CN109582915A - The non-linear observability degree adaptive filter method of improvement applied to bearingsonly tracking - Google Patents

The non-linear observability degree adaptive filter method of improvement applied to bearingsonly tracking Download PDF

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CN109582915A
CN109582915A CN201910079239.5A CN201910079239A CN109582915A CN 109582915 A CN109582915 A CN 109582915A CN 201910079239 A CN201910079239 A CN 201910079239A CN 109582915 A CN109582915 A CN 109582915A
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唐帅帅
葛泉波
何红丽
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Hangzhou Dianzi University
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Abstract

The present invention relates to a kind of non-linear observability degree adaptive filter methods of improvement applied to bearingsonly tracking, belong to observability degree adaptive-filtering field.It is theoretical in order to improve bearingsonly tracking filtering performance and improve nonlinear system observability degree, there can be a rough evaluation to the filtering accuracy of system before filtering to bearings-only system again simultaneously, a new non-linear observability degree numerical procedure is established in present invention design, and observation noise is taken into account；And the present invention designs the automatic adjusument factor based on the observability degree, regulatory factor is added in nonlinear filtering frame, judge that the adaptive-filtering effect under that mode is more preferable by comparing the size of the error covariance under different situations, it can be further improved the precision based on non-linear observability degree adaptive-filtering, so that the filtering performance of bearingsonly tracking is improved, and then bearingsonly tracking has better performance.

Description

The non-linear observability degree adaptive filter method of improvement applied to bearingsonly tracking
Technical field
The present invention relates to a kind of non-linear observability degree adaptive filter methods of improvement applied to bearingsonly tracking, belong to Observability degree adaptive-filtering field.
Background technique
Bearingsonly tracking is common technology in target tracking domain, is estimated frequently with kalman filter method it Processing, it is therefore an objective to fall the error concealment in tracking.Since bearingsonly tracking system state space model is more in practical applications For nonlinear system, bearingsonly tracking it is non-linear compared with cause by force directly by nonlinear system linearized stability it is larger, i.e., frequently with Unscented kalman filtering, volume Kalman filtering.
By classical control theory development come modern control theory, Kalman propose Kalman filtering when just by Observable Property put forward with two important properties of controllability, subsequent some scholars propose the Distinguishing theorem of ornamental and controllability, Since the precision of filtering depends on observability to a certain extent, if a real system is observable, the ability in filtering It not will lead to filtering divergence, if system is inherently unobservable, then the diverging of filter result will be caused.However existing Measure the size i.e. Observable of a mission observability degree there is no specific standard till now for control theory The size of degree, especially for for nonlinear system observability degree theoretical developments it is more difficult, about non-linear Observable The related data of topology degree is less, theoretical incomplete.Influence of the outside noise to it is very big for Kalman filtering , it is also such for observability degree.However in current non-linear observability degree research, there is no consider noise Inside, this way certainly will influence the evaluation of observability degree.
When evaluating its observability degree size in Bearing-only target tracking system, then filtering accuracy is high for observability degree height, right The tracking performance of target is better, on the contrary for the low then filtering accuracy of bearingsonly tracking system observability degree it is low to target following Performance is with regard to poor.The present invention is improved volume Kalman filtering on the basis for improving non-linear observability degree to improve filtering The purpose of precision, the present invention can filter wavefront calculations observability degree size, then to before bearingsonly tracking system filter to system Precision has a rough evaluation.And some researchers propose that observability degree is larger with Noise Correlation, and the present invention makes an uproar observation Sound considers in the calculating of non-linear observability degree, the perfect numerical procedure of non-linear observability degree.Bearingsonly tracking is filtered Have greatly improved.
Summary of the invention
In order to improve bearingsonly tracking filtering performance and improve nonlinear system observability degree theory, while again can be to pure Azimuth system has a rough evaluation to the filtering accuracy of system before filtering, present invention design establish one it is new it is non-linear can Observation degree numerical procedure, noise is taken into account；And the present invention designs the automatic adjusument factor based on the observability degree, Regulatory factor is added in nonlinear filtering frame, the filtering performance of bearingsonly tracking is improved, to improve pure side The performance of position tracking.
The method of the present invention is specifically:
Step (1) is traditional based on Lie derivatives observability degree calculation method: setting nonlinear system model are as follows:
Each rank Lie derivatives of the h along f is known by differential geometric theory are as follows:
It defines simultaneouslyThe observation of the observability of building research nonlinear system SpaceIt is the space generated by following formulaDefinitionThe Observable in space Property distribution,If dimdH (x0It is observable that)=n, i.e. full rank, which can determine that as the system,；Definition Observability degree calculating matrix based on this method are as follows:The observability degree of this method isHere δmin(Ω),δmax(Ω) respectively refers to the minimum and maximum singular value of observability degree calculating matrix； In above-mentioned expression formula, dim expression seeks order to the matrix variables,It indicatesPartial derivative is asked to variable x；It indicates to carry out matrix transposition operation, span is the expression symbol for indicating state space；
Step (2) proposes improved based on Lie derivatives observability degree numerical procedure: by observation noise in step (1) illustrates It is R in view of going its variance in observability degree calculatingk, define the observability degree calculating matrix at new k moment are as follows:
Even there is rank (QkRank of matrix is sought in)=n, rank expression；Then system is observable, and definition is improved can Observation degree numerical procedure:
Step (3) is established based on the improvement nonlinear system observability degree automatic adjusument factor: establishing adaptive adjust herein Save the factor are as follows:lmaxRefer to the maximum Observable angle value of the state component of each system Refer to the observability degree size at k moment,N value institute Some status numbers, det are indicated to the Matrix Calculating determinant.
Step (4) application background: according to Bearing-only target tracking system model, state equation is linear system, state Model equation setting are as follows:C in equationtIndicate state-transition matrix,The state variable of expression system, It is expressed as system variable xtDerivative, wtIndicate process noise；Above-mentioned shape model discretization is obtained: xk=Ckxk-1+wk,k-1, here XkFor the system state variables after discretization, xk=[xk vxk yk vyk]T,xkIndicate the position in the direction x, vxkIt is the direction x Speed, ykIt is the position in the direction y, vykIt is the speed in the direction y；CkFor the state-transition matrix after discretization, its value are as follows:wk,k-1For the system noise of discretization, its variance matrix is
Step (5) sets observational equation as y according to Bearing-only target tracking modelt=ht(xt)+υt, ytIt is observation object Variable, ht(xt) it is about xtNonlinear function, υtFor observation noise；The same sliding-model control that carries out obtains yk=hk(xk)+vk, yk,vkIt is yt,υtDiscretization as a result, wherein vkVariance matrix be R=diag ([1,1])；Following d is constant；Non-linear side Journey are as follows:
Step (6), which is established, to be carried out to using background model based on the adaptive-filtering for improving non-linear observability degree: this hair It is bright middle using volume Kalman filtering progress adaptive-filtering:
(a) volume point is calculated:
(b) the volume point propagated by state equation:
(c) status predication:
(d) covariance is predicted:
(e) it measures and updates lower calculating volume point:
(f) the volume point propagated by measurement equation: ym,k|k-1=h (xm,k|k-1) (10)
(g) predictive equation is measured:
(h) information covariance is estimated:
(i) cross covariance is estimated:
(j) filtering gain is calculated:
(k) filtering gain based on non-linear observability degree is calculated:
(l) state updates:
(m) covariance updates:
Step (6) relevant parameter is made explained below: dlRefer to the number of the volume point of volume Kalman filtering, τmIt indicates Volume point,Expression is state variable xk|k-1Estimated value, similarlyAlso illustrate that the estimated value of measured value；Sk-1Refer to Sk-1 =chol (Pk-1), chol indicates that cholesky is decomposed, and is equivalent to the extracting operation of matrix, ωmIt is corresponding weight；Pyy,k|k-1Table Show information auto-covariance, Pxy,k|k-1Indicate cross covariance,It is the filtering gain with adaptive factor；
Regulatory factor is added to the different phase of filter frame and compared by step (7):
(a) regulatory factor is added to measurement forecast period:
(b) the new breath covariance of estimation is calculated:
(c) cross covariance is calculated:
(d) new filtering gain is calculated:
(e) new state is calculated to update:
(f) estimate covariance:
Comparison step (6) and the error size of step (7) be compared with PkWithSize, choose error covariance it is small from Final scheme of the adaptive filtering variance as the final adaptive-filtering based on non-linear observability degree；In the step on character Mark mark 1 represents adaptive factor and is added to the variable obtained when the measurement stage.
Beneficial effects of the present invention: in order to improve bearingsonly tracking filtering performance and improve nonlinear system observability degree Theory, while can have a rough evaluation to the filtering accuracy of system before filtering to bearings-only system again, present invention design is built A vertical new non-linear observability degree numerical procedure, noise is taken into account；And the present invention is designed based on the Observable The automatic adjusument factor of degree, regulatory factor is added in nonlinear filtering frame, can make the filtering of bearingsonly tracking It can be improved, to improve the performance of bearingsonly tracking.
Detailed description of the invention
Fig. 1: flow diagram of the invention.
Specific embodiment
The present invention proposes a kind of non-linear observability degree adaptive filter method of the improvement applied to bearingsonly tracking, stream Journey block diagram as shown in Figure 1, including the following steps:
It is (1) traditional based on Lie derivatives observability degree calculation method: setting nonlinear system model are as follows:
Each rank Lie derivatives of the h along f is known by differential geometric theory are as follows:
It defines simultaneouslyThe observation of the observability of building research nonlinear system SpaceIt is the space generated by following formulaDefinitionThe Observable in space Property distribution,If dimdH (x0It is observable that)=n, i.e. full rank, which can determine that as the system,；Definition Observability degree calculating matrix based on this method are as follows:The observability degree of this method isHere δmin(Ω),δmax(Ω) respectively refers to the minimum and maximum singular value of observability degree calculating matrix； In above-mentioned expression formula, dim expression seeks order to the matrix variables,It indicatesPartial derivative is asked to variable x,It indicates to carry out matrix transposition operation, span is the expression symbol for indicating state space；
Make now to step (1) in order to facilitate understanding explained below: in general calculating observability degree constructs first is The ornamental matrix of system, ornamental matrix is made of state-transition matrix and observing matrix in a linear system.Nonlinear system Different from linear system, its state-transition matrix and observing matrix cannot directly write out, thus need with other methods come The observing matrix of nonlinear system is constructed, the construction method of currently used non-linear Observable matrix has Lie derivatives method and pseudo- sight Battle array and pseudo- state-transition matrix method are surveyed, the present invention improves the non-linear observability degree based on Lie derivatives.Matrix is maximum odd Different value and minimum singular value are the conditional number of matrix, can reflect the morbid state of matrix to a certain extent, indicate observability degree with this.
(2) it proposes improved based on Lie derivatives observability degree numerical procedure: considering observation noise in step (1) illustrates Into observability degree calculating, its variance is Rk, define the observability degree calculating matrix at new k moment are as follows:
Even there is rank (QkRank of matrix is sought in)=n, rank expression；Then system is observable, and definition is improved can Observation degree numerical procedure:
Now make following explanation to step (2) in order to facilitate understanding: the step proposes improved non-linear observability degree Numerical procedure, on the basis of step 1 by measure noise be added in the calculating of observability degree matrix, due to observability degree with Filtering accuracy is related, and filtering accuracy is again related to the noise of exterior, thus considers noise in non-linear observability degree Calculating in be necessary.The multiplication operation of matrix can be write as the form that the summation of matrix each element is added again, so in step Matrix element summation form is converted by matrix multiple in rapid 2, processing in this way facilitates operation.
(3) it establishes based on the improvement nonlinear system observability degree automatic adjusument factor: establishing the automatic adjusument factor herein are as follows:lmaxRefer to the maximum Observable angle value of the state component of each system Refer to the observability degree size at k moment,All status numbers of n value, det are indicated to the Matrix Calculating determinant.
Following explanation is made to step (3) below: due in volume Kalman filtering, due to the factors such as parameter is not accurate It may cause and filter the larger situation of not accurate error, increase the precision of filtering frequently with the method for the automatic adjusument factor, Step (3) is innovative to combine non-linear observability degree with the automatic adjusument factor.Establish out one based on it is non-linear can The automatic adjusument factor of observation degree improves nonlinear filtering to nonlinear filtering with the regulatory factor.
(4) application background: according to Bearing-only target tracking system model, state equation is linear system, state model Equation setting are as follows:C in equationtIndicate state-transition matrix,The state variable of expression system indicates For system variable xtDerivative, wtIndicate process noise；Above-mentioned shape model discretization is obtained: xk=Ckxk-1+wk,k-1, x herek For the system state variables after discretization, xk=[xk vxk yk vyk]T,xkIndicate the position in the direction x, vxkIt is the speed in the direction x, ykIt is the position in the direction y, vykIt is the speed in the direction y；CkFor the state-transition matrix after discretization, its value are as follows:wk,k-1For the system noise of discretization, its variance matrix is
(5) according to Bearing-only target tracking model, observational equation is set as yt=ht(xt)+υt, ytIt is observation object variable, ht(xt) it is about xtNonlinear function, υtFor observation noise；The same sliding-model control that carries out obtains yk=hk(xk)+vk, yk,vk It is yt,υtDiscretization as a result, wherein vkVariance matrix be R=diag ([1,1])；Following d is constant；Nonlinear equation are as follows:
Below to step (4) (5), make following explanation: establish the adaptive-filtering based on non-linear observability degree in order to Keep the performance of bearingsonly tracking more accurate.Parameters are as noted above, its state equation be it is linear, state model Their statistical property of the noise of noise and observation model meets Gaussian Profile, due to for being based on Kalman filtering frame Volume Kalman filtering for be directed to for discrete system, so need to carry out state equation and observational equation from Dispersion is simultaneously handled.
(6) it establishes and carries out to using background model based on the adaptive-filtering for improving non-linear observability degree: in the present invention Adaptive-filtering is carried out using volume Kalman filtering:
(a) volume point is calculated:
(b) the volume point propagated by state equation:
(c) status predication:
(d) covariance is predicted:
(e) it measures and updates lower calculating volume point:
(f) the volume point propagated by measurement equation: ym,k|k-1=h (xm,k|k-1) (10)
(g) predictive equation is measured:
(h) information covariance is estimated:
(i) cross covariance is estimated:
(j) filtering gain is calculated:
(k) filtering gain based on non-linear observability degree is calculated:
(l) state updates:
(m) covariance updates:
Step (6) relevant parameter is made explained below: dlRefer to the number of the volume point of volume Kalman filtering, τmIt indicates Volume point,Expression is state variable xk|k-1Estimated value, similarlyAlso illustrate that the estimated value of measured value；Sk-1Refer to Sk-1 =chol (Pk-1), chol indicates that cholesky is decomposed, and is equivalent to the extracting operation of matrix, ωmIt is corresponding weight；Pyy,k|k-1Table Show information auto-covariance, Pxy,k|k-1Indicate cross covariance,It is the filtering gain with adaptive factor；
Make content explained below to step (6) for convenience of understanding: the present invention considers to design using volume Kalman filtering Based on the adaptive-filtering of non-linear observability degree, volume Kalman filtering when mission nonlinear is stronger, in certain situation it Estimation performance be better than Extended Kalman filter and Unscented kalman filtering, so in the present invention use volume Kalman filtering It is applied on bearingsonly tracking as the frame of adaptive-filtering.τmIt is for volume point, volume point selection rule are as follows: volume point Quantity is 2 times of state variable dimension, and state variable is four-dimensional variable in the present invention, so volume point selection rule isωmIt is that the weight of corresponding volume point is generally chosen for volume point sum Inverse.When cholesky is decomposed, the matrix form positive definite matrix for guaranteeing to be decomposed is needed.
(7) regulatory factor is added to the different phase of filter frame and compared:
(a) regulatory factor is added to measurement forecast period:
(b) the new breath covariance of estimation is calculated:
(c) cross covariance is calculated:
(d) new filtering gain is calculated:
(e) new state is calculated to update:
(f) estimate covariance:
Comparison step (6) and the error size of step (7) be compared with PkWithSize, choose error covariance it is small from Final scheme of the adaptive filtering variance as the final adaptive-filtering based on non-linear observability degree；In the step on character Mark mark 1 represents adaptive factor and is added to the variable obtained when the measurement stage.
Evaluated error covariance is bigger to illustrate that filtering accuracy is low, on the contrary error covariance it is smaller show filtering accuracy height in order to Further increase the precision of the adaptive-filtering based on non-linear observability degree, the present invention in step (7) by observability degree from Adaptation factor is added to other stages different from filtering gain, sentences by comparing the size of the error covariance under different situations The adaptive-filtering effect broken under that mode is more preferable, can be further improved based on non-linear observability degree adaptive-filtering Precision.
In order to improve bearingsonly tracking filtering performance and improve nonlinear system observability degree theory in the present invention, simultaneously Can have a rough evaluation to the filtering accuracy of system before filtering to bearings-only system again, present invention design establish one it is new Non-linear observability degree numerical procedure, observation noise is taken into account；And the present invention designs oneself based on the observability degree Regulatory factor is adapted to, regulatory factor is added in nonlinear filtering frame, by comparing the error covariance under different situations Size judge that the adaptive-filtering effect under that mode is more preferable, can be further improved based on non-linear observability degree from The precision of adaptive filtering, so that the filtering performance of bearingsonly tracking is improved, and then bearingsonly tracking has better performance.

Claims (1)

1. being applied to the non-linear observability degree adaptive filter method of improvement of bearingsonly tracking, characterization method is this method The following steps are included:
Step (1) sets nonlinear system model are as follows:
Each rank Lie derivatives of the h along f is known by differential geometric theory are as follows:
It defines simultaneouslyThe observation space of the observability of building research nonlinear systemIt is the space generated by following formulaDefinitionThe observability in space point Cloth,If dimdH (x0It is observable that)=n, i.e. full rank, which can determine that as the system,；It defines considerable Likelihood Computation matrix are as follows:Observability degree isHere δmin(Ω),δmax(Ω) point Do not refer to the minimum and maximum singular value of observability degree calculating matrix；In above-mentioned expression formula, dim expression seeks order to the matrix variables,It indicatesPartial derivative is asked to variable x；It indicates to carry out transposition operation to matrix, Span is the expression symbol for indicating state space；
Step (2) is improved to be calculated by Lie derivatives observability degree: by observation noise in view of based on observability degree in step (1) In calculation, variance Rk, define the observability degree calculating matrix at new k moment are as follows:
Even there is rank (QkRank of matrix is sought in)=n, rank expression；Then system is observable, defines improved observability degree Numerical procedure:
Step (3) establish based on improve the nonlinear system observability degree automatic adjusument factor: establish herein automatic adjusument because Son are as follows:lmaxRefer to the maximum Observable angle value of the state component of each systemlkRefer to the observability degree size at k moment,N value institute Some status numbers, det are indicated to the Matrix Calculating determinant；
For step (4) according to Bearing-only target tracking system model, state equation is linear system, the setting of state model equation Are as follows:C in equationtIndicate state-transition matrix,The state variable of expression system is expressed as system change Measure xtDerivative, wtIndicate process noise；Above-mentioned shape model discretization is obtained: xk=Ckxk-1+wk,k-1, x herekFor discretization System state variables afterwards, xk=[xk vxk yk vyk]T,xkIndicate the position in the direction x, vxkIt is the speed in the direction x, ykIt is the side y To position, vykIt is the speed in the direction y；CkFor the state-transition matrix after discretization, its value are as follows: wk,k-1For the system noise of discretization, its variance matrix is
Step (5) sets observational equation as y according to Bearing-only target tracking modelt=ht(xt)+υt, ytIt is observation object variable, ht(xt) it is about xtNonlinear function, υtFor observation noise；The same sliding-model control that carries out obtains yk=hk(xk)+vk, yk,vk It is yt,υtDiscretization as a result, wherein vkVariance matrix be R=diag ([1,1])；Following d is constant；Nonlinear equation are as follows:
Step (6) is based on the adaptive-filtering for improving non-linear observability degree: adaptively being filtered using volume Kalman filtering Wave:
(a) volume point is calculated:
(b) the volume point propagated by state equation:
(c) status predication:
(d) covariance is predicted:
(e) it measures and updates lower calculating volume point:
(f) the volume point propagated by measurement equation: ym,k|k-1=h (xm,k|k-1) (10)
(g) predictive equation is measured:
(h) information covariance is estimated:
(i) cross covariance is estimated:
(j) filtering gain is calculated:
(k) filtering gain based on non-linear observability degree is calculated:
(l) state updates:
(m) covariance updates:
Step (6) relevant parameter is made explained below: dlRefer to the number of the volume point of volume Kalman filtering, τmIndicate volume Point,Expression is state variable xk|k-1Estimated value, similarlyAlso illustrate that the estimated value of measured value；Sk-1Refer to Sk-1= chol(Pk-1), chol indicates that cholesky is decomposed, and is equivalent to the extracting operation of matrix, ωmIt is corresponding weight；Pyy,k|k-1It indicates Information auto-covariance, Pxy,k|k-1Indicate cross covariance,It is the filtering gain with adaptive factor；
Regulatory factor is added to the different phase of filter frame and compared by step (7):
(a) regulatory factor is added to measurement forecast period:
(b) the new breath covariance of estimation is calculated:
(c) cross covariance is calculated:
(d) new filtering gain is calculated:
(e) new state is calculated to update:
(f) estimate covariance:
The error size of comparison step (6) and step (7), that is, compare PkWithSize, choose small adaptive of error covariance Answer filtering method as the final scheme of the final adaptive-filtering based on non-linear observability degree；Character subscript in the step 1 represent adaptive factor be added to obtained when the measurement stage to dependent variable.
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