CN109581356A - A kind of constraint filtering method for tracing of constant value maneuver space target - Google Patents

A kind of constraint filtering method for tracing of constant value maneuver space target Download PDF

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CN109581356A
CN109581356A CN201910006085.7A CN201910006085A CN109581356A CN 109581356 A CN109581356 A CN 109581356A CN 201910006085 A CN201910006085 A CN 201910006085A CN 109581356 A CN109581356 A CN 109581356A
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CN109581356B (en
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翟光
赵翰宇
张景瑞
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Beijing Institute of Technology BIT
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S13/00—Systems using the reflection or reradiation of radio waves, e.g. radar systems; Analogous systems using reflection or reradiation of waves whose nature or wavelength is irrelevant or unspecified
    • G01S13/66—Radar-tracking systems; Analogous systems
    • G01S13/72—Radar-tracking systems; Analogous systems for two-dimensional tracking, e.g. combination of angle and range tracking, track-while-scan radar
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S13/00—Systems using the reflection or reradiation of radio waves, e.g. radar systems; Analogous systems using reflection or reradiation of waves whose nature or wavelength is irrelevant or unspecified
    • G01S13/66—Radar-tracking systems; Analogous systems
    • G01S13/72—Radar-tracking systems; Analogous systems for two-dimensional tracking, e.g. combination of angle and range tracking, track-while-scan radar
    • G01S13/723—Radar-tracking systems; Analogous systems for two-dimensional tracking, e.g. combination of angle and range tracking, track-while-scan radar by using numerical data

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Abstract

The invention discloses a kind of constraints of constant value maneuver space target to filter method for tracing, belongs to spacecraft navigational guidance and control field.For the present invention first using the Maneuver Acceleration of extraterrestrial target as a part of state variable, construction is augmented Kalman filter；For the motor-driven extraterrestrial target of constant value, norm constraint is applied to Maneuver Acceleration, obtains optimal estimation by making to be augmented the minimization of object function；It minimizes to objective function is augmented and minimum acquisition can be carried out by the performance indicator respectively to system mode and motor-driven function；Finally, giving the process for being augmented Kalman filter tracking algorithm of local norm constraint, it is augmented Kalman filtering compared to unconfined, which can effectively improve the tracking accuracy to constant value maneuver space target.In addition, being calculated according to the algorithm, the calculated load of computer can be reduced, alleviates the limited problem of spaceborne computer resource.

Description

A kind of constraint filtering method for tracing of constant value maneuver space target
Technical field
The invention discloses a kind of constraints of constant value maneuver space target to filter method for tracing, belongs to spacecraft navigational guidance With control field.
Background technique
Target Tracking Problem is actually the tracking filter problem of dbjective state, i.e., according to the acquired target of sensor Azimuth and distance, estimate dbjective state, it is a major issue in military and civilian field.Early in last century The basic conception of the fifties, maneuvering target tracking has been formed, but until the seventies kalman filtering theory, target with Since being applied successfully in track, maneuvering target tracking technology just enters Rapid development stage.In Target Tracking Problem extensively at present The algorithm of application is, on the basis of being augmented Kalman filter, two parallel depression of order estimators is broken down into, in unknown machine It moves in the case where being constant, two depression of order estimators take turns to operate and generation system state and unknown motor-driven best estimate, together When this method solve be augmented Kalman filter filter dimension increase when, calculated load increase the problem of, but only When spacecraft is disturbed by constant value, it is augmented Kalman filtering and is only effectively.If motor-driven is time-varying, estimation will There is deviation, time-varying rate is bigger, and estimated bias is bigger.Obviously, this performance decline is that model and unknown motor-driven mismatch are made At.In this case, estimated usually using the constraint Kalman filter linearly or nonlinearly constrained is met.Pass through State is constrained, it is possible to reduce the estimated bias as caused by model mismatch.Norm holding be to system all or in part The typical non linear constraint that state applies.
Current most of spacecrafts around ground flight all have the propulsion of constant value propeller, in earth centered inertial coordinate system When interior description spacecraft moves, the acceleration direction that propeller generates is time-varying, but its size is constant.
Summary of the invention
The object of the present invention is to provide a kind of constraints of constant value maneuver space target to filter method for tracing, and this method will have The Kalman filter that is augmented of two rank formula is generalized to the system with time-varying disturbance under conditions of norm constraint, obtains two ranks The local norm holding of formula is augmented Kalman filter, when tracked with this filter with the motor-driven spacecraft of constant value, With higher precision.
The purpose of the present invention is realized by following scheme.
A kind of constraint of constant value maneuver space target disclosed by the invention filters method for tracing, initially sets up target following and asks The mathematical model of topic；Kalman filter is augmented with two rank formula secondly, establishing；Then, it is being augmented Kalman filter On the basis of apply local restriction, rebuild estimator and minimize be augmented objective function to obtain optimal estimation；Finally, asking Kalman gain and posteriority covariance out complete prediction and update step, complete a tracing process, start chasing after for subsequent time Track.
A kind of constraint of constant value maneuver space target disclosed by the invention filters method for tracing, includes the following steps:
Step 1: establishing target problem model.
In inertial system, the spacecraft of three-axis stabilization and thrust constant magnitude movement discrete time in orbit over the ground State space equation indicates:
xk+1=Akxk+Gkdk+Γkwk (1.a)
yk=Ckxk+vk (1.b)
Wherein xkIndicate system state variables, xk∈Rn, wherein ykIndicate measurement vector, yk∈Rp, process noise wkAnd survey Measure noise vkIt is all white noise and wk∈Rm、vk∈Rm, the covariance matrix of process noise is Qk, Qk∈Rm×m, measure the association of noise Variance matrix is Rk, Rk∈Rp×p, and Qk> 0, Rk> 0.Matrix AkIndicate sytem matrix, GkIt is motor-driven vector coefficients matrix, ΓkIt is noise coefficient matrix, CkIndicate output matrix；dkIndicate the motor-driven vector of constant value, dk∈Rm；Subscript k indicates etching system when k Parameters, similarly subscript k+1 indicate k+1 when the corresponding parameter of etching system.Since spacecraft relative inertness coordinate system is done Attitude motion, therefore the direction of the motor-driven vector of constant value is variation, then describes are as follows:
dk+1=(I+ δk)dk (2)
Wherein, δk∈Rm×mIt is unknown time-varying matrix, spacecraft three-axis stabilization over the ground, attitudes vibration is slow, then has
δk≈0 (3)
The motor-driven vector of constant value meets norm constraint
||dk| |=ρ wherein ρ > 0 (4)
Coefficient matrix GkAnd ΓkOrder having the same
rank[Gk]=rank [Γk]=m (5)
(Ak,Ck) there is observability, (Ak,Gk) and (Ak,Γk) there is controllability.
Kalman filter is augmented without constraint Step 2: establishing.
A. the target problem model established to step 1, i.e. formula (1.a) (1.b), are augmented, obtain being augmented system, That is formula (7.a) (7.b)；
Kalman filtering (ASKF) is augmented by unknown motor-driven a part as state, rear quantity of state is augmented and is expressed as
Wherein XkExpression is augmented state variable, Xk∈Rn×m, therefore, formula (1.a) (1.b) indicates are as follows:
Wherein subscript whippletree expression is augmented symbol,Indicate the sytem matrix after being augmented,Indicate the noise system after being augmented Matrix number,Indicate the output matrix after being augmented, andHave
ThenWithIt is expressed as
And then it obtains
B. it establishes no constraint and is augmented Kalman filter (UASKF)；
Initially motor-driven is Gaussian random variable, the motor-driven no restrained split-flow of constant valueBy following formula determine the motor-driven initial value of constant value with Its covariance without restrained split-flow value is
The cross covariance of state variable and the motor-driven variable of constant value is
It is augmented system, i.e. formula (7.a) (7.b) based on what step 2 obtained, enables δk=0, then Kalman is augmented without constraint Filter indicates are as follows:
Wherein,It is the one-step prediction of state variable,It is prior state covariance,It is posteriority covariance,It is kalman gain matrix, ηk+1Indicate residual error vector,Indicate posterior estimate:
It indicates one-step prediction measurement, steady operation and optimal result can be generated without restrained split-flow device (10) at this time.
Unconfined prior estimate errorWith Posterior estimator errorIt is defined as
Formula (7.a) (7.b) and formula (10.a) are substituted into formula (11), it is preceding to test evaluated errorIt indicates are as follows:
Formula (10.e) and formula (10.f) are updated in formula (12), Posterior estimator errorIt indicates are as follows:
Formula (13) are substituted into formula (14), are obtained:
By Posterior estimator errorAnd kalman gain matrixIt decomposes are as follows:
According to Posterior estimator errorDecomposition and kalman gain matrixDecomposition, by formula (15) rewrite are as follows:
The mathematic expectaion for asking formula (17), then has:
In formula, δk+1dk+1It indicates:
K → ∞ when the time tending to be infinite, formula (19) indicate are as follows:
Being not equal to the direction change that constant value is motor-driven it can be seen from 0 by formula (21) makes state estimation deviation occur.
Step 3: that establishes in step 2 is augmented Kalman filter without constraint, i.e., (10.a) (10.b) (10.c) (10.d) (10.e) (10.f) (10.g), middle application local restriction establish local restriction and are augmented Kalman filter, and then reduce Estimated bias improves tracking precision.
A, determine that performance index applies motor-driven norm constraint, reduction factor (3) is approximate and bring negatively affects, and rebuilds and expands It ties up Kalman filter (10.a) (10.b) (10.c) (10.d) (10.e) (10.f) (10.g).Posteriority covariance matrix will be augmented Is defined as:
Wherein local covariance is defined as
Indicate state variable posteriority covariance,Indicate the motor-driven posteriority covariance of constant value,Indicate shape State variable and the motor-driven posteriority cross covariance of constant value.
Additionally, it is contemplated that the norm constraint condition in formula (4), normalizes motor-driven estimation:
Under conditions of meeting formula (26) constraint, formula (10.e) is decomposed into
Then have
Formula (28) are substituted into formula (26), are had
Accordingly, it is considered to arrive norm constraint, objective function J is augmented by minimizingk+1To obtain optimal estimation:
Wherein
The mark of tr { } representing matrix.For convenience, by Local Property Index Definition are as follows:
WhereinIndicate Local Property index related with system state variables,It indicates and the motor-driven related office of constant value Portion's performance index, scalar lambdak+1Indicate Lagrange's multiplier.Obviously,WithIt is independent mutually, andIt is only dependent upon It is only dependent uponTherefore, Jk+1Minimum be equivalent to it is right respectivelyWithIt is minimized, is then had
B, minimum is passed through to the estimation for being augmented quantity of state applied after constrainingThe available expansion applied after constraint Tie up the optimal estimation value of quantity of state.
Constraint is augmented the predicted portions of Kalman filter are as follows:
WhereinIt is the constraint motor-driven estimation of constant value,It is the covariance of constant value ballistic error after applying constraint,It is to apply the motor-driven cross covariance between state variable of constant value after constraint.
The amendment part that constraint is augmented Kalman filter is as follows:
Wherein Θk+1For
C, to the estimation after the motor-driven application norm constraint of constant value by minimizing performance indicatorTo construct motor-driven part Estimation.Kalman filter (10) are augmented based on no constraint
It considers
Minimum to covariance is equal to the minimum of the mark of matrix, is then substituted into formula (38) in formula (33), to formula (33) it takesAnd λk+1Partial derivative, then enable partial derivative be equal to zero, then the first-order condition minimized is expressed as
Expansion (39) and formula (40)
By formula (58) it is found that det (Θ+λk+1ηk+1ηk+1 T) ≠ 0, then in formula (41)
And (Θ+λk+1ηk+1ηk+1 T) have
Wushu (44) substitutes into (43) and obtains
In addition, formula (45) are substituted into (42), obtain about λk+1Second-order equation
The value of b and c is expressed as in equation
Formula (46) is solved according to Vièta's formulas, lagrangian multiplier can be obtainedk+1
WhereinIt is expressed as
Formula (49) shows Lagrange's multiplier, and there are two possible solutions, and cannot be 0.Next, to prove, work as λk+1's Symbol is timingThere are minimum value, λk+1Symbol when being negative,There is maximum value.Due to gain matrixIt is column vector, then It can be obtained and the motor-driven related Local Property index of constant value by carrying out differential to formula (39) againHessian matrix:
Formula (51) display, Hessian matrix is diagonal matrix, according to formula (49)
According to formula (50), formula (52) is rewritten as
According to formula (35.g), formula (52) is written as:
Wherein r is the covariance of scalar measurement noise, according to formula (53) and formula (54), it is known that, work as λk+1Value be positive number When, Hessian matrix is positive definite, with the motor-driven related Local Property index of constant valueThere is minimum value, conversely, λk+1When being negative,There is maximum value.
In order to prove to work as λk+1Symbol be timing, matrix Θ+λkηk+1ηk+1 TIt is positive definite, Θ+λkηk+1ηk+1 TIt rewrites For
It is multiplied by η simultaneously in the right and leftk+1Have
According to formula (50), formula (56) the right and left simultaneously multiplied byHave
Because of Θk+1It is positive definite, thenFinally, formula (56) can be with abbreviation
Then formula (58) shows matrix Θ+λkηk+1ηk+1 TIt is positive definite, while has det (Θ+λk+1ηk+1ηk+1 T) > 0
D, the posteriority covariance matrix minimum performance index after kalman gain matrix and application constraint after applying constraint Optimal Lagrange's multiplier indicate are as follows:
Then formula (55) are substituted into formula (45) and are obtained:
WhereinBe to the motor-driven relevant local restriction kalman gain matrix of constant value, and
According to Kalman filtering algorithm, and correct the available motor-driven estimation of constraint constant value of one-step prediction:
WhereinIt is the one-step prediction of the motor-driven estimation of constant value,It is the constraint motor-driven estimation of constant value.Obviously, constant value is constrained Motor-driven estimation is normalized to meet the constraint in formula (26), then explanation meets norm constraint.Meanwhile local restriction Kalman GainPass through constraint conditionThe motor-driven estimated projection of constant value will be constrained and rotate to m dimension Euclid surface.
In addition to this, also
WhereinIt is expressed as
Formula (65), which are substituted into formula (64), to be had
Because measure noise always withWithIt is uncorrelated, therefore, ignore withWithContinuous item, then Formula (66) indicates are as follows:
Finally, it obtains
Step 4: tracing process
Table 1.
Beneficial effect
A kind of constraint of constant value maneuver space target disclosed by the invention filters method for tracing, by the expansion with two rank formula Dimension Kalman filter is generalized to the system with time-varying disturbance under conditions of norm constraint, obtains the local model of two rank formula Number keeps being augmented Kalman filter, when tracked with this filter with the motor-driven spacecraft of constant value, has higher chase after Track precision.
Detailed description of the invention
Fig. 1 is the motor-driven no restrained split-flow of embodiment 1 as a result, (a) is to estimate the orbit maneuver in x-axis direction；(b) For the motor-driven evaluated error in x-axis direction；(c) for the orbit maneuver estimation on y-axis direction；It (d) is motor-driven on y-axis direction Evaluated error；
Fig. 2 is the motor-driven estimated result after applying norm constraint of embodiment 1, and (a) is to the track machine in x-axis direction Dynamic estimation；It (b) is the motor-driven evaluated error in x-axis direction；(c) for the orbit maneuver estimation on y-axis direction；It (d) is y-axis side Upward motor-driven evaluated error；
Fig. 3 is the spacecraft of embodiment 1 before and after to the motor-driven application norm constraint of constant value, to the estimated result ratio of mobility value Compared with；
Fig. 4 be embodiment 1 Space Vehicle position and speed without restrained split-flow as a result, (a) be to the position in x-axis direction Evaluated error；It (b) is the speed estimation error in x-axis direction；(c) for the position estimation error on y-axis direction；It (d) is y-axis Speed estimation error on direction；
Fig. 5 is the estimated result of the Space Vehicle position and speed of embodiment 1 after applying norm constraint, and (a) is to x-axis side Upward position estimation error；It (b) is the speed estimation error in x-axis direction；(c) for the location estimation mistake on y-axis direction Difference；It (d) is the speed estimation error on y-axis direction；
Fig. 6 is the situation of change of performance index in x-axis direction and y-axis direction in embodiment 1.
Specific embodiment
With reference to the accompanying drawing with examples illustrate the present invention, meanwhile, in order to illustrate the present invention tracking have it is higher Precision, will be compared in the case of no restrained split-flow.
Embodiment 1
The spacecraft three-axis stabilization to orbit, and there is constant value trust engine.In track mobile process, hair Motivation always generates the acceleration of direction along ng a path.For simple computation, enable spacecraft movement on circular orbit, orbital acceleration It is unknown, and indicated by Ω, then constant value thrust acceleration can indicate in geocentric inertial coordinate system are as follows:
Wherein T indicates thrust magnitude,Indicate that initial phase, M indicate the quality of spacecraft.It will be clear that spacecraft The size of thrust acceleration is constant, but direction time changing.Do following provisions: 1) true dynamics of orbits only disturbed Journey noise；2) spacecraft can normally, accurately carry out gesture stability；3) earth station can continuous-stable pursuit spacecraft.
The spacecraft is tracked respectively with constraint postfilter and unconstrained filter, and evaluation constrains postfilter and without constraint The performance index of filter is
WhereinIndicate no restrained split-flow error,Indicate restrained split-flow error.With the ratio table of average root-mean-square error Show relative accuracy.When μ (k) is greater than 100%, show that the accuracy of restrained split-flow is high, on the contrary the accuracy without restrained split-flow It is high.
A. kinetics equation and observed quantity
1) kinetics equation describes spacecraft orbit movement with two body Models.When spacecraft receives unknown disturbance, The movement can be indicated with single-sweep polarograpy acceleration:
Wherein μeIndicate terrestrial gravitation coefficient, r=[x, y, z]TIndicate position vector of the spacecraft in track, v=[vx, vy,vz]TIndicate the velocity vector of spacecraft, w indicates the perturbation process noise of spacecraft.All vectors in formula (71) indicate In geocentric inertial coordinate system.The position vector and velocity vector of spacecraft are indicated with state vector x, then formula (71) can be write At state equation form in step 1:
Wherein
Wherein D=F=[0 I]T, D is input matrix, and F is noise matrix.Formula (73) is subjected to Taylor's exhibition in state of closing on It is provided with:
Discretization can be carried out to orbit equation according to formula (73) and formula (74), then:
xk+1=Φk+1,kxk+Bkdk+Gkwk (75)
Wherein Φk+1,kIt is state-transition matrix:
Φ(k+1,k)Φ(k+1,k)T=I6×6 (77)
Wherein Δ t is the interval updated the time.
2) observed quantity
Spacecraft is tracked in a manner of time discrete by ground phased-array radar.Phased-array radar is fixed at the earth's surface, And rotated together with the earth, measurement spacecraft is indicated in the angle and distance that can be surveyed in range, nonlinear measurement models are as follows:
zk=h (xSEZ(k))+υk (78)
WhereinIt is measurement vector comprising distance, range rate, azimuth, elevation and measurement Uncertainty.xSEZ(k) it is the position vector indicated in local east-Nan Tian top coordinate system (SEZ):
In addition to this, also
Vector rSEZ(k)=[xSEZ(k) ySEZ(k) zSEZ(k)]TCan by by vector r from geocentric inertial coordinate system (ECI) it transforms to and is obtained in ECEF coordinate system (ECEF), finally transforming to has in east-Nan Tian top coordinate system (SEZ):
Wherein TECEF→SEZIt is the rotational transformation matrix from ECEF coordinate system to SEZ coordinate system, TECI→ECEFIt is from ECI coordinate It is the rotational transformation matrix to ECEF coordinate system,Indicate position of the radar in ECEF coordinate system.Likewise, speed It can be carried out similar coordinate transform:
Then have
Wherein
Wushu (83) is updated in formula (80), and measured value then can be used x (k) to indicate, measured value near estimated state Linearisation has:
WhereinThe estimated value of expression state,It is the one-step prediction of state, Hk+1It is the Jacobin matrix of measured value, It can be calculated as follows mode and obtain:
3) numerical result
The spacecraft on equatorial orbit is considered first, and orbital eccentricity and height are respectively 0 and 600km, spacecraft mass Propeller for 1000kg, and constant size 20N is fixed on spacecraft.In simulation process, propeller is flat in x-y always Sinusoidal Maneuver Acceleration is generated in face.The primary condition of analog simulation provides as follows: initial rail state x0=[6978000m 0m 0m 0m/s 7557.9m/s 0m/s], initial motor-driven d0=[0m/s2 0m/s2 0m/s2], the covariance of original state isThe covariance of unknown Maneuver AccelerationThe covariance of process noise is set as Q=10-12 ×I3×3, measurement noise is R=diag ([40,000 400 7.62 × 10-9 7.62×10-9]).According to the arrangement process of table 1 and Given primary condition, can successfully construct filter, and the working frequency of filter is 1Hz, when emulation a length of initial circular orbit A cycle, both direction without constraining motor-driven estimation as shown in Figure 1, wherein dotted line indicates actual movement, and solid line table Show estimated result.From left figure as can be seen that acutely being vibrated without restrained split-flow device due to initial error, initial oscillation it Afterwards, the motor-driven of estimation follows true motor-driven, but has significant time delay, this can be by the fact that explain: nothing Restrained split-flow device always assumes that Maneuver Acceleration is constant, therefore motor-driven variation cannot be timely responded to without restrained split-flow device. Right figure shows motor-driven motor-driven evaluated error, the results showed that the error of both direction can all release in entire simulation process Come, worst error is both greater than 0.005m/s2。
The result that Fig. 2 is shown merits attention.Compared with the result in Fig. 1, it is clearly illustrated, once normal constraint is answered For motor-driven estimation, no restrained split-flow is normalized to keep motor-driven amplitude, and restrained split-flow it is motor-driven variation it is more sensitive. The result shows that violent concussion when starting receives significant inhibition, in addition, the time delay and evaluated error of estimation all subtract It is small.
Fig. 3 shows the size of the Maneuver Acceleration of estimation.In the case where no restrained split-flow, estimation it is motor-driven preceding It is acutely vibrated in 100 seconds, but when estimated value reaches the metastable stage, the amplitude of estimation is less than in most of time 0.02.But for restrained split-flow, because of our selection parameter KT=0.975, the size of Maneuver Acceleration is projected onto In the range of [0.0195,0.02], this makes the Maneuver Acceleration of estimation closer to true acceleration.
The evaluated error of orbital position and speed indicates in figures 4 and 5, and wherein solid line indicates evaluated error, and dotted line Indicate the boundary σ.Due to the time delay of Maneuver Acceleration estimation, the evaluated error of constrained and unconfined situation is all by first The influence of beginning vibration, as shown in the figure.After reaching the metastable stage, no restrained split-flow device is approximately reached in both directions 30m (σ) precision, and the precision of restrained split-flow device reaches about 20m (σ), and restrained split-flow device is being applied to velocity estimation Afterwards, the accuracy for the speed estimated also improves.To motor-driven norm constraint, as shown in the figure on the right in each figure.Knot Fruit clearly illustrates that after constant value norm constraint is included in unknown Maneuver Acceleration, estimated accuracy is improved.
Fig. 6 shows estimation performance of the restrained split-flow amount relative to unconstrained estimator.As can be seen that starting in simulation When, estimation performance index in both directions sharply vibrates, caused by this is mainly the initial convergence as two estimators, still After about 10s, then performance is greater than 1.2 after about 1500s always greater than 1, its final difference is in the x and y direction Reach 1.5 and 1.4 maximum value.Based in Fig. 5 as a result, can be confirmed that restrained split-flow device ratio has more preferably without restrained split-flow device Performance.
In conclusion the above is merely preferred embodiments of the present invention, being not intended to limit the scope of the present invention. All within the spirits and principles of the present invention, any modification, equivalent replacement, improvement and so on should be included in of the invention Within protection scope.

Claims (2)

1. a kind of constraint of constant value maneuver space target filters method for tracing, include the following steps:
Step 1: target problem model is established.
In inertial system, the spacecraft of three-axis stabilization and thrust constant magnitude moves in orbit over the ground for consideration one, moves It can be indicated with discrete-time state-space equation
xk+1=Akxk+Gkdk+Γkwk (1.a)
yk=Ckxk+vk (1.b)
Wherein xk∈RnIndicate system mode, dk∈RmIndicate the motor-driven vector of constant value.Since spacecraft relative inertness coordinate system does appearance State movement, therefore the direction of motor-driven vector is variation, be can be described as
dk+1=(I+ δk)dk (2)
Wherein δk∈Rm×mIt is that a unknown time-varying matrix then has because the attitudes vibration of spacecraft is slow
δk≈0 (3)
But motor-driven vector meets norm constraint
||dk| |=ρ wherein ρ > 0 (4)
And yk∈RpIndicate measurement vector, process noise wk∈RmWith measurement noise vk∈RmIt is all white noise, covariance square Battle array is respectively Qk∈Rm×m, Rk∈Rp×p, and Qk> 0, Rk> 0.Matrix Ak, Gk, ΓkAnd CkWith suitable dimension
rank[Gk]=rank [Γk]=m (5)
And (Ak,Ck) it is that can observe, (Ak,Gk) and (Ak,Γk) it is controllable.
Step 2: it establishes and is augmented Kalman filter without constraint.
A. it is augmented system
Kalman filtering (ASKF) is augmented by unknown motor-driven a part as state, rear quantity of state is augmented and is expressed as
Wherein Xk∈Rn+mExpression is augmented state vector, and therefore, system (1) is represented by
Wherein subscript whippletree expression is augmented symbol, has
ThenWithIt can be expressed as
And then it is available
B. it establishes no constraint and is augmented Kalman filtering (UASKF)
Assuming that initially motor-driven is Gaussian random variable, no restrained split-flowIt is determined by following formula
Cross covariance is
Based on the system that is augmented (7), it is assumed that constant when motor-driven and enable δk=0, then being augmented Kalman filter without constraint can be with It is expressed as
Wherein,It is one-step prediction,It is prior state covariance,It is posteriority covariance,It is kalman gain Matrix, ηk+1Indicate residual error vector
It notices if δk=0, then without restrained split-flow device (10) steady operation and optimal result can be generated.
Unconfined priori and Posterior estimator error are defined as
Formula (7) and formula (10.a) are substituted into formula (11), it can be deduced that
Likewise, formula (10.e) and formula (10.f) are updated in formula (12), Posterior estimator error can be expressed as
Further, formula (13) are substituted into formula (14) and obtained
Posterior estimator error and gain matrix are decomposed into
According to formula (16), formula (15) can be rewritten as
In view of formula (17) and formula (18), then have
When the time tending to be infinite k → ∞ then we have
Obviously, state estimation has inclined, and this is mainly due to caused by motor-driven variation.
Step 3: it is augmented in Kalman filter in no constraint and applies local restriction.
Apply motor-driven norm constraint, reduction factor (3) is approximate and bring negatively affects, and rebuilds estimator (10).
Posteriority covariance matrix will be augmented to be defined as
Wherein local covariance is defined as
Additionally, it is contemplated that meeting the motor-driven estimation of normalization constrained in formula (4)
Formula (10.e) is decomposed into
Then have
Formula (28) are substituted into formula (16), are had
Accordingly, it is considered to arrive norm constraint, objective function can be augmented by minimum to obtain optimal estimation:
The right side of formula (30) meets
The wherein mark of tr { } representing matrix.For convenience, it is by Local Property Index Definition
Wherein scalar lambdak+1Indicate Lagrange's multiplier.Clearly asWithIt is only dependent upon respectivelyWithTherefore, Jk+1Minimum can be right respectivelyWithIt is minimized, is then had
A. apply the estimation after local restriction to system mode
Pass through minimumIt is not difficult based on traditional Kalman filter example constructions partial estimation device.It saves and repeats, We directly indicate that estimation is as follows:
Predicted portions are
It is as follows to correct part
Wherein Θk+1For
B. to the estimation after motor-driven application local restriction
It is rightMinimum
By minimizing performance indicatorTo construct motor-driven partial estimation.Based on estimator (10)
It considers
Minimum to covariance is equal to the minimum of mark, then formula (38) are substituted into formula (33), formula (33) is takenWith λk+1Partial derivative, then by its be equal to zero, the first-order condition then minimized can be expressed as
Expansion (39) and formula (40)
From formula (41), due to det (Θ+λk+1ηk+1ηk+1 T) ≠ 0 (proves) in formula (58), meets
And for the latter half of equation
Wushu (44) substitutes into (43) and obtains
In addition, formula (45) are substituted into (42), obtain about λk+1Second-order equation
The value of b and c can be expressed as in equation
By using Vièta's formulas to formula (46)
WhereinIt can be expressed as
Formula (49) shows that there are two types of possible solutions for optimal Lagrange's multiplier, in order to determine the symbol of minimum, it should examine Look into second-order condition.
In formula (49), symbol is timing performance indexIt will minimize, and be maximized when selecting negative sign.Work as gainIt is When column vector, performance indicator can be obtained by carrying out differential to formula (39) againHessian matrix.
Formula (51) display, Hessian matrix is diagonal matrix, according to formula (49)
According to formula (50), formula (52) can be rewritten as
Review formula (35), it can be deduced that
Wherein r is the covariance of scalar measurement noise, further, according to formula (53) and formula (54), when selecting positive sign, Hai Sen Matrix is positive definite, and minimum performance index occurs.Opposite, it just will appear maximum performance index.
C. the update of covariance matrix
The optimal Lagrange's multiplier of minimum performance index can be expressed as
Then formula (55) are substituted into formula (45) and is obtained
WhereinIt is local kalman gain matrix relevant to disturbance, and
According to Kalman filtering algorithm, constraining motor-driven estimation can be by being corrected to following form for one-step prediction
WhereinIt is no restrained split-flow.Obviously, restrained split-flow is normalized to meet the constraint in formula (26), also, part is about Estimated value is projected to and ties up euclidean surface, rather than unconfined condition side by the m that constraint condition is crossed over by beam kalman gain To.
In addition to this, also
WhereinIt can be expressed as
Formula (61), which are substituted into formula (60), to be had
Because measure noise always withWithIt is uncorrelated, therefore, formula (66) can by omit withWith Continuous item carrys out restatement
Finally, it can be deduced that
2. a kind of constraint of constant value maneuver space target according to claim 1 filters method for tracing, it is characterised in that: such as Shown in step 3, on the basis of being augmented Kalman filter, to motor-driven application norm constraint, to reduce because of δk≈ 0 is approximate And bring negatively affects, and can then rebuild estimator, and then completes the update and iteration of algorithm.
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