Ballistic missile free section tracking method based on novel Bayesian filtering

Abstract

The invention provides a ballistic missile free section tracking method based on novel Bayesian filtering. The ballistic missile free section tracking method comprises the steps of solving random multi-step delay and measurement loss of measurement data. The method specifically comprises the following steps: firstly, establishing a new measurement model which can represent random multi-step delay and loss of measurement data; secondly, under the condition that random multi-step delay happens to measurement data, the system state is augmented, a delay variable is introduced, and an augmented Gaussian mixture filter framework is designed; then, when the measured value is lost, propagating the measured value from a previous moment to a current moment, and deriving expressions of a mean value and a covariance in a state augmentation form; and finally, tracking the free section of the ballistic missile based on the designed filtering algorithm. The method has the advantages that the accurate tracking of the ballistic missile target is realized by adopting the Bayesian estimation theory under the condition of random multi-step delay and loss of the measurement data, and the method has very important engineering significance.

Description

Technical Field

[0001] The invention relates to the field of ballistic missile free segment tracking, in particular to a ballistic missile free segment tracking method based on a novel Bayesian filter. Background Art

[0002] In the offensive and defensive confrontation, accurate tracking of enemy targets is the premise of accurate interception by anti-missile systems. However, due to the complex enemy-friendly confrontation environment, communication interference will be caused, and the limited capacity of the communication network will cause random delays and losses in the radar detection data of missile targets, resulting in large target tracking errors. In addition, according to the flight characteristics of ballistic missiles, the flight time of the active segment and the reentry segment is short and difficult to track, while the free segment accounts for about 80% of the trajectory. Therefore, it is of great significance to accurately track the free segment of ballistic missiles.

[0003] So far, many effective filtering techniques have been developed, including extended Kalman filtering (EKF), unscented Kalman filtering (UKF) based on unscented transform (UT), cubature Kalman filtering (CKF), Gauss-Hermitian filtering using quadrature rule, particle filtering, etc. A prerequisite for these filtering methods is that the measurement data can reach the filter without delay. When measurement delay or loss occurs, the performance of these algorithms will be greatly affected. In order to eliminate the problem of reduced estimation accuracy caused by measurement delay and loss, a large number of researchers have proposed various nonlinear filtering schemes. However, these methods are only applicable to measurement delays of one or two time steps and cannot be used for arbitrary delays. A small number of researchers have begun to study random multi-step delays, but they cannot solve the problem of measurement loss. Based on this, the present invention focuses on proposing a new Bayesian filter to deal with arbitrary delays and losses of measurements. Summary of the invention

[0004] The purpose of the present invention is to provide a ballistic missile free-segment tracking method based on a novel Bayesian filter to solve the above-mentioned problem of random multi-step delay and loss of radar detection data of missile targets due to communication interference and limited communication network capacity.

[0005] To achieve the above object, the present invention provides the following technical solution: a ballistic missile free segment tracking method based on a novel Bayesian filter, specifically comprising the following steps:

[0006] Step S1: Establish a new measurement model for random multi-step delay and loss of measurement data;

[0007] Step S2: By using the measurement model established in step S1, when random delay of measurement data occurs, the system state is augmented and a delay variable is introduced to design an augmented Gaussian mixture filter framework;

[0008] Step S3: Through the augmented Gaussian mixture filter framework designed in step S2, when the measurement value is lost, the measurement value is propagated from the previous moment to the current moment, and the expression of the mean and covariance is derived in the form of state augmentation;

[0009] Step S4: Based on the designed new Bayesian filtering method, the free segment of the ballistic missile is tracked according to the real-time state of the ballistic missile.

[0010] Preferably, in step S1, a new measurement model is established, which can simultaneously characterize the random multi-step delay and loss of the measurement data, and is expressed as:

[0011]

[0012] γ k is an independent Bernoulli random quantity with a value of 1 or 0; k =1 indicates that the filter receives a measurement data, i.e., y k =Z k-i , i = 0, 1, …, d, where d represents the maximum possible measurement data delay step.

[0013] Preferably, when a random step delay of the measurement data occurs, the system state is augmented in step S2;

[0014] When only random delay of measurement data occurs, the measurement equation is y = Z k-i , to augment the state, The augmented state and measurement equations are X k =F(X k-1 )+Bω k-1 , Z k =h(C i X k )+v k-i ;

[0015] in, B=[I n 0…0] T , C i X k =x k-i , C 0 =[I n 0… 0] T …C d =[0 0 … I n ] T .

[0016] Preferably, in step S2, a delay variable and its probability are introduced into the posterior probability density function of the system state;

[0017]

[0018] Among them, τ i It means that the filter receives the measurement value at time ki, and its probability is expressed as p(τ i ).

[0019] Preferably, in step S2, each item in the augmented state posterior probability density function is solved to obtain the estimated state and its covariance:

[0020] Step 1: Solve the state one-step prediction posterior probability density. The state one-step prediction posterior probability density can be decomposed into:

[0021]

[0022] Step 2: Calculate the likelihood function. According to the measurement equation after state augmentation, assuming that the measurement noise obeys Gaussian distribution, the likelihood function is:

[0023] p(Z k |X k ,τ i ,Z k-1 )≈N(Z k ; h(C i X k ),R k-i );

[0024] Step 3: Calculate the normalization coefficient and expand it into:

[0025]

[0026] So we can get the one-step forecast and its covariance:

[0027]

[0028] Step 4: Obtain state posterior probability density

[0029]

[0030] Preferably, on the basis of step S2, when the measurement value is lost, the measurement value is propagated from the previous moment to the current moment, and the mean and covariance expressions in step S3 are derived in the form of state augmentation; if the measurement value is not received at the current moment, it is compensated by using; according to the newly created measurement model, for the delayed system with measurement loss, the measurement one-step prediction can be written as:

[0031]

[0032] And the covariance in the state augmented form can be derived as:

[0033]

[0034] Compared with the prior art, the present invention has the following beneficial effects:

[0035] The present invention establishes a new measurement model, which uses a Bernoulli random variable to simultaneously characterize the random multi-step delay and loss of measurement data. When a random multi-step delay occurs during the detection process, the system state is augmented and a delay variable is introduced to marginalize the posterior probability density function, thereby capturing the relationship between the actual measurement and the delay probability, and obtaining a multi-step delay augmented Gaussian mixture filter framework. When the measurement value is lost, the measurement value of the previous moment is propagated to the current moment, and the state augmented expression of the mean and covariance is derived under the augmented Gaussian mixture filter framework. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 This is a random multi-step delay model diagram of the measurement data of the present invention.

[0037] Figure 2 It is the real trajectory and tracking trajectory diagram of the present invention.

[0038] Figure 3 This is the X-direction position tracking error diagram of the present invention.

[0039] Figure 4 This is the Y-direction position tracking error diagram of the present invention.

[0040] Figure 5 This is the Z-direction position tracking error diagram of the present invention.

[0041] Figure 6 This is a diagram showing the speed tracking effect in the X direction of the present invention.

[0042] Figure 7 This is a diagram showing the speed tracking effect in the Y direction of the present invention.

[0043] Figure 8 This is a diagram showing the speed tracking effect in the Z direction of the present invention. DETAILED DESCRIPTION

[0044] The following will be combined with the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention.

[0045] Please see attached Figure 1-8 The present invention provides a ballistic missile free segment tracking method based on a novel Bayesian filter, which specifically comprises the following steps:

[0046] Step S1: Establish a new measurement model for random multi-step delay and loss of measurement data;

[0047] Step S2: By using the measurement model established in step S1, when random delay of measurement data occurs, the system state is augmented and a delay variable is introduced to design an augmented Gaussian mixture filter framework;

[0048] Step S3: Through the augmented Gaussian mixture filter framework designed in step S2, when the measurement value is lost, the measurement value is propagated from the previous moment to the current moment, and the expression of the mean and covariance is derived in the form of state augmentation;

[0049] Step S4: Based on the designed new Bayesian filtering method, the free segment of the ballistic missile is tracked according to the real-time state of the ballistic missile.

[0050] In this embodiment, the free segment of the ballistic missile is considered to be only affected by gravity, and the influence of the earth's revolution and the influence of the gravitational perturbation of other celestial bodies are not considered. A target motion model is established in the geocentric inertial system, and the earth model adopts a standard ellipsoid model.

[0051] Select the position and velocity of the target in the geocentric inertial coordinate system as the state quantity:

[0052]

[0053] The state equation is:

[0054]

[0055] in, is the distance between the center of the earth and the target, R e is the earth's equatorial radius, J2 is the second-order principal spherical harmonic coefficient, w i is the measurement noise, μ=3.98604418×10 14 m 3 / s 2 Represents the Earth's gravitational constant.

[0056] In this embodiment, a ground-based radar is used to detect the target, and the measurement information includes the elevation angle, azimuth angle, and slant distance between the radar station and the target. The measurement equation is:

[0057]

[0058] In the formula is the transformation matrix from the geocentric inertial system to the radar system, [x r ,y r ,z r ] T is the coordinate of the radar in the geocentric system, and v is the measurement noise.

[0059] Discretize the state equation and measurement equation to obtain

[0060] x k =f(x k-1 )+ω k-1 (4)

[0061] z k =h(x k )+v k (5).

[0062] Furthermore, in step S1, a new measurement model is established, which can simultaneously characterize the random multi-step delay and loss of the measurement data, and is expressed as:

[0063]

[0064] γ k is an independent Bernoulli random quantity with a value of 1 or 0; k =1 indicates that the filter receives a measurement data, i.e., y k =Z k-i , i = 0, 1, …, d, where d represents the maximum possible measurement data delay step.

[0065] In this embodiment, it is assumed that the probability of the measured data being delayed by one time step is 0.6, the radar sampling frequency is set to T = 0.1s, the system noise equation is Q = diag(10,10,10,2,2,2), and the measurement noise variance is R = diag[100,(1 / 57.3) 2 ×10 -6 ,(1 / 57.3) 2 ×10 -6 ], the radar is located at 158°28′W and 30°15′N, and the missile’s free phase starts at a state x0=[-2.9169×10 6 4.2764×10 6 3.8831×10 6 -3423-909549].

[0066] Preferably, when a random step delay of the measurement data occurs, the system state is augmented in step S2;

[0067] When only random delay of measurement data occurs, the measurement equation is y = Z k-i , to augment the state, The augmented state and measurement equations are

[0068] X k =F(X k-1 )+Bωk-1 (7)

[0069] Z k =h(C i X k )+v k-i (8);

[0070] in, B=[I n 0 … 0] T , C i X k =x k-i , C 0 =[I n 0 … 0] T …C d =[0 0 … I n ] T .

[0071] Furthermore, based on Bayesian theory, a delay variable τ is introduced into the posterior probability density function of the system state in step S2. i and its probability p(τ i );

[0072]

[0073]

[0074] Among them, τ i It means that the filter receives the measurement value at time ki, and its probability is expressed as p(τ i ).

[0075] Preferably, in step S2, each item in the augmented state posterior probability density function is solved to obtain the estimated state and its covariance:

[0076] Step 1: Solve the state one-step prediction posterior probability density. The state one-step prediction posterior probability density can be decomposed into

[0077]

[0078] According to the augmented state equation (7), p(X k |X k-1 ,τ i ,Z k-1 )=N(X k ; F(X k-1 ),BQ k-1 B T ),but

[0079]

[0080] Step 2: Calculate the likelihood function. According to the measurement equation (8) after state augmentation, assuming that the measurement noise obeys Gaussian distribution, the likelihood function is:

[0081] p(Z k |X k ,τ i ,Z k-1 )≈N(Z k ; h(C i X k ),R k-i )(16);

[0082] Step 3: Calculate the normalization coefficient and expand it into:

[0083]

[0084] According to the above formulas (13) and (16), we can know that:

[0085]

[0086] So we can get the one-step forecast and its covariance:

[0087]

[0088] therefore

[0089]

[0090] Step 4: Estimate the state; according to the above calculations for each part, we have

[0091]

[0092] in:

[0093]

[0094] Substituting formulas (24) and (23) into (8), we get the posterior probability density function of the augmented state as a Gaussian mixture:

[0095]

[0096] Further, based on step S2, when the measurement value is lost, the measurement value is propagated from the previous moment to the current moment, and the mean and covariance expressions in step S3 are derived in the form of state augmentation;

[0097] If no measurement value is received at the current time, use y k-1 To compensate for k According to the newly established measurement model, for a delayed system with measurement loss, the measurement one-step prediction can be written as:

[0098]

[0099] And the covariance in the state augmented form can be derived as:

[0100]

[0101] Furthermore, the tracking trajectory of the free phase of the ballistic missile is obtained as follows: Figure 2 As shown in the figure, it can be seen that the present invention has good tracking performance in the case of random delay and loss of measurement. The position tracking error in each direction and the speed tracking effect in each direction are shown in the figure. Figure 3-8 As shown, it is illustrated that the present invention has a higher estimation accuracy.

[0102] Although the present invention has been described in detail with reference to the aforementioned embodiments, it is still possible for those skilled in the art to modify the technical solutions described in the aforementioned embodiments, or to make equivalent substitutions for some of the technical features therein. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A ballistic missile free-range tracking method based on a novel Bayesian filter, characterized in that: The specific steps include: Step S1: Establish a new measurement model for random multi-step delay and loss of measurement data; Step S2: Using the measurement model established in step S1, when random delays in measurement data occur, the system state is augmented and a delay variable is introduced to design an augmented Gaussian mixture filter framework; Step S3: Using the augmented Gaussian mixture filter framework designed in step S2, when the measurement value is lost, the measurement value is propagated from the previous moment to the current moment, and the expressions of the mean and covariance are derived in the form of state augmentation; Step S4: Based on the designed new Bayesian filtering method, the free segment of the ballistic missile is tracked according to the real-time status of the ballistic missile.

2. A ballistic missile free-segment tracking method based on a novel Bayesian filter according to claim 1, characterized in that: In step S1, a new measurement model is established, which can simultaneously characterize the random multi-step delay and loss of measurement data, expressed as: γ k is an independent Bernoulli random quantity with the value of 1 or 0; γ k =1 indicates that the filter receives a measurement data, i.e. y k =Z k-i , i=0,1,...,d, where d represents the maximum possible measurement data delay step.

3. The method for tracking a ballistic missile free segment based on a novel Bayesian filter according to claim 1, characterized in that: When a random step delay of the measurement data occurs, the system state is augmented in step S2; When only random delay of measurement data occurs, the measurement equation is y=Z k-i , augment the state, The augmented state and measurement equations are X k =F(X k-1 )+Bω k-1 , Z k =h(C i X k )+v k-i ; among them, B=[I n 0…0] T 、C i X k =x k-i 、C 0 =[I n 0…0] T …C d =[0 0…I n ] T 。 4. The method for tracking a ballistic missile free segment based on a novel Bayesian filter according to claim 1, characterized in that: In step S2, the delay variable and its probability are introduced into the posterior probability density function of the system state; Among them, τ i It means that the filter receives the measurement value at time ki, and its probability is expressed as p(τ i ).

5. The method for tracking a ballistic missile free segment based on a novel Bayesian filter according to claim 1, characterized in that: In step S2, each item in the augmented state posterior probability density function is solved to obtain the estimated state and its covariance: Step 1: Solve the state one-step prediction posterior probability density, the state one-step prediction posterior probability density can be decomposed into: Step 2: Calculate the likelihood function. According to the measurement equation after state augmentation, assuming that the measurement noise obeys Gaussian distribution, the likelihood function is: p(Z k |X k ,τ i ,Z k-1 )≈N(Z k ;h(C i X k ),R k-i ); Step 3: Calculate the normalization coefficient and expand it into: So we can get the one-step forecast and its covariance: Step 4: Get the state posterior probability density:

6. The method for tracking a ballistic missile free segment based on a novel Bayesian filter according to claim 1, characterized in that: On the basis of step S2, when the measurement value is lost, the measurement value is propagated from the previous moment to the current moment, and the mean and covariance expressions in step S3 are derived in the form of state augmentation; If no measurement value is received at the current moment, it is compensated by using ; according to the newly established measurement model, for a delayed system with measurement loss, the measurement one-step prediction can be written as: And the covariance in the state augmented form can be derived as:

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