A Ballistic Prediction Method for Hypersonic Reentry Vehicles Based on a Time-Varying Frequency Fitting Model

Abstract

A method for predicting the trajectory of a hypersonic reentry vehicle based on a time-varying frequency fitting model. The method is as follows: Before prediction, first obtain the estimated values of the state variables of the vehicle for a period of time before and at the current moment through trajectory tracking; analyze the maneuvering characteristics of the hypersonic reentry vehicle and design various trajectory fitting models or acceleration fitting models; considering the trajectory characteristics of the reentry skip hypersonic vehicle, design a decaying oscillation acceleration fitting model with a time-varying frequency; adopt different fitting maneuver models, design a method for solving the fitting parameters of the nonlinear model, and solve the fitting parameters of the fitting model; set different iteration numbers N to solve the model parameters, compare the predicted values of the state variables of each fitting model with the observed values of the state variables, and take the model with the smallest sum of squared residuals as the best maneuver model, and the predicted value of this model is used as the trajectory prediction result. The present invention compares various fitting models to determine the best maneuver model of the vehicle, which can improve the accuracy of trajectory prediction.

Description

Technical Field

[0001] The present invention relates to a rapid and accurate ballistic prediction method for hypersonic reentry vehicles based on a time-varying frequency fitting model, which can predict subsequent ballistic trajectories according to the estimation results of vehicle ballistic tracking. Background Art

[0002] Hypersonic reentry vehicles have advantages such as a large combat radius, high flight speed, and strong maneuverability, posing great challenges to ballistic trajectory prediction. Existing ballistic prediction methods cannot adapt to their maneuverability, and there is an urgent need to develop new vehicle ballistic prediction methods. Summary of the Invention

[0003] Aiming at the problems that existing ballistic prediction methods cannot adapt to the strong maneuverability of hypersonic reentry vehicles and cannot predict subsequent ballistic trajectories, the present invention provides a ballistic prediction method for hypersonic reentry vehicles based on a time-varying frequency fitting model.

[0004] The object of the present invention is achieved through the following technical solutions:

[0005] A ballistic prediction method for hypersonic reentry vehicles based on a time-varying frequency fitting model, the steps of the method are as follows:

[0006] Step 1: Before prediction, first obtain the estimated values of the state variables of the vehicle for a period of time before and the current moment through ballistic tracking [x 1 , y 1 , z 1 , V x1 , V y1 , V z1 ··· [x k , y k , z k , V xk , V yk , V zk (the positions and velocities of the three axes from time t = 1 to time t = k);

[0007] Step 2: Analyze the maneuver characteristics of hypersonic reentry vehicles and design various trajectory fitting models or acceleration fitting models;

[0008] Step 3: Considering the trajectory characteristics of reentry skip hypersonic vehicles, design a time-varying frequency damped oscillation acceleration fitting model;

[0009] Step 4: Adopt different fitting maneuver models and design a nonlinear model parameter solving method to solve the fitting parameter θ k ;

[0010] Step 5: Set different numbers of iterations N to solve the model parameters, and compare the predicted values of the state variables of each fitting model With the observed values of the state variables Take the model with the minimum sum of squared residuals {s 2} as the optimal maneuver model, and the predicted value of this model is used as the ballistic prediction result.

[0011] Further, the specific steps of step two are as follows:

[0012] Step 2-1: Design a constant acceleration (CA) model to match the situation when the aircraft is moving in a uniformly accelerated straight line; if the acceleration is zero, the model degenerates into a constant velocity (CV) model; the CA model is expressed as: a(t) = C 0 , where C 0 is a fitting parameter;

[0013] Step 2-2: Design a linear acceleration (LA) model to match the situation where the jerk of the aircraft is constant; if the jerk is zero, the model degenerates into a constant acceleration (CA) model; the LA model is expressed as: a(t) = C 1 t + C 0 , where C 0 、C 1 are fitting parameters;

[0014] Step 2-3: Design a quadratic curve acceleration (QA) model to match the situation where the second derivative of the acceleration of the aircraft is constant; if the second derivative is zero, the model degenerates into a linear acceleration (LA) model; the QA model is expressed as: a(t) = C 2 t 2 + C 1 t + C 0 , where C 0 、C 1 、C 2 are fitting parameters;

[0015] Step 2-4: Design a damped oscillation acceleration (DPA) model. By analyzing the maneuvering characteristics of hypersonic skip reentry vehicles, it can be seen that the altitude of the longitudinal trajectory of the vehicle can be regarded as a damped periodic oscillation function of speed; in a short time, the speed can be regarded as a linear function of time. Therefore, the periodic oscillation function can be characterized by trigonometric functions. Therefore, the DPA model can be expressed as: a(t) = C 1 f(t)sin(ωt + η 0 ), where C 1 、ω、η 0 are fitting parameters;

[0016] Two or more of the above four models can also be combined into a new comprehensive model. Among them, the most suitable model for hypersonic reentry vehicles is the damped oscillation acceleration + linear acceleration (DPA + LA) comprehensive model.

[0017] Further, the specific steps of Step 3 are as follows:

[0018] Step 3-1: Based on the combined model of decaying oscillation acceleration + linear acceleration (DPA+LA), consider the maneuvering characteristics of hypersonic reentry vehicles, design a time-varying frequency fitting model, and design the frequency of the trigonometric function in the combined model as a variable function.

[0019] Step 3-2: Considering the maneuvering characteristics of skip reentry vehicles, the variable function adopts an exponential form. Therefore, the time-varying frequency fitting model is expressed as: where θ 1 , θ 2 … θ 7 There are a total of 7 fitting parameters to be determined.

[0020] Further, the specific steps of Step 4 are as follows:

[0021] Step 4-1: Use the nonlinear least squares method to estimate the parameters of the nonlinear model. Fit a nonlinear model with n (m>n) parameters using m observation values, making the sum of the squared residuals of the fit the smallest. Its basis is to approximate the model through a linearization approximation method, and then extract the parameters through continuous iteration.

[0022] Step 4-2: Assume there are m observation data (x i , y i ), i = 1, 2, …, m. The function form to be fitted is y = f(x, θ), where y is the dependent variable of the fitted function, x is the independent variable of the fitted function, and θ is the set of parameters to be estimated, θ = [θ 1 , θ 2 , …, θ n T , where θ 1 , θ 2 , …, θ n are n parameters to be estimated. Solve the estimated value of θ to make the sum of the squared residuals S reach the minimum value, that is, satisfy the minimum sum of squared residuals formula

[0023] Step 4-3: Since the jth fitting error e j = y j - f(x j , θ), rewrite the minimum sum of squared residuals formula as This system of equations has no closed-form solution. An initial value needs to be given, and then the final estimated value can be obtained through continuous iteration.

[0024] Step 4-4: Use the Gauss-Newton iteration algorithm to solve the estimated value. Assume θ k is the kth iteration solution of the set of parameters θ to be estimated. Take the partial derivative of y = f(x, θ) with respect to θ k ​The first-order Taylor expansion is as follows: where J ij is the element in the i-th row and j-th column of the Jacobian matrix, which changes with the number of iterations. The calculation method is as follows: is the k-th iteration solution of the j-th parameter to be estimated θ j , and Δθ j is the difference of the j-th parameter to be estimated θ j ;

[0025] Steps 4 and 5: Further express the residual e i as Δy i = y i - f(x i , θ k ), where y i is the fitted value corresponding to the i-th point x i of the fitting function, and Δy i is the difference of the i-th fitted value. Substitute it into the iterative calculation and organize to get Written in matrix form:

[0026] J T JΔθ = J T Δy

[0027] According to the above formula, the fitting parameter difference Δθ is solved through the fitting value difference vector Δy, and the parameter value is updated to θ k+1 = θ k + Δθ, where θ k+1 represents the (k + 1)-th iteration solution of the set of parameters to be estimated θ. Continue the iteration until the iteration end condition is met.

[0028] The beneficial effects of the present invention compared with the prior art are as follows:

[0029] (1) Analyze the maneuvering characteristics of the reentry skip hypersonic vehicle, design a time-varying frequency fitting model for the function fitting trajectory prediction method, which has a small prediction error for the reentry skip trajectory and has good engineering practicability.

[0030] (2) Compare multiple fitting models to determine the best maneuvering model of the vehicle, which can improve the accuracy of trajectory prediction. Brief Description of the Drawings

[0031] Figure 1 is the ballistic prediction flow chart of the present invention. Detailed Embodiment

[0032] The technical solution of the present invention will be further described below in conjunction with the accompanying drawings and embodiments, but it is not limited thereto. Any modification or equivalent replacement of the technical solution of the present invention without departing from the spirit and scope of the technical solution of the present invention shall be covered by the protection scope of the present invention.

[0033] Embodiment 1:

[0034] A hypersonic reentry vehicle trajectory prediction method based on a time-varying frequency fitting model. The specific steps of trajectory prediction are as follows:

[0035] Step 1: Before prediction, first obtain the estimated values (position and velocity) of the state variables of the vehicle in a previous period and the current moment through trajectory tracking [x 1 , y 1 , z 1 , V x1 , V y1 , V z1 ··· [x k , y k , z k , V xk , V yk , V zk ;

[0036] Step 2: Analyze the maneuver characteristics of the hypersonic reentry vehicle and design various trajectory fitting models or acceleration fitting models;

[0037] Step 3: Considering the trajectory characteristics of the reentry skip hypersonic vehicle, design a time-varying frequency fitting model (attenuated oscillation acceleration model of time-varying frequency);

[0038] Step 4: Adopt different fitting maneuver models, design a non-linear model parameter solving method, and solve the fitting parameter θ of the fitting model k ;

[0039] Step 5: Set different iteration times N to solve the model parameters, compare the predicted values of the state variables of each fitting model with the observed values of the state variables Take the model with the smallest sum of squared residuals {s 2} as the best maneuver model, and the predicted value of this model is used as the trajectory prediction result.

[0040] Furthermore, the trajectory fitting / acceleration fitting model designed in Step 2 is as follows:

[0041] Step 2.1: Design a constant acceleration (CA) model to match the situation when the vehicle is moving in a straight line with uniform acceleration. If the acceleration is zero, it degenerates into a constant velocity (CV) model. The CA model can be expressed as: a(t) = C 0 , where C 0are fitting parameters;

[0042] Step 2.2: Design a linear acceleration (LA) model to match the case where the jerk of the aircraft is constant. If the jerk is zero, the model degenerates into a constant acceleration (CA) model. The LA model can be expressed as: a(t) = C 1 t + C 0 , where C 0 and C 1 are fitting parameters;

[0043] Step 2.3: Design a quadratic curve acceleration (QA) model to match the case where the second derivative of the acceleration of the aircraft is constant. If the second derivative is zero, it degenerates into a linear acceleration (LA) model. The QA model can be expressed as: a(t) = C 2 t 2 + C 1 t + C 0 , where C 0 , C 1 , and C 2 are fitting parameters;

[0044] Step 2.4: Design a damped oscillation acceleration (DPA) model. By analyzing the maneuvering characteristics of a hypersonic skip reentry vehicle, it can be seen that the altitude of the longitudinal trajectory of the vehicle can be regarded as a damped periodic oscillation function of the speed. In a short time, the speed can be regarded as a linear function of time. Therefore, the periodic oscillation function can be characterized by trigonometric functions. Thus, the DPA model can be expressed as: a(t) = C 1 f(t)sin(ωt + η 0 ), where C 1 , ω, and η 0 are fitting parameters. A combined model such as DPA + CA can also be considered.

[0045] Furthermore, when designing the time-varying frequency fitting model in Step 3, the steps are as follows:

[0046] Step 3.1: Based on the combined model of damped oscillation acceleration + linear acceleration (DPA + LA), consider the maneuvering characteristics of a hypersonic reentry vehicle and design a time-varying frequency fitting model, and design the frequency of the trigonometric function in the combined model as a variable function;

[0047] Step 3.2: Considering the maneuvering characteristics of a skip reentry vehicle, the variable function adopts an exponential form. Therefore, the time-varying frequency fitting model can be expressed as: where θ 1 , θ 2 …θ 7 There are a total of 7 fitting parameters to be determined.

[0048] Furthermore, when solving the fitting parameters of the fitting model in Step 4, the steps are as follows:

[0049] Step 4.1: Estimate the parameters of the nonlinear model using the nonlinear least squares method. Fit a nonlinear model with n (m > n) parameters using m observed values to minimize the sum of the squared residuals. The basis is to approximate the model through a linearization approximation method and then extract the parameters through continuous iteration.

[0050] Step 4.2: Assume there are observed data (x i , y i ), i = 1, 2, …, m. The functional form to be fitted is y = f(x, θ), where θ = [θ 1 , θ 2 , …, θ n is the parameter to be estimated. Solve T to minimize the sum of the squared residuals such that

[0051] Step 4.3: Since the fitting error e j = y j - f(x j , θ), rewrite the sum of the squared residuals minimum formula as This system of equations has no closed-form solution. An initial value needs to be given, and then the final estimated value can be obtained through continuous iteration.

[0052] Step 4.4: Use the Gauss-Newton iteration algorithm to solve for the estimated value. Let the fitting parameter θ k be the solution of the k-th iteration. Take the first-order Taylor expansion of y = f(x, θ) with respect to θ k , and we have where J ij is the element of the Jacobian matrix, which changes with the number of iterations. The calculation method is:

[0053] Step 4.5: Further express the residual as Δy i = y i - f(x i , θ k ). Substitute it into the iterative calculation and simplify to get Write it in matrix form:

[0054] J T JΔθ = J T Δy

[0055] Solve for Δθ from the above equation, and update the parameter value to θ k+1 = θ k + Δθ. Continue the iteration until the iteration end condition is met.

Claims

1. A hypersonic reentry vehicle trajectory prediction method based on a time-varying frequency fitting model, characterized in that: The method steps are as follows: Step 1: Before the prediction, first obtain the estimated values of the state variables of the aircraft for a period of time before and at the current moment through ballistic tracking [x 1 , y 1 , z 1 , V x1 , V y1 , V z1 ··· [x k , y k , z k , V xk , V yk , V zk ; Step 2: Analyze the maneuvering characteristics of the hypersonic reentry vehicle and design various trajectory fitting models or acceleration fitting models; the various trajectory fitting models or acceleration fitting models include (1) design a linear acceleration model LA to match the case where the vehicle acceleration is constant; if the acceleration is zero, the model degenerates into a constant acceleration model CA; the LA model is expressed as: a(t) = C 1 t + C 0 , where C 0 and C 1 are fitting parameters; (2) design a damped oscillation acceleration model DPA. Analyzing the maneuvering characteristics of the hypersonic skip reentry vehicle, it can be seen that the altitude of the vehicle's longitudinal trajectory can be regarded as a damped periodic oscillation function of speed; in a short period of time, the speed can be regarded as a linear function of time. Therefore, the periodic oscillation function can be characterized by trigonometric functions. Therefore, the DPA model can be expressed as: a(t) = C 1 f(t)sin(ωt + η 0 ), where C 1 , ω, and η 0 are fitting parameters; the linear acceleration model LA and the damped oscillation acceleration model DPA are combined into a damped oscillation acceleration + linear acceleration comprehensive model DPA + LA; Step 3: Considering the trajectory characteristics of the reentry skip hypersonic vehicle, design a decaying oscillation acceleration fitting model with a time-varying frequency; specifically, the said Step 3 is: Step 3-1: Based on the combined model of decaying oscillation acceleration + linear acceleration DPA + LA, considering the maneuvering characteristics of the hypersonic reentry vehicle, design the frequency of the trigonometric function in the combined model as a variable function; Step 3-2: Considering the maneuvering characteristics of the skip reentry vehicle, the variable function adopts an exponential form. Therefore, the time-varying frequency damped oscillation acceleration fitting model is expressed as: where θ 1 , θ 2 … θ 7 There are a total of 7 fitting parameters to be determined; Step 4: For the ballistic prediction requirements of hypersonic reentry vehicles, different fitting models designed in Steps 2 to 3 are adopted to design a solution method for nonlinear model parameters and solve the fitting parameter θ of the fitting model k ; Step 5: Set different iteration times N to solve the parameters of each fitting model, and compare the predicted values of each fitting model for the state variables with the observed values of the state variables Take the fitting model with the smallest sum of squared residuals {s 2} as the best fitting model, and use the predicted value of this best fitting model as the ballistic prediction result.

2. The hypersonic reentry vehicle trajectory prediction method based on a time-varying frequency fitting model according to claim 1, characterized in that: In step 2, the multiple trajectory fitting models or acceleration fitting models further include: (3) designing a constant acceleration model CA to match the situation when the aircraft is in a uniformly accelerated linear motion; if the acceleration is zero, the model degenerates into a constant velocity model CV; the CA model is expressed as: a(t) = C 0 , where C 0 is a fitting parameter; (4) designing a quadratic curve acceleration model QA to match the situation where the second derivative of the aircraft acceleration is constant; if the second derivative is zero, the model degenerates into a linear acceleration model LA; the QA model is expressed as: a(t) = C 2 t 2 + C 1 t + C 0 , where C 0 , C 1 , C 2 are fitting parameters; two or more of the above four models can also be combined into a new fitting model.

3. The hypersonic reentry vehicle trajectory prediction method based on a time-varying frequency fitting model according to claim 1, characterized in that: Specifically, the said Step 4 is: Step 4-1: Use the nonlinear least squares method to estimate the nonlinear model parameters, and use m observed values to fit a nonlinear model containing n parameters, where m > n, so that the sum of the squares of the fitting residuals is minimized. The basis is to approximate the model through a linearization approximation method, and then extract the parameters through continuous iteration; Step 42: Set m observed data points (x i , y i ), where i = 1, 2, …, m. The function form to be fitted is y = f(x, θ), where y is the dependent variable of the fitting function, x is the independent variable of the fitting function, and θ is the set of parameters to be estimated, θ = [θ 1 , θ 2 , …, θ n T , where θ 1 , θ 2 , …, θ n are n parameters to be estimated. Solve for the estimated value of θ such that the sum of squared residuals S reaches the minimum, i.e., satisfy the sum of squared residuals minimum formula ​ Step 43: Since the j-th fitting error e j = y j - f(x j , θ), rewrite the sum of squared residuals formula as This system of equations has no closed-form solution. An initial value needs to be given, and then the final estimated value can be obtained through continuous iteration; Step Four: Use the Gauss-Newton iterative algorithm to solve for the estimated value, and set θ k as the k-th iterative solution of the set of parameters θ to be estimated. Take the first-order Taylor expansion of y = f(x, θ) with respect to θ k to obtain where J ij is the element in the i-th row and j-th column of the Jacobian matrix, which changes with the number of iterations. The calculation method is: is the k-th iterative solution of the j-th parameter θ j to be estimated, and Δθ j is the difference of the j-th parameter θ j to be estimated; Step 4 and 5: Represent the residual e i as further Δy i = y i - f(x i , θ k ), where y i is the fitted value corresponding to the i-th point x i of the fitting function, Δy i is the difference of the i-th fitted value. Substitute it into the iterative calculation and after arrangement, we get Written in matrix form: J T JΔθ = J T Δy The fitting parameter difference Δθ is obtained by solving according to the above formula through the fitting value difference vector Δy, and the parameter value is updated to θ k+1 = θ k + Δθ, where θ k+1 represents the (k + 1)-th iterative solution result of the set of parameters θ to be estimated, and continue the iteration until the iteration end condition is met.

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