CN109582915B - Improved nonlinear observability self-adaptive filtering method applied to pure azimuth tracking - Google Patents
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Abstract
The invention relates to an improved nonlinear observability degree self-adaptive filtering method applied to pure azimuth tracking, and belongs to the field of observability degree self-adaptive filtering. In order to improve the pure azimuth tracking filtering performance and perfect the observability theory of a nonlinear system, and meanwhile, the pure azimuth system can be roughly evaluated on the filtering precision of the system before filtering, the invention designs and establishes a new nonlinear observability calculation scheme, and observation noise is taken into consideration; the invention designs the self-adaptive adjustment factor based on the observability, adds the adjustment factor into the nonlinear filter framework, judges the self-adaptive filter effect in that way by comparing the magnitudes of error covariance under different conditions, can further improve the precision of the self-adaptive filter based on the nonlinear observability, and ensures that the filter performance of pure azimuth tracking is improved, and further the pure azimuth tracking has better performance.
Description
Technical Field
The invention relates to an improved nonlinear observability degree self-adaptive filtering method applied to pure azimuth tracking, and belongs to the field of observability degree self-adaptive filtering.
Background
Pure azimuth tracking is a common technology in the field of target tracking, and is often estimated by adopting a Kalman filtering method, so as to eliminate errors in tracking. Because the state space model of the pure azimuth tracking system is mostly a nonlinear system in practical application, the strong pure azimuth tracking nonlinearity leads to large linearization error of the nonlinear system, namely unscented Kalman filtering and volumetric Kalman filtering are commonly adopted.
According to the modern control theory developed by classical control theory, two important properties of observability and controllability are proposed when Kalman is proposed by Kalman filtering, and then a plurality of scholars propose the discrimination theorem of observability and controllability, because the accuracy of the filtering depends on the observability to a certain extent, if an actual system is observable, the filtering is not caused to diverge, and if the system is not observable, the divergence of a filtering result is caused. However, in modern control theory developed to the present, no clear standard is available to measure the degree of observability of a system, i.e. the magnitude of the observability, and especially for nonlinear systems, the observability theory developed more difficultly, and the related data about the nonlinear observability theory is less and the theory is not perfect enough. The effect of external noise on it is great for kalman filtering, as well as for the observability. However, noise is not considered in current nonlinear observability studies, which tends to affect the evaluation of the observability.
When the observability of the pure azimuth target tracking system is evaluated, the high observability is high, the filtering precision is high, the tracking performance of the target is better, otherwise, the low observability of the pure azimuth target tracking system is low, the filtering precision is low, and the tracking performance of the target is poor. The invention aims to improve the volume Kalman filtering on the basis of improving the nonlinear observability, and can calculate the observability before filtering and evaluate the system accuracy before filtering of the pure azimuth tracking system. And some researchers put forward that the observability and noise have larger relativity, the invention considers the observed noise in the nonlinear observability calculation, and perfects the calculation scheme of the nonlinear observability. The method has great promotion to pure azimuth tracking filtering.
Disclosure of Invention
In order to improve the pure azimuth tracking filtering performance and perfect the observability theory of a nonlinear system, and meanwhile, the pure azimuth system can be roughly evaluated on the filtering precision of the system before filtering, the invention designs and establishes a new nonlinear observability calculation scheme, and noise is taken into consideration; the invention designs the self-adaptive adjusting factor based on the observability, and adds the adjusting factor into the nonlinear filtering framework, so that the filtering performance of pure azimuth tracking is improved, and the performance of pure azimuth tracking is improved.
The method specifically comprises the following steps:
step (1) a traditional Li Daoshu-based observability calculation method: setting a nonlinear system model as follows:
from differential geometry theory, h is known to be:
simultaneous definition ofConstruction of observational observation space for studying nonlinear systems +.>Is a space generated by the following expression +.>Definitions->Spatial observability distribution, +.>If dimdH (x) 0 ) =n, i.e. full rank can be determined that the system is observable; defining an observability calculation matrix based on the method as follows: />The method has the observability degree ofHere delta min (Ω),δ max (Ω) refer to the minimum and maximum singular values of the observability calculation matrix, respectively; in the above expression dim represents rank of the matrix variable,/for>Representation->Partial derivatives are obtained on the variable x;the representation performs transposition operation on the matrix, and span is a representation symbol representing a state space;
step (2) proposes an improved Li Daoshu-based observability calculation scheme: taking the observed noise into account in the observability calculation in the description of step (1) to give a variance R k Defining a new observability calculation matrix at the moment k as follows:
i.e. if there is rank (Q) k ) =n, rank represents the rank of the matrix; the system is observable and defines an improved observability calculation scheme:
step (3) establishing an adaptive adjustment factor based on the observability degree of the improved nonlinear system: the adaptive adjustment factors are established here as:l max refers to the maximum observable value of the state component of each system Refer to the observability level at time k,/->n values are all state numbers, det represents determinant of the matrix.
Step (4) application background: according to the pure azimuth target tracking system model, a state equation is a linear system, and the state model equation is set as follows:c in the equation t Representing a state transition matrix>State variables representing the system, expressed as system variables x t Is the derivative of w t Representing process noise; discretizing the shape model to obtain: x is x k ＝C k x k-1 +w k,k-1 Here x k Is a discretized system state variable, x k ＝[x k v xk y k v yk ] T ,x k Representing the position in the x-direction, v xk Is the velocity in the x direction, y k Is the position in the y direction, v yk Is the velocity in the y direction; c (C) k The discretized state transition matrix has the following values:w k,k-1 for discretized system noise, its variance matrix is +.>
Step (5) setting an observation equation as y according to the pure azimuth target tracking model t ＝h t (x t )+υ t ，y t Is the observed object variable, h t (x t ) Concerning x t Is a nonlinear function of v t Is observation noise; discretizing to obtain y k ＝h k (x k )+v k ，y k ,v k Is y t ,υ t Discretized result, where v k The variance matrix is r=diag ([ 1, 1)]) The method comprises the steps of carrying out a first treatment on the surface of the The following d is a constant; the nonlinear equation is:
step (6) establishing an adaptive filter based on improved nonlinear observability of an application background model: the volume Kalman filtering is adopted for self-adaptive filtering:
(a) Calculating volume points:
(b) Volume point propagated through state equation:
(c) State prediction:
(d) Covariance prediction:
(e) Calculating volume points under measurement update:
(f) Volume point propagated through measurement equation: y is m,k|k-1 ＝h(x m,k|k-1 ) (10)
(g) Measurement prediction equation:
(h) Information covariance estimation:
(i) Cross covariance estimation:
(j) Calculating a filter gain:
(k) Meter with a meter bodyCalculating a filter gain based on a nonlinear observability degree:
(l) And (5) updating the state:
(m) covariance update:
the following explanation is made on the parameters related to step (6): d, d l Refers to the number of volume points of the volume Kalman filter, τ m The point of the volume is indicated,the representation being a state variable x k|k-1 Is the same as>Also representing an estimate of the measured value; s is S k-1 Refers to S k-1 ＝chol(P k-1 ) Chol represents cholesky decomposition, which corresponds to the evolution of the matrix, ω m Is the corresponding weight; p (P) yy,k|k-1 Representing information autocovariance, P xy,k|k-1 Representing cross covariance, ++>Is a filtering gain with an adaptive factor;
step (7) adding the adjustment factors to different stages of the filter frame and comparing:
(a) Adding the adjustment factor to the metrology prediction stage:
(b) Calculating an estimated innovation covariance:
(c) Calculating cross covariance:
(d) Calculating a new filter gain:
(e) Calculating a new state update:
(f) Estimating covariance:
comparing the error magnitudes of the step (6) and the step (7), namely comparing P k And (3) withSelecting the adaptive filtering variance with small error covariance as a final scheme of the final adaptive filtering based on nonlinear observability; the letter superscript 1 in this step represents the variable that results when the adaptation factor is added to the measurement phase.
The invention has the beneficial effects that: in order to improve the pure azimuth tracking filtering performance and perfect the observability theory of a nonlinear system, and meanwhile, the pure azimuth system can be roughly evaluated on the filtering precision of the system before filtering, the invention designs and establishes a new nonlinear observability calculation scheme, and noise is taken into consideration; the invention designs the self-adaptive adjusting factor based on the observability, and the adjusting factor is added into the nonlinear filtering framework, so that the filtering performance of pure azimuth tracking can be improved, and the performance of pure azimuth tracking is improved.
Drawings
Fig. 1: the flow chart of the invention.
Detailed Description
The invention provides an improved nonlinear observability self-adaptive filtering method applied to pure azimuth tracking, which is shown in a flow chart in figure 1 and comprises the following steps:
(1) Traditional methods for calculating observability based on Li Daoshu: setting a nonlinear system model as follows:
from differential geometry theory, h is known to be:
simultaneous definition ofConstruction of observational observation space for studying nonlinear systems +.>Is a space generated by the following expression +.>Definitions->Spatial observability distribution, +.>If dimdH (x) 0 ) =n, i.e. full rank can be determined that the system is observable; defining an observability calculation matrix based on the method as follows: />The method has the observability degree ofHere delta min (Ω),δ max (omega) means respectivelyCalculating the minimum singular value and the maximum singular value of the matrix by the observation degree; in the above expression dim represents rank of the matrix variable,/for>Representation->The partial derivative is taken of the variable x,the representation performs transposition operation on the matrix, and span is a representation symbol representing a state space;
the following explanation of step (1) is now made for ease of understanding: in general, the observability is calculated by first constructing an observability matrix of the system, wherein the observability matrix is formed by a state transition matrix and an observation matrix in a linear system. Unlike linear system, the state transition matrix and the observation matrix of the nonlinear system cannot be written directly, so that other methods are needed to construct the observation matrix of the nonlinear system, and the currently commonly used construction methods of the nonlinear observable matrix include Li Daoshu method, pseudo-observation matrix and pseudo-state transition matrix method. The condition number of the matrix with the maximum singular value and the minimum singular value can reflect the condition of the matrix to a certain extent, so as to represent the observability.
(2) An improved Li Daoshu-based observability calculation scheme was proposed: taking the observed noise into account in the observability calculation in the description of step (1) to give a variance R k Defining a new observability calculation matrix at the moment k as follows:
i.e. if there is rank (Q) k ) =n, rank represents the rank of the matrix; the system is observable and defines an improved observability calculation scheme:
the following description will be made for the step (2) for the convenience of understanding: this step proposes an improved calculation scheme of the non-linear observability, and adds the measurement noise to the calculation of the observability matrix on the basis of step 1, and since the observability is related to the filtering accuracy, which in turn is related to the noise outside the system, it is necessary to consider the noise in the calculation of the non-linear observability. The multiplication operation of the matrix can be written into a form of summation and addition of all elements of the matrix, so that the matrix multiplication is converted into a form of summation of the elements of the matrix in the step 2, and the processing is convenient for operation.
(3) Establishing an adaptive adjustment factor based on the observability degree of the improved nonlinear system: the adaptive adjustment factors are established here as:l max refers to the maximum observable value of the state component of each system Refer to the observability level at time k,/->n values are all state numbers, det represents determinant of the matrix.
The following description is made for step (3): in the volume Kalman filtering, under the condition that the filtering inaccuracy error is large due to factors such as inaccurate parameters, an adaptive adjustment factor method is often adopted to increase the filtering precision, and the nonlinear observability is innovatively combined with the adaptive adjustment factor in the step (3). An adaptive adjustment factor based on a nonlinear observability is established, and nonlinear filtering is improved by using the adjustment factor.
(4) Application background: tracking system model according to pure azimuth targetThe state equation is a linear system, and the state model equation is set as follows:c in the equation t Representing a state transition matrix>State variables representing the system, expressed as system variables x t Is the derivative of w t Representing process noise; discretizing the shape model to obtain: x is x k ＝C k x k-1 +w k,k-1 Here x k Is a discretized system state variable, x k ＝[x k v xk y k v yk ] T ,x k Representing the position in the x-direction, v xk Is the velocity in the x direction, y k Is the position in the y direction, v yk Is the velocity in the y direction; c (C) k The discretized state transition matrix has the following values:w k,k-1 for discretized system noise, its variance matrix is +.>
(5) According to the pure azimuth target tracking model, setting an observation equation as y t ＝h t (x t )+υ t ，y t Is the observed object variable, h t (x t ) Concerning x t Is a nonlinear function of v t Is observation noise; discretizing to obtain y k ＝h k (x k )+v k ，y k ,v k Is y t ,υ t Discretized result, where v k The variance matrix is r=diag ([ 1, 1)]) The method comprises the steps of carrying out a first treatment on the surface of the The following d is a constant; the nonlinear equation is:
the following description is made for steps (4) and (5): the adaptive filtering based on nonlinear observability is established in order to make the performance of pure azimuth tracking more accurate. As described above, the state equation is linear, and the statistical properties of the noise of the state model and the noise of the observation model conform to gaussian distribution, and the state equation and the observation equation need to be discretized and processed because the volume kalman filter based on the kalman filter frame is specific to a discrete system.
(6) Establishing an adaptive filter based on improved nonlinear observability for an application background model: the volume Kalman filtering is adopted for self-adaptive filtering:
(a) Calculating volume points:
(b) Volume point propagated through state equation:
(c) State prediction:
(d) Covariance prediction:
(e) Calculating volume points under measurement update:
(f) Volume point propagated through measurement equation: y is m,k|k-1 ＝h(x m,k|k-1 ) (10)
(g) Measurement prediction equation:
(h) Information processing systemCovariance estimation:
(i) Cross covariance estimation:
(j) Calculating a filter gain:
(k) Calculating a filter gain based on a nonlinear observability measure:
(l) And (5) updating the state:
(m) covariance update:
the following explanation is made on the parameters related to step (6): d, d l Refers to the number of volume points of the volume Kalman filter, τ m The point of the volume is indicated,the representation being a state variable x k|k-1 Is the same as>Also representing an estimate of the measured value; s is S k-1 Refers to S k-1 ＝chol(P k-1 ) Chol represents cholesky decomposition, which corresponds to the evolution of the matrix, ω m Is the corresponding weight; p (P) yy,k|k-1 Representing information autocovariance, P xy,k|k-1 Representing cross covariance, ++>Is a filtering gain with an adaptive factor;
the following explanation is made to step (6) for easy understanding: the invention considers the volume Kalman filtering to design the self-adaptive filtering based on nonlinear observability, and when the system nonlinearity is stronger, the estimation performance of the volume Kalman filtering is better than that of the extended Kalman filtering and the unscented Kalman filtering under certain conditions, so the volume Kalman filtering is adopted as a framework of the self-adaptive filtering to be applied to pure azimuth tracking. τ m The method is characterized in that the method is a volume point, and the volume point selection rule is as follows: the number of the volume points is 2 times of the dimension of the state variable, and the state variable is a four-dimensional variable in the invention, so the volume point selection rule is thatω m The weight corresponding to the volume point is generally selected as the inverse of the total number of volume points. In cholesky decomposition, it is necessary to ensure a positive matrix of the matrix type to be decomposed.
(7) The adjustment factors were added to the different stages of the filter frame and compared:
(a) Adding the adjustment factor to the metrology prediction stage:
(b) Calculating an estimated innovation covariance:
(c) Calculating cross covariance:
(d) Calculating a new filter gain:
(e) Calculating a new state update:
(f) Estimating covariance:
comparing the error magnitudes of the step (6) and the step (7), namely comparing P k And (3) withSelecting the adaptive filtering variance with small error covariance as a final scheme of the final adaptive filtering based on nonlinear observability; the letter superscript 1 in this step represents the variable that results when the adaptation factor is added to the measurement phase.
The larger the estimated error covariance is, the lower the filtering precision is, whereas the smaller the error covariance is, the higher the filtering precision is, in order to further improve the self-adaptive filtering precision based on the nonlinear observability, the invention adds the self-adaptive factor of the observability to other stages different from the filtering gain in the step (7), judges the self-adaptive filtering effect in that mode better by comparing the magnitudes of the error covariance under different conditions, and can further improve the self-adaptive filtering precision based on the nonlinear observability.
In order to improve the pure azimuth tracking filtering performance and perfect the observability theory of the nonlinear system, and meanwhile, the pure azimuth system can be roughly evaluated on the filtering precision of the system before filtering, the invention designs and establishes a new nonlinear observability calculation scheme, and observation noise is taken into consideration; the invention designs the self-adaptive adjustment factor based on the observability, adds the adjustment factor into the nonlinear filter framework, judges the self-adaptive filter effect in that way by comparing the magnitudes of error covariance under different conditions, can further improve the precision of the self-adaptive filter based on the nonlinear observability, and ensures that the filter performance of pure azimuth tracking is improved, and further the pure azimuth tracking has better performance.
Claims (1)
1. The improved nonlinear observability self-adaptive filtering method applied to pure azimuth tracking is characterized by comprising the following steps of:
step (1) setting a nonlinear system model as follows:
from differential geometry theory, h is known to be:
simultaneous definition ofConstructing an observation space that studies observability of nonlinear systemsIs a space generated by the following expression +.>Definitions->Spatial observability distribution, +.>If dimdH (x) 0 ) =n, i.e. full rank can be determined that the system is observable; defining an observability calculation matrix as: />The observability is +.>Here delta min (Ω),δ max (Ω) refer to the minimum and maximum singular values of the observability calculation matrix, respectively; in the above expression dim represents rank of the matrix variable,/for>Representation->Partial derivatives are obtained on the variable x; />The representation performs transposition operation on the matrix, and span is a representation symbol representing a state space;
step (2) improved Li Daoshu observability-based calculation: taking the observation noise into account in the observability calculation in step (1), the variance of which is R k Defining a new observability calculation matrix at the moment k as follows:
i.e. if there is rank (Q) k ) =n, rank represents the rank of the matrix; the system is observable and defines an improved observability calculation scheme:
step (3) establishing an adaptive adjustment factor based on the observability degree of the improved nonlinear system: the adaptive adjustment factors are established here as:l max refers to the maximum observable value of the state component of each systeml k Refer to the observability level at time k,/->n refers to all state numbers, det represents solving determinant for the matrix;
and (4) according to a pure azimuth target tracking system model, the state equation is a linear system, and the state model equation is set as follows:c in the equation t Representing a state transition matrix>State variables representing the system, expressed as system variables x t Is the derivative of w t Representing process noise; discretizing the state model to obtain: x is x k ＝C k x k-1 +w k,k-1 Here x k Is a discretized system state variable, x k ＝[x k v xk y k v yk ] T ,x k Representing the position in the x-direction, v xk Is the velocity in the x direction, y k Is the position in the y direction, v yk Is the velocity in the y direction; c (C) k The discretized state transition matrix has the following values:w k,k-1 for discretized system noise, its variance matrix is +.>
Step (5) setting an observation equation as y according to the pure azimuth target tracking model t ＝h t (x t )+u t ，y t Is the observed object variable, h t (x t ) Concerning x t Is a nonlinear function of u t Is observation noise; discretizing to obtain y k ＝h k (x k )+v k ，y k ,v k Is y t ,u t Discretized result, where v k The variance matrix is r=diag ([ 1, 1)]) The method comprises the steps of carrying out a first treatment on the surface of the The following d is a constant; the nonlinear equation is:
step (6) is based on adaptive filtering to improve the non-linear observability: performing self-adaptive filtering by adopting volume Kalman filtering:
(a) Calculating volume points:
(b) Volume point propagated through state equation:
(c) State prediction:
(d) Covariance prediction:
(e) Calculating volume points under measurement update:
(f) Volume point propagated through measurement equation: y is m,k|k-1 ＝h(x m,k|k-1 ) (10)
(g) Measurement prediction equation:
(h) Information covariance estimation:
(i) Cross covariance estimation:
(j) Calculating a filter gain:
(k) Calculating a filter gain based on a nonlinear observability measure:
(l) And (5) updating the state:
(m) covariance update:
the following explanation is made on the parameters related to step (6): d, d l Refers to the number of volume points of the volume Kalman filter, t m The point of the volume is indicated,the representation being a state variable x k|k-1 Is the same as>Also representing an estimate of the measured value; s is S k-1 Refers to S k-1 ＝chol(P k-1 ) Chol represents cholesky decomposition, which corresponds to the evolution of the matrix, ω m Is the corresponding weight; p (P) yy,k|k-1 Representing information autocovariance, P xy,k|k-1 Representing cross covariance, ++>Is a filtering gain with an adaptive factor;
step (7) adding the adjustment factors to different stages of the filter frame and comparing:
(a) Adding the adjustment factor to the metrology prediction stage:
(b) Calculating an estimated innovation covariance:
(c) Calculating cross covariance:
(d) Calculating a new filter gain:
(e) Calculating a new state update:
(f) Estimating covariance:
comparing the error magnitudes of step (6) and step (7), i.e. comparing P k And (3) withSelecting an adaptive filtering method with small error covariance as a final scheme of the final adaptive filtering based on nonlinear observability; in this step, the letter superscript 1 represents the corresponding variable that is obtained when the adaptation factor is added to the measurement phase.
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