Passive sensor track optimization method based on fisher information accumulation

Abstract

The application provides a passive sensor track optimization method based on Fisher information accumulation, comprising the following steps: S1, constructing a joint variational confidence lower bound of a target state estimation value, an estimation error covariance and a heading angle; S2, constructing an optimization function with a weighted KLD as a regularization term in combination with a regularization method; S3, linearizing the optimization function; S4, calculating the partial derivative of the linearized optimization function with respect to the state estimation and the estimation error covariance, and obtaining the optimal heading angle of the sensor platform, the state estimation value and the estimation error covariance. Under the condition of the maximum variational confidence lower bound, the application mines effective measurement information to improve the observability of the system, constructs a joint optimization function of the heading angle of the sensor platform, the target state estimation and the estimation error covariance with the KLD divergence as a regularization term, makes the variational distribution approach the nonlinear state posterior distribution of the target, realizes high-precision positioning and tracking of the target, and improves the robustness of the state estimation.

Description

Technical Field

[0001] The present invention relates to the technical field of passive positioning and tracking, and in particular to a passive sensor track optimization method based on Fisher information accumulation. Background Art

[0002] Passive target positioning and tracking involves estimating variables such as the relative position and velocity between a target and a sensor, given a known measurement angle (or Doppler shift frequency). This approach has attracted widespread attention due to its strong anti-interference capabilities, excellent concealment, and long range. The use of passive sensors such as infrared and Doppler radar for navigation and target tracking holds significant theoretical and engineering application value.

[0003] However, passive sensor measurement systems typically cannot obtain the relative position information between the target and the sensor, resulting in incomplete or even unobservable systems. For example, in a single-sensor angle-only target tracking system, the fixed-position sensor lacks observability because it can only obtain angular information. This results in a non-unique solution for target state estimation, making effective target tracking impossible.

[0004] To meet the observability requirements of passive target tracking systems under single-sensor measurement conditions, the observability of passive target tracking systems typically requires that the sensor platform's track have at least one higher-order nonzero derivative greater than that of the target. For example, in a uniform linear motion target tracking system, the sensor platform must have at least acceleration information so that the target and sensor platform satisfy a triangulation relationship at different times, thereby making the system observable. Because system observability depends on the relative geometry of the sensor and target, the sensor's track directly affects the system's estimation performance. Based on the above analysis, exploring the optimal path of the sensor platform to maximize system observability and obtain more effective information will help improve target positioning and tracking accuracy under passive measurement conditions.

[0005] Research on this type of problem currently focuses on the following two areas: 1) Reward-based sensor trajectory optimization methods calculate the cumulative reward of possible sensor paths and use maximizing this cumulative reward as a constraint for sensor trajectory optimization. However, the accuracy of these target tracking systems is often limited by the choice of reward function. 2) Sensor trajectory optimization problems are modeled as stochastic control problems involving partially observable Markov decision processes. Observability is achieved by adjusting relevant sensor parameters (such as heading angle) in real time under a certain cost function or optimization criterion. These criteria are typically related to the measured Fisher information or mutual information, and the optimization process can be interpreted as solving the Euler-Lagrange equation. Dynamic programming methods are generally used to optimize the optimal implementation of sensor maneuver strategies. However, the theoretically optimal path involves numerous decision-making processes, requiring significant computational power on the hardware. Furthermore, these two approaches employ a "divide-and-conquer" approach to sensor trajectory optimization and target state estimation, failing to integrate sensor trajectory optimization and state estimation into a joint optimization framework.

[0006] Therefore, how to jointly optimize the sensor track and target state under passive measurement conditions to achieve the observability of the target system and further improve the target positioning and tracking accuracy and robustness has practical engineering significance. Summary of the Invention

[0007] In view of this, an embodiment of the present application provides a passive sensor track optimization method based on Fisher information accumulation, so as to achieve the purpose of maximizing the Fisher information matrix (FIM) and the observability of the system by optimizing the sensor track in a passive target tracking system.

[0008] The present application provides the following technical solution: a passive sensor track optimization method based on Fisher information accumulation, comprising the following steps:

[0009] S1. Construct a joint variational confidence lower bound for the target state estimate, the estimated error covariance, and the heading angle;

[0010] S2. Combine regularization methods to construct an optimization function with weighted KLD as the regularization term;

[0011] S3 linearizes the optimization function;

[0012] S4. Calculate the partial derivatives of the linearized optimization function with respect to the state estimate and the estimation error covariance to obtain the optimal heading angle, state estimate value, and estimation error covariance of the sensor platform.

[0013] According to an embodiment of the present application, step S1 specifically includes:

[0014] S101. In the variational Bayesian inference framework, the sensor platform heading angle θ k As an unknown variable, the variational hyperparameter ψ is constructed under the premise that the sensor platform rate remains unchanged. k =(x k|k ,P k|k ,θ k );

[0015] S102. Construct a Gaussian variational distribution q(x k |ψ k ), and obtain the optimal variational parameters

[0016] Among them, θ k is the heading angle of the sensor platform, x k|k is the target state estimate, P k|k is the estimated error covariance, ψ k is the variational hyperparameter, x k is the target state vector, is the optimal variational parameter.

[0017] According to an embodiment of the present application, step S2 specifically includes:

[0018] S201. Construct a weighted KLD between two iterative variational distributions as a regularization term;

[0019] S202. Construct a weighted expression for regular terms.

[0020] According to an embodiment of the present application, step S3 specifically includes:

[0021] S301. Calculate the log-likelihood expectation E during the i-th iteration q [logp(z k |x k ,θ k )] About state estimation Gradient and the estimated error covariance Gradient

[0022] S302. Calculate the analytical expression of KLD under the Gaussian assumption.

[0023] According to an embodiment of the present application, step S4 specifically includes:

[0024] S401. Calculate the linearized optimization function in step S3 with respect to x k FIM of each element

[0025] S402. Maximize the FIM Calculate the optimal heading angle

[0026] S403. Based on the optimal heading angle Calculate the target state estimate during the i+1th iteration and the estimated error covariance

[0027] Among them, FIM represents the effective measurement z k Contains the target state x k The amount of information.

[0028] According to an embodiment of the present application, in step S401, the FIM of the i+1th iteration is expressed as:

[0029]

[0030] in, and represent the FIM, filter gain and state prediction error covariance after the i-th variational iteration respectively.

[0031] According to an embodiment of the present application, in step S402, the maximum change in the heading angle of the sensor platform at time k is set to That is, the constraints are, The optimal heading angle of the sensor platform in the i+1th variational iteration is Expressed as:

[0032]

[0033] The present invention relates to a passive sensor track optimization method based on Fisher information accumulation. Under the condition of maximizing the variational confidence lower bound, the sensor platform track is optimized by calculating Fisher information accumulation, and effective measurement information is mined to the maximum extent to improve the observability of the system. At the same time, using the Kullback-Leibler (KL divergence Kullback–Leiblerdivergence, abbreviated as KLD) divergence as a regularization term, a joint optimization function of the sensor platform heading angle, target state estimation, and estimation error covariance is constructed, so that the variational distribution gradually approaches the nonlinear state posterior distribution of the target, thereby achieving high-precision positioning and tracking of the target. In addition, this regularization method can improve the robustness of state estimation. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] In order to more clearly illustrate the technical solutions of the embodiments of the present application, the following briefly introduces the drawings required for use in the embodiments. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without any creative work.

[0035] Figure 1 Flowchart of the passive sensor track optimization method according to an embodiment of the present invention;

[0036] Figure 2 Pure angle passive target tracking simulation scenario in an embodiment of the present invention;

[0037] Figure 3 Measurements of sensors and fixed track sensors for the algorithm VBAKF-KLD in an embodiment of the present invention;

[0038] Figure 4 Normalized ELBO of the VBAKF-KLD algorithm in an embodiment of the present invention;

[0039] Figure 5 KLD of the VBAKF-KLD algorithm in the embodiment of the present invention;

[0040] Figure 6 The RMSE of radial distance estimation of the VBAKF-KLD algorithm and the comparison algorithm in the embodiment of the present invention;

[0041] Figure 7 RMSE of radial velocity estimation of the VBAKF-KLD algorithm in the embodiment of the present invention and the comparison algorithm. DETAILED DESCRIPTION

[0042] The embodiments of the present application are described in detail below with reference to the accompanying drawings.

[0043] It should be noted that, in the absence of conflict, the embodiments and features in the embodiments of this application can be combined with each other. The present invention will be described in detail below with reference to the accompanying drawings and in combination with the embodiments, and the technical solutions of the present invention will be clearly and completely described. Obviously, the embodiments described 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.

[0044] like Figure 1As shown, the present invention provides a passive sensor trajectory optimization (VBAKF-KLD) method based on Fisher information accumulation. First, under the variational Bayesian optimization framework, the present invention constructs a variational confidence lower bound with the heading angle, state estimate, and its estimated error covariance as unknown parameters to build a bridge between the heading angle and FIM, and obtains the optimal solution for the heading angle when the FIM accumulation is maximized. At the same time, combined with the regularization method, an optimization function with Kullback-Leibler divergence (KLD) as the regularization term is constructed. By performing a second-order approximation on the optimization function, the partial derivative with respect to the state estimate and its estimated error covariance is calculated, thereby obtaining the optimal heading angle, state estimate, and estimated error covariance of the sensor.

[0045] In one embodiment of the present invention, the passive sensor track optimization method comprises the following steps:

[0046] S1. Construct a joint variational confidence lower bound (ELBO) on the target state estimate, the estimation error covariance, and the heading angle.

[0047] Assume that at time k the sensor observes the state vector of the platform in, and Represents the position vector and velocity vector of the observation platform on the x-axis and y-axis respectively. The sensor platform moves at a constant rate Movement, which adjusts the heading angle θk to make itself maneuverable, so that the system can reach the best position with observability and obtain effective measurement of the target z k , thereby achieving the target state The estimate of [x k y k ] T and Represents the position vector and velocity vector of the target on the x-axis and y-axis respectively. In the pure angle passive tracking system, the sensor measurement equation is expressed as:

[0048]

[0049] In the variational Bayesian inference framework, the sensor platform heading angle θk is taken as the unknown variable, and the sensor platform speed is kept constant. k , target state estimation xk|k and its estimation error covariance P k|k The joint variational ELBO transforms the above passive sensor trajectory optimization problem into a variational parameter ψ k =(x k|k ,P k|k ,θ k), the optimization of the sensor platform heading angle and the effective estimation of the target state are achieved by maximizing the ELBO, where θ k 、x k|k and P k|k are the hyperparameters of the variational confidence lower bound. Based on this, the variational ELBO for the passive target tracking problem is expressed as

[0050] L(ψ k )=Ε q [logp(z k |x k ,θ k )]-D KL [q(x k |ψ k )||p(x k )]

[0051] Among them, p(z k |x k ,θ k ) represents the measurement likelihood of the sensor, E q [logp(z k |x k ,θ k )] represents the logarithmic likelihood expectation, D KL [q(x k |ψ k )||p(x k )] represents the variational distribution q(x k |ψ k ) and the prior distribution p(x k ). In the iterative recursive estimation process, p(x k ) can be approximately expressed as the probability density function p(x k )≈p(x k |x k|k-1 ,P k|k-1 ) where x k|k-1 and P k|k-1 denote the prediction of the target state and the prediction error covariance respectively. Therefore, the optimal variational parameter It can be expressed as:

[0052]

[0053] S2. Combine regularization methods to construct an optimization function with weighted KLD as the regularization term;

[0054] (1) Construct a weighted KLD between two iterative variational distributions as a regularization term;

[0055] On the basis of satisfying the above-mentioned system observability, the filtering process in passive tracking requires the estimator to have high robustness. The regularization method is used in the variational Bayesian framework to improve the robustness of nonlinear estimation. Therefore, under the ELBO maximization condition, the weighted KLD is used as the regularization term to construct the optimization function of the state estimation and its error covariance:

[0056]

[0057] in, and Represent the state estimation and estimation error covariance in the i+1th iteration process, represents the KLD between adjacent iterative variation distributions during the iteration process, Represents the weighted parameter for adjusting the optimization step size, by adjusting The size of controls the effect of KLD in the optimization process.

[0058] (2) Constructing the expression of weighted regular terms

[0059] The value of is expressed as:

[0060]

[0061] Among them, l k represents the measurement likelihood at the kth moment, represents the determinant of the FIM of the i-th iteration. In the above regularized iterative optimization process, as the variational distribution approaches the posterior distribution, L(ψ k ) increases continuously, the regularization term Gradually decreases, that is, the KL divergence between adjacent iterative variational distributions continues to decrease.

[0062] S3 linearizes the optimization function;

[0063] (1) Calculate E separately q [logp(z k |x k ,θ k )] About the i-th iteration estimate and gradient.

[0064]

[0065]

[0066] in, represents the Jacobian matrix of the measurement function during the i-th iteration, R k represents the measurement noise variance.

[0067] (2) Calculate KL divergence.

[0068] Under the Gaussian assumption, KL divergence has an analytical expression, namely

[0069]

[0070] Where d represents x k The dimension of .

[0071] Similarly, calculate KL divergence The analytical expression of .

[0072] S4. Calculate the partial derivatives of the linearized optimization function with respect to the state estimation and the estimation error covariance to obtain the optimal heading angle of the sensor platform The estimated state value during the i+1th iteration and the estimated error covariance

[0073] (1) Compute the FIM of ELBO with respect to xk.

[0074] In the target tracking system, FIM represents the amount of information about the target state xk contained in the measurement zk. It has a non-negative property. The greater the amount of information, the more conducive it is to the effective estimation of the target state. Therefore, from the perspective of information theory, it can be intuitively understood as improving the observability of the system by increasing the Fisher information. The various expressions of FIM information increment are ELBO about The expectation of the second-order partial derivative of each element, due to the ELBO The second derivative of the velocity component is 0, and Therefore, the FIM information increment is expressed as:

[0075]

[0076] Among them, the vector represents the relative distance between the target and the sensor platform, K k Represents the filter gain.

[0077] Therefore, the FIM at time k is:

[0078] J k =J k-1 +ΔJ k

[0079] Among them, J k-1 represents the FIM at time k-1. Multiple variational iterations can obtain better heading angle and state estimation accuracy. Accordingly, the FIM at the i+1th iteration is expressed as:

[0080]

[0081] in, and represent the FIM, filter gain and state prediction error covariance after the i-th variational iteration respectively.

[0082] (2) Calculate the optimal heading angle of the sensor platform.

[0083] Maximization of FIM is approximately equivalent to maximization of its determinant. Then the optimization of FIM cumulative heading angle under the variational ELBO maximization condition is expressed as:

[0084]

[0085] At the same time, considering the maneuverability of the sensor platform, the maximum change of the heading angle of the sensor platform at time k is set to That is, the constraints are, Then the optimal heading angle of the sensor platform in the i+1th variational iteration is expressed as:

[0086]

[0087] Combined with the variational Bayesian iterative method, the principle of target state estimation and optimal heading angle optimization is as follows: under the condition of given k-1 time state estimation value, ELBO maximization is used as the criterion to derive ELBO about the measurement platform state vector The FIM accumulation is used to obtain the optimal heading angle of the measurement platform at the current moment. Further get the current measurement value z k , based on nonlinear filtering, a high-precision estimation of the passive target state at the current moment is achieved.

[0088] like Figure 2-Figure 7 As shown, Figure 2 The middle curves represent the track of the sensor, fixed track sensor, and target of the proposed optimization algorithm VBAKF-KLD, respectively. It can be seen from the figure that the sensor of the proposed algorithm can adaptively adjust its position according to the target position to achieve the best system observability. Figure 3 The figures represent the measurements of the sensor based on the proposed algorithm VBAKF-KLD and the fixed track sensor, respectively. It can be seen from the figure that the sensor measurements based on the proposed algorithm have a high frequency of change, further demonstrating that the proposed algorithm can improve the observability of the system. Figure 4 It represents the normalized ELBO of the proposed algorithm VBAKF-KLD. It can be seen from the figure that ELBO gradually increases with the increase of the number of iterations, which shows that in the VBAKF-KLD algorithm, the sensor heading angle, target state estimation value and its estimation error covariance are continuously optimized as the iteration proceeds. Figure 5 The KLD (Kullback-Leibler divergense) of the proposed algorithm VBAKF-KLD is shown in the figure. As can be seen from the figure, KLD decreases gradually with the increase of the number of iterations. Since KLD and ELBO are negatively correlated, Figure 5 Further verified Figure 4 conclusion. Figure 6 It represents the RMSE of radial distance estimation of the proposed algorithm VBAKF-KLD compared with the algorithm. RMSE is a performance indicator of the target state estimation accuracy. Figure 3 ,It can be seen that the RMSE decreases when the sensor ,measurement changes, which improves the radial distance estimation ,accuracy., when the sensor makes a large turn at 20 min, the RMSE decreases ,dramatically and converges, and the proposed VBAKF-KLD ,algorithm has the highest estimation accuracy. Figure 7 Represents the RMSE of radial velocity estimation of the proposed algorithm VBAKF-KLD compared with the algorithm. Figure 7 The RMSE curve changes and Figure 6 With similar characteristics, that is, as the optimized RMSE curve of the sensor track gradually decreases, the RMSE gradually decreases and converges, and the proposed VBAKF-KLD algorithm has the highest estimation accuracy.

[0089] The present invention makes the passive sensor tracking system observable by maximizing FIM accumulation, providing a new idea for sensor track optimization. It has theoretical reference significance for engineering applications on how to achieve observability of the target system under passive measurement conditions and further improve the accuracy and robustness of target positioning and tracking.

[0090] The above description is merely a specific embodiment of the present application, but the scope of protection of the present application is not limited thereto. Any changes or substitutions that can be easily conceived by a person skilled in the art within the technical scope disclosed in the present application should be included in the scope of protection of the present application. Therefore, the scope of protection of the present application should be based on the scope of protection of the claims.

Claims

1. A passive sensor track optimization method based on Fisher information accumulation, characterized in that: The following steps are involved: S1. Construct a joint variational confidence lower bound for the target state estimate, the estimated error covariance, and the heading angle; Step S1 specifically includes: S101. In the variational Bayesian inference framework, the sensor platform heading angle As an unknown variable, the variational hyperparameters are constructed under the premise that the sensor platform rate remains unchanged. ; S102. Constructing a Gaussian variational distribution , and obtain the optimal variational parameters ; in, is the sensor platform heading angle, is the target state estimate, is the estimated error covariance, is the variational hyperparameter, is the target state vector, is the optimal variational parameter; The joint variational confidence lower bound is expressed as: in, represents the measurement likelihood of the sensor, represents the expected logarithmic likelihood, Represents the variational distribution With prior distribution KLD between; in the iterative recursive estimation process, It can be approximately expressed as the probability density function of the target state prediction ,in, and denote the prediction and prediction error covariance of the target state, respectively; S2. Combine regularization methods to construct an optimization function with weighted KLD as the regularization term; Step S2 specifically includes: S201. Construct a weighted KLD between two iterative variational distributions as a regularization term; S202. Construct a weighted expression for the regularization term; S3 linearizes the optimization function; S4. Calculate the partial derivatives of the linearized optimization function with respect to the state estimate and the estimation error covariance to obtain the optimal heading angle, state estimate, and estimation error covariance of the sensor platform; Step S4 specifically includes: S401. Calculate the linearized optimization function in step S3 with respect to FIM of each element ; In step S401, The FIM of iterations is expressed as: in, 、 and Respectively represent FIM, filter gain and state prediction error covariance after variational iteration; S402. Maximize the FIM Calculate the optimal heading angle ; S403. Based on the optimal heading angle , calculate the The target state estimation value in the iteration and the estimated error covariance ; Among them, FIM represents effective measurement Included target status The amount of information.

2. The passive sensor track optimization method based on Fisher information accumulation according to claim 1, characterized in that: Step S3 specifically includes: S301. Calculate the The log-likelihood expectation during the iteration About state estimates Gradient and the estimated error covariance Gradient ; S302. Calculate the analytical expression of KLD under the Gaussian assumption.

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