steps:

[0013] S1: Construct a hypersonic vehicle reentry dynamics model and extract the system Jacobian linearization matrix for the initial moment of the prediction interval;

[0014] S2: Construct a homotopy-assisted linear operator based on the Jacobian linearization matrix and solve it to obtain the physical-driven zeroth-order analytical solution;

[0015] S3: Solve higher-order deformed equations and calculate analytical solutions of higher-order correction terms based on polynomial approximation;

[0016] S4: Superimpose the zero-order analytical solution and higher-order correction terms to output a high-precision predicted trajectory for the entire ballistic trajectory.

[0017] Furthermore, step S1 includes the following steps:

[0018] S101: Set the state vector of the hypersonic vehicle as follows: The control vector is ;

[0019] in: The distance from the Earth's center. Longitude Latitude For speed, For the track angle, For heading angle, For the angle of attack, The tilt angle, The rate of change of angle of attack, The rate of change of the tilt angle;

[0020] S102: Establishing the full-mass dynamic equations for the reentry phase of a hypersonic vehicle:

[0021]

[0022] Mode middle:

[0023] It is a time variable;

[0024] State vector The first derivative;

[0025] S103: The initial time of any k-th prediction interval is Given that the initial state at this moment is Calculate the Jacobian linearized matrix of the dynamic system at this initial state point. :

[0026]

[0027] Furthermore, step S2 includes the following steps:

[0028] S201: Introducing homotopy embedding parameters Construct the homotopy equation for continuous deformation:

[0029]

[0030] Mode middle:

[0031] For auxiliary linear operators;

[0032] It is a homotopy mapping function;

[0033] It is a zero-order analytical solution;

[0034] These are the convergence control parameters;

[0035] It is a nonlinear operator;

[0036] S202: Define nonlinear operators The residual form of the original dynamic equations:

[0037]

[0038] S203: To capture the local dynamic characteristics of a dynamic system, the Jacobian linearization matrix is ​​used. Constructing auxiliary linear operators :

[0039]

[0040] S204: When embedding parameters At that time, the homotopy equation simplifies to At this point, the zeroth-order analytical solution is obtained by using the matrix exponentiation method. :

[0041]

[0042] Mode middle:

[0043] It is a constant;

[0044] For linearization constant terms;

[0045] Let be the degree term of the polynomial.

[0046] Furthermore, step S3 includes the following steps:

[0047] S301: Apply homotopy mapping function Regarding embedded parameters Perform a Taylor expansion and define the first... The order correction term is ;

[0048] S302: Applying the homotopy equation to both sides with respect to the embedding parameters Taking the m-th partial derivative, we obtain the standard form of the non-homogeneous linear ordinary differential equation:

[0049]

[0050] Mode middle:

[0051] For the first Order correction term The first derivative;

[0052] For dependent on the former The source term vector of the solution of order;

[0053] S303: Transfer the source term vector Within the current prediction interval, it is approximately a K-order polynomial:

[0054]

[0055] Mode middle:

[0056] Polynomial coefficients

[0057] For the normalized time variable within the prediction interval;

[0058] S304: Using integration by parts, derive the purely algebraic analytic recurrence relation for higher-order correction terms:

[0059] (9)

[0060] In equation (9):

[0061] These are higher-order integral basis functions.

[0062] Furthermore, step S4 includes the following steps:

[0063] S401: Summing the zeroth-order analytical solution with the analytical correction terms of each order yields the high-precision analytical prediction state at any time within the current prediction interval.

[0064] (10)

[0065] In formula (10):

[0066] To truncate the total order;

[0067] Equation (10) can output the full ballistic flight trajectory under arbitrary control input in milliseconds;

[0068] S402: Loop to obtain full ballistic analytical prediction trajectory.

[0069] Compared with the prior art, the beneficial effects of the present invention are:

[0070] 1. This invention innovatively utilizes the system Jacobian linearization matrix to construct an auxiliary linear operator, making the zero-order analytical solution essentially represent the tangential linearized motion trajectory of the aircraft in the current state (including physical information such as the system's natural frequency and decay mode), which significantly improves the convergence of the homotopy series. Even in large prediction step sizes and strong nonlinear regions, it can still accurately track the oscillation characteristics of the state variables, avoiding the divergence problem of traditional methods under large step size conditions.

[0071] 2. The entire process of this invention eliminates the discrete numerical integration operation of traditional dynamic equations: when solving the zero-order analytical solution, Taylor series truncation is used to extract the integral basis function, avoiding the inversion error caused by the near singularity of the Jacobian linearization matrix in flight mechanics; when solving higher-order correction terms, polynomial approximation and analytical integration formulas are used to transform the solution of complex nonlinear differential equations into simple algebraic operations, and the calculation speed is improved by an order of magnitude compared with the traditional numerical integration method. Attached Figure Description

[0072] Figure 1 This is a flowchart of the present invention. Detailed Implementation

[0073] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the invention, not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0074] A hypersonic trajectory analytical prediction method based on matrix operator homotopy analysis is a homotopy analytical recursive trajectory prediction method based on constructing linearized operators from the system's Jacobian linearization matrix. By embedding local dynamic features into the zeroth-order analytical solution and combining it with higher-order analytical corrections, high-precision and high-efficiency online prediction of the spacecraft trajectory is achieved. The method includes the following steps:

[0075] S1: Construct a hypersonic vehicle reentry dynamics model and extract the system Jacobian linearization matrix for the initial moment of the prediction interval;

[0076] S2: Construct a homotopy-assisted linear operator based on the Jacobian linearization matrix and solve it to obtain the physical-driven zeroth-order analytical solution;

[0077] S3: Solve higher-order deformed equations and calculate analytical solutions of higher-order correction terms based on polynomial approximation;

[0078] S4: Superimpose the zero-order analytical solution and higher-order correction terms to output a high-precision predicted trajectory for the entire ballistic trajectory.

[0079] Furthermore, step S1 includes the following steps:

[0080] S101: Set the state vector of the hypersonic vehicle as follows: The control vector is ;

[0081] in: The distance from the Earth's center. Longitude Latitude For speed, For the track angle, For heading angle, For the angle of attack, The tilt angle, The rate of change of angle of attack, The rate of change of the tilt angle;

[0082] S102: Establish the full-mass dynamic equations for the reentry phase of a hypersonic vehicle, the general form of which is a continuous-time nonlinear initial value problem:

[0083]

[0084] Mode middle:

[0085] It is a time variable;

[0086] State vector The first derivative;

[0087] S103: The initial time of any k-th prediction interval is Given that the initial state at this moment is Calculate the Jacobian linearized matrix of the dynamic system at this initial state point. :

[0088]

[0089] Furthermore, step S2 includes the following steps:

[0090] S201: Introducing homotopy embedding parameters Construct the homotopy equation for continuous deformation:

[0091]

[0092] Mode middle:

[0093] For auxiliary linear operators;

[0094] It is a homotopy mapping function;

[0095] It is a zero-order analytical solution;

[0096] As the convergence control parameter, this invention sets it to... ;

[0097] It is a nonlinear operator;

[0098] S202: Define nonlinear operators The residual form of the original dynamic equations:

[0099]

[0100] S203: To capture the local dynamic characteristics of a dynamic system, the simple differential operator in the traditional homotopy analysis method is abandoned, and the Jacobian linearization matrix is ​​utilized. Constructing auxiliary linear operators :

[0101]

[0102] S204: When embedding parameters At that time, the homotopy equation simplifies to At this point, the zeroth-order analytical solution is obtained by using the matrix exponentiation method. :

[0103]

[0104] Mode middle:

[0105] It is a constant;

[0106] For linearization constant terms;

[0107] Let be the degree term of the polynomial.

[0108] Furthermore, step S3 includes the following steps:

[0109] S301: Apply homotopy mapping function Regarding embedded parameters Perform a Taylor expansion and define the first... The order correction term is ;

[0110] S302: Applying the homotopy equation to both sides with respect to the embedding parameters Taking the m-th partial derivative, we obtain the standard form of the non-homogeneous linear ordinary differential equation:

[0111]

[0112] Mode middle:

[0113] For the first Order correction term The first derivative;

[0114] For dependent on the former The source term vector of the first-order solution, according to the method of variation of constants, has a general solution in integral form;

[0115] S303: To avoid the time consumption and truncation error caused by numerical integration, the source term vector... Within the current prediction interval, it is approximately a K-order polynomial:

[0116]

[0117] Mode middle:

[0118] Polynomial coefficients

[0119] For the normalized time variable within the prediction interval;

[0120] S304: Using integration by parts, derive the purely algebraic analytic recurrence relation for higher-order correction terms:

[0121] (9)

[0122] In equation (9):

[0123] These are higher-order integral basis functions, obtained directly through algebraic operations, avoiding numerical integration.

[0124] Furthermore, step S4 includes the following steps:

[0125] S401: Summing the zeroth-order analytical solution with the analytical correction terms of each order yields the high-precision analytical prediction state at any time within the current prediction interval.

[0126] (10)

[0127] In formula (10):

[0128] To truncate the total order, we usually take... This will satisfy the accuracy requirements;

[0129] Equation (10) can output the full ballistic flight trajectory under arbitrary control input in milliseconds;

[0130] S402: Repeat this process to quickly obtain the full ballistic analytical prediction trajectory.

[0131] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above, and that the invention can be implemented in other forms without departing from its spirit or essential characteristics. Therefore, the embodiments should be considered in all respects as exemplary and non-limiting, and the scope of the invention is defined by the appended claims rather than the foregoing description. Thus, all variations falling within the meaning and scope of the equivalents of the claims are intended to be included within the present invention. No reference numerals in the claims should be construed as limiting the scope of the claims.

[0132] Furthermore, it should be understood that although this specification describes embodiments, not every embodiment contains only one independent technical solution. This narrative style is merely for clarity. Those skilled in the art should consider the specification as a whole, and the technical solutions in each embodiment can also be appropriately combined to form other embodiments that can be understood by those skilled in the art.

Claims

1. A method for analytical prediction of hypersonic trajectories based on matrix operator homotopy analysis, characterized in that: The method includes the following steps: S1: Construct a hypersonic vehicle reentry dynamics model and extract the system Jacobian linearization matrix for the initial moment of the prediction interval; S2: Construct a homotopy-assisted linear operator based on the Jacobian linearization matrix and solve it to obtain the physical-driven zeroth-order analytical solution; S3: Solve higher-order deformed equations and calculate analytical solutions of higher-order correction terms based on polynomial approximation; S4: Superimpose the zero-order analytical solution and higher-order correction terms to output a high-precision predicted trajectory for the entire ballistic trajectory.

2. The analytical prediction method for hypersonic trajectories based on matrix operator homotopy analysis according to claim 1, characterized in that: S1 includes the following steps: S101: Set the state vector of the hypersonic vehicle as follows: The control vector is ; in: The distance from the Earth's center. Longitude Latitude For speed, For the track angle, For heading angle, For the angle of attack, The tilt angle, The rate of change of angle of attack, The rate of change of the tilt angle; S102: Establishing the full-mass dynamic equations for the reentry phase of a hypersonic vehicle: Mode middle: It is a time variable; State vector The first derivative; S103: The initial time of any k-th prediction interval is Given that the initial state at this moment is Calculate the Jacobian linearized matrix of the dynamic system at this initial state point. : 。 3. The analytical prediction method for hypersonic trajectories based on matrix operator homotopy analysis according to claim 2, characterized in that: S2 includes the following steps: S201: Introducing homotopy embedding parameters Construct the homotopy equation for continuous deformation: Mode middle: For auxiliary linear operators; It is a homotopy mapping function; It is a zero-order analytical solution; These are the convergence control parameters; It is a nonlinear operator; S202: Define nonlinear operators The residual form of the original dynamic equations: S203: To capture the local dynamic characteristics of a dynamic system, the Jacobian linearization matrix is ​​used. Constructing auxiliary linear operators : S204: When embedding parameters At that time, the homotopy equation simplifies to At this point, the zeroth-order analytical solution is obtained by using the matrix exponentiation method. : Mode middle: It is a constant; For linearization constant terms; Let be the degree term of the polynomial.

4. The analytical prediction method for hypersonic trajectories based on matrix operator homotopy analysis according to claim 3, characterized in that: S3 includes the following steps: S301: Apply homotopy mapping function Regarding embedded parameters Perform a Taylor expansion and define the first... The order correction term is ; S302: Applying the homotopy equation to both sides with respect to the embedding parameters Taking the m-th partial derivative, we obtain the standard form of the non-homogeneous linear ordinary differential equation: Mode middle: For the first Order correction term The first derivative; For dependent on the former The source term vector of the solution of order; S303: Transfer the source term vector Within the current prediction interval, it is approximately a K-order polynomial: Mode middle: Polynomial coefficients For the normalized time variable within the prediction interval; S304: Using integration by parts, derive the purely algebraic analytic recurrence relation for higher-order correction terms: (9) In equation (9): These are higher-order integral basis functions.

5. The analytical prediction method for hypersonic trajectories based on matrix operator homotopy analysis according to claim 1, characterized in that: S4 includes the following steps: S401: Summing the zeroth-order analytical solution with the analytical correction terms of each order yields the high-precision analytical prediction state at any time within the current prediction interval. (10) In formula (10): To truncate the total order; Equation (10) can output the full ballistic flight trajectory under arbitrary control input in milliseconds; S402: Loop to obtain full ballistic analytical prediction trajectory.

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